Asymptotics of orthogonal polynomials with complex varying quartic weight: global structure, critical point behaviour and the first Painleve equation

Asymptotics of orthogonal p olynomials with complex v arying quartic w eigh t: global structure, critical p oin t b eha viour and the ﬁrst Painlev ´ e equation M. Bertola †‡ 1 2 , A. T o vbis ] † Centr e de r e cher ches math ´ ematiques, Universit ´ e de Montr´ eal C. P. 6128, suc c. c entr e vil le, Montr ´ eal, Qu´ eb e c, Canada H3C 3J7 ‡ Dep artment of Mathematics and Statistics, Conc or dia University 1455 de Maisonneuve W., Montr ´ eal, Qu´ eb e c, Canada H3G 1M8 ] University of Centr al Florida Dep artment of Mathematics 4000 Centr al Florida Blvd. P.O. Box 161364 Orlando, FL 32816-1364 Abstract W e study the asymptotics of recurrence co eﬃcien ts for monic orthogonal p olynomials π n ( z ) with the quartic exp onen tial w eight exp  − N  1 2 z 2 + 1 4 tz 4  , where t ∈ C and N ∈ N , N → ∞ . Our goals are: A) to describ e the regions of diﬀeren t asymptotic b eha viour (diﬀerent genera) globally in t ∈ C ; B) to iden tify all the critical p oin ts, and; C) to study in details the asymptotics in a full neighborho o d near of critical p oin ts (double scaling limit), including at and near the p oles of Painlev ´ e I solutions y ( v ) that are known to provide the leading correction term in this limit. Our results are: A) W e found global in t ∈ C asymptotic of recurrence co eﬃcien ts and of “square-norms” for the orthogonal p olynomials π n for diﬀeren t conﬁgurations of the contours of integration. Special co de was developed to analyze all p ossible cases. B) In addition to the known critical p oin t t 0 = − 1 12 , we found new critical p oin ts t 1 = 1 15 and t 2 = 1 4 . C) W e derived the leading order behavior of the recurrence co eﬃcients (together with the error estimates) at and around the poles of y ( v ) near the critical points t 0 , t 1 in what w e called the triple scaling limit. W e prov ed that the recurrence co eﬃcients hav e un b ounded O ( N − 1 )-size (in t ) “spikes” near the p oles of y ( v ) and calculated the “univ ersal” shap e of these spikes for diﬀeren t cases (dep ending on the critical p oin t t 0 , 1 and on the conﬁguration of the contours of integration). The nonlinear steep est descen t metho d for Riemann-Hilb ert Problem (RHP) is the main technique used in the pap er. W e note that the RHP near the critical p oints is very similar to the RHP describing the semiclassical limit of the fo cusing NLS near the p oin t of gradient catastrophe that the authors solv ed in [ 5 ]. Our approach is based on the tec hnique developed in [ 5 ]. 1 W ork supp orted in part by the Natural Sciences and Engineering Research Council of Canada (NSERC) 2 bertola@crm.umontreal.ca Con ten ts 1 In tro duction and main results 2 2 The Riemann–Hilb ert problem for P ainlev´ e I 8 3 The RHP for recurrence co eﬃcien ts 9 3.1 String equation for α n ( x, t ) , β n ( x, t ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 4 Steep est descent analysis of the RHP ( 3-1 ) 11 4.1 Requiremen ts on the g –function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 4.1.1 Equalit y requirements for g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 4.1.2 Inequalit y (sign) requirements (or sign distribution requirements) for h and the mo dulation equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 4.1.3 The g -function and the mo dulation equations in the genus zero case . . . . . . . . 14 4.1.4 Explicit computation of g and h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 4.2 Discussion ab out existence of h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 4.3 Sc hematic conclusion of the steep est descen t analysis . . . . . . . . . . . . . . . . . . . . . 18 4.3.1 The “gen us zero” case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 4.3.2 Recurrence co eﬃcien ts in the gen us 0 cases . . . . . . . . . . . . . . . . . . . . . . 21 4.3.3 The regions of higher genera . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 4.4 Con tour deformation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 5 Breaking curves and global phase portraits 24 5.1 Gen us 0 symmetric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 5.2 Gen us 0 non-symmetric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 5.3 Gen us 1 symmetric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 5.4 Phase p ortraits or distribution of genera in the complex t -plane . . . . . . . . . . . . . . . 27 6 Double and m ultiple scaling analysis near the Painlev ´ e I gradient catastrophe p oin ts 34 6.1 Lo cal analysis at the p oint of gradient catastrophe . . . . . . . . . . . . . . . . . . . . . . 34 6.2 Asymptotics a wa y from the p oles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 6.3 Computation of the correction near t 1 : pro of of Theorem 1.2 . . . . . . . . . . . . . . . . 39 7 Analysis near the p oles: triple scaling limit 41 7.1 The asymmetric case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 7.1.1 T riple scaling: pro of of Theorem 1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . 43 7.2 The symmetric case: pro of of Theorem 1.4 . . . . . . . . . . . . . . . . . . . . . . . . . . 47 1 1 In tro duction and main results In this pap er we consider monic polynomials π n ( z ), orthogonal with resp ect to the quartic exp onential w eight e − N f ( z ,t ) , where f ( z , t ) = 1 2 z 2 + 1 4 tz 4 , t ∈ C and N ∈ N . As z → ∞ , the w eight function is exp onen tially decaying in four sectors S j of the op ening π / 4, centered around the rays Ω j = { z : arg z = − arg t 4 + π ( j − 1) 2 } , j = 1 , 2 , 3 , 4. W e consider the most general case when the p olynomials π n ( z ) are in tegrated on the “cross” formed by the rays Ω j , where the rays Ω 1 , 2 are oriented out wards (aw ay from the origin) and the ra ys Ω 3 , 4 - in wards. The corresp onding bilinear form is h p, q i % 1 ,% 2 ,% 3 ,% 4 = 4 X j =1 % j Z Ω j p ( z ) q ( z ) e − N f ( z ,t ) dz , f ( z , t ) := 1 2 z 2 + 1 4 tz 4 (1-1) where % j are ﬁxed complex n um b ers c hosen to satisfy % 1 + % 2 = % 3 + % 4 . Moreo v er, since multiplying all the % j ’s by a common nonzero constan t does not aﬀect the families orthogonal polynomials, these parameters are only deﬁned modulo the action of the group C ∗ and hence the orthogonal polynomials are naturally parametrized b y p oin ts in CP 2 . − arg( t ) / 4 $ 2 $ 3 $ 4 $ 1 Ω 1 Ω 2 Ω 4 Ω 2 Figure 1: The contours of integration and the asymptotic directions. The con tour $ 4 is homologically equal to − $ 1 − $ 2 − $ 3 , and it is unnecessary for the deﬁnition of the pairing ( 1-2 ). Alternativ ely , the bilinear form ( 1-1 ) can b e represented as h p, q i ~ ν = 3 X k =1 ν k Z $ k p ( z ) q ( z ) e − N f ( z ,t ) dz , (1-2) ν 1 = − % 1 , ν 2 = − % 2 − % 1 , ν 3 = − % 4 (1-3) where $ j , j = 1 , 2 , 3, are simple contours emanating from ∞ along Ω j and returning to ∞ along Ω j +1 [ 3 ] (Fig. 1 ). Then in the case a) w e ha ve ν 2 6 = 0 (and w e therefore can, and will, normalize it to b e ν 2 = 1) and the following three cases are p ossible: 1. The ”generic case”: ν 1 ν 3 6 = 0, ν 1 6 = 1 6 = ν 3 , so that there are three con tours $ j in ( 1-2 ); 2. The ”consecutiv e wedges”: either ν 1 or ν 3 (but not b oth) is zero so that there are tw o adjacent con tours $ j in ( 1-2 ); 3. The ”real axis”: ν 3 = 0 and ν 1 = 1. The remaining case b): % 1 + % 2 = 0, corresp onds to ν 2 = 0, so that the follo wing three cases are p ossible: 2 1. The ”single w edge”: ν 1 = 0, so that there is only one con tour $ 3 in ( 1-2 ), for whic h we can, and will, set ν 3 = 1; 2. The ”opp osite wedges, generic”: ν 1 ν 3 6 = 0 and ν 1 6 = ν 3 ; 3. The ”opp osite wedges, symmetric”: ν 1 = ν 3 6 = 0. The orthogonalit y condition for the monic p olynomials π n ( z ) can now b e written as h π n , z k i ~ ν = h n δ nk , k = 0 , 1 , 2 , · · · , n, ~ ν = ( ν 1 , ν 2 , ν 3 ) , (1-4) where the co eﬃcien t h n can also b e written as h n = h π n , π n i ~ ν and hence is the equiv alent of the “square norm” of π n (but it is in general a complex n umber). The existence of orthogonal polynomials π n ( z ) is not a priori clear. How ev er, if three consecutive monic p olynomials exists, they are related b y a three-term recurrence relation π n +1 = ( z − β n ) π n ( z ) − α n π n − 1 ( z ) , (1-5) where α n = α n ( t, N ), β n = β n ( t, N ) are called recurrence co eﬃcients. If the bilinear pairing is inv ariant under the map z 7→ − z then it follows immediately that the orthogonal p olynomials are even or o dd according to their degree and thus β n = 0 , ∀ n ∈ N (for example in case (a1) with co eﬃcien ts ν 1 = ν 3 , and ν 2 = 0). Then the remaining recurrence co eﬃcien ts α n satisfy α n [1 + t ( α n +1 + α n + α n − 1 )] = n N , (1-6) whic h is known in literature as the string equation or the F reud equation [ 15 ]. W e are interested in the asymptotic limit of α n , β n as N → ∞ and n N = x > 0 is ﬁxed and ﬁnite, so w e will use notations α n = α n ( x, t ) , β n = β n ( x, t ) instead of α n = α n ( t, N ), β n = β n ( t, N ). In the case of ν 2 = − 1 and x = 1 (that is, n = N ) and a ﬁxed t ∈ ( − 1 12 , 0), the asymptotics of α n , β n w as obtained in [ 10 ] as α n (1 , t ) = √ 1 + 12 t − 1 6 t + O ( n − 1 ) , (1-7) and β n deca ying exp onentially as n → ∞ . (T o b e more precise, Theorem 1.1 from [ 10 ] states that there exists some n 0 = n 0 ( t ), suc h that α n (1 , t ) , β n (1 , t ) exists for all n ≥ n 0 and hav e the ab o ve men tioned asymptotics.) In the non symmetrical case, how ever, the recurrence coeﬃcients β n are, generically , diﬀeren t from zero. Then, instead of ( 1-6 ), we hav e the general F reud system (we indicate ho w to derive it in Section. 3.1 ): 0 = β n + t  (2 β n + β n +1 ) α n +1 + ( β 2 n + 2 α n (1 − δ n 0 )) β n + α n β n − 1 (1 − δ n 0 )  , (1-8) n N = α n + t  α n α n − 1 + α 2 n (1 − δ n 0 ) + α n +1 α n + β 2 n α n + α n β n − 1 ( β n + β n − 1 )  . (1-9) 3 where δ ij denotes the Kroneck er’s delta. Assuming that β n → β and α n → α as n → ∞ , we obtain tw o leading order algebraic equations β (1 + 6 tα + tβ 2 ) = 0 , α  1 + 3 tα + 3 tβ 2  = x, (1-10) whic h hav e tw o solutions β = 0 , α = √ 1 + 12 xt − 1 6 t , (1-11) and β 2 = − 6 α − 1 t , α = √ 1 − 15 xt − 1 15 t . (1-12) The dep endence on x is rather ﬁctitious: indeed, lo oking at the pairing ( 1-1 ) one sees that if 0 < x 6 = 1 then w e can rescale e z = z √ x , e t = x t and obtain the case N = n ( x = 1) without loss of any generality; w e shall assume this done throughout the pap er, but still distinguish n and N b ecause they play a slightly diﬀeren t role. W e p oin t out that the asymptotics of the recurrence co eﬃcients for orthogonal p olynomials with in- tegration on the real axis (and analytic contin uation thereof in the complex t –plane) satisﬁes identically β n = 0, and hence only the ﬁrst solution ( 1-11 ) is relev ant. Ho w ever, in view of applications to com bina- torics of maps, it is not clear what role, if any , the second solution ( 1-12 ) play . It could b e, p erhaps, of some relev ance that the critical p oin t t 1 = 1 15 of the second solution is actually closer to the origin than the critical p oin t t 0 = − 1 12 of the “standard” (ﬁrst) solution. The recurrence co eﬃcients problem was studied in [ 10 ], following earlier w ork [ 14 ], for negative v alues of t ∈ ( − 1 12 , 0). F or deﬁniteness w e shall assign arg( t ) = π , so that (refer to Fig. 1 ) the con tour $ 1 consists of the t wo rays arg( z ) = ± π 4 , $ 2 of arg( z ) = π 4 , 3 π 4 and $ 3 of arg( z ) = 3 π 4 , 5 π 4 (other determinations of arg( t 1 4 ) would only reshuﬄe the contours around). T o describ e the results of [ 10 ] and to set the stage for our results, w e need to recall that the general solution of the Painlev ´ e I (P1) equation y 00 ( v ) = 6 y 2 ( v ) − v (1-13) is parametrized by tw o parameters in terms of a certain Riemann–Hilb ert problem described in Section 2 . It is known that any solution to P1 has inﬁnitely many p oles with a Laurent expansion of the form y ( v ) = 1 ( v − v p ) 2 + v p 10 ( v − v p ) 2 + 1 6 ( v − v p ) 3 + β ( v − v p ) 4 + v 2 p 300 ( v − v p ) 6 + O (( v − v p ) 7 ) . (1-14) The Painlev ´ e prop erty asserts that the only singularities that can o ccur are of this form, that is, the p osition of these p oles dep ends on the c hosen solution, and it is largely unknown, except for some asymptotic localization of the remote poles, see, for example, [ 16 ]. In the following theorem and henceforth w e use notations α n ( t ) = α n (1 , t ) , β n ( t ) = β n (1 , t ). 4 Theorem 1.1 ([ 10 ]) L et y (0) ( v ) := y ( v ; 1 − ν 3 ) , y (1) ( v ) := y ( v ; ν 1 ) (se e Def. 2.1 b elow), b e the two solution of P1( 1-13 ) 3 . L et K ⊂ C b e a c omp act set that do es not c ontain any of the p oles of y (0) , y (1) . L et t ∈ C appr o ach the critic al value t 0 = − 1 12 in such a way that N 4 5  t + 1 12  = − v 2 9 5 3 6 5 , (1-15) wher e v ∈ K . Then, for lar ge enough n , the r e curr enc e c o eﬃcients α n ( t ) , β n ( t ) have asymptotics α n ( t ) = 2 − 2 3 5 3 2 5 ( y (1) ( v ) + y (0) ( v )) N − 2 5 + O ( N − 3 5 ) , β n ( t ) = 2 1 10 3 2 5 ( y (0) ( v ) − y (1) ( v )) N − 2 5 + O ( N − 3 5 ) (1-16) as n = N → + ∞ , which is valid uniformly in K . Mor e over, the O terms c an b e exp ande d into a ful l asymptotic exp ansion in p owers of n − 1 5 . The statement of Theorem 1.1 is an example of what is known as the double scaling limit near a critical p oin t and it is obtained using the steep est descent analysis and a special ”P ainlev´ e I parametrix” that w as ﬁrst in tro duced in [ 14 ]: it do es not address, how ev er, the asymptotics of the recurrence co eﬃcien ts when v is at or close to (in a “triple” scaling sense to b e sp eciﬁed later) a p ole of either y (1) ( v ) or y (0) ( v ). Also, no information is av ailable for the case t < − 1 / 12, as well as for general complex v alues of t . Thus, the main results of this pap er are: 1. ﬁnding the global (in t ∈ C ) leading order b ehavior of the recurrence co eﬃcients α n ( t ) , β n ( t ) and of the ”square-norms” h n for the orthogonal p olynomials π n for the cases to diﬀerent conﬁgurations of the con tours $ j as listed ab o v e; 2. deriving new critical p oin ts t 1 = 1 15 in the case ν 2 = 0 (this case w as not considered in [ 10 ]) and t 2 = 1 4 ; note that t 1 is closer to the origin t = 0 than t 0 ; 3. deriving the leading order behavior of α n ( t ) , β n ( t ) (together with the error estimates) at and around the p oles of the P1 solutions y (0) , (1) ( v ) near the critical p oin ts t 0 , t 1 ; we will see that α n ( t ) , β n ( t ) and h n ha ve un b ounded “spikes” near the poles of y (0) , (1) ( v ) and study the shap e of these spik es in certain cases. In regard to the p oint t 1 = 1 15 (whic h, to the b est of our knowledge, was not considered in the literature), w e also ﬁnd an asymptotics that is related to the same Painlev ´ e I equation in the theorem b elow. 3 The parameters that were indicated with α, β in [ 10 ] in our notations are: α = 1 − ν 1 , β = ν 3 . 