Caustics, counting maps and semi-classical asymptotics

CA USTICS, COUNTING MAPS AND SEMI-CLASSICAL ASYMPTOTICS N. M. ERCOLANI Abstra ct. This pap er devel ops a deeper understand ing o f the structure and com binato- rial signiﬁcance of the partition function for Hermitian random matrices. The coeﬃcients of the large N expansion of the logarithm of this partition function, also known as the genus exp ansion , (and its deriv a tives) are generating functions for a v ari ety of graphical enumeratio n problems. T he main results are to prov e that these generating fun ctions are in fact speciﬁc rational functions of a distinguished irrational (algebraic) function, z 0 ( t ), of the generating function parameter, t . This distinguished function is itself the generat- ing function for t he Catalan num b ers (or generalized Catalan num bers, dep en d ing on the choi ce of weigh t of th e parameter). It is also a solution of the inviscid Burgers equation for certai n initial data. The shock formation, or c austic , of the Burgers chara cteristic solution is directly related to the p oles of t he rational forms of the generating functions. As an intriguing app lication, one gains new insigh ts into the relation b etw een certain deriv ativ es of the genus expansion, in a double-sc aling lim it , and the asymptotic expansion of the ﬁrst Painlev ´ e transcend ent. This provides a precise expression of the P ainlev´ e as- ymptotic co eﬃcients directly in terms of the coeﬃcients of t h e partial fractions expansion of the rational form of the generating funct ions established in this pap er. Moreov er, these insigh ts p oin t to w ard a more general program relating the ﬁrst P ainlev ´ e hierarch y to the higher order structure of the double-scaling limit through the sp eciﬁc rational structure of generating functions in th e genus exp ansion. The pap er closes with a discussion of the relation of this work to recent developmen ts in und erstanding the asymptotics of graphical enumeratio n. As a bypro duct, these results also yield new information ab out the asymptotics of recurrence co eﬃcien ts for orthogonal p olynomials with resp ect t o exp onential weigh ts, the calculation of correlation functions for certain tied rand om walks on a 1-D lattice, and the lar ge tim e asympt otics of random matrix partition functions. 1. Introduction This pap er describ es the co eﬃcien ts in an asymptotic expansion of the logarithm of the partition function, Z N ( t ), log ( Z N ( t ) / ( Z N (0)) = N 2 e 0 ( t ) + e 1 ( t ) + 1 N 2 e 2 ( t ) + · · · + 1 N 2 g − 2 e g ( t ) + · · · for certain ensem bles of rand om Hermitian matrices. (See Theorem 1.1 for a pr ecise state- men t.) The supp ort of the Natonal Science F oundation, DMS-0808059, is gratefully acknow ledged. 1 2 N. M. ERCOLANI As is familiar fr om general con texts of statistics and p robabilit y theory , these co eﬃcient s con tain a great deal of information ab out exp ectations and correlation functions with re- sp ect to the Gibbs measure that un derlies the giv en random matrix ensemble. In our con text w e are in terested in the asymptotic b eha v ior of N − 2 log Z N ( t ) (often called the fr e e ener gy ) as the size, N , of the matrices b ecomes large. These ensem bles, and hen ce the asymptotic co eﬃcien ts e g , as well , d ep end on a coupling parameter t (the analogue of the inv erse temp erature in the classical s etting of the canonical en sem ble). It is natur al to view the e g ( t ) as generating fu nctions f or the just-ment ioned exp ectations and correla- tion f unctions and therefore it is of v alue to kno w as m uc h as p ossible ab out ho w these co eﬃcien ts dep end on the coup lin g parameters. W e will derive general closed form expressions f or al l of the co eﬃcient s, viewed as generating functions, in our asymp totic expansions as w ell as in certain deriv ativ es of these exp ansions. W e will sho w th at, with only a couple exceptions, they are all rational functions of a single irrational (algebraic) f unction of t whic h we denote b y z 0 . More succinctly , we will see that, for g ≥ 2, e g ( t ) ∈ Q ( z 0 ( − t )) , the function ﬁeld of z 0 ( − t ) o v er the rational num b ers Q . It is a standard and fairly elementa ry fact that a generating fu nction is r ational if and only if its sequ ence of co eﬃcien ts ev entually (i.e., b ey ond a certain ord er) satisﬁes a line ar recurrence relation [20]. The generating functions, suc h as e g ( t ), that we will describ e are not rational in t ; h o wev er, the result just mentio ned su ggests that if e g is indeed rational in z 0 , then it sh ould in fact b e p ossible to recursively constru ct it from z 0 . T his is, in eﬀect, the viewp oin t whic h guided our deriv ation of the rational form of the e g and other generating fun ctions that arise as co eﬃcien ts in asym p totic expansions closely related to our partition fu nction expansions (see Theorems 1.3 and 1.5). F or the f undamenta l function, z 0 ( − t ) itself, w e are able to describ e it v ery explicitly from s everal d iﬀeren t p ers p ectiv es. F or instance, it is itself th e generating function f or gener alize d Catalan numb ers wh ic h are familiar constructs in enumer ative c ombinatorics . Putting all this together, our results p ro vide a fairly detailed and sys tematic u nderstanding of the asymptotic b eha vior of correlation fun ctions for the random matrix ensem bles that w e stu dy . While this is in teresting in and of itself, there is another lev el of applications for these results. The correlation functions that w e are discussing also ha v e a combi- natorial in terpretation related to diagr ammatic exp ansions that are familiar in statistica l physic s. The e g ( t ) can b e top ologic ally identiﬁed as generating f unctions that count lab elled maps. Roughly sp eaking, a map is an em b edding of an oriented graph in to a compact, orien ted and connected su rface X with the requir emen t that the complement of the graph in the surface sh ould b e a disjoin t un ion of s imply connected op en sets. A lab elled map is a map on w hic h th e v er tices of the em b edded graph hav e b een ordered. (Tw o su c h lab elled maps are equiv alen t if the resp ectiv e graphs can b e smo othly d eformed into on e another within the su rface.) T he n th T a ylor co eﬃcien t of e g ( t ) is an in teger equal to the num b er of CAUSTICS , COUNTING MA PS AND SEMI-CLASSICAL A SYMPTOTICS 3 distinct equiv alence classes of maps to a compact, orien ted Riemann surf ace of genus g that can b e constructed using exactly n vertices and where eac h ve rtex has the same v alence. (This v alence is give n by a w eigh t asso ciated to the coupling parameter. See Section 1.2 and T h eorem 1.2.) The r esults presen ted in this pap er yield recursion f ormulae f or the co eﬃcien ts of these generating fun ctions, n o w view ed as coun ting functions for lab elled maps, that are based on the partial fr actions expansions of the rational forms of the e g and related generating fun ctions. Of particular int erest is th e new ligh t that this application sheds on a 20 y ear old pr oblem stemming f rom th e physics literature on 2D Quantum Gra vity [3, 8, 18]. In mathematical terms th is problem asks for the detailed description of a p articular double-sc aling limit of the random matrix p artition fu nction. W e refer the r eader to [14] for the history and bac kgroun d. W e can giv e an idea of th e con tribution of our w ork to this prob lem by men tioning that z 0 ( − t ) is the proﬁle of the self-similar solution to an inviscid Bur gers equation (see Section 2.3). F rom the m etho d of c h aracteristics one ma y conclude that z 0 ( − t ) has an analytic contin uation to the complex t -plane minus a branc h cut that extends from −∞ to a cr itical v alue − t c < 0 along th e negativ e real axis. This critical time is in fact the sho ck time at whic h th e Burgers caustic s tarts to form. One then deﬁ nes the double-scaling limit to b e th e limit in which t → − t c within a sector of the complex p lane with ℜ t > − t c while at the same time N → ∞ so that the similarit y v ariable ξ = N 4 / 5 ( t + t c ) − γ , remains ﬁxed at some ﬁnite, non-zero v alue ( γ is a constan t to b e determined). Because of our rationalit y result, we are, ﬁrs t of all, able to conclude that the co eﬃcien ts, e g ( t ), all ha ve the same en v elop e of h olomorph y as z 0 ( − t ), wh ic h is the ab o ve men tioned slit p lane with a common singularit y at − t c . Bu t, more strikingly , what we ﬁ nd in the double scaling limit, is that the dep endence on the com b inatorial information in z 0 ( − t ) w ashes out, so that all that remains in the limit is th e information that can b e deduced from the structure of e g as a rational fu nction of z 0 , and that rational structure is what w e describ e in this pap er. One ma y s tu dy the asymp totics of the free energy directly in the double-scaling limit. This in fact was d one, for a particular ensem ble, in [12, 13] (see also [7]) via a Riemann- Hilb ert representa tion of p olynomials orth ogonal with resp ect to the exp onentia l wei gh t that determines the Gibb s measure for the matrix ensemble under consideration. They ﬁn d that this Riemann-Hilb ert problem can b e d eformed into the Riemann-Hilb ert problem for the ﬁrst Painlev ´ e transcend ent (PI) with resp ect to the parameter ξ in the double-scaling limit. Thr ough this they are, roughly sp eaking, able to r elate the double scaling limit of N − 2 log Z N to an asymptotic expansion of PI in a sector of th e ξ plane. Our appr oac h is also based on the R iemann -Hilb ert problem for orthogonal p olynomials, but we develo p explicit expressions for the doub le scaling asymptotics of the p artition function, for a v ery general class of ensembles, in terms of our r ep resen tations of the m ap 4 N. M. ERCOLANI coun tin g functions as r ational f unctions of z 0 . This is consisten t with the r esu lts for the sp ecial case considered in [12, 13] and th erefore gives, for the ﬁrst time as far as we aw are, a dir e ct and explicit inte rpretation of the co eﬃcien ts for the P I asymp totic expansion in terms of the graphical enumeration of lab elled maps. Th is connection op ens up some inte resting new directions for inv estigatio n that w ill b e brieﬂy discussed in section 4.5. The rest of this introd uction will amplify up on and d etail the themes th at hav e ju s t b een sk etched. 1.1. P artit ion F unctions for Unitary E nsem bles. Th e ensembles of int erest in this pap er are a sub-family of what are kno wn as the Unitary Ensem b les (UE) of random matrices; the common space of these ensem bles is the space, H n , of n × n Hermitian matrices, M . The Gibbs measures for the ensem b les w e consider are giv en by the follo wing family of probabilit y measures, dµ t 2 ν = 1 e Z ( n ) N ( t 2 ν ) exp {− N T r[ V ν ( M , t 2 ν )] } dM , w h ere (1) V ν ( λ ; t 2 ν ) = 1 2 λ 2 + t 2 ν λ 2 ν (2) and the parameter t 2 ν is assumed to b e such that the integral conv erges. F or example, one ma y supp ose that ℜ t 2 ν > 0. The normalization factor e Z ( n ) N ( t 2 ν ), whic h serve s to make µ t b e a pr obabilit y measure, is called the p artition function of this unitary ensem ble. Because the trace, T r, is inv ariant under conju gation and since M is Hermitian, this p artition fun ction reduces to an expres- sion which is p rop ortional to the follo wing multiv ariate d ensit y expressed in terms of the eigen v alues { λ j } of M [23]. Z ( n ) N ( t 2 ν ) = (3) Z · · · Z exp    − N 2   1 N n X j = 1 V ν ( λ j ; t 2 ν ) − 1 N 2 X j 6 = ℓ log | λ j − λ ℓ |      d n λ. Z ( n ) N ( t ) diﬀers f r om e Z ( n ) N ( t ) only b y an o v erall factor whic h is ind ep endent of t . (Th at is b ecause th e redu ction to eigen v alues comes from a c h ange of v ariables on M that conjugates it to a d iagonal matrix; giv en the inv ariance of the trace under conjugation, the eﬀect of this v ariable c hange is only seen in dM and d o es not inv olv e t .) In all our consid erations, w e w ill alwa ys be wo rking with Z ( n ) N ( t ) / Z ( n ) N (0). So from th is p ersp ectiv e, Z ( n ) N ( t ) and e Z ( n ) N ( t ) are equiv alen t. F or the coup ling w eight V ν , we will henceforth often drop the sub script ν and just write V ; similarly , for the coupling p arameter t 2 ν w e will usually drop the sub script 2 ν and j ust CAUSTICS , COUNTING MA PS AND SEMI-CLASSICAL A SYMPTOTICS 5 write t . Which v alue of ν is int ended should b e clear from con text. (Ho w ev er s ee section 1.2 f or furth er commen t.) In stud ying asym p totic questions w e will alwa ys assume that N an d n go to inﬁn it y together suc h their ratio x = n/ N remains constant at a ﬁ n ite, non-zero v alue. When t = 0, th e Gibbs measure (1) is the measure f or the Gaussian Unitary Ensemble (GUE). W e will t yp ically wan t to work with a r escaling of our matirx ens em bles giv en by A = √ N M and whic h we will refer to as the ﬁne sc ale . In terms of this calibration, the probabilit y measure at t = 0 is dµ 0 ( A ) = 2 − n/ 2 π − n 2 / 2 exp  − 1 / 2 T r[ A 2 ]  dA. (4) The general partition fun ction asso ciated to the measure µ t , or equiv alen tly th e int egral (3), can b e expr essed as an exp ectation with r esp ect to the GUE ensem ble. First ob s erv e that e Z ( n ) N ( t ) = Z H n exp  − N T r[1 / 2 M 2 + tM 2 ν ]  dM (5) = N − n 2 / 2 Z H n exp  − T r[1 / 2 A 2 + t N ν − 1 A 2 ν ]  dA = 2 n/ 2  π N  n 2 / 2 E µ 0  exp  − t N ν − 1 T r[ A 2 ν ]  where E µ 0 denotes the exp ectation with resp ect to the measur e (4). It is then natur al to deﬁne a f u nction that inte rp olates naturally b et w een the matrix and eigen v alue represen- tations of the partition fu nctions: τ 2 n,N ( t ) = Z ( n ) N ( t ) Z ( n ) N (0) = e Z ( n ) N ( t ) e Z ( n ) N (0) = E µ 0  exp  − t N ν − 1 T r[ A 2 ν ]  , (6) where τ n,N is called the n th scaled tau function . T he u nsc ale d tau fun ction is τ n, 1 . A p rimary motiv ation for carryin g out the detailed asymptotic analysis of the in tegrals (3) is the insight it yields in to cur ren t problems of asymptotic combinato rics, sp eciﬁcally related to graphical en umeration. The work here builds on the sys tematic asymptotic analysis of (3) that was carried out in [9, 10]. In particular we tak e as our starting p oint the follo win g t wo theorems from this pr ior work. Theorem 1.1. [9] Ther e is a c onstant T > 0 such that for (c omplex) t with ℜ ( t ) ≥ 0 and | t | < T and f or x = n/ N in a neighb orho o d, U , of x = 1 , one has a uni f ormly valid 6 N. M. ERCOLANI asymptot ic exp ansion log τ 2 n,N ( t ) = n 2 e 0 ( x, t ) + e 1 ( x, t ) + 1 n 2 e 2 ( x, t ) + · · · . (7) as N → ∞ . The me aning of this exp ansion is: if you k e ep terms up to or der n − 2 h , the err or term is b ounde d by C n − 2 h − 2 , wher e the c onstant C is indep endent of x and t in the domain { x ∈ U ; t ∈ (0 ≤ ℜ t ) ∩ ( | t | < T ) } . F or e ach ℓ , the fu nction e ℓ ( x, t ) is a lo c al ly analytic function of t i n a (c omplex) ne i ghb orho o d of t = 0 . Mor e over, the asympto tic exp ansion of t - derivatives of log  Z ( n ) N  may b e c alculate d via term-by- term diﬀer entiation of the ab ove series.  