math-ph 2009-11-13 0

The Cauchy two-matrix model

We introduce a new class of two(multi)-matrix models of positive Hermitean matrices coupled in a chain; the coupling is related to the Cauchy kernel and differs from the exponential coupling more commonly used in similar models. The correlation funct…

Authors: M. Bertola, M. Gekhtman, J. Szmigielski

The Cauchy two-matrix model
The Cauc h y t w o-matrix mo del M. Bertola †‡ 1 , M. Gekh tman a 2 , J. Szmigielski b 3 † 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 E-mail: b ertola@crm.umontr e al.c a ‡ Dep artment of Mathematics and Statistics, Conc or dia University 1455 de Maisonneuve W., Montr´ eal, Qu´ eb e c, Canada H3G 1M8 a Dep artment of Mathematics 255 Hurley Hal l, Notr e Dame, IN 46556-4618, USA E-mail: Michael.Gekhtman.1@nd.e du b Dep artment of Mathematics and Statistics, University of Saskatchewan 106 Wiggins R o ad, Saskato on, Saskatchewan, S7N 5E6, Canada E-mail: szmigiel@math.usask.c a Abstract W e in tro duce a new class of tw o(multi)-matrix mo dels of p ositiv e Hermitean matrices coupled in a c hain; the coupling is related to the Cauch y k ernel and differs from the exp onen tial coupling more commonly used in similar mo dels. The correlation functions are expressed entirely in terms of certain biorthogonal p olynomials and solutions of appropriate Riemann–Hilb ert problems, th us paving the wa y to a steep est descen t analysis and univ ersality results. The in terpretation of the formal expansion of the partition function in terms of m ulticolored ribb on-graphs is pro vided and a connection to the O (1) mo del. A steep est descent analysis of the partition function rev eals that the mo del is related to a trigonal curve (three-sheeted co vering of the plane) m uch in the same w ay as the Hermitean matrix mo del is related to a hyperelliptic curve. Con ten ts 1 In tro duction 2 2 Cauc hy biorthogonal p olynomials 5 2.1 Christoffel–Darb oux Iden tities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Riemann–Hilb ert c haracterization of the integrable k ernels . . . . . . . . . . . . . . . . . . 8 1 W ork supported in part by the Natural Sciences and Engineering Researc h Council of Canada (NSERC) 2 W ork supported in part by NSF Gran t DMD-0400484. 3 W ork supp orted in part b y the Natural Sciences and Engineering Researc h Council of Canada (NSERC), Grant. No. 138591-04 1 3 Matrix Models 10 3.1 Correlation functions: Christoffel–Darb oux k ernels . . . . . . . . . . . . . . . . . . . . . . 13 3.1.1 Correlation functions in terms of biorthogonal p olynomials . . . . . . . . . . . . . 16 3.2 A multi–matrix model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 4 Diagrammatic expansion 19 5 Large N b eha viour 21 5.1 Con tinuum v ersion: cubic sp ectral curve and solution of the p oten tial problem . . . . . . 22 A A rectangular mixed 3 –matrix mo del with ghost fields 27 A.1 In teger h < N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 A.2 Half–in teger h < N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 A.3 In teger and half in teger h > N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 B Relation to the O (1) –mo del 29 1 In tro duction In the last tw o decades or so, the theory of matrix mo dels hav e been an incredibly fertile ground for a fruitful in teraction betw een theoretical ph ysics, statistics, analysis, n umber theory and dynamical systems (see the classical [ 35 ] and references therein). The interpla y with analysis has been particularly b eneficial for the Hermitean matrix mo del [ 14 ], largely due to the realization that the matrix model could be “solv ed” in terms of orthogonal p olynomials. This allo wed to translate questions ab out the spectrum of large random matrices into questions ab out the asymptotics of orthogonal p olynomials, connecting the former with a very w ell developed area of analysis. The av ailabilit y of Riemann–Hilbert metho ds [ 28 , 13 ] was the crucial ingredient in addressing questions of universalit y in the bulk and at the edge of the sp ectrum. It should b e men tioned that this fortunate symbiosis relies on the following features: • the p ossibilit y of rewriting the matrix in tegral in terms of eigenv alues and the c orrelation functions in terms of suitable orthogonal p olynomials; • the Riemann–Hilbert c haracterization of orthogonal p olynomials [ 28 ]; • the Christoffel–Darboux form ula, allowing to rewrite the k ernel of the correlation functions in terms of only tw o orthogonal p olynomials of degree N ; • the nonlinear steep est descen t metho d [ 13 ] applied to the RH problem for orthogonal p olynomials. 2 There are b y no w sev eral matrix models based on v arious ensembles of matrices. F or some (Symplectic, Orthogonal), a connection with (skew-)orthogonal polynomials can be established [ 35 , 37 ] and some of the steps in the abov e list hav e b een p erformed, but typically only for a certain restricted class of “p otentials”. F or others (lik e O ( n ) mo dels, rectangular models) the metho ds used rely on symmetries of the in tegral and dynamical system approaches (the “lo op equations”) [ 18 , 16 , 17 , 20 , 1 ] There are also the so–called multi-matrix-mo dels which inv olv e ensem bles of several matrices (t ypically Hermitean) and one of the most studied among them is the Itzykson–Zub er–Harish-Chandr a (IZHC) c hain of matrices. In its simplest form, the t wo-matrix model, it amounts to the study of the spectral prop erties of a pair of Hermitean matrices of size N × N with a measure d µ ( M 1 , M 2 ) = d M 1 d M 2 e − N T r( V 1 ( M 1 )+ V 2 ( M 2 ) − M 1 M 2 ) (1-1) where the interaction term is e N T r M 1 M 2 . The first three items of the bulleted list ab o ve can b e imple- men ted at least for p otentials whose deriv ativ e is a rational function [ 19 , 6 , 4 , 5 ] while the last item is still not fully under con trol. One of the main reasons for this difficulty is that the size of the RHP dep ends on the p otentials : for example if V j are p olynomials of degree d j then there are tw o relev ant RHPs of the same size. Clearly the question arises as to whether a Riemann–Hilb ert method can be utilized for