5 Theorem 1.2 L et y ( v ) := y (1) ( v ) = y ( v ; ν 1 ) (se e Def. 2.1 ) b e the solution of P1 ( 1-13 ); let K ⊂ C b e a c omp act set not c ontaining any p oles of y (1) . L et t dep end on N so that N 4 5 δ t := N 4 5  t − 1 15  = e − 3 iπ 5 v 3 6 5 2 1 5 5 , (1-17) wher e v ∈ K (se e 6-8 ). Then, uniformly for v ∈ K , we have α n = − 1 + i 6 2 5 e − 3 iπ 10 N 2 5 y ( v ) + O ( N − 3 5 ) , β n = − 3 i − 6 2 5 e − 3 iπ 10 N 2 5 y ( v ) + O ( N − 3 5 ) , (1-18) h n = 2 iπ ( − 1) N 1 − 3 2 5 2 3 5 e − 4 5 iπ y ( v ) N 2 5 ! exp " 9 N 4 − 195 N 4 δ t + e − 2 5 iπ 6 1 5 N 1 5 H I # (1 + O ( N − 3 5 )) , (1-19) wher e H I = 1 2 ( y 0 ) 2 + y v − 2 y 3 is the Hamiltonian of P1 ( 2-5 ) and H 0 1 ( v ) = y ( v ) . Theorem 1.2 is, of course, of the same nature as Theorem 1.1 . It is clear, though, that in order to study the full neighborho od of a critical p oint t j , j = 0 , 1, which, by analogy with the zero disp ersion limit of the fo cusing Nonlinear Schr¨ odinger equation (NLS) will b e called a p oin t of gradient catastrophe, one m ust separate the asymptotic analysis in tw o distinct regimes: • Aw ay from the p oles : the v ariable v is chosen within a ﬁxed compact set that do es not include an y p ole of the relev ant solutions to P1; • Near the p oles : the v ariable v undergo es its o wn scaling limit and approaches a giv en p ole at a certain rate. Theorems 1.1 , 1.2 are examples of the regime “aw a y from the p oles”. T o inv estigate the regime “near the p oles” we must use a nov el mo diﬁcation that we could call triple scaling . Theorem 1.3 Consider the setups as in Thm. 1.1 with ν 1 6 = 1 − ν 3 or Thm. 1.2 , with the same notation for y (0) , (1) in the former c ase and y (1) in the latter. L et v p denote any chosen p ole of y (1) (which is, if in the ﬁrst setup, not a p ole of y (0) ). L et t appr o ach t 0 = − 1 12 or t 1 = 1 15 in such a way that it satisﬁes r esp e ctively t + 1 12 = − v p N 4 5 3 6 5 2 9 5 − s 3 √ 2 N or t − 1 15 = − v p e − 3 iπ 5 3 6 5 2 1 4 5 N 4 5 − i s 2 N , (1-20) Then α n ( t ) = b 2 0 4 − 1 4 s 2 + O  N − 1 5 s − 1  , (1-21) β n ( t ) = a 0 + 1 2 s (1 − b 0 s ) + O ( N − 1 5 ) , (1-22) 6 wher e a 0 , b 0 (the limiting values of a ( t ) , b ( t ) fr om T able 2 ) ar e given by a 0 = 0 , b 0 = √ 8 and a 0 = − 3 i , b 0 = 2 i in the c ases t ∼ t 0 and t ∼ t 1 r esp e ctively. The numb ers h n satisfy: h n = π 2 N exp " − 3 N 2 + N 1 5 v p 3 1 5 2 4 5 + √ 2 s #  √ 8 − 1 s + O ( N − 1 5 s − 1 )  , t ∼ − 1 12 , (1-23) h n = π ( − 1) N exp " 9 N 4 − 13 4 N 1 5 v p e − 3 iπ 5 3 1 5 2 1 5 + 13 2 is #  2 i − 1 s + O ( N − 1 5 s − 1 )  , t ∼ 1 15 . (1-24) These formulæ hold uniformly for b ounde d values of s as long as the indic ate d err or terms r emain in- ﬁnitesimal. In p articular s c an appr o ach s = 0 or s = 1 b 0 at any chosen r ate O ( N ρ ) with ρ ∈ [0 , 1 5 ) (the c ase ρ = 0 al lowing any given ﬁxe d value s 6 = 0 , 1 b 0 ). As the reader notices, the as ymptotics has a dramatically changed form and do es not inv olve now an y transcenden tal function. Note that the scale of the phenomenon in this case is O ( N − 1 ) around the lo cation of the image of the p ole v p (see ( 6-7 ) or ( 6-8 )respectively) in the t -plane, whereas the scale at whic h the transcendental nature of the asymptotic is shown is N − 4 5 . T o study this new phenomenon, it is con venien t to set a triple scaling of the form t = t 0 + c 1 N 4 5 + c 2 N , (1-25) where the v alue of parameter c 1 corresp onds to a particular p ole of the Painlev ´ e I transcenden t. Note that, according to ( 1-21 ), ( 1-22 ), the v alues of α n , β n are un b ounded as s → 0 and s → b 0 (the latter is v alid only for β n ). A quite diﬀeren t phenomenon occurs instead if we are in the setup of Theorem 1.1 with the same triple scaling limit but with additional symmetry ν 1 = 1 − ν 3 (the case excluded from Theorem 1.3 ). In this case the tw o functions y (0) and y (1) are the same solution to P1 ( 1-13 ) and a sort of cancellation in ( 1-21 ), ( 1-22 ) o ccurs. Theorem 1.4 Consider the setup of Thm. 1.1 with t appr o aching t 0 and ν 1 = 1 − ν 3 . L et v p b e a p ole of y ( v ) := y (1) ( v ) = y (0) ( v ) and let t vary so that t = t p ( s ) = − 1 12 − v p 2 9 5 3 6 5 N 4 5 + s 2 3 3 N , (1-26) wher e s = O ( N − ρ ) with an arbitr ary ρ ∈ [0 , 1 5 ) . Then the fol lowing holds: α n = b 2 4 9 − s 2 + O ( N − 1 5 ) 1 − s 2 + O ( N − 1 5 ) , β n = 0 , (1-27) h n = π √ 8 2 N exp " − 3 N 2 + N 1 5 v p 3 1 5 2 4 5 − s 4 #  3 − s 1 + s + O ( N − 1 5 ( s 2 − 1) − 1 )  . (1-28) 7 The variable s may appr o ach the p oints s = ± 1 at some r ate (a quadruple scaling ) as long as the c orr esp onding err or indic ate d in the formulæ ab ove terms ar e inﬁnitesimal. Remark 1.1 Note that the values of α n is unb ounde d in the vicinity of s = ± 1 and h n is unb ounde d in the vicinity of s = − 1 (ther e is no information for h n in the vicinity of s = +1 ). L et us denote the Hankel determinants of the moments by ∆ n ( t, N ) (se e R emark 3.1 ) and use t p ( s ) as in ( 1-26 ): sinc e α n = ∆ n − 1 ∆ n +1 ∆ 2 n we de duc e that ∆ n ( t ( s ) , n ) vanishes at s = ± 1 (within our err or estimates), while ∆ n ± 1 ( t ( s ) , n ) vanish at s ∈ { 1 , 3 } and s ∈ {− 3 , − 1 } r esp e ctively. 2 The Riemann–Hilb ert problem for P ainlev´ e I  0 − 1 1 0   1 0 ω 0 e − 2 ϑ 1   1 0 ω − 2 e − 2 ϑ 1   1 ω − 1 e 2 ϑ 0 1   1 0 ω 2 e − 2 ϑ 1   1 ω 1 e 2 ϑ 0 1  Figure 2: The jump matrices for the P ainlev ´ e 1 RHP: here ϑ := ϑ ( ξ ; v ) := 4 5 ξ 5 2 − v ξ 1 2 . Let the in v ertible matrix-function P = P ( ξ , v ) b e analytic in eac h sector of the complex ξ -plane sho wn on Fig. 2 and satisfy the multiplicativ e jump conditions along the oriented b oundary of eac h sector with jump matrices shown on Fig. 2 . The en tries of the jump matrices satisfy 1 + ω 0 ω 1 = − ω − 2 , 1 + ω 0 ω − 1 = − ω 2 , 1 + ω − 2 ω − 1 = ω 1 , (2-1) so that the jump matrices in Fig. 2 depend, in fact, only on 2 complex parameters (that uniquely de- ﬁne a solution to P1). The matrix function P ( ξ , v ) is uniquely deﬁned b y the following RHP . Problem 2.1 (P ainlev´ e 1 RHP [ 16 ]) The ma- trix P ( ξ ; v ; ~ ω ) is lo c al ly b ounde d, admits b oundary values on the r ays shown in Fig. 2 and satisﬁes P + = P − M , (2-2) P ( ξ ; v ; ~ ω ) = ξ σ 3 / 4 √ 2  1 − i 1 i   I + O ( ξ − 1 2 )  , (2-3) wher e the jump matric es M = M ( ξ ; v , ~ ω ) ar e the matric es indic ate d on the c orr esp onding r ay in Fig. 2 , with ~ ω := ( ω − 2 , ω − 1 , ω 0 , ω 1 , ω 2 ) satisfying ( 2-1 ). 8 F or any ﬁxed v alues of the parameters ω k , Problem 2.1 admits a unique solution for generic v alues of v ; there are isolated points in the v –plane where the solv abilit y of the problem fails as stated. The piecewise analytic function Ψ( ξ , v ; ~ ω ) = P ( ξ , v ; ~ ω )e ϑσ 3 (2-4) solv es a sligh tly diﬀeren t RHP with c onstant jumps on the same rays and thus solves an ODE. Direct computations using the ODE and formal algebraic manipulations of series along the lines of [ 13 , 11 , 12 ] sho w that Ψ admits the following formal expansion Ψ( ξ , v ; ~ ω ) = ξ σ 3 / 4 √ 2  1 − i 1 i  × ×  1 − H I σ 3 √ ξ + H 2 I 1 + y σ 2 2 ξ + ( v 2 − 4 H 3 I − 2 y 0 ) 24 ξ 3 2 σ 3 + iy 0 − 2 iH I y 4 ξ 3 2 σ 1 + O ( ξ − 2 )  e ϑσ 3 , H I := 1 2 ( y 0 ) 2 + y v − 2 y 3 , ϑ := ϑ ( ξ ; v ) = 4 5 ξ 5 2 − v ξ 1 2 , as ξ → ∞ . (2-5) where y = y ( v ) solves the Painlev ´ e I equation ( 1-13 ). The matrix Ψ( ξ , v ; ~ ω ) uniquely deﬁnes a solution y ( v ; ~ ω ) of P1 ( 1-13 ), and viceversa. The family of solution we shall use consists of the choice ω 0 = 0 in ( 2-1 ). Then the constant jump matrix for Ψ dep ends only on one free parameter: we shall choose it to b e ω 1 , with ω ± 2 = − 1 and ω 1 = 1 − ω − 1 . Deﬁnition 2.1 The functions y ( v ; κ ) ar e the solutions to Painlev´ e 1, which ar e deﬁne d via Ψ ( ξ , v , ( − 1 , κ , 0 , 1 − κ , − 1)) in Pr oblem 2.1 . We shal l abbr eviate this notation by Ψ ( ξ , v ; κ ) . 3 The RHP for recurrence coeﬃcients It is w ell known ([ 7 ]) that the existence of the abov e-men tioned orthogonal p olynomials π n ( z ) is equiv alent to the existence of the solution to the follo wing RHP ( 3-1 ). More precisely , relation betw een the RHP ( 3-1 ) and the orthogonal p olynomials π n ( z ) is giv en b y the follo wing prop osition ([ 10 ]), whic h has the standard pro of (see [ 7 ]). Prop osition 3.1 L et Ω := S 3 j =1 $ j , and deﬁne ν : Ω → C by ν ( z ) = ν j when z ∈ $ j . Then the solution of the fol lowing RHP pr oblem            Y ( z ) is analytic in C \ Ω Y + ( z ) = Y − ( z )  1 ν ( z ) e − N f ( z ,t ) 0 1  , z ∈ Ω , Y ( z ) = ( 1 + O ( z − 1 ))  z n 0 0 z − n  , z → ∞ . (3-1) exists (and it is unique) if and only if ther e exist a monic p olynomial p ( z ) of de gr e e n and a p olynomial q ( z ) of de gr e e ≤ n − 1 such that h p ( z ) , z k i ν 1 ,ν 2 ,ν 3 = 0 , for all k = 0 , 1 , 2 , · · · , n − 1 , (3-2) h q ( z ) , z k i ν 1 ,ν 2 ,ν 3 = 0 , for all k = 0 , 1 , 2 , · · · , n − 2 , and h q ( z ) , z n − 1 i ν 1 ,ν 2 ,ν 3 = − 2 π i. (3-3) 9 In that c ase the solution to the RHP ( 3-1 ) is given by Y ( z ) =  p ( z ) C Ω [ p ( z ) ν ( z ) e − N f ( z ,t ) ] q ( z ) C Ω [ q ( z ) ν ( z ) e − N f ( z ,t ) ]  , z ∈ C \ Ω , where C Ω [ φ ] = 1 2 π i Z Ω φ ( ζ ) dζ ζ − z (3-4) is the Cauchy tr ansform of φ ( z ) . Remark 3.1 It fol lows imme diately that the p olynomials p, q in Pr op osition 3.1 c oincide with π n ( z ) = 1 ∆ n det        µ 0 µ 1 . . . µ n − 1 µ n µ 1 µ n +1 . . . . . . µ n − 1 . . . µ 2 n − 2 µ 2 n − 1 1 z . . . z n        q ( z ) = − 2 iπ ∆ n det        µ 0 µ 1 . . . µ n − 2 µ n − 1 µ 1 . . . µ n . . . . . . µ n − 2 . . . µ 2 n − 4 µ 2 n − 3 1 z . . . z n − 1        (3-5) r esp e ctively, wher e µ j := h z j , 1 i ν 1 ,ν 2 ,ν 3 ar e the moments and ∆ n := det [ µ i + j ] 0 ≤ i,j ≤ n − 1 . It is cle ar fr om these expr essions but it is also a wel l known fact [ 6 ] that the c ondition of existenc e of the n -th ortho gonal p olynomial p n is that ∆ n 6 = 0 ; on the other hand it is known fr om [14] that existenc e of p n is e quivalent to the solvability of the RHP and henc e the existenc e of the solution for the RHP pr oblem ( 3-1 ) is e quivalent to ∆ n 6 = 0 . This determinant is sometimes r eferr e d to as the “tau function ” of the pr oblem [ 4 ]. Note also that the ”squar e-norms” of the p olynomials ar e r atios of Hankel determinants h n := h π n ( z ) , π n ( z ) i ~ ν = h π n ( z ) , z n i ~ ν = ∆ n +1 ∆ n (3-6) If π n +1 ( z ), π n ( z ) , π n − 1 ( z ) are monic orthogonal p olynomials then they satisfy π n +1 ( z ) = ( z − β n ) π n ( z ) − α n π n − 1 ( z ) (3-7) for certain recurrence co eﬃcients α n , β n [ 18 , 6 ]. The following well kno wn statements (see, for example, [ 14 ], [ 7 ], [ 10 ]) show the connection b et ween the RHP ( 3-1 ), the orthogonal p olynomials π n ( z ) and their recurrence co eﬃcien ts. Prop osition 3.2 L et Y ( n ) ( z ) denote the solution of the RHP ( 3-1 ). If we write Y ( n ) ( z ) = 1 + Y ( n ) 1 z + Y ( n ) 2 z 2 + O ( z − 3 ) !  z n 0 0 z − n  , z → ∞ , (3-8) then h n = − 2 iπ ( Y ( n ) 1 ) 12 , α n =  Y ( n ) 1  12 ·  Y ( n ) 1  21 , β n =  Y ( n ) 2  12  Y ( n ) 1  12 −  Y ( n ) 1  22 . (3-9) Prop osition 3.3 Supp ose the RHP ( 3-1 ) has solution Y ( n ) ( z ) . L et ( 3-8 ) b e its exp ansion at ∞ and let α n , β n b e given by ( 3-9 ). If α n 6 = 0 , then the monic ortho gonal p olynomials π n +1 ( z ) , π n ( z ) , π n − 1 ( z ) exist and satisfy the thr e e term r e curr enc e r elation ( 3-7 ). 10 3.1 String equation for α n ( x, t ) , β n ( x, t ) The string equations, or F reud’s equations, for the recurrence co eﬃcien ts α n , β n are nonlinear diﬀerence equations. Assuming that the corresp onding orthogonal p olynomials exist, they can b e obtained as follo ws. On one hand we hav e z π n ( z ) = π n +1 ( z ) + β n π n ( z ) + α n π n − 1 ( z ) . (3-10) One can iterate ( 3-10 ) to ﬁnd z k π n ( z ) for any k ∈ N . On the other hand we hav e 0 ≡ 3 X j =1 ν j Z $ j ∂ z  π n π m e − N f  d z = 3 X j =1 ν j Z $ j ( π 0 n π m + π n π 0 m − N π n π m f 0 ( z )) e − N f d z = (3-11) = h π 0 n , π m i ν + h π n , π 0 m i ν − N h π n , f 0 ( z ) π m i ν (3-12) Since f 0 ( z ) is a p olynomial, the last term ab ov e can b e written as a p olynomial in the recurrence coeﬃ- cien ts using rep eatedly ( 3-10 ). F or n = m the ﬁrst t w o terms are the same and v anish b ecause π 0 n is a p olynomial of degree n − 1 and π n is orthogonal to any p olynomial of low er degree. Then ( 3-12 ) nn yields a recurrence relation. F or m = n − 1 w e hav e h π 0 n , π n − 1 i ν +  π n , π 0 n − 1  ν − N h π n , f 0 ( z ) π n − 1 i ν = n h π n − 1 , π n − 1 i ν − N h π n , f 0 ( z ) π n − 1 i ν . (3-13) Equation ( 3-13 ) yields ( 1-9 ) while ( 3-12 ) for n = m yields ( 1-8 ). If the orthogonality pairing is symmetric under z 7→ − z , that is, if h p ( z ) , q ( z ) i ν = h p ( − z ) , q ( − z ) i ν (3-14) then it follo ws easily that β n ≡ 0 and then ( 1-8 , 1-9 ) reduce simply to ( 1-6 ). 