Remark 1. In [9], the result was stated only for the sp ecial case w hen x = 1. Ho wev er, it is straigh tforward to extend this result to the m ore general setting stated here and this w as explained in [10]. F or the later parts of this pap er (sections 3 and higher) we w ill tak e x = 1 so that n = N . In this case w e will d enote the partition function Z ( N ) N b y just Z N and e g ( t ) will denote e g (1 , t ).  Our fo cus will b e on the co eﬃcient s, e g ( t ), wh ic h ha ve a direct com b inatorial meanin g that w e now b rieﬂy explain. The e g en u merate lab eled m aps. This has already b een describ ed in the op ening paragraphs of th e In tr o duction. Here we will give th e formal and more precise deﬁnition. A map D on a compact, oriented and co nnected surface X is a pair D = ( K ( D ) , [ ı ]) where • K ( D ) is a connected 1-complex; • [ ı ] is an isotopical class of inclusions ı : K ( D ) → X ; • the complement of K ( D ) in X is a disjoint un ion of op en cells (faces); • the complement of the v ertices in K ( D ) is a d isjoin t union of op en segmen ts (edges). W e introd uce the n otion of a g-map wh ic h is a m ap in w h ic h the sur f ace X is the closed, orien ted Riemann surf ace of genus g and which in addition carr ies a lab eling (ordering) of the v ertices. Theorem 1.2. [9] The c o e ﬃcients i n the asympto tic exp ansion (7) satisfy the fol lowing r elations. L et g b e a nonne gative inte ger. Then (8) e g ( t ) = X j ≥ 1 1 j ! ( − t ) j κ g ( j ) in which e ach of the c o eﬃcients κ g ( j ) is the numb er of g-maps with j 2 ν -valent vertic es. Remark 2. Due to this int erpretation, the asymp totic exp ansion (7) is often referred to as the genus exp ansion . CAUSTICS , COUNTING MA PS AND SEMI-CLASSICAL A SYMPTOTICS 7 Remark 3. It is natural to in quire whether ther e is a generating function repr esen tation for u nlab ele d maps. After all it is these ob jects that would s eem to h a ve th e more purely geometric c haracter. Indeed if it were the case that the action of relab eling a lab eled map alw ays pro du ced a d istin ct other lab eled map, th en for eac h unlab eled map with j v ertices, one w ould ha v e j ! distinct lab eled m aps and κ g ( j ) j ! (the actual T a ylor co eﬃcients of e g ( t ) up to a sign) would giv e the coun t of unlab eled maps. Ho w ev er, it can happ en that a relab eling pro d uces the same lab eled map and hence this w a y of coun ting is not generally v alid. There are some ad ho c ﬁxes for this bu t w e will not go into th ese s ince that w ould tak e u s to o far from our main thrust; b ut, it is wo rth noting that the kin d s of isotropies that cause the problem tend to b e restricted to lo w v alues of the parameters. So for certain asymptotic considerations, s u c h as th e b eha vior of these co eﬃcien ts as j → ∞ , the eﬀect of the isotropies ”w ash out”. See S ection 4.6 for a related discussion. One may also ask ab out th e consideration of more general we igh ts, for instance of the form V ( λ ; t 1 , . . . , t υ ) = 1 2 λ 2 + υ X m =1 t m λ m , whic h w ould corresp ond to allo w ing for o dd v alences and, moreo ver, for mixe d v alences when there are tw o or more n on-v anish ing coupling constan ts, t m . In fact v ers ions of b oth Theorems 1.1 and 1.2 were pro v en in this generalit y in [9]. The results presente d in th is pap er could, in p rinciple, b e deve lop ed in greater generalit y along these lines. Ho wev er, to av oid ov er-complication of the main ideas we will restrict atten tion here to the case of pure (but arbitrary), eve n v alence.  1.2. Diﬀeren t Coupling W eights. There is another (discrete) parameter, ν , in the par- tition fu nction (3) that determin es the exp onen tial weig h t V ( λ ) in the associated measur es. (Graphically it determines th e un iform v ertex v alence of the maps enumerated by the e g .) The associated coupling parameter t in the w eight should more p rop erly b e denoted by t 2 ν to d istinguish the diﬀeren t w eigh ts. Ho w ev er, as is eviden t from statemen ts in pr eviously cited theorems of th is int ro duction, we ha ve in general su ppressed the dep end ence on ν in expressions related to Z ( n ) N or d er ived f rom it. I n p art this is to av oid cumb ersome notation. In all r esults and argumen ts presente d in this pap er, ν is a ﬁxed parameter. It do es not app ear in any of the scalings w e discu ss. The role th at ν plays should alw ays b e clear from con text. But there is a m ore salien t structural reason w hy it is natural to omit the explicit ν - dep enden ce: th e formulae for co eﬃcien ts in the gen us expansion that w e derive all h a ve a universal c h aracter in ν ; th at is to s ay , the structur e of eac h of th ese formulae is that of a global mer omorp hic fun ction of an irrational algebraic fun ction z 0 ( s ) (wh ere s = − t ). Although ν d o es app ear as a discrete parameter in eac h of these fu nctions, it do es not alter that function’s essent ial meromorph ic form. F or example, we will sho w (see Theorem 1.3) 8 N. M. ERCOLANI that, for g ≥ 2, e g ( − s ) = ( z 0 ( s ) − 1) Q d ( g ) ( z 0 ( s )) ( ν − ( ν − 1) z 0 ( s )) o ( g ) , where Q is a p olynomial of degree d ( g ). I n p articular, the in teger-v alued functions d ( g ) and o ( g ) are indep en den t of ν . Moreo ver, in all cases, th e explicit dep endence on ν in suc h form ulae can b e completely determined by a ﬁ n ite num b er of sp ecializat ions at s p eciﬁc v alues of ν ( ν is a p ositive integ er). Similar structural prop erties w ill b e sho w n to hold for the co eﬃcien ts of related asymptotic exp an s ions. 1.3. The F undamen t a l Generat ing F unction. Th e generating f u nctions that w e will describ e are not r ational in s ; how ev er, as has already b een ment ioned, we will show th at they are rational f unctions of a fun damen tal generating function z 0 ( s ). Th is breaks the problem of describin g these generating fun ctions in to tw o parts. The ﬁrst is to determine the rational f u nctions as exp licitly as p ossible to the p oint that one could sa y something ab out the asso ciated recursion r elatio n or some other explicit ﬁnite description. The main p oint of the present p ap er is to mak e pr ogress on this p art of the p roblem. The second part is to describ e z 0 ( s ). Although it is n ot rational w e are actually able, based on [10], to deduce v ery explicit information ab out this algebraic fun ction. Ov er the course of this and th e next section we will p resen t four diﬀerent wa ys of d eﬁ ning z 0 ( s ), all of w hic h play a role in d escribing and app lyin g the stru ctur e of the co eﬃcient s of the gen u s expans ion and related geometric expan s ions. Th ese four d iﬀeren t c haracteri- zations are (i) as a generating fun ction for the generalized Catalan n um b ers (section 1.4.1); (ii) as the leading order term in the large N expansion of the recurr ence co eﬃcien ts for monic orthogonal p olynomials with resp ect to the exp onen tial we igh t exp( − N V ( λ )) (section 1.4.4); (iii) as the self-similar solution of an inviscid Bur gers equation (section 2.3); (iv) in terms of the endp oints of the sup p ort for th e equilibriu m measure of the large N limit of rand om Hermitian matrices (section 2.5). Based on the previous paragraphs, there are several wa ys in whic h we can talk ab out the generating functions in the genus expans ion. When w e wan t to th ink of them directly as functions of the coupling parameter w e will write e g ( t ) or equ iv alently e g ( − s ) (assuming, when we d o this, that x = 1). Wh en thinking of its structur e as a fun ction of z 0 w e will w rite e g ( z 0 ) or, sometimes, e g ( z ). Finally the t w o p ersp ectiv es are related by e g ( − s ) = e g ( z 0 ( s )). Similar con v entions will app ly to related expansion co eﬃcien ts su c h as th e z g ﬁrst describ ed in section 1.4.4. CAUSTICS , COUNTING MA PS AND SEMI-CLASSICAL A SYMPTOTICS 9 1.4. Statemen t of Results. The ﬁr st steps to ward elucidating the structur e of th e gen- erating functions e g ( t ) were take n in [10]. Th ere, d iﬀerential equations for these functions w ere der ived that made it in prin ciple p ossible to recursively solv e for the e g in terms of a distinguished com binatorial generating fu nction d en oted z 0 ( s ). The diﬀerentia l equations in question will b e review ed in section 2, but w e wan t ﬁr s t to giv e a b rief ov erview of the results of this pap er and how they d ep end on p rior w ork. 1.4.1. Gener alize d Catalan N umb ers. W e b egin with our ﬁrst description of the fun damen- tal an alytic function z 0 ( s ). z 0 ( s ) = X j ≥ 0 c j ν ζ j s j where (9) c ν = 2 ν  2 ν − 1 ν − 1  = ( ν + 1)  2 ν ν + 1  and ζ j = 1 j  ν j j − 1  = 1 ( ν − 1) j + 1  ν j j  . (Section 2.3 explains where this expansion comes from.) Wh en ν = 2, ζ j is th e j th Catalan n um b er. F or general ν these are the gener alize d Catalan numb ers wh ic h p la y a role in a wide v ariet y of en u merativ e com binatorial problems. F or a d iscussion of these applications see [24]. 1.4.2. Backgr ound for the First Main R esult. I n [10] th e ﬁ rst f ew co eﬃcien ts of (7) w ere calculate d explicitly: e 0 ( z 0 ) = η ( z 0 − 1)( z 0 − r ) + 1 2 log( z 0 ) where η = ( ν − 1) 2 4 ν ( ν + 1) and r = 3( ν + 1) ν − 1 ; e 1 ( z 0 ) = − 1 12 log ( ν − ( ν − 1) z 0 ) ; e 2 ( z 0 ) = ( z 0 − 1) Q ( z 0 ) ( ν − ( ν − 1) z 0 ) 5 where Q ( z 0 ) = ( ν − 1) 2880  ( − ν 3 + 5 ν 4 + 8 ν 5 ) + ( − ν 2 + 41 ν 3 − 24 ν 4 − 16 ν 5 ) z 0 + (44 ν − 89 ν 2 + 54 ν 3 − 17 ν 4 + 8 ν 5 ) z 2 0 + ( − 12 − 12 ν + 108 ν 2 − 132 ν 3 + 48 ν 4 ) z 3 0 + ( − 12 + 48 ν − 72 ν 2 + 48 ν 3 − 12 ν 4 ) z 4 0  . 10 N. M. ERCOLANI Remark 4. The expr ession for e 2 ( z 0 ) giv en in [10] con tained add itional terms inv olving p ossible constants of integ ration. The authors w ere able to r u le out th e pr esence of these other terms for most, b u t not all, v alues of ν . Ho w ev er, it is a consequen ce of the resu lts present ed here (Theorem 1.3) that these terms are absent for all v alues of ν and hence the expression for e 2 ( z 0 ) giv en ab o v e is un iversally v alid.  There are sev eral striking features common to th ese ﬁrst few co eﬃcien ts. First, they are all ab elian functions of the single v ariable z 0 . (An ab elian fu n ction is a function that can b e represente d as an iterated path integral of a r ational fun ction. The rational fun ctions are a minimal s u b class of the class of ab elian fun ctions.) Second, the singularities of th ese functions are restricted to z 0 = 0 , ν / ( ν − 1). Third, the co eﬃcien ts of these generating functions can b e determined by s ubstitution of the generating function for the higher Catalan n umbers, z 0 ( s ), int o these ab elian func- tions. Finally , although the coeﬃcient s of these represent ations do dep end (rationally o v er Q ) on ν , they h a ve a unive rsal c h aracter; in particular, one should note that kno wledge of the co eﬃcien ts for relativ ely few v alues of ν suﬃces to completely determine these functions for all v alues of ν . The four ab ov e-stated features w ere established in [10] (see Theorem 2.4 of this p ap er) to hold for all the gen us expansion co eﬃcien ts. But we will sho w that the co eﬃcien ts, e g , are ev en more tigh tly stru ctured than these prop erties might suggest: 1.4.3. Statement of the First Main R esult. Theorem 1.3. F or g ≥ 2 the c o eﬃcients of (7) ar e r ational functions of z 0 of the form e g ( z 0 ) = ( z 0 − 1) Q d ( g ) ( z 0 ) ( ν − ( ν − 1) z 0 ) o ( g ) wher e Q is a p olynomial of de gr e e d ( g ) whose c o eﬃcients ar e r ational functions of ν over the r ational numb ers Q . The exp onent o ( g ) and the de gr e e d ( g ) ar e non-ne gative inte gers to b e determine d. Once the rational expression for e g has b een determined , the map counts κ g ( n ) ma y b e read oﬀ d irectly b y metho ds of classical fun ction theory . This w ill b e demonstr ated for a related coun ting problem in Section 4.1. More could u n doubtedly b e s aid ab out the structure of the e g , but that will b e develo p ed elsewhere. CAUSTICS , COUNTING MA PS AND SEMI-CLASSICAL A SYMPTOTICS 11 1.4.4. R e curr enc e Co eﬃcients. W e turn next to a r elated map enumeratio n pr oblem whic h also has applications to the theory of orthogonal p olynomials w ith exp onential w eights. Let π n,N ( λ ) denote the n th monic orthogonal p olynomial with resp ect to the exp onen tial w eigh t exp ( − N V ( λ )). These p olynomials satisfy a thr ee-term recurr ence relation of the form π n +1 ,N ( λ ) = λπ n,N ( λ ) − b 2 n,N ( t ) π n − 1 ,N ( λ ) F ur ther backg round on these orthogonal p olynomials and their relation to the partition functions (3) is pr esen ted in the Ap p endix. In [10] it w as shown that, as a consequence of T heorem 1.1, Theorem 1.4. [10] The r e curr e nc e c o eﬃcients for the monic ortho gonal p olynomials with weight exp( − N V ( λ )) have a ful l asymptotic exp ansion, uniformly valid f or t ∈ (0 ≤ ℜ t ) ∩ ( | t | < T ) and x = n / N in a neighb orho o d, U , of x = 1 , of the form b 2 n,N ( t ) = x ( z 0 ( s ) + 1 n 2 z 1 ( s ) + 1 n 4 z 2 ( s ) + · · · ) (10) wher e s = − x ν − 1 t (se e Se ction 2.1.1 for the explanation of this sc aling) and b 2 n,N (0) = n/ N = x. (11) Remark 5. This result giv es us our second interpretatio n of z 0 : it is the leading ord er term in the large N exp an s ion of the r ecur rence co eﬃcient b 2 N ,N ( t ). The fact th at this leading co eﬃcient do es ind eed coincide with (9) follo ws fr om the deriv ation that will b e review ed in Section 2.3.  The relation of these expan s ion co eﬃcien ts to map en umeration is giv en by the f ollo wing [10]: z ( j ) g (0) = d j z g ds j | s =0 = # { t wo- legged g -maps with j 2 ν -v alen t vertice s } . (12) (A le g is an edge emerging from a un iv alent vertex; s o that the leg is the only edge in cident to that vertex.Th us the maps b eing counte d here h a ve exactly t wo vertice s of v alence one; all other v ertices hav e v alence 2 ν .) 