a more general class of p otentials, for example real-analytic, lik e in the case of the single-matrix mo del. The presen t pap er is a part of a larger pro ject initiated in [ 8 ], which puts forward a new multimatrix mo del (at the momen t primarily tw o matrices) that can b e completely solved along the lines describ ed ab o v e. While the new mo del is very close in spirit to the IZHC mo del, the different in teraction I ( M 1 , M 2 ) = 1 det( M 1 + M 2 ) N (1-2) links the mo del to a new class of biorthogonal p olynomials which w ere studied in [ 8 ] and termed “Cauch y biorthogonal p olynomials”. These p olynomials ha ve sev eral of the desirable features of classical orthogonal p olynomials. W e will fill in more details in Sect. 3 but here we just indicate that the mo del we wan t to study is defined on the space of pairs of p ositive Hermitean matrices of size N equipp ed with a measure d µ ( M 1 , M 2 ) := d M 1 d M 2 α ( M 1 ) β ( M 2 ) det( M 1 + M 2 ) N (1-3) for arbitrary measures α ( x )d x, β ( y )d y on R + (to b e understo o d in the formula ab o ve as conjugation- in v ariant measures on p ositive definite Hermitean matrices or, equiv alently , measures on the sp ectra of M i ’s). This immediately puts the problem at the same lev el of generalit y as the classical case. Cor- resp onding to the ab o ve positive measure is the normalizing factor, customarily called the partition function Z N := Z Z d M 1 d M 2 α ( M 1 ) β ( M 2 ) det( M 1 + M 2 ) N (1-4) 3 whose dep endence on the measures α, β carries all relev ant information about the mo del. W e will show that this model is related to Cauch y biorthogonal p olynomials Z R + Z R + p n ( x ) q m ( y ) α ( x )d x β ( y )d y x + y = δ mn (1-5) defined and studied in [ 8 ] in relation with the spectral theory of the cubic string and the Degasp eris- Pro cesi wa ve equation (see also [ 2 , 3 , 34 , 33 ]). Con trary to the biorthogonal p olynomials of [ 19 , 6 , 7 ] the prop erties of these p olynomials do not dep end on the measures α, β and the main highlights are 1. they solv e a four-term recurrence relation; 2. their zeroes are p ositiv e and simple; 3. their zeroes ha ve the in terlacing prop erty; 4. they possess Christoffel–Darb oux iden tities relev an t to matrix mo dels; 5. they can b e c haracterized by a (pair of ) Riemann–Hilb ert problem(s) of size 3 × 3; 6. the steepest descen t metho d is fully applicable. Our recent pap er [ 8 ] con tains a detailed discussion of all these points except the last one which will b e addressed in a forthcoming publication. The pap er is organized as follows; in Sect. 2 we review our previous results [ 8 ] on the Cauch y biorthogonal polynomials and the relev ant formulæ and features needed in the follo wing. In particular the Christoffell–Darb oux identities (Sec. 2.1 ), the Riemann–Hilb ert c haracterization in terms of 3 × 3 piecewise analytic matrices (Sect. 2.2 ). In Prop. 2.7 we in tro duce the 3 × 3 matrix kernel in terms of the solution of the Riemann–Hilb ert problems; this will b e used in later section to describ e the sp ectral statistics of the tw o matrices. In Sect. 3 we in tro duce in detail the tw o–matrix mo del that we ha ve outlined ab o v e and sho w how to reduce the study of its sp ectral statistics to Cauc hy biorthogonal p olynomials using a form ula app eared in [ 25 ]. W e also indicate (without details) ho w to deal with a similar mo del where R matrices are link ed in a chain (Sect. 3.2 ). In Sect. 4 we sho w ho w a formal treatment of the partition function of the mo del for large sizes of the matrices can b e used to extract com binatorial information for certain bi-colored ribbon graphs (as it has b een done for the Hermitean one-matrix model in [ 11 , 32 ] and for the multi-matrix mo dels with exp onen tial coupling in [ 16 ]). In Sect. 5 w e sho w that a saddle–p oint treatment of the partition function (see for example [ 15 ] for the one-matrix case) leads to a three –sheeted cov ering of the sp ectral plane (a trigonal curve ); this (pseudo) algebraic curve pla ys the same rˆ ole as the h yp erelliptic curve in the Hermitean matrix mo del. The App endices are devoted to the relation betw een our prop osed mo del and other matrix mo dels with rectangular matrices (and p ossibly with Grassmann entries, App. A ) and a connection (App. B with the O (1) mo del of self-a voiding loops ([ 18 ] and references therein). 4 Remark 1.1 F or a p articular c ase of me asur es α ( x ) = x a e − x and β ( y ) = y b e − y the c orr esp onding biortho gonal p olynomials app e ar e d (in a somewhat disguise d form) in the work [ 10 ]. As observe d ther ein they ar e r elate d to the classic al Jac obi ortho gonal p olynomials for the weight x a + b d x on [0 , 1] . We thank A. Bor o din for p ointing out this c onne ction. 2 Cauc h y biorthogonal p olynomials Let K ( x, y ) = 1 x + y b e the Cauc hy kernel on R + × R + . It is known that it is totally p ositive in the sense of the classical definition [ 22 , 31 ]: Definition 2.1 A totally p ositiv e kernel K ( x, y ) on I × J ⊂ R × R is a function such that for al l n ∈ N and or der e d n -tuples x 1 < x 2 < · · · < x n , y 1 < y 2 < · · · < y n we have the strict ine quality det[ K ( x i , y j )] > 0 . (2-1) It was shown in [ 8 ] that for any totally positive kernel K ( x, y ) (hence also for the Cauc hy kernel) and for an y pair of measures α ( x )d x, β ( y )d y supported in I , J ⊂ R + resp ectiv ely , the matrix of bimoments I ij = Z I Z J x i y j K ( x, y ) α ( x )d xβ ( y )d y (2-2) is a totally p ositiv e matrix , that is, ev ery square submatrix has a positive determinant. This guarantees the existence of biorthogonal p olynomials { p j ( x ) , q j ( y ) } of exact degree j such that Z I Z J p j ( x ) q k ( y ) K ( x, y ) α ( x )d xβ ( y )d y = δ j k . (2-3) The polynomials p j ( x ) , q j ( y ) are defined uniquely up to a C × action p j 7→ λ j p j ( x ) , q j 7→ 1 λ j q j and the am biguity can be disp osed of b y requiring that their leading coefficients are the same and p ositive. With this understanding we hav e p n ( x ) = 1 c n x n + . . . and q n = 1 c n y n + . . . , with the positive constant c