4 Steep est descen t analysis of the RHP ( 3-1 ) The steepest descen t analysis in general terms for these kind of orthogonal polynomials with a p olynomial external ﬁeld w as inv estigated in [ 3 ] and so we re fer the reader there for details. The schematic of the approac h is outlined here; as customary , the problem undergo es a sequence of mo diﬁcations in to equiv alent RHPs until it can b e eﬀectiv ely solved in approximate form while keeping the error terms under control. • One starts with the problem for Y ( 3-1 ) and seeks an auxiliary scalar function g ( z ), called the g – function, whic h is analytic except for a collection Σ of appropriate contours to b e describ ed subsequen tly and b eha ves like ln z + O ( z − 1 ) near z = ∞ : the contour Ω of the RHP ( 3-1 ) can b e deformed b ecause the RHP ( 3-1 ) has an analytic jump matrix. The ﬁnal c onﬁgur ation of Ω must c ontain al l the c ontours wher e g is not analytic . • Then w e introduce a new matrix T ( z ) := e − N ` σ 3 2 Y ( z )e − N ( g ( z,t ) − ` 2 ) σ 3 . (4-1) 11 As a result, T solves a new RHP          T ( z ) is analytic in C \ Ω T + ( z ) = T − ( z ) e − N 2 ( h + − h − ) ν ( z ) e N 2 ( h + + h − ) 0 e N 2 ( h + − h − ) ! , z ∈ Ω , where h ( z , t ) := 2 g ( z , t ) − f ( z , t ) − `. T ( z ) = ( 1 + O ( z − 1 )) , z → ∞ . (4-2) • At this p oint the Deift–Zhou metho d can pro ceed provided that the function g ( z ), the constant ` and the collection of contours Ω into which w e hav e deformed the problem fulﬁll a rather long collection of equalities and –most imp ortan tly– ine qualities that we set out to brieﬂy describ e [ 20 , 3 ]: we sa y here that if all these requirements are fulﬁlled the full asymptotic for the problem can be obtained in terms of Riemann Theta functions on a suitable (h yp er)elliptic Riemann surface of a p ositive genus (with the case of zero gen us not requiring any sp ecial function). 4.1 Requiremen ts on the g –function The (deformed) con tour Ω can b e partitioned into tw o disjoint subsets of oriented arcs that we shall denote b y M and term main arcs , and C or complementary arcs ; this partitioning is sub ordinated to a list of requiremen ts for g and h . 4.1.1 Equalit y requiremen ts for g 1. g ( z ) (to shorten notations, we drop the t v ariable in this subsection) is analytic in C \ ( M ∪ C ) and has the asymptotic b eha viour g ( z ) = ln z + O ( z − 1 ) , z → ∞ ; (4-3) 2. g is analytic along all the unb ounde d complementary arcs except for exactly one unb ounde d com- plemen tary arc which we will denote by γ 0 , where g + ( z ) − g − ( z ) = 2 iπ z ∈ γ 0 (4-4) (note that the function e N g ( z ) is analytic across all the un b ounded complementary arcs since N = n ∈ N b y deﬁnition); 3. on the b ounde d complementary arcs γ c , the function g has a jump g + ( z ) − g − ( z ) = 2 π iη c , η c ∈ R , z ∈ γ c , (4-5) where η c is a constan t on each connected comp onen t of the complementary arcs; 12 4. across each main arc (which are all b ounded by assumption) we hav e the jump g + ( z ) + g − ( z ) = f ( z ) + `, z ∈ M . (4-6) W e stress that the constant ` is the same for all the main arcs. Assuming that the contours M , C are kno wn, the function g ( z ) can b e considered as the solution of the scalar RHP , deﬁned by conditions 1-4. Similarly , the function h = 2 g − f can b e considered as the solution of the scalar RHP with the jumps 1 2 ( h + ( z ) − h − ( z )) = 2 π iη c , z ∈ M , 1 2 ( h + ( z ) + h − ( z )) = 0 , z ∈ C , h + ( z ) − h − ( z ) = 4 iπ , z ∈ γ 0 , (4-7) and the asymptotic b eha vior h ( z ) = − f ( z ) + 2 ln z + O ( z − 1 ) , z → ∞ . (4-8) It follo ws immediately from ( 4-7 ) that < h ( z ) is contin uous across the complemen tary arcs C . 4.1.2 Inequalit y (sign) requirements (or sign distribution requiremen ts) for h and the mo dulation equation 1. along eac h complemen tary arc γ c w e ha ve < h ( z ) ≤ 0;with the equality holding at most at a ﬁnite n umber of p oin ts. In the generic situation these would be only the endp oin ts (we shall call this case regular , with the same connotation as in [ 8 ]); 2. on b oth sides in close proximit y of each main arc γ m ⊂ M w e hav e < h ( z ) > 0 The sign distribution requirement for the main arcs implies that < h ( z ) is contin uous everywhere in C and the main arcs b elong necessarily to its zero lev el set. The main arcs γ m can b e considered as the branch-cuts of a hyperelliptic Riemann surface R ( t ), asso ciated with g and h . The n umber of main arc (the gen us of R ( t ) plus one) needs to be chosen in such a w a y that the ab o v e sign conditions will b e satisﬁed. The lo cation of the endp oin ts λ of each main arc (which are the branc h-p oin ts of R ( t )) is go verned by the requirement < h ( z ) = O ( z − λ ) 3 2 ) , z → λ, (4-9) kno wn as the mo dulation e quations . Since the jumps on the complementary arcs are constants, the ab ov e requiremen t can also b e stated as h 0 ( z ) = O ( √ z − λ ) , z → λ, (4-10) where the discon tinuit y is placed on the main arc. The logic b ehind all the ab ov e requirements and the mo dulation equations will be brieﬂy discussed in Subsections 4.2 , 4.3 . Note that the modulation equation ( 4-9 ) implies that there are three zero level curv es of < h emanating from each branch-point λ . 13 4.1.3 The g -function and the mo dulation equations in the genus zero case Due to the mo dulation equations 4-9 , solutions of the RHPs for g and for h commute with diﬀeren tiation. Th us, the (scalar) RHP for g 0 is: 1. g 0 ( z ) is analytic (in z ) in ¯ C \ M and g 0 ( z ) = 1 z + O ( z − 2 ) as z → ∞ ; (4-11) 2. g 0 ( z ) satisﬁes the jump condition g 0 + + g 0 − = f 0 on M . (4-12) Let us consider the case of a single main arc γ m with the endp oints λ 0 , λ 1 . Using the analyticity of f 0 , the solution of the latter RHP is giv en by g 0 ( z ) = R ( z ) 4 π i Z ˆ γ m f 0 ( ζ ) ( ζ − z ) R ( ζ ) + dζ , R ( z ) := p ( z − λ 1 )( z − λ 0 ) , (4-13) where the contour ˆ γ m encircles the contour γ m and has counterclockwise orientation ( z is outside ˆ γ m ). It is kno wn ([ 10 ]) that the case t ∈ ( − 1 12 , 0) is the genus zero case with real branch-points (we will derive the same result shortly). Using ( 4-13 ), the asymptotics ( 4-11 ) yields tw o equations Z ˆ γ m f 0 ( ζ ) R ( ζ ) + dζ = 0 and Z ˆ γ m ζ f 0 ( ζ ) R ( ζ ) + dζ = − 4 iπ , (4-14) called moment conditions, whic h are equiv alen t to the endp oint condition (mo dulation equation) ( 4-9 ). We use the moment c onditions ( 4-14 ) t o deﬁne the lo c ation of the endp oints λ 0 , 1 , where we put λ 0 < λ 1 . In the case of a p olynomial f ( z , t ), equations ( 4-14 ) can b e solved using the residue theorem. Setting λ 0 = a − b, λ 1 = a + b and using the residue theorem on equations ( 4-14 ) we obtain a + ta ( a 2 + 3 2 b 2 ) = 0 and a 2 + 1 2 b 2 + t ( a 4 + 3 a 2 b 2 + 3 8 b 4 ) = 2 . (4-15) There are tw o p ossibilities: a = 0 and a 6 = 0. In the ﬁrst case we obtain solutions to the system ( 4-15 ) as a = 0 b 2 = − 2 3 t (1 ∓ √ 1 + 12 t ) , (4-16) λ 0 , 1 = ∓ b = ∓ r − 2 3 t (1 − √ 1 + 12 t ) (4-17) The c hoice of the negative sign in ( 4-16 ) comes from the requiremen t that b is b ounded as t → 0. Observ e that for t > t 0 = − 1 12 the v alues ± b coincide with the branch-points, derived in [ 10 ]. At the critical p oint t = t 0 = − 1 12 (4-18) 14 the t w o pairs of roots ( 4-16 ) coincide, creating ﬁv e zero lev el curv es of < h ( z ) emanating from the endpoints ± b 0 , where b 0 = √ 8. The second pair of ro ots b = ± q − 2 3 t (1 + √ 1 + 12 t ) are sliding along the real axis from ±∞ to ± b 0 as real t v aries from 0 − to t 0 , and sliding along the imaginary axis from ± i ∞ to 0 as real t v aries from 0 + to + ∞ . In the second case a 6 = 0, the system of mo dulation equations ( 4-15 ) b ecomes  t ( a 2 + 3 2 b 2 ) = − 1 , tb 2 ( a 2 − 3 8 b 2 ) = 2 , (4-19) whic h yields b 2 = − 4 15 t (1 ± √ 1 − 15 t ) and a 2 = − 1 5 t (3 ∓ 2 √ 1 − 15 t ) . (4-20) 4.1.4 Explicit computation of g and h . Once the v alues of branc h-p oin ts (endp oin ts) λ 0 , 1 are determined, one can ca lculate explicitly g ( z ) and h ( z ) = h ( z ; t ), where h ( z ) = 2 g ( z ) − f ( z ) − ` . The expression h 0 ( z ) = R ( z ) 2 π i Z ˆ γ m f 0 ( ζ ) ( ζ − z ) R ( ζ ) + dζ , (4-21) for h 0 ( z ) is readily a v ailable from ( 4-13 ) by placing z inside the lo op ˆ γ m . Ho wev er, it seems easier to calculate h 0 ( z ) explicitly by solving the scalar RHP that h 0 ( z ) satisﬁes: 1. h 0 ( z ) is analytic (in z ) in γ m and h 0 ( z ) = − z − tz 3 + 2 z − 1 + O ( z − 2 ) as z → ∞ ; (4-22) 2. h 0 ( z ) satisﬁes the jump condition h 0 + + h 0 − = 0 on γ m , (4-23) whic h can b e easily obtained from the RHP ( 4-12 ) for g 0 ( z ). There are tw o cases, symmetric and nonsymmetric dep ending on the v alue a = 0 or a 6 = 0. Symmetric case: a = 0 . The RHP for h 0 has a unique solution (with h 0 ± ∈ L 2 ( γ m )) that is giv en b y h 0 ( z ) = − ( k + tz 2 ) √ z 2 − b 2 , where the endp oin ts ± b of γ m are kno wn and the constant k is to b e determined. Assuming that b 2 is giv en by ( 4-16 ), we obtain k = 1 + 1 2 tb 2 , so that h 0 ( z ) = −  tz 2 + 1 + tb 2 2  ( z 2 − b 2 ) 1 2 = −  tz 2 + √ 1 + 12 t 3 + 2 3   z 2 − b 2  1 2 (4-24) Since the branch-cut of the radical is [ − b, b ] w e conclude that h 0 ( z ) is an o dd function. Direct calculation yield h ( z ) = 2 ln z + √ z 2 − b 2 b − z 8 (2 tz 2 + tb 2 + 4)( z 2 − b 2 ) 1 2 . (4-25) 15 It is clear that h ( b ) = 0. There is the oriented branch-cut of h ( z ) along the ra y ( −∞ , − b ), where h + ( z ) − h − ( z ) = 4 π i . Combined with ( 4-24 ), that implies h ( z ) = O ( z − b ) 3 2 (4-26) (or a higher pow er of ( z − b )). At the p oin t of gradient catastrophe t 0 = − 1 12 , the order O ( z − b ) 3 2 in ( 4-26 ) should b e replaced by O ( z − b ) 5 2 . Because of the 4 π i jump along ( −∞ , − b ), the function h ( z ) do es not ha ve O ( z + b ) 3 2 b eha vior near z = − b ; how ev er, < h ( z ) do es hav e O ( z + b ) 3 2 b eha vior near z = − b . Non-symmetric case: a 6 = 0 . F ollowing the same lines (the algebraic computation b eing a bit more in volv ed) one obtains h 0 ( z ) = − t  z 2 + az − b 2  p ( z − a ) 2 − b 2 (4-27) where a, b are giv en by ( 4-20 ). A direct computation yields in w = z − a as (using tb 2 a 2 − 3 8 tb 4 = 2) h ( z ) = Z z a − b h 0 ( ζ ) δ ζ = − p w 2 − b 2  t 4 ( w 2 − b 2 )( w + 4 a ) + 2 b 2 w  + 2 ln w + √ w 2 − b 2 b ! . (4-28) Remark 4.1 One c an verify dir e ctly that h ( z ) satisﬁes the fol lowing RHP: 1. h ( z ) is analytic (in z ) in C \ { γ m ∪ ( −∞ , λ 0 ) } and h ( z ) = − 1 4 tz 4 − 1 2 z 2 + 2 ln z − ` + O ( z − 1 ) as z → ∞ , (4-29) wher e ` = ln b 2 4 − b 2 8 − 1 2 ; (symmetric c ase, with b given by ( 4-16 ).) (4-30) ` = ln b 2 4 − 2 a 2 + b 2 8 − 1 2 , (nonsymmetric c ase, with a, b given by ( 4-20 ) (4-31) 2. h ( z ) satisﬁes the jump c ondition h + + h − = 0 on γ m and h + − h − = 4 π i on ( −∞ , λ 0 ) , (4-32) Remark 4.2 As it was mentione d ab ove, the solution to the sc alar RHP for h c ommutes with diﬀer enti- ation in z ; on the same b asis, it c ommutes with diﬀer entiation in t as wel l. Thus, we obtain the fol lowing RHP for h t (symmetric al c ase): 1. h t ( z ) is analytic (in z ) in ¯ C \ γ m and h t ( z ) = − 1 4 z 4 − ` t + O ( z − 1 ) as z → ∞ , (4-33) wher e ` t = 3 32 b 4 = 2 + 12 t − 2 √ 1 + 12 t 24 t 2 ; (4-34) 16 2. h t ( z ) satisﬁes the jump c ondition h t + + h t − = 0 on γ m . (4-35) These RHP has the unique solution h t ( z ) = − z 8 (2 z 2 + b 2 ) p z 2 − b 2 (4-36) that c an b e veriﬁe d dir e ctly. In the non-symmetric al c ase, h t c an b e c alculate d in a similar way. Remark 4.3 The g -function g t ( z ) was deﬁne d in [ 10 ], e q. (3.2), as g t ( z ) = Z b − b ln( z − ξ ) dµ t ( ξ ) , (4-37) wher e µ t is the e quilibrium me asur e in the external ﬁeld f ( z , t ) and x = n/ N = 1 . In the c ase t ≥ 0 , the e quilibrium me asur e µ t is the unique Bor el pr ob ability me asur e on R that minimizes the functional I f ( µ ) = Z Z ln 1 | x − y | dµ ( x ) dµ ( y ) + Z f ( x, t )) dµ ( x ) (4-38) among al l Bor el pr ob ability me asur es on R . (Her e the subindex t do es not me an diﬀer entiation.) The me asur e µ t c an b e c alculate d explicitly. It turns out to b e supp orte d on the interval I = [ − b, b ] , wher e b is deﬁne d by ( 4-16 ), and it has a density given by dµ dx = t 2 π  x 2 + √ 1 + 12 t + 2 3 t  p b 2 − x 2 , x ∈ [ − b, b ] . (4-39) (In fact, the c ase t ∈ ( t 0 , 0) , the e quilibrium me asur e µ t minimizes ( 4-38 ) among al l Bor el pr ob ability me asur es with the supp ort on [ − b, b ] .) The function g t ( z ) satisﬁes the r e quir ements of Se ction 4.1.1 b e c ause of ( 4-37 ). Sinc e the RHP for g has a unique solution, we have g t ( z ) = g ( z ) . 