12 N. M. ERCOLANI 1.4.5. Backgr ound for the Se c ond Main R esult. In [10], th e ﬁr st f ew h igher ord er terms in the expansion (10) were calculated explicitly: z 1 ( z 0 ) = z 0 ( z 0 − 1) n ( ν − 1) ν 12  ( ν 2 + ν − 2) z 0 − ν 2  o ( ν − ( ν − 1) z 0 ) 4 (13) = z 0  ν ( ν + 2) / 12 ( ν − ( ν − 1) z 0 ) 2 + − ν (3 ν + 2) / 12 ( ν − ( ν − 1) z 0 ) 3 + ν 2 / 6 ( ν − ( ν − 1) z 0 ) 4  z 2 ( z 0 ) = z 0 ( z 0 − 1) P 4 ( z 0 ) ( ν − ( ν − 1) z 0 ) 9 where (14) P 4 ( z 0 ) = 1 1440 ( ν − 1) ν  (2 ν 6 − 14 ν 7 + 24 ν 8 ) + ( − 12 ν 3 + 148 ν 4 − 546 ν 5 + 758 ν 6 − 252 ν 7 − 96 ν 8 ) z 0 + (264 ν 2 − 1510 ν 3 + 25551 ν 4 − 500 ν 5 − 1789 ν 6 + 840 ν 7 + 144 ν 8 ) z 2 0 + ( − 536 ν + 1396 ν 2 + 912 ν 3 − 4596 ν 4 + 2492 ν 5 + 1296 ν 6 − 868 ν 7 − 96 ν 8 ) z 3 0 + (168 + 234 ν − 1467 ν 2 + 558 ν 3 + 1902 ν 4 − 1446 ν 5 − 267 ν 6 + 294 ν 7 + 24 ν 8 ) z 4 0  z 3 ( z 0 ) = z 0 ( z 0 − 1) P 7 ( z 0 ) ( ν − ( ν − 1) z 0 ) 14 where (15) P 7 ( z 0 ) is a p olynomial of degree 7 (for the explicit expression, see [10], p p. 66-67) . 1.4.6. Statement of the Se c ond Main R esult. W e will sho w that the pattern whic h app ears to b e emerging in the preceding examples is in fact the general structure for the co eﬃcients z g . Note that we adopt the same notational con ven tions f or fu nctional dep endence of the z g that w ere describ ed for the e g at the end of section 1.3. Theorem 1.5. z g ( z 0 ) = z 0 ( z 0 − 1) P 3 g − 2 ( z 0 ) ( ν − ( ν − 1) z 0 ) 5 g − 1 = z 0 ( a ( g ) 0 ( ν ) ( ν − ( ν − 1) z 0 ) 2 g + a ( g ) 1 ( ν ) ( ν − ( ν − 1) z 0 ) 2 g +1 + · · · + a ( g ) 3 g − 1 ( ν ) ( ν − ( ν − 1) z 0 ) 5 g − 1 ) , (16) wher e P 3 g − 2 is a p olynomial of de gr e e 3 g − 2 in z 0 whose c o eﬃcients ar e r ational functions of ν over the r ational numb ers Q . This theorem has a num b er of remark able corollaries. CAUSTICS , COUNTING MA PS AND SEMI-CLASSICAL A SYMPTOTICS 13 1.4.7. R esidue F ormulae for Gr aphic al Enumer ation. F or instance, we ﬁnd that # { t wo- legged g -maps w ith j 2 ν -v alen t vertice s } = c j ν 3 g − 1+ j X ℓ =0 a ( g, j ) ℓ ( ν ) where a ( g, j ) ℓ ( ν ) = [( j − 1) ν − (2 g + ℓ + ( j − 2))] a ( g, j − 1) ℓ ( ν ) + ν [2 g + ℓ + ( j − 2)] a ( g, j − 1) ℓ − 1 ( ν ) and a ( g, 0) ℓ ( ν ) = a ( g ) ℓ ( ν ) , wh ic h is implicitly deﬁned as the ℓ th residue in (16) . 1.4.8. R elation with the First Painlev´ e T r ansc endent. When ν = 2 we w ill sh o w that a ( g ) 3 g − 1 = − 2 5 g − 1 (2 / 3) g / 2 α g for g ≥ 1 a (0) ℓ = δ ℓ, 0 , where the { α g } are the coeﬃcients in the asymptotic expansion of a class of solutions, (70), to the ﬁ rst Pai nlev ´ e equation. As will b e discu ssed in section 4.5, this p oints to a broader connection of these en umerativ e co eﬃcient s, for general ν , to the ﬁrst P ainlev´ e hierarch y and of th e other fun d amen tal residues, a ( g ) ℓ ( ν ), in (16), to the higher order terms in the expansion of the double scaling limit of the recurrence co eﬃcien ts. 1.5. Outline. In Section 2 w e will review the d iﬀeren tial equations for the generating functions z g ( s ) and also provide additional c h aracteriza tions of the fund amen tal generating function z 0 ( s ). In particular we w ill describ e the relation b et w een the p olar singularit y of z g ( z 0 ), at z 0 = ν / ( ν − 1), and caustic form ation in a Burgers equation for z 0 . With this bac kgroun d, we give the pr o of of Th eorem 1.5 in S ection 3. The diﬀeren tial equations for the generating fu nctions e g are reviewe d in Section 4.2 and then T h eorem 1.3 is pr o ved. Applications of and corollaries to Theorem 1.5 are pr esen ted in S ection 4. T h ese include those results men tioned at the end of the p revious su bsection. Finally , in S ection 4.6 w e dra w some analogies with and p ossible connections to other recen t work in the theory of map en umeration. App end ix A p resen ts th e essen tial elements that we u se related to orthogonal p olynomials with exp on ential weig h ts, their recurrence co eﬃcien ts and the T o da Lattice equations whic h describ e how these recurr ence co eﬃcient s transform w hen the p arameters in the w eigh ts are v aried. W e also indicate h o w the d iﬀeren tial equations reviewe d in section 2 are r elated to a con tinuum limit of the T o da lattice. In Section A.3 a new com bin atorial formula is deriv ed f or the forcing co eﬃcients that app ear in the diﬀerentia l equations for z g . These ha ve relev ance for calculating the correlation fu nctions for certain tied random w alks on a one-dimensional lattice. In App en d ix B w e b rieﬂy review the one-p oint correlation functions for eigen v alues of th e UE ensembles and the Riemann-Hilb ert m etho ds that were used to d educe the s tructure of their asymptotic expansions. Th e pu rp ose of this is to gi v e p oin ts of reference for some of the theorems quoted in the int ro duction (particularly T heorems 1.1 and 1.4). 14 N. M. ERCOLANI The App endix also outlines an extension of th ese prior results to describ e th e large time b eha vior of the co eﬃcien ts z g and e g . This extension is n eeded in the pr o ofs of Theorems 1.5 and 1.3. 2. Ba ckgr ou nd on Different ial Equa t ions for Higher Genus Coefficients A complete diﬀeren tial charact erization of the asymptotic co eﬃcien ts, z g , app earing in (10 ) is giv en by the ﬁrs t three theorems b elo w. This c haracterization is f ou n ded on a con tin u um limit of the well-kno wn sys tem of diﬀerent ial equations (the T o da L attic e equations) which, in this s etting, describ e ho w the recurr en ce co eﬃcien ts ev olv e with t . In Ap p endix A the reader will ﬁnd the p ertinen t b ac kground on T o da equations and a su mmary of h o w their con tinuum limit (22) is derived. 2.1. Scalings and Notation. The pr incipal resu lts of th is p ap er all in v olv e stud ying the b ehavi or of certain fu nctions of the p artition function in the asymptotic limit as the parameters N , n and /or t tend to inﬁnity (or some critical v alue), t ypically with some com bin ation of these parameters b eing h eld ﬁxed. In this su bsection w e summ arize the v arious scalings that will b e considered. W e will also take th is opp ortunit y to s tate some notational con ven tions that will b e us ed s ystematica lly throughout the pap er . 2.1.1. Fine Sc aling. F or sev eral reasons, and in particular wh en we discuss, in section 2 the deriv ation of the diﬀerent ial equations which yield explicit expressions for e g and related generating functions, it will b e most eﬀectiv e to u se the ﬁn e scaling r epresen tation of the partition function in terms of the GUE exp ectation that was giv en in (6). There the natural scaled v ariable to consider wa s t N ν − 1 . There is a closely r elated ﬁn e scaling that will pla y a role in our second main r esult r elated to recur rence co eﬃcien ts (see section 1.4.4). The scaling v ariable in this con text will b e giv en b y s n ν − 1 , wh ere s serve s as a similarit y v ariable relating x and t (see (18)). Recall that n and N gro w at the same r ate so that x = n / N remains ﬁxed at a ﬁ nite, n on-zero v alue. The relations b et ween all these scalings ma y b e succinctly summarized as follo ws: − 1 2 s n ν − 1 = θ = 1 2 t N ν − 1 (17) s = − x ν − 1 t. (18) The macr osc opic v ariable θ will serv e to relate the partition fu nction to standard expres- sions for tau f u nctions and Hir ota formulae that pla y a role in th e d ev elopment s describ ed in s ections 1.4.4 and 2. Note that when x = 1 , s = − t . Th e minus sign b etw een s and t has b een in tro duced b ecause, although the cur ren t usage of t is standard for most repre- sen tations of rand om matrix partition fu nctions, s (= − t ) is the natur al v ariable for the generating fun ctions that app ear in the gen us expansion. See, for ins tance, form ula (8) ab o ve. The factors of 1 2 are introd uced in (17 ) in ord er to remo v e an o verall f actor of 1 2 in the con tin u um T o da equ ations (22). CAUSTICS , COUNTING MA PS AND SEMI-CLASSICAL A SYMPTOTICS 15 2.1.2. Sp atial Fine Sc aling. W e will in addition sometimes need to compare asymptotic expansions in { n − r } to those in { ( n + ℓ ) − r } where ℓ is small (and b ounded) in comparison to n . Of course, these are equiv alen t asymp totic gauges and it is natural to relate them b y a p ow er of the sp atial scaling v ariable w =  1 + ℓ n  (19) whic h we think of as a quant it y close to 1. This is similar to x bu t arises in a diﬀerent con text. This v ariable also serv es to mediate b et w een the asso ciated ﬁn e scaling v ariables. F or instance if s n ν − 1 = − 2 θ = ˜ s ( n + ℓ ) ν − 1 , then , (20) ˜ s = w ν − 1 s. (21) 2.2. Con t in uum Limit s and C luster Expansions. Theorem 2.1. [10] The c ontinuum limit of the T o da L attic e e quations (94 ) with r esp e ct to the time sc aling (17) and the lattic e sc aling (19) as n → ∞ is giv e n by the fol lowing inﬁnite or der p artial diﬀer ential e quation f or f ( s, w ) : f s = F ( ν )  n − 1 ; f , f w , . . . , f w m , . . .  . = c ν f ν f w + 1 n 2 F ( ν ) 1 ( f , f w , f w w , f w w w ) + · · · (22) + 1 n 2 g F ( ν ) g ( f , f w , f w (2) , · · · , f w (2 g +1) ) + · · · for ( s, w ) ne ar (0 , 1) and initial data gi ven by f (0 , w ) = w. F ( ν ) g = X λ : | λ | =2 g +1 ∋ ℓ ( λ ) ≤ ν +1 d ( ν,g ) λ Q j r j ( λ )! f ν − ℓ ( λ )+1 Y j  f w ( j ) j !  r j ( λ ) (23) wher e λ = ( λ 1 , λ 2 , . . . ) is a p artition, with λ 1 ≥ λ 2 ≥ λ 3 ≥ · · · , of 2 g + 1 ; r j ( λ ) = # { λ i | λ i = j } ; ℓ ( λ ) = P j r j ( λ ) is the length of λ ; and | λ | = P i λ i is the size of λ ; and d ( ν,g ) λ ar e c o eﬃcie nts to b e describ e d in the next pr op osition. The follo wing prop osition give s an explicit closed form expression for the co eﬃcien ts d ( ν,g ) λ . This is a new result. I ts pro of will b e giv en in A.3. Prop osition 2.2. d ( ν,g ) λ = X ( ν + 1 , ν, . . . , 2 , 1) ⊆ µ ⊆ (2 ν , 2 ν − 1 , . . . , ν ) µ ∈ R 2 m λ ( µ 1 − η 1 , . . . , µ ν +1 − η ν +1 ) 16 N. M. ERCOLANI where R is the set of r estricte d p artitions (meaning that µ 1 > µ 2 > · · · > µ ν +1 ), ( η 1 , . . . , η ν +1 ) = (2 ν, 2 ν − 2 , . . . , 2 , 0), and m λ ( x 1 , . . . , x ν +1 ) is the m onomial symmetric p olynomial asso ci- ated to λ [21]. The r elation of inclusion b et wee n partitions, ρ ⊆ µ means that µ j ≥ ρ j for all j . Remark 6. As a p olynomial in the w -deriv ativ es of f , F ( ν ) g is r eminiscen t of the p artial Bel l p olynomials , B | λ | ,ℓ ( λ ) ; ho wev er, it is d iﬀerent from these in that it dep en d s on an additional parameter, ν + 1, and the co eﬃcien ts, whic h d ep end on d ( ν,g ) λ , are combinatoria lly more complex. W e n ote also that th ese co eﬃcien ts diﬀer slightly from those present ed in [10] in that the analogous co eﬃcien ts in the former pap er incorp orated the parts m ultiplicities factor, 1 / Q j r j !, in to the co eﬃcien t itself. In common parlance, one w ould tak e the cont in uum limit in T heorem 2.1 to b e the limit of these equations w h en n → ∞ ; i.e., the le ading or der terms which constitute a ﬁ nite order (in f act ﬁ rst order) p de. W e will in fact b e fo cus sing on this leading order equation in the next t w o subsections. How ev er, for the general analysis w e wan t to carry out in this pap er, w e will n eed to lo ok at the successiv e higher ord er corrections pr esen t in (22). In fact we will n eed to consider these corrections to all orders. So w e are int erested in th e hierarch y of ﬁnite order p d es generated b y this expansion. Th is is completely in the spirit of the asymptotic analysis of w eakly n onlinear d iﬀeren tial equations. As in that s etting, w e p osit the solution to hav e the form of an asymptotic exp ansion: f ( s, w ) = f 0 ( s, w ) + 1 n 2 f 1 ( s, w ) + · · · + 1 n 2 g f g ( s, w ) + · · · . (24) The hierarch y of equations in question m a y then b e dev elop ed by inserting (24) int o (22 ) and then su ccessiv ely , as g increases, collecti ng term s of order in n − 2 g . It sh ould b e clear that at eac h order th ese equations will b e partial diﬀerent ial equations, but in several dep en- den t v ariables: f 0 , f 1 , . . . , f g − 1 , f g . I t should also b e clear that th is hierarc hy is a tr iangular system in these d ep endent v ariables: at order 2 g the p de is an equ ation for f g ( s, w ) for c e d b y diﬀeren tial terms in f 0 , f 1 , . . . , f g − 1 whic h should already b e kno w n functions having b een determined by solving the lo w er order equations in the hierarch y . T hese statemen ts and the structure of the hierarch y are m ad e precise in the next theorem. Theorem 2.3. [10] The or der 2 g e quation, for g > 0 , in the asymptotic exp ansion of (22) on (24) is d f g ds = c ν  ( f 0 ) ν ( f g ) w + ν ( f 0 ) ν − 1 ( f 0 ) w f g  + F orcing g , wher e (25) F orcing g =        c ν ν + 1 ∂ ∂ w X 0 ≤ k j < g k 1 + · · · + k ν +1 = g f k 1 · · · f k ν +1        (26) + F ( ν ) 1 [2 g − 2] + F ( ν ) 2 [2 g − 4] + · · · + F ( ν ) g [0] . CAUSTICS , COUNTING MA PS AND SEMI-CLASSICAL A SYMPTOTICS 17 F ( ν ) ℓ [2 r ] denotes the c o eﬃcient of n − 2 r in F ( ν ) ℓ . Note, in p articular, that such a term c annot involve f k if k ≥ r . In (26) it is natur al, and it wil l b e c onvenient, to denote the ﬁrst gr oup of terms by F ( ν ) 0 [2 g ] . Also, for g 6 = 0 , f g (0 , w ) = 0 and, at le ading or der, f 0 (0 , w ) = w . The structure of (25) and (26 ) suggests that these equations ma y b e solv ed r ecursiv ely unless a r esonanc e arises at some s tage that obs tructs the solution of the linear equation (25) and requ ires the int ro duction of a solv ability condition in order to con tinue. Absent suc h resonances, one exp ects to ﬁ n d exp ressions for eac h f g , for g > 0, as fu nctions, in some appropr iate function class, of just f 0 . Generally , in the asymptotic analysis of w eakly nonlinear diﬀerent ial equations, one do es not exp ect to get aw a y without runn ing in to resonances. Ho w ev er, in our sp ecial case w e do in fact av oid resonances at all stages. This w as established in [10]. In f act more w as s h o wn th ere: th e formal s er ies (24) that can b e bu ilt from these r ecursiv ely constructed co eﬃcien ts is in fact an asymp totic expansion uniformly v alid for s negativ e and near 0. The pr ecise statemen ts are giv en in th e n ext theorem. App endix A.2 outlines the essen tial element s un derlying these r esults. Theorem 2.4. [10] (i) f g ( s, w ) = w 1 − 2 g z g ( w ν − 1 s ) (27) f ( s, 1) = z 0 ( s ) + 1 n 2 z 1 ( s ) + 1 n 4 z 2 ( s ) + · · · . (ii) The c o eﬃcient z g is an ab elian function of z 0 with singularities only p ossible at z 0 = 0 and z 0 = ν / ( ν − 1) . (iii) The c o eﬃcient z g is mor e explicitly pr esente d as a f unction of z 0 thr ough the fol- lowing i nte gr al e quation: z g ( s ) = z g ( z 0 ( s )) = z 0 ( s ) 2(1 − g ) ν − ( ν − 1) z 0 ( s ) Z z 0 ( s ) 1 ( ν − ( ν − 1) z ) c ν z ν +3 − 2 g F orcing g ( z ) dz , wher e F orcing g ( z 0 ) = F orcing g   w =1 . The terms i n F orcing g | w =1 dep end only on z k ( z 0 ) , k < g and their derivatives and these derivatives c an in turn b e r e-expr esse d in terms of z k ( z 0 ) for k < g . (Se e se ctions 3.1.2 and 3.1.3). Note that b 2 n,N (see (10)) and x f ( s, 1) p ossess the same asymptotic expansion. Note also that since f (0 , w ) = w , and consistent w ith (11), z 0 (0) = 1 z g (0) = 0 for g ≥ 1 . 