n giv en b y [ 8 ] c n = r D n +1 D n , D n := det  I j k  0 ≤ j,k ≤ n − 1 , I j k := Z R + Z R + x j y k K ( x, y ) α ( x )d xβ ( y )d y . (2-4) The determinantal expressions for the BOPs in terms of bimoments can b e obtained using Cramer’s rule [ 8 ]. Prop osition 2.1 (Thm. 4.5 in [ 8 ]) F or any total ly p ositive kernel K ( x, y ) the zer o es of the biortho go- nal p olynomials p j ’s ( q j ’s) ar e simple, r e al and c ontaine d in the c onvex hul l of the supp ort of the me asur e α ( β r esp e ctively). In the case K ( x, y ) = 1 x + y w e named the corresp onding p olynomials { p j ( x ) , q j ( y ) } Cauch y BOPs and prov ed, in addition, 5 Prop osition 2.2 (Thm. 5.2 in [ 8 ]) The r o ots of adjac ent p olynomials in the se quenc es { p j ( x ) } , { q j ( y ) } ar e interlac e d. Cauc hy BOPs enjoy more structure than the BOPs asso ciated to a generic kernel: they solve a four term recurrence relation of the form x ( π n − 1 p n ( x ) − π n p n − 1 ( x )) = a ( − 1) n p n +1 ( x ) + a (0) n p n ( x ) + a (1) n p n − 1 ( x ) + a (2) n p n − 2 ( x ) y ( η n − 1 q n ( y ) − η n q n − 1 ( y )) = b ( − 1) n q n +1 ( y ) + b (0) n q n ( y ) + b (1) n q n − 1 ( y ) + b (2) n q n − 2 ( y ) (2-5) where π n := Z R + p n ( x ) α ( x )d x , η n := Z R + q n ( y ) β ( y )d y . (2-6) As prov ed in [ 8 ] π n , η n are strictly p ositive . The other co efficients app earing in the recurrence relations are describ ed in lo c. cit. W e also need to introduce certain auxiliary p olynomials, b q n , b p n whose defining prop erties are 1. deg b q n = n + 1, deg b p n = n ; 2. Z b q n d β = 0 or b q n ( y ) = q n +1 ( y ) η n +1 − q n ( y ) η n ; 3. Z Z b p n ( x ) b q m ( y ) d α d β x + y = δ mn or 1 η n ( b p n − 1 − b p n ) = p n . In addition b p n , b q n admit the determinantal represen tations: b q n ( y ) = 1 η n η n +1 p D n D n +2 det        I 00 . . . I 0 n +1 . . . . . . I n − 1 0 . . . I n − 1 n +1 β 0 . . . β n +1 1 . . . y n +1        (2-7) b p n ( x ) = 1 D n +1 det        I 00 . . . I 0 n 1 . . . . . . I n − 1 0 . . . I n − 1 n x n − 1 I n 0 . . . I n n x n β 0 . . . β n 0        (2-8) After lengthy manipulations one obtains sev eral Christoffel–Darboux–like identities (CDIs) for Cauch y BOPs which pla y a crucial rˆ ole in what follo ws and hence will b e carefully describ ed. 2.1 Christoffel–Darb oux Iden tities Using recurrence co efficien ts featured in eqs. ( 2-5 ) we define the follo wing 3 × 3 matrices A n ( x ) =    0 0 b (2) n +1 − b ( − 1) n − 1 − b (0) n + x η n +1 0 0 − b ( − 1) n 0    , B n ( y ) =    0 0 a (2) n +1 − a ( − 1) n − 1 − a (0) n + y π n +1 0 0 − a ( − 1) n 0    (2-9) 6 In addition we define the follo wing integral transforms q (1) n ( w ) := Z R + q n ( ζ ) β d ζ w − ζ ; q (2) n ( w ) := Z R + q (1) n ( − x ) w + x α ( x )d x (2-10) p (1) n ( z ) := Z R + p n ( ξ ) α d ξ z − ξ ; p (2) n ( z ) := Z R + p (1) n ( − y ) z + y β ( y )d y , (2-11) and the following v ectors ~ q ( µ ) n ( x ) =    q ( µ ) n − 2 ( x ) q ( µ ) n − 1 ( x ) q ( µ ) n ( x )    , ~ p ( µ ) n ( x ) =    p ( µ ) n − 2 ( x ) p ( µ ) n − 1 ( x ) p ( µ ) n ( x )    (2-12) where µ = 0 , 1 , 2 and q (0) n ≡ q n , p (0) n ≡ p n . F or α ( x ) , β ( y ) w e define α ? ( x ) = α ( − x ) and β ? ( y ) = β ( − y ); next we define the W eyl functions (or Marko v-functions) as W β ( z ) = Z 1 z − y β ( y )d y = − W β ? ( − z ) , W α ∗ ( z ) = Z 1 z + x α ( x )d x = − W α ( − z ) , W α ∗ β ( z ) = − Z Z 1 ( z + x )( x + y ) α ( x ) β ( y )d x d y W β α ∗ ( z ) = Z Z 1 ( z − y )( y + x ) α ( x ) β ( y )d x d y (2-13) Prop osition 2.3 (Thm 7.3 in [ 8 ]) The fol lowing Christoffel–Darb oux-like identities hold ( z + w ) n − 1 X j =0 q ( µ ) j ( w ) p ( ν ) j ( z ) = ~ q ( µ ) n ( w ) · A ( − w ) · c p n ( ν ) ( z ) − F ( w , z ) µ,ν (2-14) F ( w , z ) =   0 0 1 0 1 W β ∗ ( z ) + W β ( w ) 1 W α ( z ) + W α ∗ ( w ) W α ∗ ( w ) W β ∗ ( z ) + W α ∗ β ( w ) + W β ∗ α ( z )   (2-15) wher e the auxiliary ve ctors marke d with a hat ar e char acterize d by a Riemann–Hilb ert pr oblem describ e d b elow. Corollary 2.1 (Thm 7.4 in [ 8 ]) Evaluating ( 2-14 ) on the “antidiagonal” z = − w gives the p erfect dualit y ~ q ( µ ) n ( w ) · A ( − w ) · c p n ( ν ) ( − w ) = J µν J :=   0 0 1 0 1 0 1 0 0   (2-16) Remark 2.1 The pr op osition ab ove defines, in fact, 9 identities, but we wil l only ne e d the 4 identies c orr esp onding to µ, ν = 0 , 1 (i.e. the princip al submatrix of size 2 × 2 ). 7 2.2 Riemann–Hilb ert c haracterization of the in tegrable kernels The sums app earing on the left hand side in Prop. 2.3 are all examples a general framew ork of “in tegrable k ernels” that w ere studied in great generality in [ 30 , 27 , 29 , 26 , 24 ]. Prop osition 2.4 ( Prop. 8.1 in [ 8 ]) Consider the R iemann–Hilb ert pr oblem (RHP) of finding a ma- trix Γ( w ) such that 1. Γ( w ) is analytic on C \ ( supp ( β ) ∪ supp ( α ? )) 2. Γ( w ) satisfies the jump c onditions Γ( w ) + = Γ( w ) −   1 − 2 π iβ 0 0 1 0 0 0 1   , w ∈ supp ( β ) ⊂ R + Γ( w ) + = Γ( w ) −   1 0 0 0 1 − 2 π iα ∗ 0 0 1   , w ∈ supp ( α ∗ ) ⊂ R − (2-17) 3. its asymptotic b ehavior at w = ∞ = ( w ) 6 = 0 is Γ( w ) = ( 1 + O ( w − 1 ))   w n 0 0 w − 1 0 0 0 w − n +1   (2-18) Then such a Γ( w ) exists and is unique. Mor e over Γ( w ) c an e quivalently b e written as: Γ( w ) =    c n η n 0 0 0 1 η n − 1 0 0 0 ( − 1) n − 1 η n − 2 c n − 2       b q (0) n − 1 b q (1) n − 1 b q (2) n − 1 q (0) n − 1 q (1) n − 1 q (2) n − 1 b q (0) n − 2 b q (1) n − 2 b q (2) n − 2    . (2-19) or also Γ( w ) :=   1 − c n η n 0 0 1 0 0 ( − 1) n − 1 η n − 2 c n − 2 1      0 0 c n 0 1 η n − 1 0 ( − 1) n c n − 2 0 0    := Y ( w ) z }| { [ ~ q (0) n ( w ) , ~ q (1) n ( w ) , ~ q (2) n ( w )] (2-20) wher e the normalization c onstants η n , c n have b e en intr o duc e d in ( 2-4 , 2-6 ). Remark 2.2 The quantities r eferring to the letter q on the right hand side in Pr op. 2.3 c an b e extr acte d fr om the RHP involving Γ . The next pr op osition wil l achieve the same go al for the r emaining quantities. Prop osition 2.5 (Prop. 8.2 in [ 8 ]) Consider the Riemann–Hilb ert pr oblem (RHP) of finding a matrix b Γ( z ) such that 1. b Γ( z ) is analytic on C \ ( supp ( α ) ∪ supp ( β ? )) 8 2. b Γ( z ) satisfies the jump c onditions b Γ( z ) + = b Γ( z ) −   1 − 2 π iα ( z ) 0 0 1 0 0 0 1   , z ∈ supp ( α ) ⊆ R + b Γ( z ) + = b Γ( z ) −   1 0 0 0 1 − 2 π iβ ∗ 0 0 1   , z ∈ supp ( β ∗ ) ⊆ R − , (2-21) 3. its asymptotic b ehavior at z = ∞ = ( z ) 6 = 0 is b Γ( z ) =  1 + O  1 z    z n 0 0 1 0 0 0 1 z n   . (2-22) Then such a b Γ( z ) exists and is unique. Mor e over b Γ( z ) c an e quivalently b e written as: b Γ( z ) =   c n 0 0 0 − 1 0 0 0 ( − 1) n c n − 1     p 0 ,n p 1 ,n p 2 ,n b p 0 ,n − 1 b p 1 ,n − 1 b p 2 ,n − 1 p 0 ,n − 1 p 1 ,n − 1 p 2 ,n − 1   . (2-23) or also b Γ( z ) =    0 0 − c n η n 0 − 1 0 ( − 1) n c n − 1 η n − 1 0 0      1 − 1 0 0 1 0 0 − 1 1   := b Y ( z ) z }| {  ~ b p (0) n ( z ) , ~ b p (1) n ( z ) , ~ b p (2) n ( z )  . (2-24) wher e the normalization c onstants η n , c n have b e en intr o duc e d in ( 2-4 , 2-6 ). Remark 2.3 These p olynomials b p n ( z ) := b p (0) n ( z ) and the auxiliary functions b p (1) n ( z ) , b p (2) n ( z ) wer e intr o- duc e d in [ 8 ] indep endently fr om a RHP formulation, but for al l pr actic al purp oses the formulation ab ove is sufficiently explicit. Finally the RHPs ab ov e allo w us to reconstruct the ratio of tw o consecutive principal minors of the bimomen t matrix (this will b ecome relev ant when discussing the partition function of the matrix mo del) Prop osition 2.6 (Corollary 8.1 in [ 8 ]) If D n is a le ading princip al n × n minor of the bimoment matrix I , then D n D n − 1 = ( − 1) n lim w →∞ w 2 n − 1 Γ 2 , 3 ( w ) Γ 2 , 1 ( w ) = c 2 n − 1 > 0 (2-25) The CDIs in Prop. 2.3 can be written in a very simple form in terms of the solutions of the Riemann– Hilb ert problems. T o sho w this let us momentarily denote by Y ( w ) and b Y ( z ) the matrices with columns giv en by the ~ q ( µ ) n ( w ) and ~ p ( ν ) n ( z ) resp ectiv ely . Then ( 2.3 ) can b e reform ulated as ( z + w ) H n ( z , w ) := ( z + w )   n − 1 X j =0 q ( µ ) j ( w ) p ( ν ) j ( z )   µ,ν + F ( w , z ) = Y t ( w ) A ( − w ) · b Y ( z ) (2-26) 9 No w the p erfect dualit y of Cor. 2.1 implies that Y ( w ) A ( − w ) = J b Y ( − w ) − 1 . (2-27) Note that the solutions of the RHPs Γ( w ) , b Γ( z ) differ from Y ( w ) , b Y ( z ) only b y some constant in vertible left multipliers, and hence Y ( w ) · A ( − w ) · b Y ( z ) = J · b Y ( − w ) − 1 · b Y ( z ) = J · b Γ( − w ) − 1 · b Γ( z ) . (2-28) W e collect this into the follo wing prop osition for later reference Prop osition 2.7 The matrix k ernel H n ( z , w ) [ H n ] µν ( z , w ) :=   n − 1 X j =0 q ( µ ) j ( w ) p ( ν ) j ( z ) + F µν ( w , z ) z + w   (2-29) is given in terms of the solution of the Riemann–Hilb ert pr oblem ( 2-24 ) - ( 2-22 ) as H n ( z , w ) := J · b Γ( − w ) − 1 · b Γ( z ) z + w (2-30) The relev ance of the matrix k ernel H n ( z , w ) will b ecome clear when we will discuss the sp ectral statistics of the matrix mo del. 3 Matrix Mo dels Consider the vector space H N of Hermitean matrices of size N endow ed with the U ( N )–inv ariant Lebesgue measure d M := Y i 0 } (3-2) with the induced measure. Let α, β b e tw o p ositive densities of finite mass supp orted on the p ositiv e real axis: α ( M ) will simply mean the pro duct measure on the eigen v alues of M . Define now the measure on H + N × H + N = { ( M 1 , M 2 ) } d µ ( M 1 , M 2 ) := d M 1 d M 2 α ( M 1 ) β ( M 2 ) det( M 1 + M 2 ) N (3-3) Definition 3.1 F or the p ositive finite–mass me asur e ( 3-3 ) we define the partition function as the inte gr al Z N := Z H + N ×H + N d µ ( M 1 , M 2 ) (3-4) 10 The resulting random matrix mo del falls into the general class of two–matrix–mo dels although the “coupling” term is not the most common one (whic h is e T r ( M 1 M 2 ) ). Ho wev er these and m uch more general couplings hav e b een considered relativ ely recently in [ 25 ]. As customary , we re-express the Leb esgue measures d M i in terms of the normalize d Haar measure d U i of the unitary group U ( N ) and the Leb esgue measure on the cone R N + d M 1 = G N d U Y d x j ∆ 2 ( X ) , d M 2 = G N d V Y d y j ∆ 2 ( Y ) , (3-5) where ∆( X ) := det[ x j − 1 i ] is the V andermonde determinant asso ciated with the n -tuple X of ordered eigen v alues x 1 ≤ . . . ≤ x n of M 1 and ∆( Y ) is defined in the same wa y relative to the matrix M 2 . The constan t G N is not of muc h relev ance. It dep ends only on N but not on the densities. The resulting measure inv olves t wo copies of the unitary group and one of the in tegrals can b e p erformed leading (up to a multiplicativ e constan t dep ending on the normalizations of the Haar measures and the size N ) to the measure b elo w on the sp ectra of M 1 , M 2 d µ ( X, Y ) = G 2 N ∆ 2 ( X )∆ 2 ( Y ) Z U ( N ) d U det( X + U Y U † ) N ! α ( X )d X β ( Y )d Y (3-6) α ( X ) := N Y j =1 α ( x j ) , d X := N Y j =1 d x j β ( Y ) := N Y j =1 β ( y j ) , d Y := N Y j =1 d y j . (3-7) A t this p oin t w e need to compute the in tegral o ver U ( N ); w e can use the result on pages 23-24 of [ 25 ] for the sp ecial example (A-28) Z U ( N ) det( 1 − z AU B U † ) − r d U = C N ,r det  (1 − z a i b j ) N − r − 1  1 ≤ i,j ≤ N ∆( A )∆( B ) (3-8) C N ,r := Q N − 1 k =1 k ! z N ( N − 1) / 2 Q N − 1 k =1 ( r − N + 1) k (3-9) where ( r − N + 1) k := k Y j =1 ( r − N + j ) (3-10) is the Pochammer sym b ol. Setting z = − 1 , A = X − 1 , B = Y in ( 3-8 ) we obtain Z U ( N ) d U det( 1 + X − 1 U Y U † ) r = C N ,r det h (1 + y j x i ) N − r − 1 i 1 ≤ i,j ≤ N ∆( X − 1 )∆( Y ) (3-11) Multiplying b oth sides b y det( X ) − r w e then obtain Z U ( N ) d U det( X + U Y U † ) r = ( − 1) N ( N − 1) / 2 C N ,r det  ( x i + y j ) N − r − 1  1 ≤ i,j ≤ N ∆( X )∆( Y ) (3-12) 11 The case of main relev ance to us is r = N , which yields ( C N ,N = ( − 1) N ( N − 1) / 2 ) Z U ( N ) d U det( X + U Y U † ) N = det h 1 x i + y j i ∆( X )∆( Y ) (3-13) This shows that the measure d µ ( M 1 , M 2 ) can b e reduced to a measure on the sp ectra of the tw o matrices (w e use the same symbol for the measure on the eigenv alues) d µ ( X, Y ) = G 2 N ∆ 2 ( X )∆ 2 ( Y ) det[ K ( x i , y j )] ∆( X )∆( Y ) α ( X )d X β ( Y )d Y K ( x, y ) = 1 x + y . (3-14) In fact we could hav e used the general formula ( 3-8 ) for