4.2 Discussion ab out existence of h T o the reader it could b e a little bit of a mystery as to why there exists any function h ( z , t ) satisfying the ab ov e long list of conditions. How ev er, this result was pro ven in a general setting, that is, for any p olynomial f ( z , t ) and for any t ∈ C in [ 2 ]. The idea of the pro of is quite simple. Suppose that we ha ve our contours Ω and we wan t to ﬁnd the h ( z , t ) function for a sp eciﬁc v alue t ∈ C . Assume, on the other hand, that for a certain v alue of the parameter t (for example, for t ∗ ∈ ( − 1 12 , 0),) one can someho w ﬁnd h ( z , t ∗ ), satisfying all of the ab ov e requiremen ts (for example, h can b e calculated directly using the residue theory , as ab ov e, or b y use of the potential theory). Then one chooses a path in the parameter space ( t -plane) that connects t ∗ to t and shows that the requirements can b e maintained throughout the path; we shall call this the con tinuation principle in the parameter space . This idea (implemen ted 17 in slightly diﬀeren t form) w as at the basis of the discussion of [ 20 ] and [ 2 ]. In a general situation with f ( z ) b eing an arbitrary p olynomial, the existence of a suitable h ( z , t ∗ ) was established in [ 2 ], but in the presen t pap er w e will pro ve all the inequalities for h ( z , t ∗ ), t ∗ ∈ ( − 1 12 , 0), directly . In fact, the con tin uation principle is not limited to the p olynomial or even rational potentials f . F or example, in the context of the semiclassical limit of the fo cusing NLS, the contin uation principle for a large class of analytic f , w as stated and pro ven in [ 19 ] T o indicate the obstacles that make the con tinuation principle nontrivial we p oin t out that, as w e follo w our path in t –plane, it may happ en that the regions where −< h < 0 (the sea ) mov es in such a w ay to either pinch oﬀ one of the complemen tary arcs or to “exp ose” one of the main arcs (or causew a ys ); in that case we can use lo cal analysis to guaran tee that a new main arc (causewa y) or complemen tary arc respectively can b e “sewn in” in order to adjust the situation; suc h an adjustmen ts increases the gen us of the solution. In fact it is quite a daunting task to try and describ e in w ords this pro cess; w e in vite the reader to hav e a close lo ok at the pictures of the “phase diagrams” Figs. 5 , 6 , 7 , 8 , 9 , 10 . The reader should try and imagine how the main arcs and complementary arcs (which are not marked in the pictures) deform as w e cross the phase-transition curves, also known as breaking curves, indicated there. In fact an in teractive exploration to ol w as designed in Matlab and it is av ailable up on request. 4.3 Schematic conclusion of the steep est descent analysis The ﬁnal steps in the steep est descent analysis inv olve adding additional contours, the lenses , which enclose eac h main arc and lie entirely within the −< h < 0 region (the sea). One then re-deﬁnes T ( z ) within the regions b et w een the main arc and its corresp onding lens b y using the factorization  a d 0 a − 1  =  1 0 a − 1 d − 1 1   0 d − d − 1 0   1 0 ad − 1 1  (4-40) of the jump matrices of T so that T + ( z ) = T − ( z )  1 0 ν − 1 m e − N h − 1   0 ν m − ν − 1 m 0   1 0 ν − 1 m e − N h + 1  , (4-41) where ν m is the (constan t!) v alue of ν ( z ) on the main arc under consideration. Therefore, deﬁning b T ( z ) as T outside of the lenses and by b T ( z ) := T ( z )  1 0 ∓ ν − 1 m e − N h 1  (4-42) in the regions within the lenses and adjacent to the ± sides of γ m one achiev es a new problem with jumps that are constan t on the main arcs and exp onen tially close to the iden tit y or constant jumps on the lenses and complemen tary arcs. W e sp end a few more words for the “gen us zero” case, namely , when there is a single main arc γ m connecting t wo endp oin ts λ 0 , λ 1 , since this is the situation mostly relev an t to the analysis here; the case 18 with several arcs, for the case of real p oten tials on the real line was fully treated in [ 8 ] and in the complex plane in [ 3 ]; while not b eing conceptually more diﬃcult, it requires the introduction and use of special functions called Theta functions . 4.3.1 The “genus zero” case This is the case when there is a single main arc γ m that connects tw o endpoints λ 0 , λ 1 ; since the coeﬃcients ν j are deﬁned up to multiplicativ e constant, we can and will assume without loss of generality that they ha ve b een normalized so that the ν m on the main arc satisﬁes ν m = 1. Then the RHP for b T is                b T + ( z ) = b T − ( z )  1 0 e − N h 1  on the upp er and low er lips resp ectiv ely , b T + ( z ) = b T − ( z )  0 1 − 1 0  = b T − ( z ) iσ 2 on γ m . b T + ( z ) = b T − ( z )  1 e N h 0 1  on Ω \ γ m (4-43) Due to the sign requiremen ts, the oﬀ–diagonal entries of the jumps on the lenses and complementary axis tend to zero exp onentially fast in an y L p -space of the resp ective arcs, p ≥ 1, but not in L ∞ b ecause at the endp oints λ 0 , λ 1 w e necessarily hav e < h = 0. Near these p oin ts one has to construct explicit lo cal solutions of the RHP called p ar ametric es [ 8 ]. The t yp e of lo cal RHP dep ends on the b ehavior of h ( z ) near the endp oin ts. In a generic situation one has e N h ( z ) = e N C ( j ) 0 ( t )(( z − λ j ) 3 2 (1+ O ( z − λ j )) , j = 0 , 1 with C ( j ) 0 ( t ) some nonzero constan t. The critical case (or ”gradien t catastrophe” case) corresp ond to those sp ecial case whereb y C ( j ) 0 ( t ) = 0 at one or the other or b oth endp oints, and thus e N h ( z ) = e N C ( j ) 1 ( t c )(( z − λ j ) 5 2 (1+ O ( z − λ j )) (4-44) where C ( j ) 1 ( t c ) is now nonzero (nondegenerate gradien t catastrophe). In the former case the local parametrix can b e constructed in terms of Airy functions and its construction is very well known since [ 8 ] (see also [ 10 ], [ 3 ]). The latter case requires the solution of a sp ecial RHP which can b e reduced to an instance of the RHP for the P ainlev´ e I Problem 2.1 . This was done in [ 10 ] and will not b e rep eated here. W e p oint out that one of the main distinctive features is that • in the generic case there are three level curves < h = 0 that emanate from the corresp onding endp oin t λ j (one of them b eing the main arc), see Fig. 3 ; • in the critical case there are ﬁv e level curves < h = 0, one of them b eing the main arc (see Fig. 3 ) The ﬁnal steps in the appro ximation mandates that we ﬁx t wo disks D 0 , D 1 (small enough not to enclose an y other endp oin t) around the endp oin ts λ 0 , λ 1 and deﬁne a suitable appro ximate solution Φ( z ) :=    Φ ext ( z ) for z outside D 0 , 1 Φ ext ( z ) P 0 ( z ) inside D 0 Φ ext ( z ) P 1 ( z ) inside D 1 (4-45) 19 γ m γ c γ m γ c γ c  1 (1 − κ )e N h 0 1   0 1 − 1 0   1 0 e − N h 1   1 0 e − N h 1   1 κ e N h 0 1  Figure 3: The t wo typical conﬁgurations of lev el curves and sign distributions near the endp oint in the generic case (left) and critical case (right). Indicated are the complementary arcs γ c (there might b e only one complemen tary arc in the critical case, dep ending on the function ν ( z )) and the lenses. The blue (dark er) color corresp onds to the region where −< h < 0 (the sea). The parameter κ equals ν 1 or 1 − ν 3 dep ending on the endp oint under consideration (see Fig. 11 and parameters therein). suc h that the error matrix E ( z ) := b T ( z )Φ − 1 ( z ) solves a small–norm Riemann–Hilb ert problem (as N → ∞ ) and th us can be -in principle- be completely solved in Neumann series. Here b y P 0 , 1 ( z ) we denote the parametrices near the endp oin ts λ 0 , 1 resp ectiv ely . In all situations the matrix Φ ext (“mo del solution” or “exterior parametrix”) solves a RHP of the form (mo del problem) Φ ext ( z ) + = Φ ext ( z ) − iσ 2 , z ∈ γ m = [ λ 0 , λ 1 ] , , Φ ext ( z ) = 1 + O ( z − 1 ) , z → ∞ (4-46) with some particular growth b eha vior near the endpoints which dep end on the scaling limit under con- sideration. In the usual case it satisﬁes Φ ext ( z ) = O ( z − λ j ) − 1 4 , z → λ j , (4-47) but in sp ecial cases the b ehavior needs to b e mo diﬁed. A t any rate, once w e hav e ac hieved a suitable approximation for b T ( z ), the recurrence co eﬃcients for the orthogonal p olynomials can and will b e recov ered via the formulae h n = − 2 iπ e N ` ( T 1 ) 12 , α n = ( T 1 ) 12 ( T 1 ) 21 , β n = ( T 2 ) 12 ( T 1 ) 12 − ( T 1 ) 22 (4-48) where b T ( z ) near ∞ equals T ( z ) (since we are in the exterior region) and has expansion b T ( z ) = T ( z ) = 1 + T 1 z + T 2 z 2 + . . . , z → ∞ . (4-49) 20 The latter co eﬃcien t matrices can b e obtained from the corresp onding expansion of Φ ext ( z ) near inﬁnity , to within the error determined b y E ; in the generic ( r e gular ) case ( t 6 = t 0 , t 1 , t 2 and not on the breaking curv es) the parametrices P 0 , P 1 are the w ell–known Airy parametrices and the standard error analysis (whic h we do not rep ort here) sho ws that E in tro duces an error of order O ( N − 1 ). In this case the exterior parametrix (mo del solution) Φ ext in the gen us 0 region is the “standard” solution (that w e shall denote by Ψ 0 ) to the follo wing “mo del RHP”:        Ψ 0 ( z ) is analytic in C \ [ λ 0 , λ 1 ] , Ψ 0+ ( z ) = Ψ 0 − ( z ) iσ 2 on [ λ 0 , λ 1 ] , Ψ 0 ( z ) = 1 + O ( z − 1 ) as z → ∞ , Ψ 0 ( z ) = O ( z − λ 0 , 1 ) − 1 4 , z → λ 0 , 1 . (4-50) The solution to the RHP ( 4-50 ) is giv en by Ψ 0 ( z ) = ( σ 3 + σ 2 ) 2  z − λ 1 z − λ 0  σ 3 4 ( σ 3 + σ 2 ) =  z − λ 1 z − λ 0  σ 2 4 , (4-51) whic h has expansion (recall our notation λ 1 = a + b , λ 0 = a − b ) Ψ 0 ( z ) = 1 − b 2 z σ 2 + b 2 8 z 2 1 − abσ 2 2 z 2 + O ( z − 3 ) , z → ∞ . (4-52) Th us, near z = ∞ , one ﬁnds T ( z ) =  1 + 1 z O ( N − 1 )  Ψ 0 ( z ) ⇒ T j = ( 1 + O ( N − 1 ))Ψ 0 ,j , (4-53) where Ψ 0 ,j denote the T aylor co eﬃcients of Ψ 0 ( z ) at inﬁnity . 4.3.2 Recurrence co eﬃcients in the genus 0 cases As explained in Section 4.1.3 there are tw o types of gen us zero solutions and hence the ﬁnal formulæ are diﬀerent. Using ( 4-48 ), the appro ximation ( 4-53 ), the explicit form of Ψ 0 ( 4-51 ) and the explicitly calculated expressions for λ 1 , λ 2 , one ﬁnds the results summarized in T able 1 . 4.3.3 The regions of higher genera F rom the global analysis rep orted in Figures 5 , 6 , 7 , 8 , 9 , 10 , the reader can see that there are regions where the hyperelliptic surface of h 0 ( z ) has genus 1 or 2. In this case, while the general scheme of the steep est descen t analysis remains intact, the solution of the relev an t model problem for Ψ 0 requires Riemann–Theta functions. F ormulæ can be found in [ 8 , 3 ]. The recurrence coeﬃcients also are expressible in terms of Theta functions. In fact the form ulæ in [ 8 ] could be directly applied here, simply b y mo difying the choice of the a, b -cycles (in the standard lore of Riemann surfaces) as describ ed extensively in [ 3 ]. W e shall not write explicit formulæ here since it would require setting up a go od deal of additional notation. Suﬃce it to say that the nature of the resulting expressions is one of rapidly oscillating functions of N and t , with amplitude that dep ends only on t . 21 Symmetric gen us 0 ( δ t := t + 1 12 ) Non symmetric gen us 0 ( δ t := t − 1 15 ) h n = 2 π √ 12 δ t − 1 6 t ! 1 2 + N exp " N 4 − ( √ 12 δ t + 1) 2 24 t # α n = b 2 4 = √ 12 δ t − 1 6 t β n = a = 0 h n = 2 π i √ 15 δ t − 1 15 t ! 1 2 + N exp " N 9 + 4 i √ 15 δ t − 30 δ t 60 t # α n = b 2 4 = i √ 15 δ t − 1 15 t β n = a = − i  1 5 t  3 + 2 i √ 15 δ t   1 2 T able 1: The leading order appro ximations of the ”square-norms” and recurrence co eﬃcien ts in the tw o gen us-zero cases: all expressions are understo o d to within an error term of O ( N − 1 ). T he expressions for h n = π b e N ` and ` ’s are in ( 4-30 , 4-31 ). In fact a more careful analysis shows that β n in the symmetric case is exp onentially small [ 10 ]. The reason is that the RHP can b e seen to b e close exp onentially to a RHP with a symmetry z 7→ − z , for which the expression for β n automatically yields zero. Note that there are tw o c hoices of signs for a in ( 4-20 ) (the c hoice of signs for b amounts only in exc hanging the lab els of the branch-points) that lead to diﬀerent (but quite similar) formulæ and results; we will formulate all the results for this particular c hoice whereby a ' − 3 i . 4.4 Contour deformation. A general discussion of contour deformations once the appropriate g –function has b een found can b e read in [ 3 ] and [ 2 ]; we give here a brief sketc h. W e advise the reader to accompany this part with the pictures that are pro vided plentiful. In general, the contours of in tegration for the pairing h p, q i ~ ν can b e deformed b y use of the Cauc h y theorem: an y deformation that we shall allow must be such that the deformed con tour approaches ∞ along the same direction arg( z ) = − 1 4 arg( t ) + k 2 π of the original contour, so as to preserv e integrabilit y (w e may ev en mandate that each contour is a straight line outside of a suﬃciently large circle). Indeed, from < h ( z ) = −< ( f ( z ) − 2 g ( z ) + ` ) and from the fact that g ( z ) is b ounded by a logarithm, we see that for | z | large enough the sign of < h is the same as of −< f , for whic h the abov e directions are the directions of the steep est descent. The ﬁnal deformation of the con tours must fulﬁll the follo wing requiremen ts, that w e describe referring to the regions −< h > 0 as (dry) land , −< h < 0 the sea (or other watery expression) and the main arcs (where < h ≡ 0) as bridges or causewa ys : • Along each contour −< h is alwa ys nonnegative, −< h ≥ 0, i.e. each contour do es not get wet; • if tw o or more (oriented) con tours hav e b een deformed so that they go through the same bridge (main arc), then the traﬃc (i.e. the weigh t of that part of con tour) is the (signed) sum of all the 22 traﬃcs. F or example if $ 1 , $ 2 are deformed so that they pass trough the same main arc, then the w eight of that arc shall b e ν 1 − ν 2 ; • e ach bridge (main arc) must carry a nonzero traﬃc, or else one needs to ﬁnd a diﬀerent g –function; • the precise form of the deformed con tours as they en ter/lea ve a bridge (i.e. the complemen tary arcs) is largely irrelev an t, but for deﬁniteness w e shall stipulate that they pro ceed for a short distance along the steep est ascent line or −< h . In order to oﬀer some rigorous study we consider in more detail the symmetric case of genus 0. Lemma 4.1 In the c ase ν 2 = − 1 and t ∈ ( − 1 12 , 0) , the function < h ( z ) , wher e h ( z ) is given by ( 4-25 ), satisﬁes the sign c onditions along the c ontour Ω . Pro of. First, it follo ws from ( 4-25 ) that < h ( z ) = 0 on [ − b, b ]. T o show that < h ( z ) > 0 immediately ab o v e the main arc [ − b, b ], it is suﬃcien t, b y the Cauc h y-Riemann equations, to show that = h 0 ( z ) < 0 on the upper shore of [ − b, b ]. The latter follows directly from ( 4-24 ). W e can now use the o ddness of h 0 ( z ) to show that < h ( z ) > 0 also b elo w the main arc. So, the correct signs around γ m are prov en. The correct distribution of signs of < h along the complemen tary arcs follows from the top ology of zero level curv es of < h . Because < h ( z ) is ev en (   z + √ z 2 − b 2   is even), it is suﬃcien t to consider level curves only in the right halfpane. Direct c heck shows that b oth terms in ( 4-25 ) hav e p ositiv e real part on i R + . There are t wo legs of zero level curv es of < h emanating from z = b and four legs coming from inﬁnit y with asymptotes ± π 8 , ± 3 π 8 , see ( 4-29 ). Denote these legs as χ ± j , j = 1 , 2 resp ectiv ely . Since < h ( z ) < 0 as real z → b + 0 and < h ( z ) > 0 as real z → + ∞ , we conclude that < h ( z ∗ ) = 0 at some z ∗ ∈ ( b, + ∞ ). So, the only p ossible top ology of the level curves χ ± j is that χ 1 is connected with χ − 1 through z ∗ and χ 2 , χ − 2 are connected to b (since < h ( z ) is a harmonic function, its lev el curv es cannot form b ounded lo ops). Thus, one can choose as complementary arcs any smo oth curves “on the land” b et w een χ 1 and χ 2 and b et w een χ − 1 and χ − 2 , i.e., in the region where < h ( z ) < 0 that connect b and ∞ . Fig. 5 shows (in red) main arcs γ m , but not complementary arcs γ c . Ho wev er, level curves χ j can b e visualized in the “snapshots” of z -plane that corresp ond to t ∈ ( − 1 12 , 0). Q.E.D. Remark 4.4 Similarly to L emma 4.1 , it is e asy to establish the c orr e ct sign distribution outside γ m in the c ase when h is given by ( 4-28 ), which is valid, for example, when ν 1 = ν 2 = 0 , ν 3 6 = 0 (Single we dge) and t ∈ (0 , 1 15 ) , se e Fig. 7 . F or t ∈ (0 , 1 15 ) b oth b 2 and a 2 ar e ne gative, so that a, b ∈ i C . F r om ( 4-28 ) it fol lows that < h ( a ) = 0 and setting the orientation of γ m upwar ds, we se e that < h 0 ± ( a ) ≶ 0 r esp e ctively. Thus, we have the c orr e ct sign distribution outside γ m . 