18 N. M. ERCOLANI A principal result of th is pap er is to reﬁne part (ii) of th e previous theorem to state that z g is a r ational fu nction of z 0 for g > 1. The rational functions are a sub class of the ab elian functions. Another wa y to sa y this is that w e establish solv abilit y , without resonances, of the diﬀerential hierarc h y (25 ), b ey ond g = 1, within the function class of rational f u nctions. There is a solv abilit y condition at g = 1 whic h is to c h o ose a branch of the logarithm at z 0 = ν / ( ν − 1). 2.3. Leading Order. The leading equation in the PDE scheme (22 ) is f s = c ν f ν f w (28) f (0 , w ) = w , whic h is an instance of the inviscid Bur gers e quation , well-kno wn fr om the theory of sho c k w a ve s. A t leading order in n this b ecomes d f 0 ds = c ν ( f 0 ) ν ( f 0 ) w . Substituting th e s elf-similar scaling (27) and then ev aluating at w = 1 it furth er r ed uces to the ODE z ′ 0 ( s ) = c ν z 0 ( s ) ν  z 0 ( s ) + ( ν − 1) sz ′ 0 ( s )  (29) with initial condition z 0 (0) = 1. T he s olution is given imp licitly b y (30) 1 = z 0 ( s ) − c ν sz 0 ( s ) ν , whic h is our third in terpretation of z 0 and the one f r om whic h our ﬁ rst one, (9), wa s deriv ed. Remark 7. W e mentio n that, in th e case ν = 2, (30) has a r elation to the follo wing discrete equation that play ed a k ey role in the early stud ies of 2D quantum gravit y in the physic s literature. 4 tb 2 n,N ( t )  b 2 n − 1 ,N ( t ) + b 2 n,N ( t ) + b 2 n +1 ,N ( t )  + b 2 n,N ( t ) = n N . (31) In the literature on orthogonal p olynomials, this relation is one of the basic examples of what are kno wn as F r e ud e quations [22]. In the p h y s ics literature it was referred to as the string e qu ation . It is an example of a discrete P ainlev ´ e equation and, in an app ropriate sense, is a discretization of the ordinary diﬀerential equation that d eﬁnes the (con tinuous) ﬁrst P ainlev ´ e transcendent [12, 13]. With r esp ect to the con tinuum limit describ ed in the previous subsection and based on Section A.2, (31) limits at leading ord er to (30) ev aluated at ν = 2. 2.4. The Burgers Caust ic. As illus trated in Figure 1, the solution, z 0 dev elops a c austic where z ′ 0 b ecomes und eﬁned. (This inﬁn ite slop e corresp ond s to the p oin t, in un scaled v ariables, where c h aracteristics for (28) would ﬁr st fo cu s or sho ck .) CAUSTICS , COUNTING MA PS AND SEMI-CLASSICAL A SYMPTOTICS 19 1 0 0 2 -0.05 -0.1 -0.15 -0.2 s z 4 3 5 Figure 1. Burgers Caustic F rom (29), and using (30), one can expr ess th e deriv ativ e pur ely in terms of z 0 as z ′ 0 = c ν z ν +1 0 ν − ( ν − 1) z 0 . (32) F rom this one sees that the sho ck coincides w ith the p rop osed uniqu e p ole lo cation, ν − ( ν − 1) z 0 = 0 , in the rational expressions for th e higher generating functions. F or futu r e use we wan t to calculate here the sho ck time s c whic h corresp onds to z 0 = ν / ( ν − 1); s c = s  z 0 = ν ν − 1  = z 0 − 1 c ν z ν 0 | z 0 = ν ν − 1 = ( ν − 1) ν − 1 c ν ν ν . (33) F rom (32) it is straigh tforward to derive the leading lo cal b eha vior near the caustic,  z 0 − ν ν − 1  = − s − 2 c ν ν ν +1 ( ν − 1) ν ( s − s c ) { 1 + O ( s − s c ) } . (34) In the sp ecial case of ν = 2, s c = 1 48 and one can solv e globally for z 0 from (30) u sing the quadratic formula, z 0 = 1 − √ 1 − 48 s 24 s . Remark 8. It follo w s from these observ ations and Theorems 1.3 and 1.5 that the maximal domain of holomor phy , conta ining t = 0, for b oth e g and z g is the set of t ∈ C \ ( − s c , ∞ ). 20 N. M. ERCOLANI 2.5. Another Interpretation of z 0 . The four th and ﬁnal interpretation of z 0 is in terms of the sp ectral densit y for r andom Hermitean matrices w ith resp ect to th e probabilit y distribution (1). F or th e class of weig h ts, V ν , w e are considering and for all t ≥ 0 (see section B.1), th e mean density of eigen v alues f or this ensemble limits to a cont in uous density supp orted on a ﬁnite interv al [ − β , β ]. This d ensit y deﬁ n es a p robabilit y measure, usu ally referred to as the e quilibriu m me asur e , w hic h h as the follo wing c haracterizatio n. Theorem 2.5. [5] The e q uilibrium me asur e µ V is absolutely c ontinuous with r esp e ct to L eb esgue me asur e, and dµ V = ψ dλ, ψ ( λ ) = 1 2 π χ ( − β ,β ) ( λ ) p ( λ + β )( β − λ ) h ( λ ) , (35) wher e h ( λ ) is a p olynomial of de gr e e 2 ν − 2 , which is strictly p ositive on the interval [ − β , β ] . The endp oin ts of supp ort ± β ( t ) v ary sm o othly with t (or equiv alen tly s ) and z 0 ( s ) = β 2 ( − s ) 4 . (36) The fact that this c haracterization of z 0 coincides with the other deﬁn itions giv en in th is and the previous section, follo ws from the requirement th at the equilibr ium measure should b e a probability measure. This is an algebraic cond ition on the end p oin ts of supp ort that is equ iv alent to (30). A d etailed explanation of this relation is given in [10]. The equilibr ium measur e is the unique s olution of the follo wing v ariational p roblem for logarithmic p otentia ls in the p resence of an external ﬁeld [25]. (In this setting, V is the external ﬁeld.) sup µ ∈ A  − Z V ( λ ) dµ ( λ ) + Z Z log | λ − η | dµ ( λ ) dµ ( η )  , (37) where A is the s et of all p ositiv e Borel measur es on the r eal axis with un it mass. This v ariational pr oblem, and hence the d eﬁnition of µ V , can b e extended to t ∈ ( − s c , 0) [7]. It follo ws that this inte rpretation of z 0 extends as w ell to this negativ e inte rv al and coincides, there, with its other inte rpretations giv en in this section. 3. Ra tional ity of the As ymptotic Recurren ce Coefficients The ﬁr st step to w ard pro ving Theorem 1.5 is to sh o w that z g m u st b e a rational function of z 0 . Give n parts (ii) and (iii) of Theorem 2.4, it suﬃces to show that the integ rand in Theorem 2.4 (iii) h as n o r esidues at z = 0 and z = ν / ( ν − 1). In [10] this v anish ing of residues w as established for z 1 , z 2 and z 3 (in fact, Th eorem 1.5 wa s fully established in these three cases b y direct calculation). If th ere is a term of F orcing g whic h generates a non-zero residu e in the inte grand of (2) w e call th at term r esonant . W e will sho w b y induction that for all g , F orcing g can b e decomp osed as a sum of terms eac h of wh ic h is non-resonan t. CAUSTICS , COUNTING MA PS AND SEMI-CLASSICAL A SYMPTOTICS 21 Prop osition 3.1. (i) z g is regular at z 0 = 0 and in f act v anish es at least simply there. (ii) F orcing g ma y b e w ritten as a sum of terms eac h of wh ich h as th e form f ν − m +1 0 ( f k 1 ) w ( j 1 ) ( f k 2 ) w ( j 2 ) · · · ( f k m ) w ( j m ) where 0 < m ≤ ν + 1 and j 1 + · · · + j m = 2( g − k 1 − k 2 − · · · − k m ) + 1, with k i < g . (iii) F or ( k , n ) 6 = (0 , 0) , and assuming that Theorem 1.5 h olds f or all 0 < k < g , ( f k ) w ( n ) ( s, 1) = z 0 ( b ( k, n ) 0 ( ν − ( ν − 1) z 0 ) 2 k + n + · · · + b ( k, n ) 3 k + n − 1 ( ν − ( ν − 1) z 0 ) 5 k + 2 n − 1 ) . The pr o of of the rational form (16) for z g follo ws fairly straigh tforw ardly from this p rop o- sition. Pro of (of The orem 1.5 ). Inser tin g expansions of the form (iii), from Pr op osition 3.1, for eac h f actor of the pro duct app earing in (ii), fr om Prop osition 3.1 , one derives the r ational form (as a f u nction of z 0 ) of eac h term in F orcing g (see Section 3.1.2 b elo w for fu rther explanation). One ma y then deduce, fr om Theorem 2.4(iii), that z g is a sum of terms, with K = k 1 + k 2 + · · · + k m and J = j 1 + j 2 + · · · + j m , of the form z 2 − 2 g 0 ν − ( ν − 1) z 0 Z z 0 1 ( ν − ( ν − 1) z ) c ν z ν +3 − 2 g · z ν +1  D 0 ( ν − ( ν − 1) z ) 2 g +1 + · · · . . . + D 3 K + J − m − 1 ( ν − ( ν − 1) z ) 5 K +2 J − m  dz = z 2 − 2 g 0 c ν ( ν − ( ν − 1) z 0 ) Z z 0 1 z 2 g − 2  D 0 ( ν − ( ν − 1) z ) 2 g + · · · . . . + D 3 K + J − m − 1 ( ν − ( ν − 1) z ) 5 K +2 J − m − 1  dz = z 2 − 2 g 0 c ν ( ν − ( ν − 1) z 0 ) Z z 0 1  ν − ( ν − ( ν − 1) z )) ν − 1  2 g − 2  D 0 ( ν − ( ν − 1) z ) 2 g + · · · . . . + D 3 K + J − m − 1 ( ν − ( ν − 1) z ) 5 K +2 J − m − 1  dz = z 2 − 2 g 0 c ν ( ν − ( ν − 1) z 0 ) Z z 0 1  E 0 ( ν − ( ν − 1) z ) 2 + · · · + E 5 K +2 J − m − 3 ( ν − ( ν − 1) z ) 5 K +2 J − m − 1  dz = 1 z 2 g − 2 0 ( ˆ E − 1 ( ν − ( ν − 1) z 0 ) + ˆ E 0 ( ν − ( ν − 1) z 0 ) 2 + · · · + ˆ E 5 K +2 J − m − 3 ( ν − ( ν − 1) z 0 ) 5 K +2 J − m − 1 ) (38) where the leading p ole term, with co eﬃcien t ˆ E − 1 , arises f rom the low er b oun d of inte gration whic h insur es that this expr ession v anishes at z 0 = 1. The in tegrand in the p en u ltimate line ab ov e is clearly non-resonant and it th er efore follo ws ind uctiv ely that z g is rational as 22 N. M. ERCOLANI a function of z 0 . Ho wev er, in order to establish the form for z g stated in equation (16) and to complete th e induction we n eed to make fu rther use of this r esult. Th e tota l expression for z g is comprised of a s um of series of the form (38), eac h ser ies coming from one of the terms in to which F orcing g ma y b e d ecomp osed (see Prop osition 3.1(ii)). W e determine next the maximal p ole order of this total expression. T he p ole order of eac h comp onen t s er ies as given by the formula ab o v e, is 5 K + 2 J − m − 1. By Prop osition 3.1: (ii), ther e is one constr aint, J = 2( g − K ) + 1, and t wo b oun dary conditions, 0 < m ≤ ν + 1 and 0 ≤ K ≤ g . It is straigh tforward to chec k that this linear pr ob lem is maximized o ver in teger v alues by ( K, m ) = ( g , 2) or ( K , m ) = ( g − 1 , 1), so that, resp ectiv ely , J = 1 or J = 3. Hence the maximal p ole order can b e at m ost 5 g − 1. When J = 1, the terms that con tr ibute to th e maximal p ole are of the form (39) ν c ν f ν − 1 0 g − 1 X m =1 f m ( f g − m ) w as can b e seen from (22). When J = 3, a term yielding this maximal p ole order could only come f rom the F ( ν ) 1 [2 g − 2] comp onen t of F orcing g as can b e s een from (26). Moreo ver, since m = 1, the only term within this comp onent giving th is maximal ord er comes from the summand asso ciated to the partition λ = (3). More will b e said later ab out the structure of th e co eﬃcien t of the highest order p ole (Section 4.4). F or n o w it will suﬃce to conclude that z g has the f orm z g = 1 z 2 g − 2 0  C 0 ( ν − ( ν − 1) z 0 ) + C 1 ( ν − ( ν − 1) z 0 ) 2 + · · · + C 5 g − 2 ( ν − ( ν − 1) z 0 ) 5 g − 1  (40) for all ν ≥ 2. By Prop osition 3.1 (i), z g v anishes at z 0 = 0. F or this to b e p ossible, th e Laurent ser ies in (40) m ust ha v e a factor of z 2 g − 1 0 and h en ce th is expr ession f or z g b ecomes z g = z 0 ( a ( g ) 0 ( ν − ( ν − 1) z 0 ) 2 g + a ( g ) 1 ( ν − ( ν − 1) z 0 ) 2 g +1 + · · · + a ( g ) 3 g − 1 ( ν − ( ν − 1) z 0 ) 5 g − 1 ) = z 0 ( z 0 − 1) P 3 g − 2 ( z 0 ) ( ν − ( ν − 1) z 0 ) 5 g − 1 (41) where the f actor of z 0 − 1 in the last line is inf er r ed from the fact that eac h term cont ributing to the ﬁn al expression v anishes at z 0 = 1. Th is completes the indu ction and th e pro of of Theorem 1.5.  3.1. Pro of of Prop osition 3.1 . CAUSTICS , COUNTING MA PS AND SEMI-CLASSICAL A SYMPTOTICS 23 3.1.1. R e gularity at z 0 = 0 . Recall from (32) that z 0 ( s ) satisﬁes the ﬁrs t order o d e z ′ 0 = c ν z ν +1 0 ( ν − ( ν − 1) z 0 ) . It follo ws from this d y n amical rep resen tation that if z 0 is initialized at 1, then its tra jectory , as s → −∞ , limits monotonically and asymp totical ly to th e ﬁxed p oint at z 0 = 0. Hence, to stud y the b ehavior of z g as z 0 → 0, it suﬃces to consider the limit of z g ( s ) as s → − ∞ . By Theorem B.3 one has an asym p totic expansion for the orthogonal p olynomial recurr ence co eﬃcien ts that is un if orm ly v alid for s in a neighborho o d of the half-line ( −∞ , 0) in th e complex s -plane, so that | b 2 N ,N ( − s ) − z 0 ( s ) − N − 2 z 1 ( s ) − · · · − N − 2 g z g ( s ) | < C N 2 g +2 (42) where C is indep endent of s in a neighborho o d of the negativ e s -axis. W e p r o ceed ind uc- tiv ely and so can assu me that z j for j < g v anishes as s → −∞ (w e kno w, to b egin with, that z 0 v anishes in this limit). If one r estricts to s on the negativ e real half-line, then (42) ma y b e re-expressed as z g − C N 2 < N 2 g  b 2 N ,N ( − s ) − z 0 ( s ) − N − 2 z 1 ( s ) − · · · − N − 2 g +2 z g − 1 ( s )  and (43) z g + C N 2 > N 2 g  b 2 N ,N ( − s ) − z 0 ( s ) − N − 2 z 1 ( s ) − · · · − N − 2 g +2 z g − 1 ( s )  . (44) W e no w c h o ose and ﬁx N so that C N 2 < ǫ 2 . Next w e note that b 2 N ,N ( − s ) ≤ 1 N Z ∞ −∞ λ 2 p 2 N ( s, λ ) e − N V ( s, λ ) dλ → 0 as s → −∞ , where p n is th e n th orthonormal p olynomial with resp ect to the exp onentia l weigh t exp ( − N V ( s, λ )) = exp  N ( 1 2 λ 2 − sλ 2 ν )  . T h e inequalit y is a direct consequence of th e recursion formula for these orthonormal p olynomials. The limit in the second lin e follo ws b ecause the family of densities { p 2 N ( s, λ ) e − N V ( s, λ ) } constitutes a d elta sequen ce as s → − ∞ . As a consequence of this observ ation and the ind u ction, one may c ho ose | s | su ﬃcien tly large so that the right-hand side of (43) is less than ǫ/ 2 and the right-hand side of (44) is greater than − ǫ/ 2. T herefore, | z g ( s ) | < C N 2 + ǫ 2 < ǫ for s suﬃ cien tly negativ e. Sin ce ǫ was arbitrary , this establishes statemen t (i) of Prop osition 3.1. 