an y r := N + h ; note, how ever, that for h in teger and less than − 1 Harnad–Orlov’s form ula in the form presented abov e cannot b e used, since the determinan t in the n umerator v anishes (for N ≥ − h ) and so do some denominators in the definition of the constants C N ,r . In other words, one should take an appropriate limit and use de l’Hˆ opital’s rule. F or an y h ≥ 0 or any h / ∈ − N , tracing the steps ab ov e one could obtain general mo dels where the reduced measures on the sp ectra hav e a form d µ h ( X, Y ) = G 2 N C N ,r ∆ 2 ( X )∆ 2 ( Y ) det( K h ( x i , y j )) ∆( X )∆( Y ) α ( X )d X β ( Y )d Y K h ( x, y ) = 1 ( x + y ) 1+ h , (3-15) corresp onding to the unreduced measure d µ h ( M 1 , M 2 ) = d M 1 d M 2 α ( M 1 ) β ( M 2 ) det( M 1 + M 2 ) N + h . (3-16) It will b e imp ortan t for us in what follows that for any v alue of h 6∈ − N the kernel K h is totally p ositiv e or at least sign r e gular on R + × R + . (Sign-regularity means that determinan ts in Def. 2.1 are nonzero and their sign dep ends only on their size.) Lemma 3.1 Consider the kernel K h = 1 ( x + y ) 1+ h , r estricte d to ( x, y ) ∈ R + × R + . Then, 1. for 1 + h > 0 , K h is total ly p ositive, 2. for 1 + h ∈ R − \ − N , K h is sign-r e gular. Pro of. T o prov e this assertion define for 0 < y 0 < · · · < y n functions k i ( x ) = 1 ( x + y i ) s , i = 0 , . . . , n, s = 1 + h . Denote for j = 1 , 2 , ... , c j ( s ) = s ( s + 1) . . . ( s + j − 1) and c 0 ( s ) = 1. Then the W ronskian W ( k 0 , . . . , k n )( x ) of k 0 , . . . , k n is equal to det  ( − 1) j c j ( s ) 1 ( x + y i ) s + j  n i,j =0 = ( − 1) n ( n − 1) 2 n Y j =0 c j ( s ) ( x + y i ) s det  1 ( x + y i ) j  n i,j =0 (3-17) 12 Th us, W ( k 0 , . . . , k n )( x ) is a nonzero multiple of ( − 1) n ( n − 1) 2 ∆( z 0 , . . . , z n ), where z i = 1 ( x + y i ) and ∆( z 0 , . . . , z n ) is the V andermonde determinant constructed out of the z i ’s. Since z 0 > . . . > z n , we see that ( − 1) n ( n − 1) 2 ∆( z 0 , . . . , z n ) = ∆( z n , . . . , z 0 ) > 0 (3-18) Th us W ( k 0 , . . . , k n )( x ) is p ositiv e for an y n if s > 0 since all the c j ( s ) are p ositiv e num b ers; if s < 0 then - denoting by [ s ] the greatest integer less than s (hence negative) - w e see that the sign of W is sig n ( W ) = ( − 1) − [ s ] n − [ s ](1 − [ s ]) 2 (3-19) for any n ≥ 0 and x ≥ 0. T ogether these observ ations imply (Thm 2.3, ch. 2 of [ 31 ]), that K h is totally p ositiv e for s > 0 on R + × R + or at least sign regular for s ∈ R − \ − N . Q.E.D. In App. A we will show that for h integer or half–integer, the mo del ( 3-16 ) is the reduction of a 3–matrix mo del, where a “ghost” gaussian matrix A has b een integrated out. Dep ending on the range h < N or h > N the matrix A consists of ordinary v ariables (b osons) or Grassmann v ariables. Remark 3.1 The fact that the me asur e d µ h dep ends on N (in the determinant in the denominator) is a fe atur e of the mo del r ather than a pr oblem; inde e d, in studying the lar ge N limit it is natur al to make the str ength of the inter action incr e ase at the same r ate as the size of the matric es. F or example, in the “standar d” two–matrix mo del one c onsiders the inter action e cN T r ( M 1 M 2 ) 3.1 Correlation functions: Christoffel–Darb oux kernels In this section we will compute the correlation functions of the mo del. More precisely , Definition 3.2 The c orr elation functions of the mo del ar e define d as R ( r,k ) ( x 1 , . . . x r ; y 1 , . . . , y k ) := N ! Q r j =1 α ( x j ) Q k j =1 β ( y j ) ( N − r )!( N − k )! Z N Z N Y ` = r +1 α ( x j )d x ` N Y j = k +1 β ( y j )d y j ∆ 2 ( X )∆ 2 ( Y ) det[ K ( x i , y j )] ∆( X )∆( Y ) (3-20) wher e Z N := 1 N ! R ∆( X )∆( Y ) det[ K ( x i , y j )] i,j ≤ N α ( X ) β ( Y )d X d Y . These functions allow one to compute the probabilit y of having r eigenv alues of the first matrix and k eigen v alues of the second matrix in measurable sets of the real axis. The computations of these correlation functions in term of biorthogonal functions reported below follo ws the general approach in [ 19 ] and [ 23 ]. Using the well–kno wn formula for the Cauc hy determinan t det  1 x i + y j  = ∆( X )∆( Y ) Q i,j ( x i + y j ) (3-21) 13 w e obtain the measure ∆( X ) 2 ∆( Y ) 2 Q i,j ( x i + y j ) α ( X ) β ( Y )d X d Y (3-22) W e will b e using the correlation functions only to compute exp ectations of sp ectral functions, namely functions of X , Y which are separately symmetric in the p erm utations of the x j ’s or y j ’s. Lemma 3.2 Supp ose F ( X ) is a symmetric function under the action of the symmetric gr oup S N . Then 1 N ! Z F ( X )∆( X ) det[ K ( x j , y j )] α ( X )d X = Z F ( X )∆( X ) N Y j =1 1 x j + y j α ( X )d X Pro of. Under the assumption F ( X ) = F ( X σ ) for any σ ∈ S N w e hav e Z F ( X )∆( X ) N Y j =1 1 x j + y j α ( X )d X = = 1 N ! X σ ∈ S N Z F ( X )∆( X σ ) N Y j =1 1 x σ ( j ) + y j α ( X )d X = = 1 N ! Z F ( X )∆( X )   X σ ∈ S N  ( σ ) N Y j =1 K ( x σ ( j ) , y j )   α ( X )d X = = 1 N ! Z F ( X )∆( X ) det[ K ( x j , y j )] α ( X )d X . Q.E.D. Corollary 3.1 L et F ( X, Y ) b e symmetric with r esp e ct to either set of variables X or Y . Then 1 N ! Z F ( X , Y )∆( X )∆( Y ) det[ K ( x j , y j )] α ( X ) β ( Y )d X d Y = Z F ( X , Y )∆( X )∆( Y ) N Y j =1 1 x j + y j α ( X ) β ( Y )d X d Y (3-23) Since w e are in terested in the unor der e d spectrum of the matrices M 1 , M 2 , in view of the ab o ve Cor. 3.1 w e will fo cus henceforth on the following unnormalized measure d e ν ( X , Y ) := ∆( X )∆( Y ) N Y j =1 α ( x j ) β ( y j ) x j + y j d X d Y = ∆( X )∆( Y ) α ( X ) β ( Y ) det( X + Y ) d X d Y (3-24) F rom the prop erties of the V andermonde determinan t we can write ∆( X )∆( Y ) = det [ e p i ( x j )] det [ e q i ( y j )] (3-25) where e p i , e q i are any monic polynomials of exact degree i in the resp ectiv e v ariables ( i = 0 , . . . , N − 1). 