23 5 Breaking curv es and global phase p ortraits A breaking curv e Λ in the complex t -plane separates the regions of diﬀerent genera in the asymptotic b eha vior of the recurrence co eﬃcien ts, or regions of the same gen us but with diﬀeren t num ber of main arcs (see, for example, the breaking curve that joins t = − 1 12 to t = 1 4 in Fig. 5 , whic h separates tw o regions of gen us 2). It satisﬁes the system of equations h 0 k ( z ) = 0 and < h k ( z ) = 0 , (5-1) whic h is the system of 3 real equations for tw o com plex v ariables z and t . Here the subindex k in h k indicate the gen us of the Riemann surface R ( t ) where h k is deﬁned. In our cases the gen us can b e 0 , 1 , 2. T o simplify notations, we will drop the subidnex k whenever the genus of h is obvious. W e will consider the breaking curves in the t plane where the sign requirements fail b ecause a saddle p oin t ˆ z of < h (a point satisfying h z ( ˆ z , t )=0) collided with the con tour Ω = M ∪ C . That means that either a complementary arc is pinched by the rising “sea” or a main arc (causew a y) is touc hed b y the dry land b ecause of the receding “sea”. In any case, equations ( 5-1 ) will b e satisﬁed at z = ˆ z . There are three cases of breaking that we consider: • genus 0 symmetric, i.e., h 0 ( z ) is given by ( 4-25 ); • genus 0 non-symmetric, i.e., h 0 ( z ) is given by ( 4-28 ); • genus 1 (symmetric). The resulting equations when plugging the expression for h into the system ( 5-1 ) are relatively simple and could be analyzed analytically; we ﬁnd it muc h more eﬀectiv e and informativ e to study and plot them n umerically . 5.1 Genus 0 symmetric Using ( 4-24 ), w e obtain the following equation for the saddle p oin t: tz 2 = − (1 + 1 2 tb 2 ) = − 1 3  2 + √ 1 + 12 t  or z = ± s 2 + √ 1 + 12 t − 3 t . (5-2) Substituting this into < h ( z ) = 0 and using ( 4-25 ) after some algebra yields the following implicit equation for the breaking curv e <   ln 1 + 2 √ 1 + 12 t + q 3[( √ 1 + 12 t + 1) 2 − 1] 1 − √ 1 + 12 t + q 3[( √ 1 + 12 t + 1) 2 − 1] 12 t   = 0 (5-3) 24 or ϕ ( u ) := < " ln 1 + 2 u + √ 3 u 2 − 6 u 1 − u + √ 3 u 2 − 6 u u 2 − 1 # = 0 , (5-4) where u = √ 1 + 12 t . Note that according to ( 5-4 ) Λ is a b ounded curve that starts at the p oin t of gradien t catastrophe u = 0 b ecause for large u the expression in ( 5-4 ) tends to ln(2 + √ 3). T o obtain the asymptotics of the breaking curve Λ near t = t 0 , we use expansion ( 6-2 ) of h ( z , t ) near the branc h-p oin t z = b , where the co eﬃcients C 0 , C 1 are giv en in T able 2 , left column, to write h ( z ) = C 0 ( z + b ) 3 2 ( z 2 − b 2 ) 3 2 + C 1 ( z + b ) 5 2 ( z 2 − b 2 ) 5 2 + · · · . (5-5) This expansion is a direct consequence of the mo dulation equation. Using ( 5-2 ), we calculate z 2 − b 2 = √ 1+12 t − t = √ 12( t − t 0 ) − t . Since for t close to t 0 the saddle p oin t ˆ z ( t ) (that satisﬁes h z ( ˆ z ( t ) , t ) ≡ 0) is close to b , w e hav e h ( z ( t ) , t ) = C 0 [12( t − t 0 )] 3 4 ( − 2 bt ) 3 2 + C 1 [12 x ( t − t 0 )] 5 4 ( − 2 bt ) 5 2 + O (( t − t 0 ) 7 4 ) = [12( t − t 0 )] 5 4 15(2 b ) 3 ( − t ) 5 2 (10 tb 2 + 4) + O (( t − t 0 ) 7 4 ) , (5-6) where w e utilized the formulae for C 0 , C 1 . Now the requirement < h = 0 yields 5 4 arg( t − t 0 ) = ± π 2 + π k , k ∈ Z , so that the breaking curve Λ near the p oin t of gradient catastrophe t 0 is tangen tial to arg( t − t 0 ) = ± 2 π 5 + 4 π 5 k . (5-7) Note that there are v arious branches to keep trac k of: the principal branc h of the radical √ 1 + 12 t leads to the curve joining t = − 1 12 to t = 0 ( u = 0 to u = 1 corresp ondingly), light curve from − 1 12 to 0 on Fig. 4 ; the secondary branch leads to the curve that joins t = − 1 12 to t = 1 4 ( u = 0 to u = 2 corresp ondingly), on Fig. 4 . 5.2 Genus 0 non-symmetric In this case w e are lo oking for zero es of h 0 ( z ) satisfying z 2 + az − b 2 = 0, see ( 4-27 ). They are given by z = − 1 2 ± s 1 4 −  b a  2 . (5-8) Substituting ( 5-8 ) in < h ( z ) = 0, where h is deﬁned by ( 4-28 ), and rep eating the previous arguments, we obtain an implicit equation for the additional breaking curves. Leaving the lengthy but straightforw ard details aside, w e obtain the curves on Fig. 4 that join t = 1 15 to t = 0 and t = − 1 12 to 1 15 resp ectiv ely . 25 5.3 Genus 1 symmetric F or the case of genus 1 there are 4 branch-points; the only situation where w e can hav e the saddle p oin t h 0 ( z ) = 0 on the zero-lev el set is when the saddle p oint is b etw een to distinct connected comp onen ts of the zero lev el-set of < h ( z ) = 0. It is seen from the mo dulation equations that h 0 ( z ) 2 is alwa ys a p olynomial of degree 6; we lo ok here for solutions where ( h 0 ( z )) 2 is an ev en p olynomial. Since we are seeking a solution of gen us 1, there must b e a single double ro ot. By the symmetry this ro ot must b e at the origin; this allo ws us to write h 0 ( z ; t ) = − tz s  z 2 + 1 + 2 √ t t   z 2 + 1 − 2 √ t t  (5-9) Indeed a simple Laurent expansion at ∞ yields h 0 ( z ; t ) = − tz 3 − z + 2 z + O ( z − 2 ), and eviden tly h 0 ( − z ; t ) = − h 0 ( z ; t ). Although the curve is of genus 1, the integral of h 0 ( z ) is elementary and a direct computation yields (recall that h ( z ) v anishes at one of the branch-points) Z 0 λ 0 h 0 ( z ; t )d z = h (0; t ) = − √ 1 − 4 t 4 t + ln  1 + √ 1 − 4 t 2 √ t  (5-10) W e lea ve it to the reader to v erify that < h (0; t ) is contin uous at t = 1 4 b y using the iden tit y 1 − √ 1 − 4 t 2 √ t 1+ √ 1 − 4 t 2 √ t ≡ 1. The implicit equation of this breaking curve is then simply < ( h (0; t )) = 0. The curv e is the one joining t = 1 4 to the t = 0 in Figs. 4 , 5 , 6 , 8 , 9 , 10 . − 0.3 − 0.2 − 0.1 0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4 − 1 12 1 15 1 4 Figure 4: All breaking curv es, summarized: they are symmetric ab out the real t –axis. Dep ending on the case under study , some of them ma y not b e “active”, namely they b elong to diﬀerent sheets of the phase p ortraits. W e refer to Figs. 5 , 6 , 7 , 8 , 9 , 10 for the sp eciﬁcs. 26 5.4 Phase p ortraits or distribution of genera in the complex t -plane There are six diﬀerent situations dep ending on the v alues of ν j ’s in the deﬁnition of the bilinear form eq. ( 1-2 ). Note that only their v alues up to common multiplication by nonzero constant is relev ant, i.e., the orthogonal p olynomials are parametrized by p oints [ ν 1 : ν 2 : ν 3 ] ∈ CP 2 . So, we hav e: 1. ”Generic” case: ν j 6 = 0, ν 1 6 = ν 2 , ν 2 6 = ν 3 ; 2. ”Real axis”: ν 2 = ν 1 , ν 3 = 0; 3. ”Single W edge”: ν 1 = 0 = ν 2 , ν 3 6 = 0. 4. ”Consecutive W edges”: ν 3 = 0, ν 2 6 = ν 1 , ν 2 6 = 0, ν 1 6 = 0; 5. ”Opp osite W edges, generic”: ν 2 = 0, ν 3 6 = ν 1 , ν 1 ν 3 6 = 0; 6. ”Opp osite W edges, symmetric”: ν 2 = 0, ν 3 = ν 1 6 = 0. W e provide the results of the computer-assisted inv estigation for all six cases in the tables that follo w (Figures 5 , 6 , 7 , 8 , 9 , 10 ). The common feature is the following: as we mo v e around the origin t = 0 coun terclo c kwise the asymptotic directions of the integration contours mov e clo ckwise b y arg ( t ) / 4. Therefore, a counterclockwise loop around t = 0 yields a new conﬁguration of con tours obtained by a clo c kwise rotation of π / 2 of the initial one. In general, th us, we can exp ect that our phase portraits to hav e four sheets . In the ”Generic” and ”Opp osite W edges, symmetric” case, ho wev er, these four sheets are actually identical, and in the case of ”Real Axis” t wo of them are equal. In the pictures that follow the cut (if necessary) is alwa ys along the negative t -axis and the gluing is the top of the negative axis of sheet j is glued to the b ottom of the negative axis of sheet j + 1 (mo d 4). W e hop e that the pictorial representation will serve more than many pages of verbal explanation. W e rather explain brieﬂy the algorithm used to inv estigate the phase p ortraits; in [ 2 ] an algorithm to ﬁnd n umerically ”Boutroux curv es” w as explained. The algorithm pro duces a solution of the ”modulation equations” (for branc h-p oin ts) in high genus, but will not enforce the sign distribution (sign conditions for h ( z )) needed to ha v e an appropriate g -function. In terms of the Remark 4.3 it ma y yield a signed equi- librium measure. Plotting the level curv es allows one to decide unambiguously whether the numerically pro duced g -function satisﬁes the sign distribution. The pictures b elow are pro duced by some co de written in Matlab which is av ailable up on request; the co de will allow ”interactiv e exploration” of the t –plane and to pro duce the pictures interactiv ely . Remark 5.1 (Zero es of the orthogonal p olynomials) In al l situations c onsider e d b elow, se e Fig- ur es 5 - 10 , the main ar cs c onsist of al l the r e d ar cs that ar e surr ounde d by the shade d (light blue) r e gions on b oth sides. These ar cs, as is wel l known (se e for example [ 3 , 2 ]), also r epr esent the limiting ar cs wher e the ro ots of the ortho gonal p olynomials ac cumulate, and the (we ak) limit of their density c an b e r e c over e d fr om the jump of h 0 ( z ) . 27 − 0.2 −0.1 0 0.1 0.2 0.3 0.4 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 − 1 1 2 1 4 Figure 5: Generic . The region inside the curv e joining 1 4 to 0 is of genus 1; inside the curve that joins t = − 1 12 and t = 0 it is of genus 0 (symmetric about z = 0). Everywhere else it is of genus 2, except for the degeneration to genus 0 o ccurring on the curve that joins t = − 1 12 and t = 1 4 , and to genus 1 on the ra y [ 1 4 , ∞ ). There is a Painlev ´ e I transition at t = − 1 12 and a P ainlev´ e I I transition at t = 1 4 . 28 −0.2 −0.1 0 0.1 0.2 0.3 0.4 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 − 1 12 −0.2 −0.1 0 0.1 0.2 0 .3 0.4 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 − 1 12 1 4 Figure 6: Real Axis . There are t wo sheets glued along t ∈ R − ; the level curv es are alwa ys symmetric ab out z = 0. On the ﬁrst sheet the solution is alwa ys of gen us 0. On the second sheet it is of gen us 0 outside of the curv e connecting t = − 1 12 and t = 1 4 . The region inside the curve joining 1 4 to 0 is of genus 1; inside the curv e that joins t = − 1 12 and t = 0 it is of gen us 0 (symmetric ab out z = 0). In the remaining part it is of gen us 2. 29 −0.2 −0.15 −0.1 −0.05 0 0.0 5 0.1 0.1 5 0.2 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 − 1 12 1 15 −0.2 −0.15 −0.1 −0. 05 0 0.05 0.1 0.15 0.2 − 0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 − 1 12 1 15 Figure 7: Single W edge . There are four sheets glued along t ∈ R − ; sho wn here are only sheet 1 and 2, b ecause the sheets 3 , 4 are copies of sheet 1 , 2 where the function h ( z ) has undergone z 7→ − z . Note that there at the critical point t = 1 15 on all four sheets we hav e a transition of t yp e P ainlev´ e I (and also at t = − 1 12 ). 30 −0.2 −0.1 0 0.1 0.2 0.3 0.4 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 − 1 12 1 15 t=0.27951+0.413 01i −0.2 −0.1 0 0.1 0 .2 0.3 0.4 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 − 1 12 1 4 t=0.89118−0.40569i Figure 8: Consecutive W edges . There are four sheets glued along t ∈ R − ; shown here are only sheet 1 and 2, b ecause the remaining tw o sheets are copies of sheet 1 where the function h ( z ) has undergone z 7→ − z . Note the P ainlev´ e I transition at b oth t = − 1 12 and t = 1 15 . 31 −0. 3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 − 1 12 1 4 Figure 9: Opp osite W edges, generic . There are t wo sheets glued along t ∈ R − ; sho wn here are only sheet 1, b ecause sheet 2 is a copy of sheet 1 where the function h ( z ; t ) has undergone h ( z, t ) 7→ h ( z ; t ). Note the Painlev ´ e I transition at t = − 1 12 and P ainlev´ e I I transition at t = 1 4 . −0.2 −0.1 0 0.1 0.2 0.3 0.4 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 1 4 Figure 10: Opposite W edges, symmetric . There is only one sheet. Note the P ainlev´ e I I transi- tion at t = 1 4 . The sp ectral curve is alw ays of gen us 1 except at the p oint t = 1 4 . 