3.1.2. The Structur e of F or cing g . By (26), F orcing g is a graded sum F ( ν ) 0 [2 g ] + F ( ν ) 1 [2 g − 2] + · · · + F g [0] 24 N. M. ERCOLANI where F ( ν ) r [2 g − 2 r ] is the co eﬃcien t of n − 2 g +2 r in F ( ν ) r . T he terms in F ( ν ) r are in 1 − 1 corresp ondence with the partitions of 2 r + 1 as d escrib ed in (23); i.e., the terms in this comp onen t are eac h prop ortional to an expression of the form (45) f ν − m +1 0 ( f k 1 ) w ( j 1 ) ( f k 2 ) w ( j 2 ) · · · ( f k m ) w ( j m ) with 0 ≤ k i < g and w h ere P m i =1 j i = 2 r + 1. The fu rther requir emen t that these terms are co eﬃcien ts of the p o wer n − 2 g +2 r in F ( ν ) r means th at P m i =1 k i = 2 g − 2 r . Eliminating r fr om these tw o equations we ma y conclude that eac h term of F orcing g is prop ortional to an exp r ession of the form (45 ) wh ere j 1 + · · · + j m = 2( g − k 1 − k 2 − · · · − k m ) + 1. This establishes statemen t (ii) of Pr op osition 3.1. 3.1.3. L aur ent Exp ansions of F or c i ng g . In this section w e set up and pro v e the main lemma needed to establish the p ole expans ions, Prop osition 3.1 (iii), of the fund amen tal factors, ( f k ) w ( n ) ( s, 1). F rom (27) one has f k ( s, w ) = w 1 − 2 k z k ( w ν − 1 s ) . One can calculate the ﬁr st couple deriv ativ es d irectly: ( f k ) w = (1 − 2 k ) w − 2 k z k + ( ν − 1) w ν − 1 − 2 k sz ′ k ( f k ) w w = − 2 k (1 − 2 k ) w − (2 k +1) z k + ( ν − 1)( ν − 4 k ) w ν − 2 − 2 k sz ′ k + ( ν − 1) 2 w 2 ν − 3 − 2 k s 2 z ′′ k The general form of the higher order w -deriv ativ es is giv en by ( f k ) w ( n ) ( s, w ) = n X j = 0 ( ν − 1) j P ( n,k ) j ( ν ) w ( ν − 1) j − (2 k + n − 1) s j z ( j ) k Note that the co eﬃcien t of w ( ν − 1) j − (2 k + n − 1) s j z ( j ) k has t w o sour ces in ( f k ) w ( n − 1) . T hese tw o sources are ( ν − 1) j − 1 P ( n − 1 ,k ) j − 1 ( ν ) w ( ν − 1)( j − 1) − (2 k + n − 1)+1 s j − 1 z ( j − 1) k and ( ν − 1) j P ( n − 1 ,k ) j ( ν ) w ( ν − 1) j − (2 k + n − 1)+1 s j z ( j ) k . Giv en this, one can r e ad oﬀ a r ecur sion f orm u la for the co eﬃcient s P ( n,k ) j ( ν ) that is give n in the next lemma. Lemma 3.1. ( f k ) w ( n ) ( s, 1) = n X j = 0 ( ν − 1) j P ( n,k ) j ( ν ) s j z ( j ) k (46) CAUSTICS , COUNTING MA PS AND SEMI-CLASSICAL A SYMPTOTICS 25 where P ( n,k ) j ( ν ) = P ( n − 1 ,k ) j − 1 ( ν ) + { ( ν − 1) j − (2 k − 2 + n ) } P ( n − 1 ,k ) j ( ν ) (47) with P ( n,k ) 0 ( ν ) = (1 − 2 k )( − 2 k ) · · · (1 − 2 k − n + 1) (48) P ( n,k ) n ( ν ) = 1 and (49) P ( n,k ) j ( ν ) = 0 for j > n and j < 0 .. (50) W e next turn to stu dying th e Laurent expansion of z k around z 0 = ν ν − 1 . T his analysis is sub ord inate to the indu ctiv e assump tion that f or 0 < k < g , z k ( z 0 ) = z 0 ( z 0 − 1) P 3 k − 2 ( z 0 ) ( ν − ( ν − 1) z 0 ) 5 k − 1 . It f ollo ws that z k = z 0 ( a ( k, 0) 0 ( ν ) ( ν − ( ν − 1) z 0 ) 2 k + a ( k, 0) 1 ( ν ) ( ν − ( ν − 1) z 0 ) 2 k + 1 + · · · + a ( k, 0) 3 k − 1 ( ν ) ( ν − ( ν − 1) z 0 ) 5 k − 1 ) . Moreo v er, diﬀerentia ting this expr ession j times, w ith r esp ect to s , one arriv es at z ( j ) k = c j ν z j ν +1 0 3 k − 1+ j X ℓ =0 a ( k, j ) ℓ ( ν − ( ν − 1) z 0 ) 2 k + ℓ + j ! . (51) 26 N. M. ERCOLANI T o s ee th is, n ote that z ( j ) k = dz ( j − 1) k ds = c j − 1 ν z ( j − 1) ν 0 z ′ 0 3 k − 1+ j − 1 X ℓ =0 [( j − 1) ν + 1] a ( k, j − 1) ℓ ( ν − ( ν − 1) z 0 ) 2 k + ℓ + j − 1 + ( ν − 1) z 0 3 k − 1+ j − 1 X ℓ =0 (2 k + ℓ + ( j − 1)) a ( k, j − 1) ℓ ( ν − ( ν − 1) z 0 ) 2 k + ℓ + j ! = c j − 1 ν z ( j − 1) ν 0 z ′ 0 3 k + j − 2 X ℓ =0 [( j − 1)( ν − 1) − 2 k − ℓ + 1] a ( k, j − 1) ℓ ( ν − ( ν − 1) z 0 ) 2 k + ℓ + j − 1 + ν 3 k + j − 2 X ℓ =0 (2 k + ℓ + ( j − 1)) a ( k, j − 1) ℓ ( ν − ( ν − 1) z 0 ) 2 k + ℓ + j ! = c j ν z j ν +1 0 3 k + j − 2 X ℓ =0 [( j − 1)( ν − 1) − 2 k − ℓ + 1] a ( k, j − 1) ℓ ( ν − ( ν − 1) z 0 ) 2 k + ℓ + j + ν 3 k + j − 2 X ℓ =0 (2 k + ℓ + ( j − 1)) a ( k, j − 1) ℓ ( ν − ( ν − 1) z 0 ) 2 k + ℓ + j + 1 ! = c j ν z j ν +1 0 3 k − 1+ j X ℓ =0 [( j − 1) ν − (2 k + ℓ + ( j − 2)] a ( k, j − 1) ℓ + ν (2 k + ℓ + ( j − 2)) a ( k, j − 1) ℓ − 1 ( ν − ( ν − 1) z 0 ) 2 k + ℓ + j ! where, in the fourth equation, the identit y (32), z ′ 0 = dz 0 ds = c ν z ν +1 0 ν − ( ν − 1) z 0 , was applied. F rom this one deduces the follo wing recursion for the su ccession of derive d Laurent co- eﬃcien ts Lemma 3.2. a ( k, j ) ℓ = [( j − 1) ν − (2 k + ℓ + ( j − 2))] a ( k, j − 1) ℓ + ν [2 k + ℓ + ( j − 2)] a ( k, j − 1) ℓ − 1 with a (0 , 0) 0 = 1 a (0 , 0) ℓ = 0 for ℓ > 0 a ( k, j ) ℓ = 0 for ℓ < 0 and ℓ ≥ 3 k + j. (52) Finally we consider the form of the expressions s j z ( j ) k whic h app ear in (46). Recall, from (30), that s = z 0 − 1 c ν z ν 0 , (53) CAUSTICS , COUNTING MA PS AND SEMI-CLASSICAL A SYMPTOTICS 27 so one ma y deduce from (3.1.3) that s j z ( j ) k = z 0 ( z 0 − 1) j 3 k − 1+ j X ℓ =0 a ( k, j ) ℓ ( ν − ( ν − 1) z 0 ) 2 k + ℓ + j = z 0 (1 − ( ν − ( ν − 1) z 0 )) j ( ν − 1) j 3 k − 1+ j X ℓ =0 a ( k, j ) ℓ ( ν − ( ν − 1) z 0 ) 2 k + ℓ + j = z 0 ( ν − 1) j j X r =0 3 k − 1+ j X ℓ =0 ( − 1) j − r  j j − r  a ( k, j ) ℓ ( ν − ( ν − 1) z 0 ) 2 k + ℓ + r = z 0 ( ν − 1) j j X r =0 3 k + j + r − 1 X m = r ( − 1) j − r  j r  a ( k, j ) m − r ( ν − ( ν − 1) z 0 ) 2 k + m where m = ℓ + r . In s erting this last exp ansion int o the expansion for ( f k ) w ( n ) ( s, 1) giv en b y (46), and app lyin g the v an ish ing conditions (52) to extend the b ounds of the in ner summation, yields ( f k ) w ( n ) ( s, 1) = z 0 n X j = 0 P ( n,k ) j j X r =0 3 k + 2 j − 1 X m =0 ( − 1) j − r  j r  a ( k, j ) m − r ( ν − ( ν − 1) z 0 ) 2 k + m = z 0 3 k + 2 n − 1 X m =0  P n j = 0 P ( n,k ) j ( ν ) P j r =0 ( − 1) j − r  j r  a ( k, j ) m − r ( ν )  ( ν − ( ν − 1) z 0 ) 2 k + m = z 0 3 k + 2 n − 1 X m =0  P n j = 0 P ( n,k ) j ( ν ) P m r =0 ( − 1) j − r  j r  a ( k, j ) m − r ( ν )  ( ν − ( ν − 1) z 0 ) 2 k + m . (54) In the second line w e applied (52 ) again to extend the up p er b ound of the innermost summation on the ﬁr st line in order to b e ab le to pull this s u mmation to the outside. In the last line, th e upp er b ou n d of the innermost sum mation was c han ged from j in the previous line to m . T o jus tify this note that if m < j , then the terms with ind ex greater than m in the original su m v anish anyw a y b y (52); on the other hand, if m > j then terms greater than j in the new sum will v anish since  j r  = 0 if r > j p er the usu al con ven tions for binomial co eﬃcien ts. The pr o of of Pr op osition 3.1(iii) thus comes do wn to establishing the follo wing v anishing lemma. 28 N. M. ERCOLANI Lemma 3.3. n X j = 0 P ( n,k ) j ( ν ) m X r =0 ( − 1) j − r  j r  a ( k, j ) m − r ( ν ) = 0 (55) for m = 0 , 1 , . . . n − 1. Pro of. Fix k > 0. The argument p ro ceeds ind uctiv ely . Th e base step is for n = 1 which implies that m = 0. So the expression to consider is 1 X j = 0 P (1 ,k ) j ( ν )( − 1) j  j 0  a ( k, j ) 0 ( ν ) = P (1 ,k ) 0 ( ν ) a ( k, 0) 0 ( ν ) − P (1 ,k ) 1 ( ν ) a ( k, 1) 0 ( ν ) =  P (1 ,k ) 0 ( ν ) + P (1 ,k ) 1 ( ν )(2 k − 1)  a ( k, 0) 0 ( ν ) by Lemma 3.2 = [1 − 2 k + 1 · (2 k − 1)] a ( k, 0) 0 ( ν ) by (46) = 0 . F or the indu ction step, we assume the lemma is tru e for n − 1; i.e., n − 1 X j = 0 P ( n − 1 ,k ) j ( ν ) m X r =0 ( − 1) j − r  j r  a ( k, j ) m − r ( ν ) = 0 CAUSTICS , COUNTING MA PS AND SEMI-CLASSICAL A SYMPTOTICS 29 for m = 0 , 1 , . . . n − 2, and consider th e expressions on the left-hand sides of the putativ e equations in Lemma 3.3: n X j = 0 ( − 1) j P ( n,k ) j m X r =0 ( − 1) r  j r  a ( k, j ) m − r = − (2 k − 2 + n ) n X j = 0 ( − 1) j P ( n − 1 ,k ) j m X r =0 ( − 1) r  j r  a ( k, j ) m − r + n X j = 0 ( − 1) j h P ( n − 1 ,k ) j − 1 + { ( ν − 1) j } P ( n − 1 ,k ) j i m X r =0 ( − 1) r  j r  a ( k, j ) m − r applying (48) = − (2 k − 2 + n ) n − 1 X j = 0 ( − 1) j P ( n − 1 ,k ) j m X r =0 ( − 1) r  j r  a ( k, j ) m − r − n − 1 X j = 0 ( − 1) j P ( n − 1 ,k ) j m X r =0 ( − 1) r  j + 1 r  a ( k, j +1) m − r + ( ν − 1) n − 1 X j = 0 ( − 1) j P ( n − 1 ,k ) j m X r =0 ( − 1) r j  j r  a ( k, j ) m − r b y (50) and sh ifting j in the midd le sum = − (2 k − 2 + n ) n − 1 X j = 0 ( − 1) j P ( n − 1 ,k ) j m X r =0 ( − 1) r  j r  a ( k, j ) m − r − n − 1 X j = 0 ( − 1) j P ( n − 1 ,k ) j m X r =0 ( − 1) r  j + 1 r  h { j ν − (2 k + m − r + ( j − 1)) } a ( k, j ) m − r + ν { 2 k + m − r + ( j − 1) } a ( k, j ) m − r − 1 i + ( ν − 1) n − 1 X j = 0 ( − 1) j P ( n − 1 ,k ) j m X r =0 ( − 1) r j  j r  a ( k, j ) m − r b y L emma 3.2 = − (2 k − 2 + n ) n − 1 X j = 0 ( − 1) j P ( n − 1 ,k ) j m X r =0 ( − 1) r  j r  a ( k, j ) m − r + (2 k − 1 + m ) n − 1 X j = 0 ( − 1) j P ( n − 1 ,k ) j m X r =0 ( − 1) r  j + 1 r  n a ( k, j ) m − r − ν a ( k, j ) m − r − 1 o + n − 1 X j = 0 ( − 1) j P ( n − 1 ,k ) j m X r =0 ( − 1) r ( j − r )  j + 1 r  n a ( k, j ) m − r − ν a ( k, j ) m − r − 1 o − ν n − 1 X j = 0 ( − 1) j P ( n − 1 ,k ) j m X r =0 ( − 1) r j  j + 1 r  a ( k, j ) m − r + ( ν − 1) n − 1 X j = 0 ( − 1) j P ( n − 1 ,k ) j m X r =0 ( − 1) r j  j r  a ( k, j ) m − r . 30 N. M. ERCOLANI W e n ext r earrange th e terms in the last expression and mak e use of P asc al’s identity ,  j + 1 r  =  j r  +  j r − 1  to ac hieve some redu ctions. n X j = 0 ( − 1) j P ( n,k ) j m X r =0 ( − 1) r  j r  a ( k, j ) m − r = ( m − ( n − 1)) n − 1 X j = 0 ( − 1) j P ( n − 1 ,k ) j m X r =0 ( − 1) r  j r  a ( k, j ) m − r + (2 k − 2 + m ) n − 1 X j = 0 ( − 1) j P ( n − 1 ,k ) j m X r =0 ( − 1) r  j r − 1  n a ( k, j ) m − r − ν a ( k, j ) m − r − 1 o + n − 1 X j = 0 ( − 1) j P ( n − 1 ,k ) j m X r =0 ( − 1) r j  j r  n a ( k, j ) m − r − ν a ( k, j ) m − r − 1 o − ν n − 1 X j = 0 ( − 1) j P ( n − 1 ,k ) j m X r =0 ( − 1) r j  j r − 1  a ( k, j ) m − r − n − 1 X j = 0 ( − 1) j P ( n − 1 ,k ) j m X r =0 ( − 1) r j  j r  a ( k, j ) m − r = ( m − ( n − 1)) n − 1 X j = 0 ( − 1) j P ( n − 1 ,k ) j m X r =0 ( − 1) r  j r  a ( k, j ) m − r + (2 k − 2 + m ) n − 1 X j = 0 ( − 1) j P ( n − 1 ,k ) j m X r =1 ( − 1) r  j r − 1  n a ( k, j ) m − r − ν a ( k, j ) m − r − 1 o − ν n − 1 X j = 0 ( − 1) j P ( n − 1 ,k ) j m − 1 X r =0 ( − 1) r j  j r  a ( k, j ) m − r − 1 − ν n − 1 X j = 0 ( − 1) j P ( n − 1 ,k ) j m X r =1 ( − 1) r j  j r − 1  a ( k, j ) m − r = ( m − ( n − 1)) n − 1 X j = 0 ( − 1) j P ( n − 1 ,k ) j m X r =0 ( − 1) r  j r  a ( k, j ) m − r − (2 k − 2 + m ) n − 1 X j = 0 ( − 1) j P ( n − 1 ,k ) j ( m − 1 X r =0 ( − 1) r  j r  a ( k, j ) m − r − 1 − ν m − 2 X r =0 ( − 1) r  j r  a ( k, j ) m − r − 2 ) . CAUSTICS , COUNTING MA PS AND SEMI-CLASSICAL A SYMPTOTICS 31 The second lin e of this ﬁnal expression v anishes by ind uction for all m ≤ n − 1 since the upp er b ou n ds of the internal summations, m − 1 or m − 2, are ≤ n − 2 in th is r ange. Similarly , the ﬁr st line v anish es, for m ≤ n − 2, by induction. F or m = n − 1 it also v anishes b ecause the leading co eﬃcien t, m − ( n − 1), v anish es for this v alue of m . This completes the induction and the pro of of Lemma 3.3.  4. Applica t ions 4.1. In t egral and Recursion F orm ulas for Map Coun t s. In this section we will give sev eral representat ions of the count for tw o-legged m aps (12). The most d irect one will in v olve the Laurent expansions of z ( j ) g that w ere deve lop ed in S ection 3.1. W e mak e use of the deﬁn ing imp licit relation, (30), for z 0 giv en b y (56) 1 = z ( s ) − αz ( s ) ν where α = c ν s and c ν = 2 ν  2 ν − 1 ν − 1  . The j th co eﬃcien t of the T a ylor expansion of z g as a f unction of α near 0, z g = P j ≥ 0 ζ ( g ) j ( ν ) α j , is n aturally giv en by ζ ( g ) j ( ν ) = 1 2 π i I z g ( α ) α j + 1 dα. (57) One can c hange v ariables from α near 0, in th is int egral, to z near 1 by diﬀerent iating the relation (56) dz dα = z ν 1 − ν αz ν − 1 and then using the relation again to eliminate α in the d iﬀerential dα dz = 1 − ν αz ν − 1 z ν = ν − ( ν − 1) z z ν +1 , so that ζ ( g ) j ( ν ) = 1 2 π i I z ∼ 1 ( ν − ( ν − 1) z ) z ν j − 1 z g ( z ) ( z − 1) j + 1 dz = 1 2 π i I z ∼ 1 z ν j P 3 g − 2 ( z ) ( z − 1) j ( ν − ( ν − 1) z ) 5 g − 2 dz . 32 N. M. ERCOLANI In the second line w e ha v e rewr itten th e int egrand using the more explicit form of z g ( z ) giv en by Theorem 1.5. An alternativ e expr ession can b e f ound by usin g instead the p artial fractions expansion of z g giv en by (41). ζ ( g ) j ( ν ) = 3 g − 1 X i =0 1 2 π i I z ∼ 1 a ( g, 0) i ( ν ) z ν j ( y − 1) j + 1 ( ν − ( ν − 1) z ) 2 g + i − 1 dz where P 3 g − 1 i =0 a ( g, 0) i ( ν ) = 0 since z g ( z ) v anishes at z = 1 for g > 0. These in tegral formulas f or the generating fun ction co eﬃcients ma y b e re-expressed as recursion formulas. T o accomplish th is transformation we return to the α con tour int egral (57) whic h ma y b e recast in terms of higher deriv ativ es as ζ ( g ) j ( ν ) = 1 2 π i 1 j ! c j ν I s ∼ 0 z ( j ) g ( s ) s ds = 1 2 π i 1 j ! c j ν I z ∼ 1 ( ν − ( ν − 1) z ) z ( j ) g ( z ) z ( z − 1) dz (58) = 1 2 π i 1 j ! I z ∼ 1 z j ν z − 1 3 g − 1+ j X ℓ =0 a ( g, j ) ℓ ( ν ) ( ν − ( ν − 1) z ) 2 g − 1+ j + ℓ ! dz . (59) where in the second line we app lied the Laur ent exp ansion (51). Corollary 4.1. z ( j ) g (0) = # { t wo- legged g -maps w ith j 2 ν -v alen t vertice s } = j ! c j ν ζ ( g ) j ( ν ) = c j ν 3 g − 1+ j X ℓ =0 a ( g, j ) ℓ ( ν ) and a ( g, j ) ℓ ( ν ) = [( j − 1) ν − (2 g + ℓ + ( j − 2))] a ( g, j − 1) ℓ ( ν ) + ν [2 g + ℓ + ( j − 2)] a ( g, j − 1) ℓ − 1 ( ν ) b y L emma 3.2 with a (0 , 0) 0 ( ν ) = 1 a (0 , 0) ℓ ( ν ) = 0 for ℓ > 0 a ( g, j ) ℓ ( ν ) = 0 for ℓ < 0 and 3 g − 1 X ℓ =0 a ( g, 0) ℓ ( ν ) = 0 . CAUSTICS , COUNTING MA PS AND SEMI-CLASSICAL A SYMPTOTICS 33 The co eﬃcien ts a ( g, j ) ℓ ( ν ) app earing in this ev aluation are th us recursive ly expressible in term of the 3 g co eﬃcien ts, a ( g, 0) ℓ ( ν ), of the partial fractions expans ion of z g . These latter co eﬃcien ts are rational fu nctions of the d ( ν,k ) λ , for k ≤ g , ov er the rational n um b ers Q . 4.2. Rationalit y of the Asymptotic Partition F unction Co e ﬃcients. W e w ill no w mak e use of the follo wing recurs iv e integ ral form ulae for the fu ndament al coeﬃcients e g . (The deriv ation of these formulae is acco mplished by a pro cess s imilar to that describ ed in section A.2 for the co eﬃcien ts z g ; for the details we refer the reader to [10 ].) Theorem 4.2. [10] e g ( − s ) = − 1 (2 − 2 g )(1 − 2 g ) driv ers g ( z 0 ( s )) (60) − 1 2 − 2 g  z 0 ( s ) − 1 c ν z 0 ( s ) ν  (2 g − 2) / ( ν − 1) Z z 0 ( s ) 1  c ν z ν z − 1  (2 g − 2) / ( ν − 1) (driv ers g ( z )) • dz + 1 (1 − 2 g )  z 0 ( s ) − 1 c ν z 0 ( s ) ν  (2 g − 1) / ( ν − 1) Z z 0 ( s ) 1  c ν z ν z − 1  (2 g − 1) / ( ν − 1) (drivers g ( z )) • dz + K 1 s (2 g − 2) / ( ν − 1) + K 2 s (2 g − 1) / ( ν − 1) when g 6 = 1 , wher e driv ers g ( z 0 ( s )) = − g X ℓ =1 2 (2 ℓ + 2)! ∂ (2 ℓ +2) ∂ w (2 ℓ +2) h w 2 − 2( g − ℓ ) e g − ℓ ( − w ν − 1 s ) i     w =1 (61) + the n − 2 g term of log ∞ X m =0 1 n 2 m z m ( s ) ! . We denote by (drivers g ( z )) • the derivative of drive rs g ( z ) with r esp e ct to z . K 1 and K 2 ar e c onstants of inte gr ation either determine d by the r e quir ement that e g b e a lo c al ly analytic function of s or by the e v aluation of e g for low values of ν thr ough its c ombinatorial char acterization. When g = 1 , e 1 ( − s ) = 1 ( ν − 1) "  z 0 ( s ) − 1 c ν z 0 ( s ) ν  1 / ( ν − 1) Z z 0 ( s ) 1  c ν z ν z − 1  ν / ( ν − 1) ( ν − ( ν − 1) z ) c ν z ν +1 driv ers 1 ( z ) dz − Z z 0 ( s ) 1 ( ν − ( ν − 1) z ) z ( z − 1) driv ers 1 ( z ) dz # = − 1 12 log ( ν − ( ν − 1) z 0 ( s )) , (62) wher e we have chosen the princip al br anch of the lo garithm. 