14 It is therefore natural [ 19 ] to choose the sets of monic p olynomials { e p j ( x ) , e q j ( y ) } j ∈ N to be biorthog- onal with resp ect to the bi-measure α ( x ) β ( y ) x + y d x d y . This means that Z R + Z R + e p k ( x ) e q ` ( y ) α ( x ) β ( y ) x + y d x d y = c k 2 δ k` . (3-26) The constan t c k w as defined in ( 2-4 , with K ( x, y ) = 1 / ( x + y )). The normalization constant for d e ν is th us Z N = Z Z d e ν ( X , Y ) = N ! N − 1 Y k =0 c 2 k ! = N ! det  I j k  0 ≤ j,k ≤ N − 1 (3-27) If we in tro duce the orthonormal biorthogonal p olynomials p n := 1 c n e p n , q n := 1 c n e q n (3-28) then we can write the normalize d measure as ν ( X , Y ) = 1 N ! det [ p i ( x j )] det [ q i ( y j )] N Y j =1 α ( x j ) β ( y j ) x j + y j d N X d N Y . (3-29) Notice now that the pro duct of determinants in ( 3-29 ) is a determinant of the pro duct of the tw o matrices indicated; hence det [ p i − 1 ( x j )] i,j ≤ N det [ q j − 1 ( y i )] i,j ≤ N = det K N ( x i , y j ) (3-30) K N ( x, y ) := N − 1 X j =0 p j ( x ) q j ( y ) (3-31) The kernel K N ( x, y ) is a “repro ducing” k ernel K N ( x, y ) = Z Z K N ( x, z ) K N ( w , y ) α ( w ) β ( z )d w d z z + w (3-32) whic h follows immediately from the biorthogonality . In addition we ha ve Z Z K N ( x, y ) α ( x ) β ( y )d x d y x + y = N . (3-33) Summarizing, in the new notation w e hav e Prop osition 3.1 The pr ob ability me asur e on X × Y induc e d by ( 3-3 ) is given by: 1 ( N !) 2 det[ K N ( x i , y j )] det[ K ( x i , y j )] α ( X ) β ( Y )d X d Y (3-34) while the c orr elation functions ar e: R ( r,s ) ( x 1 , . . . x r ; y 1 , . . . , y s ) = (3-35) Q r j =1 α ( x j ) Q s j =1 β ( y j ) ( N − r )!( N − s )! Z det[ K N ( x i , y j )] det[ K ( x i , y j )] N Y ` = r +1 α ( x j )d x j N Y j = s +1 β ( y j )d y j (3-36) 15 Example 3.1 Consider r = 1 , s = 0 . In this c ase, with the help of L emma 3.2 , we obtain: R (1 , 0) = 1 ( N − 1)! N ! α ( x 1 ) Z det[ K N ( x i , y j )] det[ K ( x i , y j )] N Y ` =2 α ( x ` )d x ` N Y j =1 β ( y j )d y j = 1 ( N − 1)! α ( x 1 ) Z det[ K N ( x i , y j )] N Y i =1 1 x i + y i N Y ` =2 α ( x ` )d x ` N Y j =1 β ( y j )d y j = 1 ( N − 1)! α ( x 1 ) X σ,σ 0 ∈ S N  ( σ )  ( σ 0 ) Z p σ (1) − 1 ( x 1 ) . . . p σ ( N ) − 1 ( x N ) q σ 0 (1) − 1 ( y 1 ) . . . q σ 0 ( N ) − 1 ( y N ) N Y i =1 1 x i + y i N Y ` =2 α ( x ` )d x ` N Y j =1 β ( y j )d y j = α ( x 1 ) Z K N ( x 1 , y 1 ) x 1 + y 1 β ( y 1 )d y 1 (3-37) 3.1.1 Correlation functions in terms of biorthogonal p olynomials In the pap er by Eynard and Mehta [ 19 ] (generalized in [ 23 ]) they never used any sp ecific information ab out the model they w ere considering (with the Itzykson–Zub er in teraction) but only the fact that the matrices were coupled in a c hain. W e recall the relev ant result here 4 . Define H 00 ( x, y ) := K N ( x, y ) H 01 ( x, x 0 ) := Z H 00 ( x, y ) β ( y )d y x 0 + y , H 10 ( y , y 0 ) := Z H 00 ( x, y 0 ) α ( x )d x x + y H 11 ( y , x ) := Z Z H 00 ( z , w ) α ( z )d z β ( w )d w ( z + y )( x + w ) − 1 x + y . (3-38) Since H 00 is a repro ducing k ernel Z H 00 ( x, y ) H 11 ( y , x 0 ) β ( y )d y = H 01 ( x, x 0 ) − H 01 ( x, x 0 ) = 0 (3-39) and similarly Z H 11 ( x, z ) H 00 ( z , x 0 ) α ( z )d z = H 10 ( x, x 0 ) − H 10 ( x, x 0 ) ≡ 0 . (3-40) In tegrating these tw o equations against ( x + y 0 ) − 1 α ( x )d x (the first) or ( x 0 + y 0 ) − 1 β ( y 0 )d y 0 (the second) w e find also Z H 11 ( y , x ) H 10 ( y , y 0 ) β ( y )d y = Z Z H 11 ( x, y ) H 01 ( y , y 0 ) α ( y )d y ≡ 0 (3-41) The correlation functions for r eigen v alues x 1 , . . . , x r of M 1 and s eigenv alues y 1 , . . . , y s of M 2 w ere computed in [ 19 ] and are giv en by R ( r,s ) ( x 1 , . . . , x r ; y 1 , . . . , y s ) = r Y j =1 α ( x j ) s Y k =1 β ( y k ) × 4 It could be extended to the chain of matrices (see below) but it becomes a bit cumbersome to describ e. 16 × det     H 01 ( x i , x j )  1 ≤ i,j ≤ r  H 00 ( x i , y j )  1 ≤ i ≤ r, 1 ≤ j ≤ s  H 11 ( y i , x j )  1 ≤ i ≤ s,j ≤ r  H 10 ( y i , y j )  1 ≤ i,j ≤ s    (3-42) where 0 ≤ r ≤ N , 0 ≤ s ≤ N , 1 ≤ r + s , with the understanding that if either r or s is 0 then the corresponding blo cks lab eled b y r, s resp ectiv ely , are absent. Th us, for example R (1 , 0) ( x 1 ) = α ( x 1 ) H 0 , 1 ( x 1 , x 1 ). This is a nontrivial result and perhaps one can get a bit of an insight b y considering the sp ecial case r = s = N . W e kno w that in this case R ( N ,N ) ( x 1 , . . . x N ; y 1 , . . . , y N ) = (3-43) det[ K N ( x i , y j )] det[ K ( x i , y j )] α ( X ) β ( Y ) (3-44) so we ha v e the follo wing not at all obvious iden tity: Lemma 3.3 det[ K N ( x i , y j )] det[ K ( x i , y j )] = det     H 01 ( x i , x j )  i,j ≤ N  H 00 ( x i , y j )  i,j ≤ N  H 11 ( y i , x j )  i,j ≤ N  H 10 ( y i , y j )  i,j ≤ N    (3-45) Pro of It suffices to observe that the 2 N × 2 N matrix on the right hand side has the following blo ck structure: K = " − P 0 ( x ) T Q 1 ( − x ) P 0 ( x ) T Q 0 ( y ) P 1 ( − y ) T Q 1 ( − x ) − K ( x , y ) − P 1 ( − y ) Q 0 ( y ) # (3-46) where P 0 ( x ) = [ p i − 1 ( x j )] 1 ≤ i,j ≤ N , P 1 ( y ) = [ p (1) i − 1 ( y j )] 1 ≤ i,j ≤ N and, after exchanging p i s with q i s, w e define the remaining sym b ols (see equations ( 2-11 ) and ( 2-10 ) for definitions) accordingly . Moreo ver, K ( x , y ) = [ K ( x i , y j ] 1 ≤ i,j ≤ N . With this notation in place it is clear that K admits the following (Bruhat) decomp osition: K = " P 0 ( x ) T 0 − P 1 ( − y ) T I #  0 I I 0   − K ( x , y ) 0 − Q 1 ( − x ) Q 0 ( y )  . (3-47) Th us det K = det P 0 ( x ) T det K ( x , y ) det Q 0 ( y ) = det[ K N ( x i , y j )] det[ K ( x i , y j )]. Q.E.D. A comparison of the definitions ( 3-38 ) with the en tries of the Christoffel–Darboux identities of Prop. 2.3 shows that they are in timately related, in fact H µ,ν ( x, y ) = ( − ) µ + ν N − 1 X j =0 q ( µ ) j  ( − ) µ x  p ( ν ) j  ( − ) ν y  − δ µ, 1 δ ν, 1 x + y (3-48) and we recognize that (up to some signs) these are precisely the entries of the H n k ernel of Prop. 2.7 : H µ,ν ( x, y ) = ( − ) µ + ν H n,µ,ν (( − ) µ x, ( − ) ν y ) (3-49) More explicitly we ha ve 17 Prop osition 3.2 The kernels of the c orr elation functions ar e given in terms of the solution of the RHP in Pr op. 2.5 as fol lows H 00 ( x, y ) = H N , 00 ( x, y ) = h b Γ − 1 ( − y ) b Γ( x ) i 3 , 1 x + y H 01 ( y 0 , y ) = − H N , 01 ( − y 0 , y ) = h b Γ − 1 ( − y ) b Γ( − y 0 ) i 3 , 2 y 0 − y H 10 ( x, x 0 ) = − H N , 10 ( x, − x 0 ) = h b Γ − 1 ( x 0 ) b Γ( x ) i 2 , 1 x 0 − x H 11 ( y , x ) = H N , 11 ( − y , − x ) = h b Γ − 1 ( x ) b Γ( − y ) i 2 , 2 x + y (3-50) The potential relev ance of these formulæ is that it allo ws one to compute the large N asymptotic b e- ha viour of the correlation functions in terms of the asymptotic b eha viour of only 3 consecutiv e biorthog- onal p olynomials and auxiliary functions asso ciated with them that enter in the Riemann–Hilb ert for- m ulation of Props. 