32 Remark 5.2 Although the r esults, pr esente d in Figur es 5 - 10 ar e numeric al, ther e is a str aightforwar d way for their analytic justiﬁc ation. Consider, for example, the Generic c ase shown in Figur es 5 . A c c or ding to L emma 4.1 , the genus zer o r e gion c ontains the interval ( − 1 12 , 0) . Sinc e a br e ak c an only o c cur at one of the curves deﬁne d by ( 5-3 ), the r e gion c ontaine d inside the black curve on Figur es 5 is the genus zer o r e gion. A c c or ding to the c ontinuation principle in the p ar ameter sp ac e (se e [ 3 ] and [ 19 ]), for any t on the four-she ete d Riemann surfac e Ξ (with br anch-p oints t = 0 and t = ∞ ) ther e exists a c ontour Ω = M ∪ C and h n ( z ) , wher e M is the union of al l the br anch-cuts of h n , such that < h n satisﬁes the sign c onditions on Ω . Sinc e ther e ar e 8 le gs of zer o level curves of < h , the genus of the solution for any t c annot b e gr e ater than two (as ther e c an b e no b ounde d close d lo ops of < h = 0 ). L et us take a p oint t ∗ , = t ∗ > 0 , on the main br anch Λ of the br e aking curve ( 5-3 ), se e Subse ction 5.1 , that c ontains genus zer o r e gion inside (the curve fr om − 1 12 to 0 ). Sinc e t ∗ is on the br e aking curve, ther e exists a z ∗ ∈ C , such that the p airs ( t ∗ , ± z ∗ ) satisfy ( 5-1 ) (her e we use the evenness of < h ( z ) ). Cho ose z ∗ so that = z ∗ > 0 . If we c an show that < h t ( z ∗ ; t ∗ ) > 0 (5-11) as t cr osses Λ along < t = < t ∗ going up, then we c an pr ove that the genus of h is changing fr om zer o to two as t cr osses Λ . A c c or ding to the Cauchy-Riemann c onditions, ( 5-11 ) is e quivalent to = h t ( z ∗ ; t ∗ ) < 0 , (5-12) wher e = t = = t ∗ . Using ( 4-36 ) and the fact that 1 + t ∗ b 2 2 + t ∗ z 2 ∗ = 0 (which fol lows fr om h 0 ( z ∗ ; t ∗ ) = 0 ), we obtain z 2 ∗ = − 1 t − b 2 2 , so that h t ( z ∗ ; t ∗ ) = 1 4 t ∗ s  − 1 t ∗ − b 2 2   − 1 t ∗ − 3 b 2 2  (5-13) T o c alculate the br anch of the squar e r o ot in ( 5-13 ) we take t ∗ → t 0 . As shown in subse ction 5.1 , in this limit arg ( t ∗ − t 0 ) → 2 π 5 , so that, using ( 4-16 ), we obtain arg h t ( z ∗ ; t ∗ ) → 11 π 10 . (5-14) That pr oves ine quality ( 5-12 ) when t ∗ is close d to t 0 . Mor e over, for any t ∗ ∈ Λ we obtain h t ( z ∗ ; t ∗ ) = − 1 4 √ 3 t 2 ∗ q 2 √ 1 + 12 t ∗ − (1 + 12 t ∗ ) or h ( u ) = − 12 √ 6 u − u 2 ( u 2 − 1) 2 , (5-15) wher e u = √ 1 + 12 t ∗ . It is e asy to se e that the upp er halfplane p art of the genus zer o r e gion (b etwe en Λ and R ) is c ontaine d in the semistrip 0 ≤ < u ≤ 1 , = u ≥ 0 of the u -plane. Dir e ct c alculations show that: = h ( u ) = 0 on [0 , 1] , and; = h ( u ) < 0 on i R + , on 1 + i R + and on any se gment = u = y , < u ∈ [0 , 1] , wher e y > 0 is suﬃciently lar ge. Thus, using the maximum principle for = h ( u ) and the fact that dh du 6 = 0 on [0 , 1] , we c onclude that = h ( u ) < 0 inside the semistrip. So, we pr ove d the tr ansition fr om genus zer o to genus two acr oss Λ . Similar c onsider ations wil l le ad to rigor ous pr o ofs of tr ansitions thr ough other level curves. 33 6 Double and m ultiple scaling analysis near the P ainlev ´ e I gra- dien t catastrophe p oin ts Ha ving disp osed of the global analysis of the problem in the complex t –plane w e now fo cus on the so– called double (and multiple) scaling analysis near the tw o p oints of gradient catastrophe that are related to the P ainlev´ e I transcendents. These are t 0 := − 1 12 , and t 1 := 1 15 . (6-1) There is another p oint of gradient catastrophe at t 2 := 1 4 whic h –how ev er– inv olv es the Painlev ´ e I I transcenden t and should b e analyzed in a separate w ork. The tw o points t 0 , t 1 can b e analyzed muc h in a parallel fashion: they b oth necessitate of the same t yp e lo cal parametrix near one -or b oth- endp oints. The diﬀerences b et ween the t wo cases app ear by insp ection of the ﬁgures: indeed • near t = t 0 the genus zero function h ( z ; t ) = h 0 ( z , t ) alw ays has symmetric level-curv es and hence the Painlev ´ e I parametrix (ﬁrst introduced in [14]) is needed near b oth endp oin ts λ 0 = − b = − λ 1 (see Figs. 5 , 6 , 7 , 8 , 9 , vignettes near t 0 ) ; • near t = t 1 the genus zero function h ( z ; t ) = h 0 ( z , t ) do es not hav e any sp ecial symmetry and the PI parametrix is needed only near one endp oint (see Figs. 5 , 6 , 7 , 8 , 9 , vignettes near t 1 ). 6.1 Lo cal analysis at the p oint of gradien t catastrophe Near an endp oin t the gen us-zero h –function has necessarily an expansion of the following form 1 2 h ( z ) = C ( j ) 0 ( z − λ j ) 3 2 + C ( j ) 1 ( z − λ j ) 5 2 + C ( j ) 2 ( z − λ j ) 7 2 + · · · = C ( j ) 0 ( z − λ j ) 3 2 1 + C ( j ) 1 C ( j ) 0 ( z − λ j ) + C ( j ) 2 C ( j ) 0 ( z − λ j ) 2 + · · · ! , j = 0 , 1 , (6-2) where the co eﬃcien ts C ( j ) k = C ( j ) k ( t ) dep end on t . The gradient-catastrophe o ccurs when the leading co eﬃcien t C ( j ) 0 ( t ) v anishes at one or b oth endp oin ts λ 0 , 1 of the main arc γ m , while (in general) the next co eﬃcien t C ( j ) 1 ( t ) do es not. F or our f ( z , t ), the gradient catastrophe p oint is either t 0 or t 1 . Elementary singularit y theory[ 1 ] guarantees the v alidit y of the follo wing deﬁnition. Deﬁnition 6.1 (Scaling co ordinate) The scaling co ordinate ζ ( z ) = ζ ( z ; t, N ) and the exploration parameter τ = τ ( t, N ) ar e deﬁne d by N 2 h ( z ; t ) = 4 5 ζ 5 2 ( z ; t, N ) + τ ( t, N ) ζ 3 2 ( z ; t, N ) , (6-3) wher e ζ ( b ; t, N ) ≡ 0 , ζ ( z ; t, N ) is analytic al ly invertible in z in a ﬁxe d smal l neighb orho o d D j of z = λ j and τ is analytic in C ( j ) 0 at C ( j ) 0 = 0 , wher e j = 0 , 1 . 34 Let us consider the endp oin t λ 1 near the p oin t of gradient catastrophe t ∗ , where t ∗ = t 0 or t ∗ = t 0 . The expression ( 6-3 ) is the normal form of the singularit y deﬁned b y h ( z ; x, t ) (in the sense of singularit y theory [ 1 ]). The lo cal b eha viour (we suppress the superscripts) ζ = N 2 5  5 4 C 1  2 5  1 − 6 C 0 C 2 25 C 2 1 + O ( C 2 0 )  ( z − λ 1 )(1 + O ( z − λ 1 )) , (6-4) τ = N 2 5 C 0  4 5 C 1  3 5 (1 + O ( C 0 )) (6-5) w as calculated in [ 5 ]. The determination of the ro ot is ﬁxed uniquely by the requirement that the image of the main arc γ m , where < h ≡ 0, b e mapp ed to the negative real ζ –axis. F ollowing [ 5 ], we deﬁne: Deﬁnition 6.2 The double sc aling ne ar t = t ∗ shal l b e deﬁne d as the appr opriate dep endenc e of t such that the variable v = v ( t, N ) := 3 8 τ 2 ( t, N ) = 3 8 N 4 5 C 2 0  5 4 C 1  − 6 5 (1 + O ( C 0 )) (6-6) is kept within a disk of arbitr ary but ﬁxe d (in N ) r adius ar ound v = 0 . The variable v shal l b e r eferr e d to as the Painlev ´ e co ordinate . Lemma 6.1 In the double sc aling ne ar t = t 0 for the symmetric genus zer o c ase or ne ar t = t 1 for the non-symmetric c ase, the Painlev ´ e c o or dinate v has the fol lowing exp ansion v ( t ) = 3 8 τ 2 ( t, N ) = − 3 6 5 2 9 5  t + 1 12  N 4 5 (1 + O ( √ t − t 0 )) , (6-7) v ( t ) = 3 8 τ 2 ( t, N ) = 3 6 5 2 1 5 5 e 3 iπ 5  t − 1 15  N 4 5 (1 + O ( √ t − t 1 )) . (6-8) In either c ases the function v ( t ) is a c onver gent series in √ t − t j ; if v is kept b ounde d as N → ∞ then t − t j = O ( N − 4 5 ) . Ther efor e fr om ( 6-7 , 6-8 ) it fol lows imme diately that with ac cur acy O ( N − 2 5 ) the map v ( t ) is line ar in t − t j N 4 5 . The pro of is a direct computation with the help of T able 2 and Def. 6.1 . 6.2 Asymptotics a w a y from the p oles The asymptotic analysis no w dep ends on the regions in the P ainlev´ e v ariable v ( 6-7 ) or ( 6-8 ) that we are in vestigating. W e will split this analysis into the following tw o cases. • Aw ay from the p oles : the v ariable v is c hosen within a ﬁxed compact set K , that do es not con tain any p ole of the relev ant solutions to P1; • Near the p oles : the v ariable v undergo es its o wn scaling limit and approaches a giv en p ole at a certain rate. 35 Near t 0 = − 1 12 , δ t := t − t 0 Near t 1 = 1 15 δ t := t − t 1 a = 0 a = − 3 i q 1 + 2 i 3 √ 15 δ t √ 1 + 15 δ t = − 3 i + √ 15 δ t + O ( δ t ) b = √ 8 p 1 + √ 12 δ t = √ 8 − √ 8 2 √ 12 δ t + O ( δ t ) b = 2 i p 1 + i √ 15 δ t = 2 i + √ 15 δ t + O ( δ t ) C 0 = − 1 6  2 + 3 tb 2  √ 2 b = − 2 4 √ 2 √ 12 δ t 3 4 p 1 + √ 12 δ t C 0 = − 1 3 t √ 2 ba (2 a + 3 b ) = 2 3 e 3 iπ 4 √ 15 δ t + O ( δ t ) C 1 = − 19 tb 2 + 2 20 √ 2 b = 4 p 1 + √ 12 δ t 60 4 √ 2  16 − 19 √ 12 δ t  C 1 = − 1 5 t (2 a 2 + 15 ab + 8 b 2 ) 2 √ 2 b = 2e 3 iπ 4 15 + O ( √ δ t ) N − 2 5 ζ 0 ( λ 1 ) = 3 − 2 5 2 − 1 10 + O ( N − 2 5 ) N − 2 5 ζ 0 ( λ 1 ) = 6 − 2 5 e 3 iπ 10  1 + O ( N − 2 5 )  τ ζ 0 ( λ 1 ) = 4 C 0 5 C 1 (1 + O ( C 0 )) = − √ 8 √ 12 δ t + O ( δ t ) τ ζ 0 ( λ 1 ) = 4 C 0 5 C 1 (1 + O ( C 0 )) = 4 √ 15 δ t + O ( δ t ) ` = − 3 2 + ln 2 − 6 δ t + 2 3 (12 δ t ) 3 2 + O ( δ t 2 ) ` = 9 4 + ln( − 1) − 13 4 (15 δ t ) − 2 3 i (15 δ t ) 3 2 + O ( δ t 2 ) T able 2: The explicit expressions and relev ant expansions of the indicated quan tities: these are the result of straigh tforward algebraic manipulations using ( 4-16 , 4-20 , 4-25 , 4-28 , 4-30 , 4-31 , 6-2 , 6-3 , 6-4 , 6-5 ). Eac h of these t wo cases requires a sligh tly diﬀerent analysis dep ending on the nature of the gradient catastrophe p oin t, b e it t 0 = − 1 12 or t 1 = 1 15 . In the former case the analysis was carried out in full in the regime ”Aw ay from the p oles” by [ 10 ] and the relev an t theorem is Thm. 1.1 : w e will not add anything to it. The only case that is not cov ered by the mentioned theorem in the same regime is when t undergo es a double scaling limit near t 1 and a sp ecial Painlev ´ e parametrix is needed only at one endp oin t, sa y , at λ 1 . Of course, one ma y still use the results of [ 10 ] with minor mo diﬁcations to cov er this new case, but since we will need some preparatory material, w e brieﬂy analyze this case b elo w. W e shall construct an appro ximation to the matrix b T ( z ; t, N ) app earing in ( 4-42 ) in the form b T ( z ) =        E ( z )Ψ 0 ( z ) for z outside of the disks D 1 , D 0 , E ( z )Ψ 0 ( z ) P 1 ( z ) for z inside of the disk D 1 , E ( z )Ψ 0 ( z ) P 0 ( z ) for z inside of the disk D 0 , (6-9) 36 where D 0 , D 1 are small ﬁxed disks centered at λ 0 and λ 1 resp ectiv ely , see Fig. 11 and Ψ 0 as in ( 4-51 ). Here E ( z ) is the so-called error matrix that will b e shown to b e close to the identit y matrix 1 and P 0 , 1 ( z ) are lo cal parametrices at z = λ 0 , λ 1 , resp ectively , that will be constructed through the matrix Ψ( ξ , v ) deﬁned by ( 2-4 ). A lo cal parametrix P ( z ) (we drop the indices for con venience) must hav e a certain 0 1 -1 0 1 -1 λ 1 λ 0 1 − ν 1 ν 1 1 − ν 3 ν 1 ν 3 1 − ν 1 -1 1 0 1 -1 0 λ 0 λ 1 Figure 11: The deformation of the contours and the partitioning in complemen tary (black) and main arcs (blue). Shown also are the lenses and the disks near the tw o branch-points λ 1 , λ 0 . The left frame refers to the case t ∼ 1 15 , the right frame to t ∼ − 1 12 . The thin lines in the shaded area on the left frame are the level-curv es passing through the saddle p oin t. F or the case on the left ( t = 1 15 ) we hav e ν 2 = ν 1 ∈ C , and ν 3 can b e normalized to ν 3 = 1; the deformed contour $ 3 consists of the complementary arc on the imaginary axis, the left one and the main arc (thick). The con tour $ 2 + $ 1 (homological sum) consists of the tw o left/righ t complemen tary arcs. F or the right frame, w e ha ve t = e iπ 1 12 and the weigh ts are ν 2 = 1 and ν 1 , ν 3 ∈ C : the contour $ 1 is deformed to go through the tw o complementary arcs on the b ottom and top right, and $ 2 runs along the top righ t, top left complementary arcs and the main arc, while $ 3 runs along the tw o complementary arcs on the left. The weigh ts of the v arious complementary arcs are indicated in the ﬁgure and determine the parameters of the P ainlev´ e parametrices to be used according to Deﬁnition 2.1 . The level-curv es in these pictures are numerically accurate. n umber of prop erties (see Theorem 6.1 ), one of them b eing the restriction P ( z )     z ∈ ∂ D = 1 + o ε (1) (6-10) 37 on the b oundary of the resp ectiv e disk D , where o ε (1) denotes some inﬁnitesimal of ε = 1 N , uniformly in z ∈ ∂ D and in t ∈ ˆ K = v − 1 ( K ). If the local parametrices P 0 , 1 ( z ) satisfying ( 6-10 ) can be found then the “error matrix” E ( z ) is seen to satisfy a smal l–norms RHP and, thus, b e uniformly close to the iden tity . More precisely , the matrix E has jumps on: (a) the parts of the lenses and of the complementary arcs that lie outside of the disks D 0 , D 1 , and; (b) on the b oundaries of the tw o disks D 0 , D 1 . The jumps in (a) are exp onen tially close to the identit y in any L p norm (including L ∞ ) while (b) on the b oundary of the disks D 0 , 1 w e hav e E + ( z ) = E − ( z )Ψ 0 ( z ) P − 1 0 , 1 ( z )Ψ − 1 0 ( z )     z ∈ ∂ D 0 , 1 = E − ( 1 + o ε (1)) . (6-11) F rom the analysis in [ 8 ] it follows that, for | z | large enough, kE ( z ) − 1 k ≤ o ε (1) | z | (with the p oin t wise matrix norm) and that the rate of con vergence is estimated as the same as the o ε (1) that app ears in ( 6-10 ) as ε → 0. In the case at hand we keep in mind that near t = t 1 = 1 15 the endp oint λ 0 requires the standard Airy parametrix and that the corresp onding error term arising on the b oundary of D 0 is of order O ( N − 1 ). Deﬁnition 6.3 (Lo cal parametrix a w ay from the p oles) L et ζ ( z ; ε ) b e the lo c al c onformal c o or di- nate ne ar λ 1 intr o duc e d in Def. 6.1 so that N 2 h ( z ; x, t ) = θ ( ζ ; τ ) = 4 5 ζ 5 2 + τ ζ 3 2 . (6-12) L et Ψ( ξ ; v ; κ ) denote the Psi–function of the Painlev´ e I Pr oblem 2.1 ac c or ding to Def. 2.1 . The p ar ametrix P ( z ; κ ) is deﬁne d by P ( z ; κ ) = 1 √ 2  1 1 i − i  ζ − σ 3 4 Ψ  ζ + τ 2 ; 3 8 τ 2 ; κ  e − θ ( ζ ; τ ) σ 3 , ζ := ζ ( z ) . (6-13) Theorem 6.1 The matrix P 1 ( z ) := P ( z ; ν 1 ) satisﬁes: 1. Within D 1 , the matrix P 1 ( z ) solves the exact jump c onditions on the lenses and on the c omplemen- tary ar c; 2. On the main ar c (cut) P 1 ( z ) satisﬁes P 1 , + ( z ) = σ 2 P 1 , − ( z ) σ 2 , (6-14) so that Ψ 0 P 1 within D 1 solves the exact jumps on al l ar cs c ontaine d ther ein (the left-multiplier in the jump ( 6-14 ) c anc els against the jump of Ψ 0 ); 38 3. The pr o duct Ψ 0 ( z ) P 1 ( z ) (and its inverse) ar e –as functions of z – b ounde d within D 1 , namely the matrix P 1 ( z ) c anc els the gr owth of Ψ 0 at z = λ 1 ; 4. The r estriction of P 1 ( z ) on the b oundary of D 1 is P 1 ( z )     z ∈ ∂ D 1 = 1 −  H I + τ 3 16  σ 3 √ ζ + 1 2 ζ "  H I + τ 3 16  2 1 +  y + τ 4  σ 2 # + O ( ζ − 3 2 ) , (6-15) wher e v = 3 8 τ 2 , y ( v ) = y (1) ( v ) and H I = 1 2 ( y 0 ) 2 + y v − 2 y 3 = R y ( s )d s . The pro of of Theorem 6.1 can b e found in Theorem 5.1 of [ 5 ] (although it was for the tritronqu ´ ee solution, the pro of is identical for the general case). 