34 N. M. ERCOLANI F or c onvenienc e we also r e c or d her e the planar ( g = 0 ) r esult, (63) e 0 ( − s ) = 1 2 log( z 0 ( s )) + ( ν − 1) 2 4 ν ( ν + 1) ( z 0 ( s ) − 1)  z 0 ( s ) − 3( ν + 1) ν − 1  . W e can no w extend this c haracterizatio n b y pro ving Theorem 1.3. F or g ≥ 2 , the c o eﬃc i ent e g ( − s ) = e g ( z 0 ( s )) is a r ational fu nction of z 0 with p oles only at z 0 = ν / ( ν − 1) and which vanishes at le ast simply at z 0 = 1 . Pro of. W e p ro ceed inductiv ely assumin g th e th eorem to hold f or all e r with 2 ≤ r < g and with e 1 and e 0 as give n, resp ectiv ely , by (62) and (63). T o start, one needs to examine the terms of drivers g ( z ) for g ≥ 2. Note that for 1 ≤ ℓ ≤ g − 2, ∂ ∂ w h w 2 − 2( g − ℓ ) e g − ℓ ( − w ν − 1 s ) i (64) = (2 − 2( g − ℓ )) w 1 − 2( g − ℓ ) e g − ℓ  − w ν − 1 s  + w ν − 2( g − ℓ ) se • g − ℓ  − w ν − 1 s  = (2 − 2( g − ℓ )) w 1 − 2( g − ℓ ) e g − ℓ  − w ν − 1 s  + w ν − 2( g − ℓ ) z 0 − 1 c ν z ν 0 e • g − ℓ  − w ν − 1 s  where in the thir d lin e we ha v e used the id en tit y (53). The ﬁ nal expression for (64) h as three prop erties that will b e u seful to note: It v anishes at z 0 = 1 . (65) Its minimal p ole ord er at z 0 = ν /ν − 1 is not less than that of e g − ℓ ( z ) . (66) It ma y also ha v e p oles at z 0 = 0 but no where else . (67) It is straigh tforward to see that these three prop erties are maint ained un der fu rther diﬀer- en tiation with resp ect to w . W e next separately c hec k that these same prop erties hold for w -deriv ativ es of e g − ℓ when ℓ = g − 1 or g , whic h corresp ond resp ectiv ely to e 1 and e 0 . F or e 1 w e hav e ∂ ∂ w  e 1 ( − w ν − 1 s )  = ν − 1 12 · w ν − 2 ν − ( ν − 1) z 0 · z 0 − 1 c ν z ν 0 . It is clear from this th at the 2 g th w -d eriv ative of e 1  − w ν − 1 s  app earing in drivers g has the three prop erties (65 - 67). Similarly one can see from (63) that the (2 g + 2) th w -d eriv ative of w 2 e 0  − w ν − 1 s  app earing in d riv ers g has the th r ee pr op erties (65 - 67). Prop ert y (66) is v acuous in this case since the expression has no p oles at z 0 = ν /ν − 1 ; we tak e the minimal p ole ord er in this case to b e ∞ . Lastly w e consider the last collection of terms in drivers g . log ∞ X m =0 1 n 2 m z m ( s ) ! = log( z 0 ( s )) + log 1 + ∞ X m =1 1 n 2 m z m ( s ) z 0 ( s ) ! . CAUSTICS , COUNTING MA PS AND SEMI-CLASSICAL A SYMPTOTICS 35 The n − 2 g term of this expression has terms of the form, for m 1 + · · · + m r = g , z m 1 . . . z m r z r 0 = ( z 0 − 1) r Q 3 g − 2 r ( z 0 ) ( ν − ( ν − 1) z 0 ) 5 g − r where Q 3 g − 2 r is a p olynomial of degree 3 g − 2 r . The form of the righ t h an d side follo ws directly f r om Theorem 1.5 and sh o ws that th is part of d riv ers g v anishes at z 0 = 1, has minim um p ole order of 2 g at z 0 = ν ν − 1 and of order ∞ at z 0 = 0. W e m a y thus conclude that Lemma 4.1. F or g ≥ 2, drivers g ( z ) v anishes at least simply at z 0 = 1, and its only singularities are p oles restricted to o ccur at either z = 0 or z = ν / ( ν − 1). Also, driv ers g ( z ) • has only p oles for singularities wh ic h are restricted to o ccur at either z = 0 or z = ν / ( ν − 1) with min imal p ole ord er at least 2 at b oth lo cations. W e also observ e, as was n oted in [10], that the terms of (60) inv olving K 1 and K 2 can only ha ve the p ossibilit y to con tribute if (2 g − 2) / ( ν − 1) or (2 g − 1) / ( ν − 1) are in tegers. But in that case it follo ws fr om (53) that they are terms which v anish simply at z 0 = 1 and ha ve p oles on ly at z 0 = 0. As a consequence of this observ ation and the lemma one ma y conclude from the repr esen tation (60) that e g , for g ≥ 2, is lo cally r ational at z 0 = ν ν − 1 . It th u s suﬃces to consider the b ehavior of (60) near z 0 = 0 , 1 in order to establish Theorem 1.3. Returning to Theorem 1.1, note that when t = 0 , log  Z ( k ) N ( t ) Z ( k ) N ( 0 )  = 0; therefore, e g = 0 when s = 0 (or equ iv alently wh en z 0 = 1) for all g . F or the in tegral terms in (60), since b y the p receding lemma driv ers g ( z ) • is r egular at z = 1, one m a y write th e integrand in eac h of th ese terms as a Puiseux series around z = 1. The integ ral can th en b e carried out term by term on this series with constan t of integrat ion set to zero (n ote that the constan ts of int egration are already captured in the terms cont aining the constan ts K 1 and K 2 . Ho w ev er, the prefactor in front of eac h of these in tegrals reduces the total result to a Lauren t series at z 0 = 1 except for the p ossibilit y of one term prop ortional to ( z 0 − 1) m log( z 0 − 1). Ho wev er, since by Theorem 1.1 e g is analytic near z 0 = 1, su c h a term must either cancel with other suc h terms in the full expr ession for e g , or it cann ot exist. It follo ws th at (60) has a regular Laur ent expan s ion at z 0 = 1 which in fact must b e a T a ylor exp an s ion with v anishing constan t term since e g = 0 at z 0 = 1. W e apply Theorem B.3 once more to conclud e that e g is regular at z 0 = 0: e g ( t ) = N 2 g  N − 2 log  Z N ( t ) Z N (0)  − e 0 ( t ) − · · · − N − (2 g − 2) e g − 1 ( t )  − N 2 g  N − 2 log  Z N ( t ) Z N (0)  − e 0 ( t ) − · · · − N − 2 g e g ( t )  = ⇒ | e g ( t ) | ≤ C g − 1 + C g N 2 36 N. M. ERCOLANI uniformly as t → ∞ and hence e g remains b ounded as z 0 approac h es 0. It follo ws, in exactly the same man n er as was done for z 0 near 1, that all singular parts of (60) at z 0 = 0 m u st cancel eac h other.  Corollary 4.3. The integ ration constan ts K 1 and K 2 can b e non-zero only if (2 g − 2) / ( ν − 1) or (2 g − 1) / ( ν − 1) are in tegers and in addition the com bination of the other terms in (60 ) has a p ole at z 0 = 0. 4.3. The Double Scaling Limit for ν = 2 a nd P a inle v´ e I. The topic of the so- called double-sc aling limit of the random matrix partition fu n ction (3) h as b een extensively discussed in b oth the ph ys ics and the mathematical physics lite rature related to t wo- dimensional quan tum gravit y . F or a concise history of these dev elopmen ts and relev an t references w e refer the reader to [14]. In this lit erature atten tion w as focused on the sp ecial case of the weig h t V ( λ ) = 1 2 λ 2 + tλ 4 whic h corresp onds to the case of ν = 2 in our more general treatmen t. In this case the double-scaling limit refers to the regime in parameter space where t → t c (= − 1 48 ) and N → ∞ simultaneo usly so that N 4 / 5  t + 1 48  = − γ 1 ξ (68) remains ﬁxed. The authors in [3 , 8, 18] arrived at this p r escription of the limit through a formal scaling argument based on the F reud equation (31). Basically they p osited a general scaling form, in n, N and t , for th e recurrence co eﬃcient s b 2 n,N ( t ) and d etermined that (68 ) w as the u nique scaling that would (formally) asymp totica lly b alance (31 ). Realizi ng th at this r elation, w hic h they called the string equation, could b e tak en to b e a d iscrete form of the ﬁrst Pa inlev ´ e equation y ′′ = 6 y 2 + ξ , (69) they to ok (69) to rep resen t the exact non-p erturb ative string the ory for tw o-dimensional euclidean quantum gra v ity . The corresp onding mathematical conjecture was that the large N expansion of the recur- rence coeﬃcients, under the scaling (68 ), has a limiting form y ( ξ ), at leading order in N , that solv es (69) which hereafter will b e r eferred to as PI. The pr op osition was ﬁ rst placed on a rigorous fo oting, based on Riemann-Hilb ert analysis, in the pap ers [12, 13]. A complete and self-con tained pr o of wa s recent ly d etailed in [7 ]. The F reud equation and its relation to PI pla y no role in our descrip tion of the double scaling limit for general ν w h ic h w e take up in the n ext su bsection. The r eason that w e ha ve d w elled on them for the case ν = 2 in this su b section, apart from their historical relev ance for our topic, is that this sp ecial case pro vid es a concrete p oint of comparison (see Section 4.5) for ou r m ore general results and a jumpin g oﬀ p oin t for p ossible f uture in v estigations that might inv olv e a b r oader connection to P ainlev ´ e transcendents. F or these reasons we will brieﬂy review, h ere, the essen tial prop erties of the relev an t PI solutions. F or a systematic devel opmen t and fur ther details w e refer the reader to [19]. CAUSTICS , COUNTING MA PS AND SEMI-CLASSICAL A SYMPTOTICS 37 There is a family of solutions to (69) characte rized by having the common asymptotic expansion y ( ξ ) ∼ r − ξ 6   1 + ∞ X g =1 α g ( − ξ ) − 5 g / 2   as ξ → −∞ (70) for particular co eﬃcien ts α g . These expansions are v alid in angu lar s ectors, t ypically of width 2 π 5 in the complex ξ plane. Th ey diﬀer fr om one another by ju mps across s ectors (Stok es lines) wh ich are exp onen tially sm all. When n eeded, for sp eciﬁcit y , w e will tak e y ( ξ ) to b e that solution with the ab o ve asymp totic expansion v alid in the full sector 3 π 5 < arg ξ < 7 π 5 as | ξ | → ∞ . The α g satisfy the follo wing quadr atic recur sion f ormula α g +1 = 25 g 2 − 1 8 √ 6 α g − 1 2 g X m =1 α m α g +1 − m (71) α 0 = 1 . 4.4. The Double Scaling Limit for General ν . In this sec tion w e will describ e a double scaling limit for the free energy of the u n itary ensem b les (1) in terms of the rational represent ation of the z g . This will b e done in th e setting of general v alues of ν whic h should enable one to see the un iversal charact er of these results. W e r eturn to the asymptotic expansion for the orthogonal p olynomial recursion coeﬃcient s (Theorem 1.4 or Theorem B.3) and substitute the p olar expansions found in Theorem 1.5. b 2 N ,N = z 0 + ∞ X g =1 z g N − 2 g (72) b 2 N ,N − ν ν − 1 =  z 0 − ν ν − 1  + ∞ X g =1 z g N − 2 g =  z 0 − ν ν − 1  (73) + z 0 ∞ X g =1 ( a ( g ) 0 ( ν ) ( ν − ( ν − 1) z 0 ) 2 g + a ( g ) 1 ( ν ) ( ν − ( ν − 1) z 0 ) 2 g +1 + · · · + a ( g ) 3 g − 1 ( ν ) ( ν − ( ν − 1) z 0 ) 5 g − 1 ) N − 2 g . W e ﬁrst p ro ceed formally to d etermine what the double s caling balance should b e; i.e., we c ho ose δ in ν − ( ν − 1) z 0 ∼ N δ suc h th at the h ighest ord er terms of (73) at th e p ole ha ve a common factor inv olving N that is indep endent of g . In other words δ s hould b e c hosen so that ⌊ (5 g − 1) δ ⌋ = − 2 g . 38 N. M. ERCOLANI Therefore, δ = − 2 5 . Combining this with (34 ) one is led to tak e the doub le scaling ans atz for general ν to b e N 4 / 5 ( s − s c ) = γ ( ν ) 1 ξ (74) where, b y (33), s c = ( ν − 1) ν − 1 c ν ν ν and γ ( ν ) 1 is a constant d ep ending only on ν . W e will in general notationally supp ress the explicit dep end ence of γ 1 on ν since this should b e clear from con text. Also, by (34),  z 0 − ν ν − 1  = − s − 2 c ν ν ν +1 ( ν − 1) ν ( s − s c ) { 1 + O ( s − s c ) } = − s − 2 c ν ν ν +1 ( ν − 1) ν γ 1 ξ N − 2 / 5 + O ( N − 4 / 5 ) b y (74) . Substituting the ansatz (74) in to (73), one ma y formally conclude that b 2 N ,N − ν ν − 1 = − s − 2 c ν ν ν +1 ( ν − 1) ν γ 1 ξ N − 2 / 5 + ν ν − 1 ∞ X g =1      a ( g ) 0 / ( ν − 1) 2 g  − 2 c ν ν ν +1 ( ν − 1) ν γ 1 ξ  g N − 4 g / 5 + · · · + a ( g ) 3 g − 1 / ( ν − 1) 5 g − 1  − 2 c ν ν ν +1 ( ν − 1) ν γ 1 ξ  (5 g − 1) / 2 N − 2 g +2 / 5      N − 2 g + O ( N − 4 / 5 ) = − s − 2 c ν ν ν +1 ( ν − 1) ν γ 1 ξ N − 2 / 5 + ν ν − 1 ∞ X g =1 a ( g ) 3 g − 1  − 2 c ν ν ν +1 ( ν − 1) ν +2 γ 1 ξ  (5 g − 1) / 2 N − 2 / 5 + O ( N − 4 / 5 ) = − s − 2 c ν ν ν +1 ( ν − 1) ν γ 1 ξ N − 2 / 5 + ν ν − 1 ∞ X g =1 a ( g ) 3 g − 1  − 2 c ν ν ν +1 ( ν − 1) ν +2 γ 1 ξ  (5 g − 1) / 2 N − 2 / 5 + O ( N − 4 / 5 ) = −  − 2 c ν γ 1 ν ν +1 ( ν − 1) ν +2 ξ  1 / 2    ( ν − 1) − ν ν − 1 ∞ X g =1 a ( g ) 3 g − 1 ( ν )  − 2 c ν γ 1 ν ν +1 ( ν − 1) ν +2 ξ  − 5 g / 2    N − 2 / 5 (75) + O ( N − 4 / 5 ) . W e n ext study what can b e said ab out the leading ord er, for large N , term in th e formal expansion (75). Th e co eﬃcien ts of the series at this leading ord er are essentia lly giv en by CAUSTICS , COUNTING MA PS AND SEMI-CLASSICAL A SYMPTOTICS 39 the a ( g ) 3 g − 1 ( ν ) which are, s equ en tially , the highest order co eﬃcien ts of the p ole at ν / ( ν − 1) for the sequence of z g . W e d eriv e a recurs ion formula for these co eﬃcien ts. Prop osition 4.4. a g +1 3( g +1) − 1 ( ν ) = ν 3  25 g 2 − 1  6 a ( g ) 3 g − 1 ( ν ) + ν 2 g X m =1 a ( m ) 3 m − 1 ( ν ) a ( g − m +1) 3( g − m +1) − 1 ( ν ) (76) a (1) 2 ( ν ) = ν 2 6 (77) Pro of. First recall from the p ro of of T heorem 1.5, giv en in Section 3 that the co eﬃcient , that a ( g ) 3 g − 1 ( ν ), has (indu ctiv ely) tw o sources. T he ﬁrst of these comes from forcing the in tegral equation for z g (see Theorem 2.4:(iii)) with terms of the form " ν c ν f ν − 1 0 g − 1 X m =1 f m ( f g − m ) w # w =1 (the J = 1 case). Th e other comes from forcing with terms of the form h ν c ν 6 f ν 0 ( f g − 1 ) w w w i w =1 (the J = 3 case; n ote that here we h a ve used Prop osition 2.2 to calculate that d ( ν,g ) (3) = ν c ν ). Inserting the partial fractions expansion of these terms as th e forcing in the in tegral equa- tion for z g and then collec ting the co eﬃcien ts of the terms p rop ortional to 1 / ( ν − ( ν − 1) z 0 ) 5 g − 1 one der ives an equation in v olving a m ixture of co eﬃcien ts of the form a ( k, j ) ℓ ( ν ). Ho wev er, b y rep eated application of the linear r ecursions fr om Lemma 3.2, su p plemen ted b y (48) - (50), one arr iv es at the follo wing expression in volving only the maximal p ole co eﬃcien ts of un diﬀeren tiated z n , ν ν − 1 a g 3 g − 1 ( ν ) = ν 2 2( ν − 1) g − 1 X m =1 a m 3 m − 1 ( ν ) a g − m 3( g − m ) − 1 ( ν ) + ν 4 (5 g − 4)(5 g − 6) 6( ν − 1) a g − 1 3( g − 1) − 1 ( ν ) . Making some obvious cancella tions and sh ifting from g to g + 1 one thus ded u ces (76). T he initial condition (77) follo ws d ir ectly fr om examination of (13).  