2.4 , 2.5 . The steep est–descen t analysis of these problems which will appear in a forthcoming pap er. W e expect that, after a complete description of the asymptotics of the BOPs is obtained, these formulæ can b e used to address the issue of univ ersality for this matrix mo del via the Riemann–Hilb ert approach. Remark 3.2 Note also that the formula ab ove is not symmetric inasmuch as the BOPs p , q play differ ent r oles in the Christoffel l-Darb oux the or em; we c ould however r ewrite the the or em as N − 1 X j =0 p j ( x ) q j ( y ) = ~ p t 0 ,N ( x ) B N ( − x ) ~ ˇ q 0 ,N ( y ) x + y . (3-51) wher e B ( z ) was define d in ( 2-9 ). The auxiliary ve ctor ˇ q 0 ,N enters in a similar R iemann–Hilb ert formu- lation with the rˆ oles of the densities α and β inter change d. 3.2 A m ulti–matrix mo del It is p ossible to extend the mo del ( 3-3 ) to a chain of matrices with the nearest neighbor interaction 1 det( M i + M i +1 ) N + h i (3-52) The “strength” of the in teraction (i.e. h j ’s) may depend on a site along the chain. Sp ecifically , consider the space  H + N  × R and a collection of R positive densities α 1 , . . . , α R on R + . Define the finite mass measure d µ ( M 1 , . . . , M R ) = R − 1 Y ` =1 1 det( M ` + M ` +1 ) N + h ` R Y ` =1 α ` ( M ` )d M ` (3-53) 18 Using the Harnad–Orlov formula one we obtain, up to normalization constan t, the following measure whic h we denote by the same sym b ol d µ ( X 1 , . . . , X R ) = R Y ` =1 α ` ( X ` )∆( X ` ) 2 d X ` R − 1 Y ` =1 det  ( x `,i + x ` +1 ,j ) − 1 − h `  1 ≤ i,j ≤ N ∆( X ` )∆( X ` +1 ) (3-54) F ollowing the same steps that led to the expression ( 3-24 ) as p ositiv e density for ( S N ) 2 –in v ariant observ ables, we obtain the following reduced measure on the sp ectra (up to a normalization dep ending only on h ` ’s but not on the measures) d e ν ( X 1 , . . . , X R ) = ∆( X 1 )∆( X R ) R − 1 Y ` =1 det( X ` + X ` +1 ) − 1 − h ` R Y ` =1 α ` ( X ` )d X ` (3-55) In the case where all in teractions are the same h ` = 0 we ha v e d e ν ( X 1 , . . . , X R ) = ∆( X 1 )∆( X R ) R − 1 Y ` =1 1 det( X ` + X ` +1 ) R Y ` =1 α ` ( X ` )d X ` (3-56) whic h can b e seen as a generalization of ( 3-24 ). This model too can b e treated with the aid of biorthogonal p olynomials that will now satisfy a R + 3 recurrence relation; note that the length of the recurrence relations do es not dep end on the “p oten tials”, or densities, α j . This is in sharp contrast with the usual m ulti-matrix mo del with interaction e c T r( M j M j +1 ) [ 19 , 6 ]. T o c haracterize these biorthogonal p olynomials, an ( R + 2) × ( R + 2) Riemann–Hilb ert problem can b e set up and the strong asymptotics can be dealt with, but the complexity of this problem definitely warran ts a separate pap er. 4 Diagrammatic expansion In parallel with the 1 N 2 – expansion for the Hermitean matrix model and the IZHC tw o-matrix model, w e w ould lik e to sk etch the similar formal expansion of the mo del in terms of colored ribb on graphs. The w eights α and β en tering the definition ( 3-3 ) are assumed to b e of the form α ( M 1 ) = e − N T r( U ( M 1 )) , β ( M 2 ) = e − N T r( V ( M 1 )) . W e p erform a shift and a rescaling of the matrices so that det( M 1 + M 2 ) 7→ det( 1 − ζ M 1 − η M 2 ). Of course the v alues of ζ and η are suitably restricted to a neighborho od of the origin: how ever this restriction is irrelev ant since the manipulations b elo w are in the sense of formal p o wer series. The pro cedure amounts to a p erturbativ e T a ylor–expansion around a Gaussian integral. In other w ords we will b e considering a partition function in the form Z N := Z Z d M 1 d M 2 e − N 2 T r ( M 1 2 + M 2 2 ) + N P ∞ ` =1 T r( ζ M 1 + η M 2 ) ` − N T r U p ( M 1 ) − N T r V p ( M 2 ) . (4-1) where U p , V p are the p erturbations of the Gaussian (including a quadratic term as well) which, for con venience, w e parametrize as the following formal series U p ( x ) := − ∞ X j =1 u j − ζ j j x j , V p ( y ) := − ∞ X j =1 v j − η j j y j (4-2) 19 W e th us hav e Z N = * exp N T r   ∞ X j =1 u j − ζ j j M 1 j + ∞ X j =1 v j − η j j M 2 j + ∞ X ` =1 1 `  ( ζ M 1 + η M 2 ) `    + (4-3) where the av erage is taken w.r.t. the underlying ( unc ouple d! ) Gaussian measure 1 Z (0) N d M 1 d M 2 exp  − N 2 T r  M 2 1 + M 2 2   , Z (0) N := Z d M 1 d M 2 exp  − N 2 T r  M 2 1 + M 2 2   (4-4) Note that, due to the shifts in u j , v j , the ` = 1 term in the third sum ab o ve cancels exactly against the shifts in the first tw o sums; similarly , only the term ζ η 2 T r( M 1 M 2 ) remains in the quadratic part: Z N = D exp N T r  u 1 M 1 + u 2 2 M 1 2 + v 1 M 2 + v 2 2 M 2 2 + ζ η M 1 M 2 + . . . E (4-5) Using Wic k’s theorem for the ev aluation of Gaussian integrals and the frequently used com binatorial in terpretation (see [ 15 ] for an excellen t in tro duction) one sees that the partition function is the sum ov er all p ossible bi-colored ribb on graphs resp ecting rules whic h we no w sp ecify • There are tw o colors for v ertices/edges (say , red/blue); v ertices are distinguished in monochromatic and bichromatic. • Eac h mono c hromatic vertex of v alency j enters with a weigh t N u j /j (blue) or N v j /j (red). • Eac h edge (red or blue) enters with a w eight 1 N . • Eac h bichromatic v ertex of v alency ` enters with a weigh t ζ k η ` − k N ` where 1 ≤ k ≤ ` − 1 is the num b er of blue half-edges and ` − k the num b er of red ones and app ears in with a multiplicit y  ` k  corre- sp onding to all p ossible arrangemen ts of k blue legs amongst ` , up to cyclic reordering. In particular there is only one bic hromatic biv alent v ertex (up to automorphism) whic h en ters with