4 6.3 Computation of the correction near t 1 : pro of of Theorem 1.2 Pro of of Theorem 1.2 . According to ( 6-9 ), we hav e E + = E − Ψ 0 P − 1 0 , 1 Ψ − 1 0 = E − ( 1 + ∆ M ( z )) on ∂ D 0 , 1 . (6-16) In particular, according to ( 6-15 ), Ψ 0 P − 1 1 Ψ − 1 0 = 1 +  H I + τ 3 16   σ 3 − iσ 1 2 √ ζ p + r p ζ σ 3 + iσ 1 2  + + 1 2 ζ  H I + τ 3 16  2 1 −  y + τ 4  σ 2 ! + O ( N − 3 5 ) , p := z − λ 1 z − λ 0 . (6-17) Using 1 + ∆ M ( z ) to denote the jump-matrix of E on all the contours (see b elow), we can rewrite ( 6-16 ) as the in tegral equation E ( z ) = 1 + 1 2 iπ Z E − ( s )∆ M ( s )d s s − z , (6-18) where the in tegral is taken along all the jumps of E , that is, along the parts of the lenses and the complemen tary arcs that lie outside D 0 ∪ D 1 as w ell as along the boundaries of D 0 , D 1 . How ev er, the con tribution to E coming from the in tegrals along all these contours, except for ∂ D 1 , are of order not exceeding O ( N − 1 ) (note that the parametrix in D 0 is the standard Airy parametrix). Therefore, to obtain the leading order solution, we consider ( 6-18 ) with the contour ∂ D 1 . This integral equation will b e solved b y iterations. The ﬁrst iteration yields E (1) ( z ) = 1 + 1 λ 1 − z (  H I + τ 3 16  σ 3 − iσ 1 2 p ζ 0 ( λ 1 ) / ( λ 1 − λ 0 ) ! + 1 2 ζ 0 ( λ 1 )  H I + τ 3 16  2 1 −  y + τ 4  σ 2 !) . (6-19) Retaining only the terms up to order O ( N − 2 5 ) in the second iteration, w e obtain E (2) − ( z ) = E (1) − ( z ) + res s = λ 1  H I + τ 3 16  2 σ 3 − iσ 1 2 p ζ 0 ( λ 1 ) / ( λ 1 − λ 0 ) ! 1 ( λ 1 − s )( s − z ) s p ( s ) ζ ( s ) σ 3 + iσ 1 2 ! d s = 4 The parametrix P 1 coincides with the parametrix considered in [ 5 ] up to conjugation by σ 2 . 39 = 1 + E (1) 1 λ 1 − z −  H I + τ 3 16  2  1 ζ 0 ( λ 1 )  1 λ 1 − z  1 − σ 2 2  = = 1 + 1 λ 1 − z (  H I + τ 3 16  σ 3 − iσ 1 2 p ζ 0 ( λ 1 ) / ( λ 1 − λ 0 ) ! − 1 2 ζ 0 ( λ 1 ) y + τ 4 −  H I + τ 3 16  2 ! σ 2 ) . (6-20) Therefore, using the fact that λ 1 = a + b, λ 0 = a − b , w e hav e T ( z ) =  1 + E 1 λ 1 − z + O ( N − 3 5 )   1 − ( λ 1 − λ 0 ) σ 2 4 z + ( λ 1 − λ 0 ) 2 − 4( λ 2 1 − λ 2 0 ) σ 2 32 z 2  = = 1 − 2 E 1 + bσ 2 2 z − ( a + b ) E 1 z 2 + bE 1 σ 2 2 z 2 + b 2 − 4 abσ 2 8 z 2 . (6-21) F rom this we can read oﬀ the relev an t matrix entries: ( T 1 ) 22 =  H I + τ 3 16  p 2 ζ 0 ( λ 1 ) /b =: G , (6-22) ( T 1 ) 12 = i G + ib 2 − i 2 ζ 0 ( λ 1 )  y + τ 4  + i b G 2 , (6-23) ( T 1 ) 21 = i G − ib 2 + i 2 ζ 0 ( λ 1 )  y + τ 4  − i b G 2 , (6-24) ( T 2 ) 12 = iab 2 − ( a + b )  i G + i 2 ζ 0 ( λ 1 )  y + τ 4  − i b G 2  − ib G 2 , (6-25) where all the terms ha ve accuracy O ( N − 3 5 ). Direct computation using ( 4-48 ) shows α n = b 2 4 − b 2 ζ 0 ( λ 1 )  y + τ 4  + O ( N − 3 5 ) , β n = a − y + τ 4 ζ 0 ( λ 1 ) + O ( N − 3 5 ) . (6-26) Using T able 2 , we see that a = a 0 + τ 4 ζ 0 ( λ 1 ) + O ( N − 4 5 ) , b = b 0 + τ 4 ζ 0 ( λ 1 ) + O ( N − 4 5 ) , (6-27) where a 0 = − 3 i , b 0 = 2 i , and, th us α n = b 2 0 4 − b 0 2 ζ 0 ( λ 1 ) y + O ( N − 3 5 ) = − 1 + i 6 2 5 e − 3 iπ 10 N 2 5 y ( v ) + O ( N − 3 5 ) , (6-28) β n = a 0 − y ζ 0 ( λ 1 ) + O ( N − 3 5 ) = − 3 i − 6 2 5 e − 3 iπ 10 N 2 5 y ( v ) + O ( N − 3 5 ) . (6-29) T o compute h n w e use ( 4-48 ). Noticing that G = O ( N − 1 5 ), w e can rearrange ( T 1 ) 12 as follo ws: ( T 1 ) 12 = ib 0 2 − iy 2 ζ 0 ( λ 1 ) + i G + i b 0 G 2 + O ( N − 3 5 ) = ib 0 2  1 − y b 0 ζ 0 ( λ 1 ) + 2 G b 0 + 2 G 2 b 2 0 + O ( N − 3 5 )  = ib 0 2  1 − y b 0 ζ 0 ( λ 1 )  e 2 G b 0 (1 + O ( N − 3 5 )) . (6-30) 40 Therefore h n = − 2 iπ ( T 1 ) 12 e N ` = π b 0  1 − y b 0 ζ 0 ( λ 1 )  e N ` + 2 G b 0 (1 + O ( N − 3 5 )) . (6-31) Utilizing ( 6-22 ) and the v alues in T able 2 we ﬁnd: h n = 2 iπ  1 − y 2 iζ 0 ( λ 1 )  exp " N ` − i H I p − iζ 0 ( λ 1 ) + τ 3 16 ζ 0 ( λ 1 ) 3 √ − i ( ζ 0 ( λ 1 ) 5 ) 1 2 !# (1 + O ( N − 3 5 )) = 2 iπ  1 − y 2 iζ 0 ( λ 1 )  exp " N ` − i H I p − iζ 0 ( λ 1 ) − 4(15 δ t ) 3 2 √ − i N 6 e − iπ 4 !# (1 + O ( N − 3 5 )) = 2 iπ ( − 1) N 1 − e − 4 5 iπ 3 − 2 5 2 3 5 y ( v ) N 2 5 ! exp " 9 N 4 − 13 N 4 (15 δ t ) − N 2 3 i (15 δ t ) 3 2 − i e 1 10 iπ 6 1 5 H I N 1 5 − 2 N 3 (15 δ t ) 3 2 !# (1 + O ( N − 3 5 )) (6-32) = 2 iπ ( − 1) N 1 − 3 2 5 2 3 5 e − 4 5 iπ y ( v ) N 2 5 ! exp " 9 N 4 − 195 N 4 δ t + e − 2 5 iπ 6 1 5 N 1 5 H I # (1 + O ( N − 3 5 )) . (6-33) Q.E.D. 7 Analysis near the p oles: triple scaling limit The analysis in [ 10 ] was carried through under the assumption that –in the double scaling limit– the P ainlev´ e co ordinate is chosen in an arbitrary compact set that does not contain any of the p oles of the functions y (0) , y (1) (see Theorem 1.1 ). Our sp ecial interest now is the analysis in the vicinity of any one of suc h p oles. T o set the stage in general terms, w e shall consider the case where the Painlev ´ e v ariable v under go es its own sc aling . If v p is the pole under scrutiny , we shall consider the follo wing triple scaling limit , whereb y , in addition to N → ∞ and N 4 5 ( t − t ∗ ) b eing b ounded, we also imp ose v − v p = O  N − 1 5 − ρ  , (7-1) where ρ ≥ 0 (dep ending on the situation, it may b e b ounded ab o v e). There are tw o distinct scenarios dep ending on whether the coalescence of the saddle p oints (zero es of h 0 ( z )) with the branc h-p oin ts λ 0 , 1 o ccurs at both branch-points or only at one, sa y , at λ 1 . These scenarios corresp onds to the analysis near the critical p oints t 0 = − 1 12 , and t 1 = 1 15 resp ectiv ely . W e recall that t 0 = 1 12 is a p oint of gradient catastrophe in all the situations discussed in Sect. 5.4 , with the exception of situation ”Opp osite W edges, symmetric” (Fig. 10 ). Vicev ersa, the gradien t catastrophe p oin t t 1 = 1 15 o ccurs only in ”Single W edge” (Fig. 7 ) and ”Consecutive W edges” (Fig. 8 ). 41 7.1 The asymmetric case Under this title we treat b oth the case where t is near t 1 (whic h requires a sp ecial parametrix only near one endp oin t, say λ 1 , and the standard Airy parametrix near the other) and the case of t near t 0 but with ν 1 6 = 1 − ν 3 The latter case requires some special parametrix at both endp oin ts; but a giv en v alue of v , generically , can b e near the p ole o f only one of the t wo sp ecial solution y (0) ( v ) , y (1) ( v ) of the Painlev ´ e I equation that enter in Theorem 1.1 . Below, we assume that v is close to the pole v p of y (1) ( v ). The case when a p ole v p of y (1) ( v ) is sim ultaneously a p ole of y (0) ( v ) even though ν 1 6 = 1 − ν 3 and, th us, y (0) 6≡ y (1) , could b e treated as the symmetric case (Subsection 7.2 ) with minor modiﬁcations (but w e shall not consider it here for simplicit y). W e deﬁne the approximate solution to the RHP ( 4-2 ) with the jump matrix ( 4-43 ) as Φ( z ) =        E ( z )Ψ 0 ( z ) for z outside of the disks D 0 , D 1 E ( z )Ψ 0 ( z ) P 0 ( z ) for z inside of the disk D 0 , E ( z )Ψ 0 ( z ) ˆ P 1 ( z ) for z inside of the disk D 1 . (7-2) where the matrix E ( z ), discussed b elow, is needed to “adjust” the situation due to the p ole v p . Here the parametrix P 0 ( z ) is the Airy parametrix if we are near t 1 . If we are near t 0 , the parametrix P 0 ( z ) is giv en by P 0 ( z ) := σ 3 P ( − z ; 1 − ν 3 ) σ 3 , (7-3) where P ( z ; κ ) was introduced in Deﬁnition 6.3 . T o introduce the parametrix ˆ P 1 ( z ), w e ﬁrst deﬁne b Ψ by the Maso ero factorization ([ 17 ]) Ψ( ξ ; v ; κ ) = ( ξ − y ) − σ 3 / 2 " 1 2  y 0 + 1 2( ξ − y )  1 1 0 # b Ψ( ξ ; v ; κ ) (7-4) with Ψ as in Def. 6.3 and y = y ( v ; κ ) (prime denotes deriv ativ e in v ). Deﬁnition 7.1 (Lo cal parametrix near the p oles.) The p ar ametrix b P 1 ( z ) is deﬁne d in D 1 as b P 1 ( z ) = b P 1 ( z ; ν 1 ) = 1 √ 2  i i − 1 1  ζ 3 4 σ 3 b Ψ  ζ + τ 2 ; 3 8 τ 2 ; ν 1  e − θ ( ζ ; τ ) σ 3 , (7-5) where ζ ( z ; ε ) is the lo cal conformal co ordinate in D 1 , see Deﬁnition 6.3 . W e can then formulate the statemen t corresp onding to Theorem 6.1 for the new lo cal parametrix. Theorem 7.1 (Theorem 6.1 in [ 5 ]) The matrix b P 1 satisﬁes: 1. Within D 1 , the matrix b P 1 ( z ) solves the exact jump c onditions on the lenses and on the c omplemen- tary ar cs; 42 2. On the main ar c (cut) b P 1 ( z ) satisﬁes b P 1+ ( z ) = σ 2 b P 1 − ( z ) σ 2 , (7-6) so that Ψ 0 b P 1 within D 1 solves the exact jumps on al l ar cs c ontaine d ther ein (the left-multiplier in the jump ( 7-6 ) c anc els against the jump of Ψ 0 ); 3. The pr o duct Ψ 0 ( z )( z − λ 1 ) − σ 2 b P 1 ( z ) (and its inverse) ar e –as functions of z – b ounde d within D 1 , namely the matrix b P 1 ( z ) c anc els the gr owth of Ψ 0 ( z )( z − λ 1 ) − σ 2 at z = λ 1 ; 4. The r estriction of b P 1 ( z ) on the b oundary of D 1 is b P 1 ( z )     z ∈ ∂ D α =  1 + O ( ζ − 1 2 )  p 1 − ζ /y 1 + p ζ /y ! − σ 3 , (7-7) wher e O ( ζ − 1 2 ) is uniform w.r.t. v in a smal l, c omp act neighb orho o d of a p ole v p that do es not c ontain any zer o of y ( v ) . The statemen ts in [ 5 ] w ere tailored to the case of the tritronqu ´ ee solution and there was a slightly diﬀeren t normalization, but the proof go es through in id entical fashion. Also note that the parametrix in [ 5 ] diﬀers from b P 1 b y a conjugation by σ 2 . 7.1.1 T riple scaling: pro of of Theorem 1.3 Before delving in to the pro of we mak e some preparatory remarks: ﬁrst oﬀ, recall that w e are c ho osing v so that v − v p = O ( N − 1 5 − ρ ), ρ ≥ 0; this means that y ( v ) = 1 ( v − v p ) 2 + O ( v − v p ) 2 also grows at a rate y ( v ) = O ( N 2 5 +2 ρ ). Recall also that for z ∈ ∂ D 1 w e hav e ζ ( z ) = O ( N 2 5 ); ther efor e ζ ( z ) y ( v ) = O ( N − 2 ρ ) , z ∈ ∂ D 1 , ρ ≥ 0 . (7-8) In the case ρ = 0 the disk around λ 1 shall be chosen suﬃcien tly small so that | ζ /y | < 1 − δ for some δ > 0; this means that the rightmost factor in ( 7-7 ) is a uniformly smo oth and b ounded matrix on ∂ D 1 . In fact it also tends to the identit y if ρ > 0, but in general it do es so very slowly (in N ) or not at all (if ρ = 0, which is the most interesting case). Therefore we can mov e the rightmost factor in ( 7-7 ) to the left at “no cost”. So, we can write ˆ P 1 ( z )     z ∈ ∂ D 1 =  1 + O ( ζ − 1 2 )  p 1 − ζ /y 1 + p ζ /y ! − σ 3 = p 1 − ζ /y 1 + p ζ /y ! − σ 3  1 + O ( ζ − 1 2 )  . (7-9) If ρ = 0, the ab o v e mentioned factor do es not tend to iden tity . W e require that the approximate solution Φ( z ) from ( 7-2 ) satisﬁes Φ( z ) = ( Φ + ( z ) = Φ − ( z )( 1 + o (1)) uniformly in z on ∂ D 0 ∪ ∂ D 1 Φ( z ) is b ounded for z inside the disks D 0 ∪ D 1 , (7-10) 43 In particular, in view of p oin t 3 in Theorem 7.1 , the requirements of ( 7-10 ) will b ecome true if the matrix E ( z ), introduced in ( 7-2 ), would satisfy the following RHP problem for E ( z ). Problem 7.1        E + ( z ) = E − ( z )Ψ 0 ( z )  √ 1 − ζ /y 1+ √ ζ /y  σ 3 Ψ − 1 0 ( z ) on ∂ D 1 , E ( z ) = O (1)( z − λ 1 ) − σ 3 ( σ 2 + σ 3 ) as z → λ 1 , E ( z ) = 1 + O ( 1 z ) as z → ∞ , (7-11) wher e O (1) me ans an invertible matrix analytic at z = λ 1 , b ounde d to gether with its inverse, and the cir cle ∂ D 1 has p ositive orientation. Note that the second condition of ( 7-11 ) is equiv alent to E ( z )Ψ 0 ( z )  i i − 1 1  ζ ( z ) 3 4 σ 3 = O (1) as z → λ 1 , (7-12) giv en that ζ ( z ) = O ( z − λ 1 ). Equation ( 7-12 ) together with Theorem 7.1 , item 3, guarantee the b ound- edness of Φ( z ) = E ( z )Ψ 0 ( z ) ˆ P 1 ( z ) = E ( z )Ψ 0 ( z ) 1 √ 2  i i − 1 1  ζ 3 4 σ 3 | {z } = O (1) = O (1) z }| { b Ψ  ζ + τ 2 ; 3 8 τ 2  e − θ ( ζ ; τ ) σ 3 , (7-13) within the disc D 1 . Pro of of solution of the Problem 7-11 Let b E ( z ) = 1 2 ( σ 2 + σ 3 ) E ( z )( σ 2 + σ 3 ). Then        b E + ( z ) = b E − ( z ) M ( z ) on ∂ D 1 , b E ( z ) = O (1)( z − λ 1 ) − σ 3 as z → λ 1 , b E ( z ) = 1 + O ( 1 z ) as z → ∞ , (7-14) where M = 1 2 ( σ 2 + σ 3 )Ψ 0 p 1 − ζ /y 1 + p ζ /y ! σ 3 Ψ − 1 0 ( σ 2 + σ 3 ) . (7-15) Using ( 4-51 ) and the fact that S σ 2 = e ln S σ 2 = cosh(ln S σ 2 ) + sinh(ln S σ 2 ) = 1 2 ( S + S − 1 ) 1 + 1 2 ( S − S − 1 ) σ 2 , (7-16) w e calculate M ( z ) = [ A ( z ) , B ( z )] = 1 q 1 − ζ y 1 + i s ζ p y σ + − i s ζ py σ − ! , where p := z − λ 1 z − λ 0 (7-17) 44 and σ ± = 1 2 ( σ 1 ± iσ 2 ). W e mak e the Ansatz that b E − ( z ) = 1 + L z − λ 1 : then the (constant in z ) matrix L m ust b e c hosen so that b E + ( z ) = b E − ( z ) M ( z ) satisﬁes b E + ( z )( z − λ 1 ) σ 3 =  1 + L z − λ 1  [ A, B ]( z − λ 1 ) σ 3 = O (1) , z ∈ D 1 . (7-18) In ligh t of ( 6-4 ) we see that A ( λ 1 ) = O (1), and th us w e need to consider only the second column of ( 7-18 ):  1 + L z − λ 1  B ( z ) z − λ 1 = B ( z ) z − λ 1 + LB ( z ) ( z − λ 1 ) 2 = O (1) (7-19) or ( LB ( λ 1 ) = 0 B ( λ 1 ) + LB 0 ( λ 1 ) = 0 . (7-20) Calculating B ( λ 1 ) = (0 , 1) T , B 0 ( λ 1 ) = i s ζ 0 ( λ 1 ) 2 by , ζ 0 ( λ 1 ) 2 y ! T , (7-21) w e see that L = i s ( λ 1 − λ 0 ) y ζ 0 ( λ 1 ) σ − (7-22) solv es the system ( 7-20 ). Thus              b E − ( z ) = 1 + i q y ( λ 1 − λ 0 ) ζ 0 ( λ 1 ) σ − z − λ 1 , b E + ( z ) =   1 + i r y ( λ 1 − λ 0 ) ζ 0 ( λ 1 ) σ − z − λ 1   M ( z ) (7-23) solv es the RHP ( 7-14 ). Since i 2 ( σ 2 + σ 3 ) σ − ( σ 2 + σ 3 ) = 1 2 ( σ 3 − iσ 1 ), w e obtain that                E − ( z ) = 1 + q y ( λ 1 − λ 0 ) ζ 0 ( λ 1 ) ( σ 3 − iσ 1 ) 2( z − λ 1 ) , E + ( z ) =   1 + q y ( λ 1 − λ 0 ) ζ 0 ( λ 1 ) ( σ 3 − iσ 1 ) 2( z − λ 1 )   Ψ 0 p 1 − ζ /y 1 + p ζ /y ! σ 3 Ψ − 1 0 (7-24) solv es the RHP ( 7-11 ). Q.E.D. Error analysis. The error matrix E ( z ) = T ( z )Φ − 1 ( z ) has jumps on the lenses and on the complemen- tary arcs outside the disks D 0 , D 1 , as well as on the b oundary of these disks. The jump matrices on the lenses and on the complementary arcs approac h 1 exp onentially fast in