Using (77) one can rewrite (76) as a g +1 3( g +1) − 1 ( ν ) = ν 3  25 g 2  6 a ( g ) 3 g − 1 ( ν ) + ν 2 g − 1 X m =2 a ( m ) 3 m − 1 ( ν ) a ( g − m +1) 3( g − m +1) − 1 ( ν ) It is immediate from this th at a ( g ) 3 g − 1 ( ν ) > 0 and h en ce, 40 N. M. ERCOLANI Corollary 4.5. The p ole order of z g at z 0 = ν / ( ν − 1) is exactly equal to 5 g − 1 for all g ≥ 1. It also follo ws from this form of th e recurs ion that a g +1 3( g +1) − 1 ( ν ) /a ( g ) 3 g − 1 ( ν ) > C g 2 and therefore the series in the leading order term of (75) is divergen t as ξ → −∞ . O ne exp ect this series to b e an asymptotic and their are v arious approac hes one could tak e to determine to what fun ctions (as ν v aries) it migh t b e asymptotic (generalized Borel summation, seeking an o d e with a solution whose asymp totic co eﬃcien ts satisfy Prop osition 4.4, etc.). Ho wev er, as we will suggest in the next section, th e form of our recursion suggests another approac h b oth to charac terizing the leading order series and to establishin g the v alidit y of the doub le scaling expansion (75). 4.5. Univ ersality and the P ainlev´ e I Hiera rc hy . W e w ill no w restrict our stud y of (75) to the sp ecial case ν = 2 where it sp ecialize s to the follo w ing, still formal, expansion: b 2 N ,N − 2 = − ( − 192 γ 1 ξ ) 1 / 2    1 − 2 ∞ X g =1 a ( g ) 3 g − 1 ( − 192 γ 1 ξ ) − 5 g / 2    N − 2 / 5 + O ( N − 4 / 5 ) . (78) W e quote a recent r esult of Duits and Kuijlaars whic h will enable us to giv e a p recise c haracterization of (78). Theorem 4.6. [7] Ther e ar e c onstants γ 1 –se e (68)– and γ 2 such that b 2 N ,N − 2 = − γ 2 ( y α ( ξ ) + y β ( ξ )) N − 2 / 5 + O ( N − 3 / 5 ) as N → ∞ , (79) and wher e y α and y β ar e two memb ers of the family of PI solutions having the c ommon asymptot ics (70). The c onstants γ 1 and γ 2 ar e indep endent of the choic e of α and β . This exp ansion holds uniformly for ξ in c omp act subsets of R not c ontaining any of the p oles of y α and y β and c an in fact b e extende d to a ful l asymptotic exp ansion in p owers N − 1 / 5 . The deriv ation of the ab o v e result d ep ends fun damen tally on an extension of the deﬁni- tion of orth ogonal p olynomials with exp onen tial we igh ts to a more general class of non- Hermitean orthogonal p olynomials which corresp onds, in an appropriate sense, to taking t in th e weig h t V to lie in the in terv al  − 1 48 , 0  . There is a corresp onding extended notion of the equilibrium measure ment oned in Section 2.5 for these negativ e v alues of t . I n [7 ] this measure is deﬁned thr ough a v ariational problem along a deformed contour in the complex λ plane. These orthogonal p olynomials are charac terized by a R iemann -Hilb ert (R-H) problem and their asymptotics ma y b e stu d ied via the Deift-Zhou metho d of nonlin- ear steep est d escen t [6]. In particular, this inv olv es the construction of lo cal parametrices in term s of th e Riemann -Hilb ert p roblem for the PI equation. (Th e parameters α and β in th e ab ov e Theorem extend through th is pr o cess from the con tour deformations to the CAUSTICS , COUNTING MA PS AND SEMI-CLASSICAL A SYMPTOTICS 41 R-H pr oblem for the orthogonal p olynomials to the R-H problem for PI to the family of PI solutions. A key p oint for relating this an alysis to ou r analysis in the case of (78) is that for − 1 48 < t < 0 there is an N 0 ( t ) s u c h that f or all N > N 0 ( t ) the recurrence co eﬃcient s ˆ b 2 N ,N ( t ) for the non-Hermitean p olynomials exist ([7], Theorem 1.1). Moreo ve r, ˆ b 2 N ,N ( t ) has a full asymp totic expansion, ind ep endent of the choice of α and β , in inv erse p o wers of N . Ind eed the construction of this expansion is mo deled on that of [9]. It follo ws that ˆ b 2 N ,N ( t ) = b 2 N ,N ( t ) for − 1 48 < t < 0. W e can use these obser v ations to directly relate th e h ighest order p olar expansion co ef- ﬁcien ts, a ( g ) 3 g − 1 of z g for ν = 2 that app ear in (78) to the co eﬃcien ts in the asymptotic expansion (70) of the PI solution. First w e need to determine the p inning constan ts γ 1 and γ 2 . Matc hing the co eﬃcien ts of N − 2 / 5 b et w een (78), (79) and (70) requires − ( − 192 γ 1 ξ ) 1 / 2    1 − 2 ∞ X g =1 a ( g ) 3 g − 1 ( − 192 γ 1 ξ ) − 5 g / 2    = − γ 2 ( y α ( ξ ) + y β ( ξ )) = − 2 γ 2 r − ξ 6   1 + ∞ X g =1 α g ( − ξ ) − 5 g / 2   . (80) Comparison of the ﬁrs t tw o terms give s 192 γ 1 = 4 6 γ 2 2 − 2 a (1) 2 (192 γ 1 ) 5 / 2 = α 1 α 1 = − 1 8 √ 6 b y (71) a (1) 2 = 2 3 b y (77) . F rom this one immediately s ees that γ 2 = 2 3 / 5 3 2 / 5 (81) γ 1 = 2 − 9 / 5 3 − 6 / 5 4 = 1 4 γ − 3 2 (82) These v alues agree with the pinning constan ts stated in [7] (Theorem 1.2) mo dulo the factor of 1 / 4 in (82). Ho wev er, in that reference the form of the weigh t is take n to b e V = λ 2 / 2 + tλ 4 / 4 w hic h diﬀers from the f orm of the w eigh t used here eﬀectiv ely b y s caling the time t by 1 / 4; hence, the ﬁr st pinning constan t deﬁn ed in (68 ) should diﬀer in fact from that app earing in [7] by a factor of 1 / 4. With these constants determined one may n o w us e (80) to express the highest ord er p ole co eﬃcien ts (for ν = 2) in terms of the PI asymptotic co eﬃcients. 42 N. M. ERCOLANI Corollary 4.7. a ( g ) 3 g − 1 = − 2 5 g − 1 (2 / 3) g / 2 α g for g ≥ 1 . (83) Substitiuting (83) into the nonlinear recusr s ion (71) one deduces the follo w ing quadr atic recursion b et wee n th e co eﬃcien ts of the h ighest order p oles of the z g in the case of ν = 2. a ( g + 1) 3( g +1) − 1 = 4 3  25 g 2 − 1  a ( g ) 3 g − 1 + g X m =1 a ( m ) 3 m − 1 a ( g + 1 − m ) 3( g +1 − m ) − 1 . This is in complete agreemen t with Prop osition 4.4 for the case ν = 2 and so we can ﬁnally state Prop osition 4.8. The leading order s eries in (78) coincides with the asymptotic expansion of the PI solution sp eciﬁed in T h eorem 4.6 . Moreo v er, based on the v alidit y of the expansion (79), the expansion (78) is also v alid for large N . In addition, from the deriv ation of the expansion d ispla yed in (75) one deduces an imp ro ve- men t of the result stated in Th eorem 4.6. Corollary 4.9. Th e double-scaling limit of b 2 N ,N has a full asymptotic expansion in even p o w ers of N − 1 / 5 . In p articular, the next order correction in (79), after leading ord er, is O ( N − 4 / 5 ). The form of the expansion (75) suggests that Prop osition 4.8 , for the sp ecial case of ν = 2, should extend to h a ve a u niv ers al c h aracter for general ν . By univ e rsal here w e mean that the v arious constan ts and expressions w e h a ve b een discussin g wh en ν = 2 shou ld b e replaced in the general case by ﬁ xed rational functions of the p arameter ν . Although w e d o not carry out the details here, this extension should f ollo ws straigh tforwardly by generalizing the Riemann-Hilb ert problem for PI to the R-H problem for the PI hierarc h y . (F or a d escription of this hierarc h y we refer the r eader to [26].) Sp eciﬁcally one needs to replace the Hamiltonian for PI app earing in the RH p roblem by the Hamiltonian for the higher order equations in the P I h ierarc hy . Another in triguing problem is to stud y the ﬁne structure of the higher order terms in the d ou b le scaling expansion (79); i.e., the co eﬃcien ts of N − 2 h/ 5 for inte ger h > 1. This should lead to extensions of Prop osition 4.4 that enable one to recur siv ely determine the fundamental co eﬃcients a ( g ) ℓ ( ν ) for low er v alues of ℓ .. 4.6. Relation to Ot he r Enumerativ e Results. The sub ject of m ap enumeration has b een m u c h studied going bac k at least to th e early w ork of T utte [28], in the ’60s, who in tro duced the notion of a ro oted map. A map is said to b e r o ote d if a v ertex of the map together w ith an ed ge adjacen t to it and a side of th at edge are d istinguished. Since then, m u c h work b een done on coun ting v arious classes of ro oted planar ( g = 0) maps . The cases CAUSTICS , COUNTING MA PS AND SEMI-CLASSICAL A SYMPTOTICS 43 of higher genus map s ha ve b een more c hallenging and therefore the main emphasis has b een placed on trying to determine the asymptotic b eha vior of v arious t yp es of enumeratio ns for large v alues of the discrete parameters.. A particular example of this is giv en by the problem of coun ting the n umber, M n,g , of ro oted maps on a gen u s g Riemann surface with exactly n edges. Bender and Canﬁeld [1] show ed M n,g ∼ t g n 5( g − 1) / 2 12 n as n → ∞ . (84) Man y other map classes h a ve similar asymptotics [15] with th e same sequence of constan ts t g . In [1] a recurs ion is giv en for these constan ts the ﬁr st f ew of wh ic h are: t 0 = 2 √ π t 1 = 1 24 t 2 = 7 √ 4320 π Ho wev er, th er e h a ve b een some recent improv emen ts in the recursion form ula for th e t g [2] based on conn ections to the enumeratio n of branched co verings of Riemann su r faces [17]. More recen tly [16] it h as b een observed that this n ew recursion is closely related to the recursion for PI asymptotic co eﬃcient s (71). The up shot is that t g = − 1 2 g − 2 6 g 2 · 1 Γ( 5 g − 1 2 ) α g (85) = 1 2 7 g − 3 · 1 Γ( 5 g − 1 2 ) a ( g ) 3 g − 1 b y (83) , whic h rev eals an inte resting alb eit myste rious link b etw een t w o prima f acie quite diﬀerent classes of map enumeratio n pr oblems. Appendix A. Recurrenc e Coe fficients an d their Continuum Limits This app en d ix pro vides a reasonably self-con tained presenta tion of the relev an t results (and outline of their pro ofs) from [9, 10]. It also giv es th e pr o of of Prop osition 2.2: a new c haracterization of the co eﬃcien ts in th e conti n uum T o da equ ations. A.1. F ull Asymptotics of the Recurrence Co eﬃcients. Reca ll from Section 1.4.4 that we hav e deﬁned π n,N ( λ, t ) to b e the n th monic orthogonal p olynomial w ith resp ect to th e exp onen tial weigh t exp( − N V ( λ )) and that these p olynomials s atisfy a thr ee-term recurrence relation of the form π n +1 ,N ( λ ) = λπ n,N ( λ ) − b 2 n,N ( t ) π n − 1 ,N ( λ ) . (86) There is a basic relation b et ween these recurrence co eﬃcient s and the tau functions deﬁned in (6) giv en by: 44 N. M. ERCOLANI b 2 n, 1 ( θ ) = 1 2 d 2 dθ 2 1 log  τ 2 n, 1 ( θ 1 , θ )  θ 1 =0 where (87) τ 2 n, 1 ( θ 1 , θ ) = Z ( n ) 1 ( θ 1 , θ ) / Z ( n ) 1 (0 , 0) and (88) Z ( n ) N ( t 1 , t ) = Z · · · Z exp    − N n X j = 1  1 2 λ 2 j + tλ 2 ν j + t 1 λ j     V ( λ ) d n λ , (89) where V ( λ ) = Y j < l | λ j − λ l | 2 . The relation (87 ) is called a H ir ota formula [11] and and stems fr om the in tegrable sys tems theory asso ciated to the T o da Lattice (see Section A.2). The conn ection to orthogonal p olynomials comes from the represent ation of th e partition fun ctions (3) as Hank el deter- minan ts of momen ts for exp onential weigh ts [9, 10, 27]. The d eﬁnitions on the next tw o lines are consisten t extensions of our earlier deﬁnitions. I n deed when θ 1 , t 1 = 0, (88) and (89) are equiv alen t to (6) and (3) resp ectiv ely . The Hirota formula m a y b e extended to more generally scaled recurren ce co eﬃcien ts by making use of the s calings introd uced in Section 2.1.1. I n deed, the tau functions are con v ariant w ith r esp ect to these scalings so that τ n, 1 ( θ 1 , θ ) = τ n,N ( t 1 , t ) = τ n,n ( − s 1 , − s ) where we deﬁne (90) t 1 = 2 θ 1 / √ N consisten t with (17), and (91) s 1 = − t 1 / √ x consistent with (18) . (92) The ﬁnal transf orm ation to τ n,n is realized b y a v ariable c hange λ = √ x ˆ λ of the eigen v alues in (89). It th en f ollo ws fr om (87 ) that 1 n b 2 n, 1 ( θ ) = 1 2 n d 2 dθ 2 1 log  τ 2 n,n ( − s 1 , − s )  s 1 =0 = 1 n 2 d 2 ds 2 1 log  τ 2 n,n ( − s 1 , − s )  s 1 =0 ; moreo v er, N n b 2 n,N ( t ) = 1 n b 2 n, 1 ( − s 2 n ν − 1 ) = 1 n 2 d 2 ds 2 1 log  τ 2 n,n ( − s 1 , − s )  s 1 =0 . (93) Theorem 1.1 may no w b e applied to the righ t h and side of (93) to establish th e expansion (10). A.2. The Con tin uum Limit of the T o da Lat tice E quations for the Recurrence Co eﬃcien ts. As the external w eigh t, V ν ( λ ; t ) c hanges with t , the corresp onding orthogo- nal p olynomials, π n,N ( λ, t ), also ev olv e. Ho w th ey ev olve is go v ern ed b y h o w the r ecurrence co eﬃcien ts, wh ic h also d ep end on t , c hange. It is w ell-kno wn that the t dep end en ce of the recurrence co eﬃcien ts is go v erned [4] b y a system of nonlinear diﬀerential equations kno wn CAUSTICS , COUNTING MA PS AND SEMI-CLASSICAL A SYMPTOTICS 45 as the T o da L attic e Equations . Th er e is a diﬀerent system for eac h ν . T hough diﬀeren t, these d iﬀeren t n onlinear s ystems hav e a common general form . Th e diﬀerent asso ciated ﬂo w s on the recurr ence co eﬃcien ts that these diﬀerent systems ind u ce comm u te with one another and, in fact, this commutat ivit y is related to the c omplete inte gr ability of the T o da Lattices. Th is feature do es n ot p la y a direct role in what we present in th is pap er bu t it is und oubtedly related to the ”univ ersalit y in ν ” that h as already b een mentioned sev eral con texts. In our setting, the T o da Lattice equations tak e the follo wing form, for eac h ν : 1 2 db 2 n dθ = X { w } " ν +1 Y m =1 b 2 n + ℓ m ( w )+1 − ν +1 Y m =1 b 2 n + ℓ m ( w ) # (94) where the dep en den t v ariable b 2 n stands for b 2 n, 1 ( θ ) as deﬁned in (87). (A d etailed deriv a- tion of th e form (94) of the T o da equations from their more s tandard presenta tion in the in tegrable systems literature is giv en in Section 4.1 of [10].) T h e summation in (94) is tak en ov er the set of w alks, d enoted { w } , of length 2 ν on Z th at start at +1 and en d at − 1. If one visualizes the Z lattice along whic h the walk tak es place as a vertic al axis and the steps of th e w alk as discrete equally spaced p oints along a horizonta l axis wh ic h serves to order those steps, then the walk is graphed as a zig-zag path compr ised of line segmen ts of slop e +45 ◦ (upturn ) or − 45 ◦ (do wnturn). Suc h a walk is completely determined b y sp ecifying where its down turn s (of w hic h there are exactly ν + 1) o ccur. If the discrete v ariable ℓ denotes lo cations on th e v ertical axis then, on a giv en w alk, ℓ can only range o ver the in terv al [ − ν , ν ]. Finally we let ℓ m ( w ) den ote the vertica l axis lo cation on the w alk w after its m th do w n tu rn; then ℓ m ( w ) + 1 denotes the lo cation b efor e this do wnt urn. In order to study the con tinuum limit of (94) w e note, from (93) and (10), that these equations con tain terms with pr ima facie diﬀeren t asymp totic gauges: b 2 n = b 2 n  − s 2 n ν − 1  = n  z 0 ( s ) + 1 n 2 z 1 ( s ) + 1 n 4 z 2 ( s ) + · · ·  (95) b 2 n + ℓ = b 2 n + ℓ  − ˜ s 2( n + ℓ ) ν − 1  = ( n + ℓ )  z 0 ( ˜ s ) + 1 ( n + ℓ ) 2 z 1 ( ˜ s ) + 1 ( n + ℓ ) 4 z 2 ( ˜ s ) + · · ·  = n  wz 0 ( sw ν − 1 ) + · · · + 1 n 2 g w 1 − 2 g z g ( sw ν − 1 ) + · · ·  , (96) where w = 1 + ℓ n as deﬁned in (19). Th e in tro d uction of th e latttice scaling v ariable w in (96) allo ws u s to analyze all th e terms app earing in (94) with resp ect to the same asymptotic gauge. Since ℓ ∈ [ − ν , ν ] and ν is ﬁxed, one may assu me that ℓ << n . Then one can stud y th e limit as n → ∞ of the equations (94), with the substitutions (96), as one w ould study the contin uum limit of a n u merical sc heme or a molecular c hain with analytic 46 N. M. ERCOLANI p oten tial. T o carry this out, w e introd uce a more compact notation for (96 ) as f ( s, w ) = f 0 ( s, w ) + 1 n 2 f 1 ( s, w ) + · · · + 1 n 2 g f g ( s, w ) + · · · , wher e f g ( s, w ) = w 1 − 2 g z g ( w ν − 1 s ) so th at b 2 n + ℓ = nf  s, 1 + ℓ n  and in particular (97) n − 1 b 2 n = f ( s, 1) = z 0 ( s ) + 1 n 2 z 1 ( s ) + 1 n 4 z 2 ( s ) + · · · and 1 2 db 2 n dθ = 2 n ν − 1 1 2 db 2 n ds = n ν d ds f ( s, w )     w =1 (98) Finally , substituting (97 ) and (98 ) in to (94) and T aylo r expanding in ℓ m n (and ℓ m +1 n ) one arriv es at the basic con tin u um equations d ds f ( s, w )     w =1 = X { w } nf ν +1 ( ν +1 Y m =1 " 1 + f w f  ℓ m + 1 n  + f w (2) 2 f  ℓ m + 1 n  2 + · · · + f w ( h ) h ! f  ℓ m + 1 n  h + · · · # − ν +1 Y m =1 " 1 + f w f ℓ m n + f w (2) 2 f  ℓ m n  2 + · · · + f w ( h ) h ! f  ℓ m n  h + · · · #) (99) from whic h Theorems 2.1 and 2.4 can b e d ed uced. A.3. Co eﬃcien t F ormulae for the Con t in uum T o da Equations. W e now pro vide a pro of of Pr op osition 2.2. Recall from T h eorem 2.1 that, u sing multi-i ndex n otation, F ( ν ) g = X | λ | = 2 g + 1 ℓ ( λ ) ≤ ν + 1 d ( ν,g ) λ Q j r j ( λ )! f ν − ℓ ( λ )+1 1 λ ! ∂ | λ | f ∂ w λ These co eﬃcien ts h a ve the charac ter of correlation fun ctions for certain tied random w alks. A description based on this p er s p ectiv e is presente d in [10]. In the follo wing deriv ation, that description will b e us ed but not explained in full detail. Pro of of Prop osition 2.2. The coeﬃcients d ( ν,g ) λ are dir ectly related to th e expression of the T o d a Lattice equations for the b 2 k , in terms of tied wal ks on a one dimensional lattice as describ ed in [10]. The wal ks in question can b e redu ced to considering w alks of length 2 ν on Z that start at +1 and end at − 1. If one visualizes the Z lattice along wh ic h the w alk tak es place as a vertic al axis and the time steps as d iscrete equ ally sp aced p oint s along a horizonta l axis, th en the w alk is graphed as a zig-zag path comprised of line segmen ts of slop e +45 ◦ (upturn ) or − 45 ◦ (do wnturn). W e lab el the lo cations on the vertica l axis by ℓ and those on the horizon tal axis by i . F or the wa lks u nder consideration, ℓ ranges o ver the in terv al [ − ν, ν ] and i ranges fr om 1 to 2 ν . Su c h a walk is completely determined by sp ecifying when its do wnturns (of which there are exactly ν + 1) o ccur. W e will let i j , for CAUSTICS , COUNTING MA PS AND SEMI-CLASSICAL A SYMPTOTICS 47 j ∈ { 1 , . . . ν + 1 } , denote the step at whic h the j th do w n tu rn tak es place. W e also let ℓ j denote the lo cation of the path after the j th do w n tu rn (then ℓ j + 1 denotes the lo cation of the path b efor e the j th do w n tu rn). Giv en this set-up, and based on the deriv ation given in [10 ], the co eﬃcien ts can b e expr essed as d ( ν,g ) λ = X w alks [ m λ ( ℓ 1 + 1 , . . . , ℓ ν +1 + 1) − m λ ( ℓ 1 , . . . , ℓ ν +1 )] (100) = X i 1 < ··· 0 as long as the equ ilibrium measure µ V t for the weigh t V t at that v alue of t , satisﬁes the follo w ing conditions: (a) The supp ort of µ V t is a single in terv al of the form [ − β , β ]; (b) µ V t satisﬁes the v ariational equations of (37 ); (c) µ V t v anishes lik e a square ro ot at b oth end p oin ts of its supp ort. In Section B.1 w e will show that these conditions are in deed satisﬁed for all t > 0 by explicitly writing do wn the equilibriu m measure for all these v alues of t . It then follo ws, from analytic contin uation and uniqu eness of p o wer series expansions for analytic fu nctions, that the co eﬃcien ts of the large N expansions of logarithms of tau f unctions (6) and their deriv ativ es, at these v alues of t , coincide with those constructed near t = 0 (such as e g and z g ). Ho wev er, for our applications we r equ ire m ore. W e need to kn o w the limiting v alues of these co eﬃcien ts as t → ∞ . T o accomplish this one sh ould actually bu ild the asymp totic expansions of correlatio n fu nctions in a neigh b orho o d of t = ∞ . In section B.2 w e ind icate ho w th e constructions of [9] may b e adapted to th is large-time regime. B.1. The Equilibrium Measure for t > 0 . W e present here an explicit expression for the equ ilibrium measure (35) th at is u niformly v alid f or all t ≥ 0. T o accomplish this, w e rescale the domain v ariable λ so that the su p p ort of the m easur e r emains ﬁxed as t v aries: λ = 2 √ z 0 η wh ere (101) z 0 = z 0 ( − t ) as deﬁned in (9) . CAUSTICS , COUNTING MA PS AND SEMI-CLASSICAL A SYMPTOTICS 49 Prop osition B.1. The equilibr ium measure for t ≥ 0 is giv en, und er the v ariable d ilation (101), by the follo wing exp r ession w h ic h is a con v ex linear com bination, for z 0 ∈ [0 , 1], of t wo probabilit y measures on [ − 1 , 1]: dµ V t ( η ) = 2 π χ ( − 1 , 1) ( η )        z 0 + (1 − z 0 )     (2 η ) 2 ν − 2  2 ν − 1 ν − 1  + ν − 1 X j = 1 2  2 j − 1 j − 1   2 ν − 1 ν − 1  (2 η ) 2 ν − 2 − 2 j            p ( η + 1)(1 − η ) dη . (102) This expr ession dep end s on t only through the parameter z 0 . Wh en z 0 = 1 this corresp onds to the weigh t for t = 0 and the measure has the dens it y of the Wigner semicircle la w ; when z 0 = 0 this corresp onds to t = ∞ which w e think of as th e limiting asymptotic measur e for large t . T h is limiting density also giv es a probabilit y measure s u pp orted on [ − 1 , 1]. It is in fact the density of the equilibr ium measure for the pu r e monomial we igh t V ∞ ( η ) = 2 2 ν c ν η 2 ν . (103) Pro of. Pinning d own an explicit expression for (35) amoun ts to d etermining the co eﬃ- cien ts of the p olynomial h ( λ ). W e already k n o w β , from (36), explicitly in terms of z 0 . The essen tial ingredients for calculating (104) h ( λ ) = 1 + ν − 1 X j = 0 h j λ 2 j w ere dev elop ed in section 3 of [10]. W e recall those elemen ts here. Deﬁne the sequence { v j } ∞ j = 0 b y p λ 2 − β 2 λ = 1 − ∞ X i =0 v i 1 λ 2 i +2 ; whose T a ylor co eﬃcien ts can b e computed to b e (105) v i = 1 4 i  2 i − 1 i − 1  β 2 i +2 i + 1 , for i > 0 with v 0 deﬁned to b e β 2 / 2. T h e co eﬃcien ts of h are directly expressed in term s of the { v j } as 50 N. M. ERCOLANI h j = 4 ν ( ν − j ) t v ν − 1 − j β 2 (106) = 4 ν ( ν − j ) 1 − z 0 c ν z ν 0 v ν − 1 − j β 2 = 2  2 ν − 2 j − 3 ν − j − 2   2 ν − 1 ν − 1  (1 − z 0 ) z j + 1 0 , for j < ν − 1; h ν − 1 = 1  2 ν − 1 ν − 1  (1 − z 0 ) z ν 0 . Equation (30) w as used to eliminate t (= − s ) and c ν w as replaced by its deﬁnition f rom (9). The exact expression (102) now follo ws by substituting (104) with the co eﬃcien ts (106 ) in to (35 ), using (36) to re-express β in terms of z 0 , c h anging v ariables according to (101) and, ﬁnally , c h anging the index of summ ation f r om j to ν − j . It is an exercise to c h ec k that µ V ∞ is a probabilit y measure by directly inte grating th e densit y (102), for z 0 = 0, o v er [ − 1 , 1]. (It is manifest that th is density is p ositive on ( − 1 , 1).) The equilibriu m measure for m onomial weigh ts such as (103) is explicitly calculated on page 183 of [4 ]. One ma y c hec k, by dir ect comparison, that (102 ), with z 0 = 0, is ind eed the equilibrium measure for (103). The n ext section will give another p ersp ectiv e on why this is th e case.  B.2. The Riemann-Hilb ert Problem at I nﬁnit y. The pr ior results (Theorems 1.1, 1.4, 2.1) cited in S ections 1 and 2 , and on whic h the results of this pap er fundamentall y dep end, all deriv e f rom a detailed asymptotic analysis of the sp ectral density for eigen v alues of r an d om Hermetian matrices that was carried out in [9]. Exp licitly this sp ectral measure is d eﬁned as the exp ectation ρ ( N ) 1 ( t, λ ) = d dλ E µ t  1 N # { j : λ j ∈ ( −∞ , λ ) }  w.r.t. the probabilit y density (107) dµ t = 1 Z N ( t ) exp    − N 2   1 N N X j = 1 V t ( λ j ) − 1 N 2 X j 6 = ℓ log | λ j − λ ℓ |      d N λ. This h as a remark able expr ession as a one-p oint c orr elation function in terms of the W r on- skian determinan t ρ ( N ) 1 ( λ ) = K N ( λ, λ ) = e − N V t ( λ ) − 2 π iN  Y ′ 11 ( λ ) Y 21 ( λ ) − Y 11 ( λ ) Y ′ 21 ( λ )  (108) of the follo wing Riemann-Hilb ert (RH) problem f or th e 2 × 2 matrix Y ( λ ): • Y analytic in C \ R , CAUSTICS , COUNTING MA PS AND SEMI-CLASSICAL A SYMPTOTICS 51 • Y =  I + O ( 1 λ )  λ N σ 3 , • Y has H¨ older con tin u ous b ound ary v alues Y ± , fr om ab o v e and b elo w resp ective ly , along λ ∈ R , • Y + = Y −  1 e − N V t ( λ ) 0 1  , for λ ∈ R , where σ 3 is the diagonal Pauli matrix r eferred to by this standard notation. This problem is directly related to a RH problem for orthonormal p olynomials with w eight V t . W e will not need to go into that here, but refer the reader to [9] for m ore details and references. The relev ance of the one-p oin t fun ction to the problems discussed in this pap er can b e seen from the fact that Z N ( t ) = Z N (0) exp  − N 2 Z t 0 Z R ρ ( N ) 1 ( t, λ ) λ 2 ν dλdt  . (109) This falls within the more general class of p r oblems which study th e large N limits of in tegrals of the form Z R ρ ( N ) 1 ( t, λ ) f ( λ ) dλ (110) where f ( λ ) is a general C ∞ with no worse than p olynomial gro wth at in ﬁ nit y . W e n o w w an t to study ho w the RH problem f or Y b eha ves in the limit as t → ∞ . Under the c hange of v ariables (101) the pr oblem for Y tr ansforms to an equiv alen t problem • Z analytic in C \ R , • Z =  I + O ( 1 η )  η N σ 3 , • Z h as H¨ older conti n uous b oun dary v alues Z ± along η ∈ R , • Z + = Z −  1 e − N ˜ V t ( η ) 0 1  , for λ ∈ R , where ˜ V t ( η ) = 2 z 0 η 2 + 2 2 ν tz ν 0 η 2 ν = 2 z 0 η 2 + 2 2 ν 1 − z 0 c ν z ν 0 z ν 0 η 2 ν = z 0 2 η 2 + (1 − z 0 ) 2 2 ν c ν η 2 ν . (111) Equation (30) w as u sed to eliminate t (= − s ). W e see that as t → ∞ , z 0 → 0 and the wei gh t limits to 2 2 ν c ν η 2 ν , consisten t with (103 ). Th is establishes that the equilibr ium measures (35), for th e v ariational pr oblems (37), con verge as t → ∞ to the equilibrium measure for the v ariational problem w ith m onomial weigh t (103) th at is asso ciated to the RH prob lem for Z at z 0 = 0. Giv en this, all of the analysis of [9] to derive the full asym p totic expansion 52 N. M. ERCOLANI for ρ ( N ) 1 uniformly v alid for p ositiv e t near 0, carries o ver mutatis mutandis to giv e the asymptotic expansion of this one-p oin t function near t = ∞ : Theorem B.2. Ther e is a z ∗ 0 ∈ [0 , 1] such that for al l z 0 ∈ [0 , z ∗ 0 ) , the fol lowing asymptotic exp ansion is valid uniformly in z 0 : Z ∞ −∞ f ( λ ) ρ ( N ) 1 ( t ( z 0 ) , λ ) dλ = f 0 + N − 2 f 1 + N − 4 f 2 + · · · pr ovide d that the function f ( λ ) i s C ∞ and gr ows no faster than a p olynomial as | λ | → ∞ . The c o eﬃcients dep end analytic al ly on t for z 0 ( − t ) ∈ [0 , z ∗ 0 ) , and the asymptot ic exp ansion may b e diﬀer entiate d term by term. This construction is reminiscen t of the small amplitud e limit for action-angle v ariables in Hamiltonian mec hanics. It is, in eﬀect, a one-p oin t compactiﬁcation at inﬁnit y of the family of v ariational prob lems (37), th eir resp ectiv e uniqu e solutions, and asso ciated RH problems all parametrized by t ∈ [0 , ∞ ). This p arametrization is now naturally referenced to the compact in terv al 0 ≤ z 0 ≤ 1. Finally w e apply this r esu lt to extend the domain of un iform v alidit y in t for Theorems 1.1 and 1.4. Theorem B.3. Ther e is a c onstant ∆ > 0 such that for (c omplex) t with ℜ ( t ) ≥ 0 , |ℑ ( t ) | < ∆ one has a unif ormly valid asympto tic e xp ansion log τ 2 N ,N ( t ) = N 2 e 0 ( t ) + e 1 ( t ) + 1 N 2 e 2 ( t ) + · · · (112) as N → ∞ . Also, the r e curr enc e c o eﬃcients for the monic ortho gonal p olynomials with weight exp( − N V ( λ )) ha ve a fu l l asymp totic exp ansion, uniformly valid for (c omplex) t with ℜ ( t ) ≥ 0 , |ℑ ( t ) | < ∆ , of the form b 2 N ,N ( t ) = z 0 ( − t ) + 1 N 2 z 1 ( − t ) + 1 N 4 z 2 ( − t ) + · · · (113) as N → ∞ . The me aning of these exp ansions is: if you ke e p terms up to or der N − 2 h , the err or term is b ounde d by C N − 2 h − 2 , wher e the c onstant C is indep endent of t in the domain { ( ℜ t ≥ 0; − ∆ < ℑ t < ∆ } . M or e over, in this domain, for e ach ℓ , the functions e ℓ ( t ) and z ℓ ( − t ) ar e analytic functions of t and the asympto tic exp ansion of derivatives of log ( Z N ( t )) and b 2 N ,N ( t ) may b e c alculate d via term-by-term diﬀer entiation of the series. Pro of. F or an y t ∈ [0 , ∞ ) the m etho d s of [9] enable one to construct asymptotic expansions of the form (112) and (113) which are uniformly v alid in a complex neighborh o o d of that t in tersected with the half plane ℜ t ≥ 0. By Th eorem B.2 w e also h a ve suc h a neighborh o o d around t = ∞ . C onsider an op en co ve ring of [0 , ∞ ] by s u c h complex neigh b orho o ds which corresp onds to an op en cov er of z 0 ∈ [0 , 1]. By compactness of th is latter interv al there is a ﬁnite sub-co ver w hic h in tur n corresp onds to a ﬁnite op en co ver of t ∈ [0 , ∞ ) for which CAUSTICS , COUNTING MA PS AND SEMI-CLASSICAL A SYMPTOTICS 53 the genus expansion (7) and th e recurren ce co eﬃcien t expansion (10) is u niformly v alid. Since the co ver is ﬁnite, a wid th ∆ in Th eorem B.3 can b e determined.  The uniform v alidity of these expansions in a semi-inﬁnite strip of the t -plane is im p ortan t for the results of this p ap er. See, in particular, section 3.1.1 and the end of section 4.2. Referen ces 1. E. A. Bender and E. R. Canﬁeld. The asymp totic number of maps on a surface. J. Combin. The ory, Ser. A 43 , 244-257 (1986) 2. E. A . Bend er, Z. C. Gao, and L. B. Richmond. The map asymp totics constant t g . Ele ctr on. J. Combin. 15 , no. 1, R esearc h pap er 51, (2008) 3. E. Br ´ ezin and V.A. Kazako v. Exactly Solv able ﬁeld t heories of closed strings. Phys. L etts. B 236 (1990) 144-150. 4. P . Deift. 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On th e enumeration of planar maps. Bul l. Amer. Math. So c. 74 , 64-74 (1968) Dept. of Ma th, Univ. of Arizona, 520-621-2713, F A X: 520-626-51 86 E-mail addr ess : ercolani@math.ariz ona.edu

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