N − 1 and uniformly in z . It is also clear that E → 1 as z → ∞ since b oth T and Φ = E Ψ 0 do so. So, it remains only to pro v e the 45 uniform conv ergence to 1 of the jump matrix on ∂ D 1 (con vergence on ∂ D 0 w as established in Subsection 6.2 ). Indeed, using ( 7-2 ), ( 7-11 ), ( 7-7 ), ( 7-9 ) and ( 7-24 ), we hav e E + = T Φ − 1 + = T ˆ P − 1 1 Ψ − 1 0 E − 1 + = T Φ − 1 − E − Ψ 0 ˆ P − 1 1 Ψ − 1 0 Ψ 0 p 1 − ζ /y 1 + p ζ /y ! − σ 3 Ψ − 1 0 E − 1 − = E − E − Ψ 0 ( 1 + O ( ζ − 1 / 2 ))Ψ − 1 0 E − 1 − = E − E − ( 1 + O ( ζ − 1 / 2 )) E − 1 − . (7-25) On the b oundary z ∈ ∂ D 1 w e ha ve ζ = O ( N 2 5 ) and in our triple scaling y = O ( N 2 5 +2 ρ ) with ρ ≥ 0. Then L is of the order O ( N − 1 5 √ y ) = O ( N ρ ). Thus, E − O ( ζ − 1 / 2 ) E − 1 − = O ( ζ − 1 / 2 ) + [ L, O ( ζ − 1 / 2 )] z − λ 1 − L O ( ζ − 1 / 2 ) L ( z − λ 1 ) 2 = O ( N − 1 5 ) + O ( N − 1 5 + ρ ) + O ( N − 1 5 +2 ρ ) . (7-26) So, it is the last term that contr ibutes the slow est decay . Therefore, we obtain E + = E − ( 1 + O ( N − 3 5 y )) , z ∈ ∂ D 1 . (7-27) The latter estimate sho ws that we can control the error provided y = y (1) = O ( N 2 5 + ρ ) , or , equiv alently , v − v p = O ( N − 1 5 − ρ ) , (7-28) where 0 ≤ ρ < 1 5 . Computation of the recurrence coeﬃcients: W e need to use ( 3-9 ) and ( 7-27 ). Using ( 7-2 ), ( 7-24 ) and the expansion of Ψ 0 ( 4-52 ) w e obtain Φ( z ) = E − ( z )Ψ 0 ( z ) =  1 + 1 2 k ( σ 3 − iσ 1 ) z − λ 1   1 − b 2 z σ 2 + b 2 8 z 2 1 − abσ 2 2 z 2 + O ( z − 3 )  =  1 + 1 2 k ( σ 3 − iσ 1 )  1 z + λ 1 z 2   1 − b 2 z σ 2 + b 2 8 z 2 1 − abσ 2 2 z 2 + O ( z − 3 )  (7-29) = 1 + 1 z  k 2 ( σ 3 − iσ 1 ) − b 2 σ 2  + 1 z 2  b 2 1 8 − ab 2 σ 2 + k 2  a + 1 2 b  ( σ 3 − iσ 1 )  (7-30) k := s ( λ 1 − λ 0 ) y ζ 0 ( λ 1 ) = s 2 by ζ 0 ( λ 1 ) (7-31) W e introduce s := s ζ 0 ( λ 1 ) 2 by ( v ) = s ζ 0 ( λ 1 ) 2 b 0 ( v − v p ) + O  N 1 5 ( v − v p ) 5  , (7-32) where the latter expression follows from ( 1-14 ). Here and henceforth a 0 , b 0 denote the v alues of a, b calculated exactly at one of the critical p oin ts t 0 or t 1 . Assuming in ρ = 0 in ( 7-28 ), w e obtain s = O (1) as N → ∞ . On the other hand, if ρ ∈ (0 , 1 5 ), then, consequently , s scales as O ( N − ρ ). Thus, w e hav e the triple scaling limit t − t j = v κN 4 5 = v p κN 4 5 + q 2 b 0 ζ 0 ( λ 1 ) κN 4 5 s, (7-33) 46 where κ is the constant in front of δ t = t − t j app earing in formulæ ( 6-7 , 6-8 ). Explicitly , using T able 2 , w e obtain t + 1 12 = − v p 3 6 5 2 9 5 N 4 5 − s 3 √ 2 N , t − 1 15 = v p e − 3 iπ 5 3 6 5 2 1 5 5 N 4 5 − i 2 s 15 N . (7-34) No w, according to ( 4-48 ), ( 4-30 ), ( 4-31 ) and ( 7-27 ), we obtain: α n = ( T 1 ) 12 ( T 1 ) 21 = b 2 4 − by ( v ) 2 N 2 5 C + O ( N − 3 5 y ) = b 2 0 4 − 1 4 s 2 + O ( N − 3 5 y , N − 2 5 ) , (7-35) β n = ( T 2 ) 12 ( T 1 ) 12 − ( T 1 ) 22 = a 0 + 1 2 s (1 − b 0 s ) + O ( N − 3 2 y ) + O ( N − 3 2 y , N − 2 5 ) , (7-36) h n = − 2 iπ ( T 1 ) 12 e N ` =  π  b 0 − 1 s  + O ( N − 3 2 y , N − 2 5 )  e h N ln b 2 4 − N 2 a 2 + b 2 8 − N 2 i . (7-37) Here O ( N − 2 5 ) error term comes from replacing a, b with their resp ective v alues a 0 , b 0 considered at the critical p oin t t 0 or t 1 . Note, how ev er, that in the regime ( 7-1 ), the O ( N − 2 5 ) term is of a smaller order than the O ( N − 3 2 y ) term. Therefore, in all these expressions, in the regime ( 7-1 ), the error is at b est O ( N − 1 5 ) (recall that y = O ( N 2 5 +2 ρ ) ρ ∈ [0 , 1 5 )). Thus in the exp onent e N ` w e can use the expansion in T able 2 up to order δ t included. So, h n = π  b 2 0 4  N exp  − N 2 a 2 0 + b 2 0 + 4 8 + cN δ t   b 0 − 1 s + O ( N − 3 2 y )  , (7-38) where c = − 6 for the case t ∼ t 0 and c = − 13 · 15 4 for the case t ∼ t 1 . One has then to replace δ t by the expressions in ( 7-34 ). So, in the leading order, h n = π  √ 8 − 1 s  2 N exp " − 3 N 2 + N 1 5 v p 3 1 5 2 4 5 + √ 2 s # , t ∼ − 1 12 , (7-39) h n = π  2 i − 1 s  ( − 1) N exp " 9 N 4 − 13 4 N 1 5 v p e − 3 iπ 5 3 1 5 2 1 5 + i 13 2 s # , t ∼ 1 15 . (7-40) It is remark able to note that the genus zero leading order asymptotics α n ( t ) ∼ b 2 4 and β n ( t ) ∼ a are v alid as long y = o ( N 2 5 ) with the accuracy O ( y N 2 5 ). Ho w ever, when y = O ( N 2 5 ), b oth terms in ( 7-35 ), ( 7-36 ), contribute to the leading order, whereas, when y = O ( N 2 5 +2 ρ ) with ρ ∈ [0 , 1 5 ), the asymptotics are determined by the latter terms of ( 7-35 ), ( 7-36 ). In this case, b oth α n and β n are unbounded as N → ∞ . So, the pro of of Theorem 1.3 is completed. Q.E.D. 7.2 The symmetric case: pro of of Theorem 1.4 W e are now in the symmetric situation and hence the critical p oin t to consider can only b e t 0 = − 1 12 , where λ 1 = b , λ 0 = − b and a ≡ 0. This case is signiﬁcantly diﬀerent from the previous inasm uch as the t wo Painlev ´ e parametrices in D 0 , 1 are identical: in particular y (1) = y (0) = y . Th us, if the double scaling 47 is such that we are close to a p ole v p of y ( v ), this will simultane ously aﬀect the b oth parametrices and, as we shall see, will hav e a signiﬁcant eﬀect on the asymptotics of α n . On the other hand, due to the exact symmetry of the bilinear pairing, the orthogonal p olynomials hav e the same parity of their degree and th us automatically β n ≡ 0. It will b e adv antageous for us to use a diﬀer ent solution to the mo del problem ( 4-50 ), which has a diﬀeren t growth rate near the branch-points: suc h mo diﬁcation (see [ 11 ]) is called a discr ete Schlesinger tr ansformation . In terms of the RHP ( 4-50 ), this amounts to replacing the solution Ψ 0 ( 4-51 ) with Ψ 1 ( z ) := 1 2 ( σ 3 + σ 2 )  z − b z + b  − 3 4 σ 3 ( σ 3 + σ 2 ) =  z − b z + b  − 3 4 σ 2 . (7-41) This matrix satisﬁes all the conditions of the RHP ( 4-50 ) except the last one, as it clearly has a diﬀeren t gro wth b eha viour near the endp oin ts ± b . W e then shall construct an approximate solution Φ( z ) =        E ( z )Ψ 1 ( z ) for z outside of the disks D 0 , D 1 E ( z )Ψ 1 ( z ) b P 0 ( z ) for z inside of the disk D 0 , E ( z )Ψ 1 ( z ) ˆ P 1 ( z ) for z inside of the disk D 1 , (7-42) where ˆ P 1 ( z ) is deﬁned by ( 7-5 ) and b P 0 ( z ) = σ 3 b P 1 ( − z ) σ 3 . (7-43) Due to the fact that we are using Ψ 1 instead of Ψ 0 , the b oundedness of the pro duct Ψ 1 b P 0 , 1 at λ 0 , λ 1 follo ws immediately (see also Theorem 7.1 , item 3). Hence, the requirements on the left multiplier E ( z ) are no w diﬀerent compare with the asymmetric case studied ab ov e (we reuse the same symbol E with a new meaning relativ e to the previous section). Problem 7.2 Find the matrix E ( z ) is analytic (to gether with its inverse) on C \ ( ∂ D 0 ∪ ∂ D 1 ) and satisﬁes    E + ( z ) = E − ( z )Ψ 1 ( z )  √ 1 − ζ /y 1+ √ ζ /y  σ 3 Ψ − 1 1 ( z ) on ∂ D 0 , 1 , E ( z ) = 1 + O ( 1 z ) as z → ∞ , (7-44) wher e the c ontours ∂ D 0 , 1 have p ositive orientation. Pro of of solution of Problem 7.2 . Note that an y solution to this RHP has unit determinan t and hence its in verse is also analytic and b ounded. As b efore, we ﬁnd it more conv enien t to solv e the RHP for b E ( z ) = F E ( z ) F instead of the RHP ( 7-44 ). Here F = σ 2 + σ 3 √ 2 . The jump matrix for the new RHP is M ( z ) = F − 1 Ψ 1 ( z ) Q − 1 ( z )Ψ − 1 1 ( z ) F , (7-45) where Q := p 1 − ζ /y 1 + p ζ /y ! − σ 3 , (7-46) 48 y = y ( v ), v was deﬁned by ( 6-7 ) and the lo cal scaling co ordinate ζ = ζ ( z , N ) near z = b was introduced in ( 6-4 ). Direct calculations yield M ( z ) = 1 p 1 − ζ ( z ) /y   1 i q ζ ( z )( z + b ) 3 y ( z − b ) 3 − i q ζ ( z )( z − b ) 3 y ( z + b ) 3 1   . (7-47) Similarly , near z = − b we obtain ˜ M ( z ) = 1 q 1 − ˜ ζ ( z ) /y   1 i q ˜ ζ ( z )( z + b ) 3 y ( z − b ) 3 − i q ˜ ζ ( z )( z − b ) 3 y ( z + b ) 3 1   , (7-48) where ˜ ζ = ˜ ζ ( z ) = ζ ( − z ) , z ∈ D 0 . (7-49) Note that the orthogonal p olynomials in this case are ev en/o dd and the symmetry of the RHP implies (whic h can b e v eriﬁed directly from the ab o ve formulæ and also as a consequence of ( 7-45 )) Ψ 1 ( z ) = σ 3 Ψ 1 ( − z ) σ 3 ⇒ f M ( z ) = σ 2 M ( − z ) σ 2 (7-50) Using ( 7-47 ), ( 7-48 ), ( 6-4 ), ( 7-49 ), w e obtain M ( z ) − 1 = η σ + z − b + O 1 ( z ) , ˜ M ( z ) − 1 = η σ − z + b + O 0 ( z ) , (7-51) where η = iN 1 5 s (2 b ) 3 C y . (7-52) Here C = N − 2 5 ˜ ζ 0 ( b ) and ˜ C = N − 2 5 ˜ ζ 0 ( − b ) = − C by the symmetry of the problem. Note that the O 0 , 1 terms in ( 7-51 ) are analytic at z = λ 0 , 1 and when ev aluated at z = λ 0 , 1 = ± b are prop ortional to σ ± (resp ectiv ely). The matrix b E ( z ) satisﬁes b E ( z ) = 1 + I | s − b | = r b E − ( s )( M ( s ) − 1 ) s − z d s 2 iπ + I | s + b | = r b E − ( s )( ˜ M ( s ) − 1 ) s − z d s 2 iπ . (7-53) W e p ose the Ansatz b E − ( z ) = 1 + A z − b + ˜ A z + b , (7-54) and obtain A z − b + ˜ A z + b = − Aη σ + ( z − b ) 2 − η σ + z − b − ˜ Aη σ + 2 b ( z − b ) − ˜ Aη σ − ( z + b ) 2 − AO 1 ( b ) z − b − ˜ AO 0 ( − b ) z + b − η σ − z + b − Aη σ − 2 b ( z + b ) . (7-55) That leads to the follo wing system for the unknown A, ˜ A (recall that O 1 ( b ) ∝ σ + , O 0 ( − b ) ∝ σ − ): Aσ + = 0 , ˜ Aσ − = 0 , 49 A + η 2 b ˜ Aσ + = − η σ + , ˜ A + η 2 b Aσ − = − η σ − . (7-56) This system has the solution A = 1 1 + η 2 (2 b ) 2  0 − η 0 η 2 2 b  , ˜ A = − 1 1 + η 2 (2 b ) 2  η 2 2 b 0 η 0  = − σ 2 Aσ 2 (7-57) So, w e found b E ( z ) and, thus, E ( z ). Note that the function E ( z ) in the region outside of the disks is a rational function with p oles at ± b , while, inside the disks, it is analytic and given b y formula ( 7-53 ). Q.E.D. Error analysis. The error matrix E ( z ) = T ( z )Φ − 1 ( z ) has jumps on the lenses and on the complemen- tary arcs outside the disks D 0 , D 1 , as well as on the b oundary of these disks. The jump matrices on the lenses and on the complementary arcs approach 1 exp onentially fast in N and uniformly in z . It is also clear that E → 1 as z → ∞ . So, it remains only to pro v e the uniform conv ergence to 1 of the jump matrix on ∂ D 0 , 1 : the computations are absolutely parallel and we rep ort only the one for ∂ D 1 . Using ( 7-2 ), the solution to Problem 7.2 and eq. ( 7-7 ), we hav e E + = T Φ − 1 + = T ˆ P − 1 1 Ψ − 1 1 E − 1 + = T Φ − 1 − E − Ψ 1 ˆ P − 1 1 Ψ − 1 1 Ψ 1 p 1 − ζ /y 1 + p ζ /y ! − σ 3 Ψ − 1 1 E − 1 − = E − E − Ψ 1 ( 1 + O ( ζ − 1 / 2 ))Ψ − 1 1 E − 1 − = E − E − ( 1 + O ( ζ − 1 / 2 )) E − 1 − . (7-58) On the boundary z ∈ ∂ D 1 w e ha ve ζ = O ( N 2 5 ) and in our double scaling y = O ( N 2 5 +2 ρ ), where ρ ∈ [0 , 1 5 ). Moreo ver, E − = 1 + F AF z − b + F ˜ AF z + b , where A, ˜ A are of the same order. That creates the situation that is drastically diﬀerent from the previous: for example, the matrices A, ˜ A ( 7-57 ) remain b ounded no matter ho w fast y grows (and hence η → 0 ( 7-52 )). The only un b oundedness o ccurs when the denominators in ( 7-57 ) v anish, which means that η has a ﬁnite v alue η 2 = − 4 b 2 or, equiv alently , N − 2 5 y ( v ) = 2 C b. (7-59) Condition ( 7-59 ) iden tiﬁes tw o p oin ts near the p ole v = v p at a distance of order O ( N − 1 5 ). Thus, in ( 7-58 ) w e hav e E − O ( ζ − 1 / 2 ) E − 1 − = O ( ζ − 1 / 2 ) + O ζ − 1 2  1 + η 2 4 b 2  − 1 ! = O ( N − 1 5 ) + O N − 1 5  1 + η 2 4 b 2  − 1 ! (7-60) The very last contribution to the error term comes from the denominators of the matrices A, ˜ A ( 7-57 ) and prev ents us from getting close “to o fast” to the p oin ts where they v anish. 50 Computation of the recurrence co eﬃcients: F ollowing [ 5 ], w e ﬁnd the expansion of the matrix Φ( z ) = E ( z )Ψ 1 ( z ) at z = ∞ : E Ψ 1 = F ˆ E ( z )  z − b z + b  − 3 4 σ 3 F − 1 = 1 + F ( A + ˜ A ) F − 1 + 3 2 bσ 2 z + b 2 F ( ˜ A − A ) F − 1 + 9 8 b 2 1 z 2 + O ( z − 3 ) . (7-61) Using ( 7-57 ), w e obtain ˜ A + A = − 1 1 + η 2 (2 b ) 2  η 2 2 b σ 3 + η σ 1  , ˜ A − A = − 1 1 + η 2 (2 b ) 2  η 2 2 b 1 − iη σ 2  , (7-62) (7-63) so that F ( ˜ A + A ) F − 1 = 1 1 + η 2 (2 b ) 2  − η 2 2 b σ 2 + η σ 1  , F ( ˜ A − A ) F − 1 = 1 1 + η 2 (2 b ) 2  − η 2 2 b 1 + iη σ 3  . (7-64) It follo ws from ( 7-61 ) and ( 7-64 ) that the residue of Φ at inﬁnity , which we denote by Φ 1 , is Φ 1 = F ( A + e A ) F − 1 + 3 2 bσ 2 = b 1 1 + η 2 4 b 2  − η 2 2 b 2 σ 2 + η b σ 1  + 3 2  1 + η 2 4 b 2  σ 2  = = b 1 1 + η 2 4 b 2  − η 2 8 b 2 σ 2 + η b σ 1  + 3 2 σ 2  (7-65) W e note in passing that Φ 1 is oﬀ-diagonal and Φ 2 is diagonal (whic h implies β n = 0, which -of course- is iden tity and not just an approximation due to the sp ecial symmetry of this case). Then (Φ 1 ) 12 = b 2 − 3 i + iη 2 4 b 2 + 2 η b 1 + η 2 (2 b ) 2 = − i b 2 3 + iη 2 b 1 − iη 2 b (7-66) (Φ 1 ) 21 = b 2 3 i − iη 2 4 b 2 + 2 η b 1 + η 2 (2 b ) 2 = i b 2 3 − iη 2 b 1 + iη 2 b (7-67) Using ( 7-60 ), w e can now calculate the (leading order) ﬁnal expressions α n = ( T 1 ) 12 ( T 1 ) 21 = b 2 4 9 + η 2 4 b 2 1 + η 2 4 b 2 , β n = 0 (7-68) where it is understo o d that b oth expressions (also in the denominators) are aﬀected by an error of the order indicated in ( 7-60 ). Introducing s = − iη 2 b = N 1 5 s 8 b 3 C y , (7-69) 51 w e note that O ( N − 1 5 (1 + η 2 / (4 b 2 )) − 1 ) = O ( N − 1 5 ( s 2 − 1) − 1 ) and w e ﬁnd ﬁnally (using T able 2 for the symmetric case) 5 α n = b 2 4 9 − s 2 + O ( N − 1 5 ) 1 − s 2 + O ( N − 1 5 ) , β n = 0 , h n = 2 N π √ 8 exp  − 3 N 2 − 6 N 1 5 δ t N − 4 5  3 − s 1 + s + O N − 1 5 1 − s 2 !! . (7-70) Using ( 6-7 ) and T able 2 to relate s and t , we can write ( 7-70 ) as h n = π √ 8 2 N exp " − 3 N 2 + N 1 5 v p 3 1 5 2 4 5 − s 4 # 3 − s 1 + s + O N − 1 5 1 − s 2 !! . (7-71) Q.E.D. References [1] V. I. Arnol’d, S. M. Guse ˘ ın-Zade and A. N. V archenk o, Singularities of diﬀer entiable maps. Vol. I , volume 82 of Mono gr aphs in Mathematics . Birkh¨ auser Boston Inc., Boston, MA, 1985. The classiﬁcation of critical p oints, caustics and w a v e fron ts, T ranslated from the Russian b y Ian P orteous and Mark Reynolds. [2] M. 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Asymptotics of orthogonal polynomials with complex varying quartic weight: global structure, critical point behaviour and the first Painleve equation

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