weigh t ζ η / N . In general, since eac h such vertex corresp onds to a trace of the form T r( M 1 a 1 M 2 b 1 M 1 a 2 . . . ) and in view of the cyclicity of the trace, there are precisely ` equiv alent v ertices obtained b y cyclically p erm uting the matrices in the sum, which corresp onds diagrammatically to a rotation of the colors of the legs of the v ertex. Hence there are in fact 1 `  ` k  = ( ` − 1)! ( ` − k ) ! k ! inequiv alent bicolored ` –v alent v ertices in eac h diagram con tributing with a weigh t ζ k η ` − k / N . • Eac h connected F eynman diagram Γ constributing to the p erturbative sum has a p o wer N F − E + V = N 2 − 2 g where g = g (Γ) is the gen us of the surface ov er which the graph can b e dra wn. 20 Figure 1: An example of a c on- ne cte d diagr am c ontributing to the p artition function. Summing up o ver all p ossible lab eling of the bicolored fat-graphs Γ leav es a factor | Aut (Γ) | in the partition function (see [ 15 ]) and the result is hence ln Z N = X Connected bic hromatic fatgraphs Γ N 2 − 2 g Γ | Aut (Γ) | ∞ Y j =1 u n j j v m j j ∞ Y ` =2 ` − 1 Y k =1 ζ ( ` − k ) r ` k η kr ` k (4-6) where n j = n j (Γ) is the num b er of blue vertices with v alency j , m j = m j (Γ) the num b er of j –v alent red v ertices and r ` k = r ` k (Γ) is the n umber of ` –v alent bichromatic v ertices with k red (and hence ` − k blue) legs. 5 Large N b eha viour Consider tw o densities α ( x ) , β ( y ) of the form α ( x ) = α ~ ( x ) = e − 1 ~ V ( x ) , β ( y ) = β ~ ( y ) = e − 1 ~ U ( y ) , ~ = T N , T > 0 . (5-1) Here we ha ve in tro duced a dep endence on the small parameter ~ on the measures (but we will not emphasize this dep endence in the notation). F rom the exp erience amassed in the literature on the ordinary orthogonal polynomials w e start by considering the heuristic “saddle–p oint” for the partition function Z N , namely the total mass of the reduced measure ( 3-24 ). The heuristics calls for a scaling approac h where we send the size of the matrices N to infinity and the scaling parameter ~ to zero as O (1 / N ), namely that w e v ary the densities α , β by raising them to the pow er 1 ~ . W e thus hav e Z N = Z ∆( X )∆( Y ) det  1 x i + y j  i,j ≤ N α ( X ) β ( Y )d X d Y . (5-2) Using 3-22 we ha ve Z N ∝ Z ∆( X ) 2 ∆( Y ) 2 Q i,j ( x i + y j ) exp " − 1 ~ N X i =1 U ( x i ) + V ( y j ) # d X d Y (5-3) T aking 1 N 2 times the logarithm of the in tegrand we ha ve the expression S ( X , Y ) := 1 T N N X j =1 U ( x j ) + V ( y j ) − 1 N 2 X j 6 = k ln | x j − x k | − 1 N 2 X j 6 = k ln | y j − y k | + 1 N 2 X j,k ln | x j + y k | = = S U ( X ) + S V ( Y ) + J ( X, Y ) . (5-4) It is con v enient –in order to deal with a more standard potential-theoretic problem– to map Y → − Y and define V ? ( y ) = V ( − y ) so that we can rewrite the action as S ( X , Y ) = S U ( X ) + S V ? ( Y ) + 1 N 2 X j,k ln | x j − y k | (5-5) 21 whic h describ es the energy of a gas of 2 N particles of charge +1 (for the x j ’s) and − 1 (for the y j ’s) separated b y an imp enetrable, electrically neutral, partition confining the p ositively charged particles to the p ositiv e real axis under the p oten tial U and the negatively charged particles to the negative axis under the p oten tial V ? . The usual argumen t is that the configuration of the minimum contribute the most to the integral and –under suitable assumptions of regularit y for the p oten tial– one wan ts to sho w that the sequence of these minima configurations tends in-measure to some probability distributions. W e define ρ = 1 N N X k =1 δ x k , µ = 1 N N X j =1 δ y j (5-6) and rewrite the action as S [ ρ, µ ] := 1 T Z R + U ( x ) ρ ( x )d x − Z R + Z R + ρ ( x ) ρ ( x 0 ) ln | x − x 0 | d x d x 0 + + 1 T Z R − V ? ( y ) µ ( y )d y − Z R − Z R − µ ( y ) µ ( y 0 ) ln | y − y 0 | d y d y 0 + Z R + Z R − ρ ( x ) µ ( y ) ln | x − y | d x d y (5-7) 5.1 Con tin uum v ersion: cubic sp ectral curv e and solution of the p oten tial problem W e immediately rephrase the ab ov e minimization problem in a contin uum version. In order to emphasize the symmetry of the problem it is con venien t to denote U ( x ) b y V 1 ( x ) and V ? ( y ) = V ( − y ) by V 2 ( y ) and denote the corresp onding equilibrium densities by ρ 1 ( x ) , ρ 2 ( y ). The main point of this section is Thm. 5.1 whic h states that the resolven ts (Mark o v functions) of the equilibrium distributions are solutions of a cubic equation that defines a trigonal curv e 5 ; this result is the analogue of the better-known result for the equilibrium measure app earing in the one–matrix model ([ 12 ] and references therein). The deriv ation that we present here is of a formal heuristic nature inasmuc h as w e discoun t sev eral imp ortant issues about the regularity of the equilibrium measures. How ever this sort of manipulations is quite common and they can b e obtained also from the lo op e quations as done in [ 18 ] for the O ( n ) mo del. Rewriting the functional in the new notation for the p otentials w e find S [ ρ 1 , ρ 2 ] := 1 T Z R + V 1 ( x ) ρ 1 ( x )d x − Z R + Z R + ρ 1 ( x ) ρ 1 ( x 0 ) ln | x − x 0 | d x d x 0 + + 1 T Z R − V 2 ( y ) ρ 2 ( y )d y − Z R − Z R − ρ 2 ( y ) ρ 2 ( y 0 ) ln | y − y 0 | d y d y 0 + 5 Namely a curve of the general form w 3 + Aw 2 + B w + C = 0, with A, B , C smooth functions of the sp ectral parameter z . 22 + Z R + Z R − ρ 1 ( x ) ρ 2 ( y ) ln | x − y | d x d y (5-8) W e will b e studying the measures ρ 1 , ρ 2 that minimize the ab o ve functional; here w e will assume their regularit y (which shall b e prov ed in a separate pap er). Also, the p oten tials V 1 , V 2 will b e assumed real analytic . More precisely , w e make the following assumption on the nature of the p oten tials Assumption 5.1 The two p otentials V 1 ( x ) , V 2 ( y ) ar e r estrictions to the p ositive/ne gative axis of two r e al–analytic functions such that • lim x → + ∞ V 1 ( x ) ln x = + ∞ , and similarly lim y →−∞ V 2 ( y ) ln | y | = + ∞ ; • the derivatives V 0 j ( x ) have at most finitely many p oles in a strip of finite width ar ound R ; • final ly, V 1 ( x ) > C 1 | ln( x ) | for some c onstant C 1 > 0 and x > 0 and V 2 ( y ) > C 2 | ln | y | | for y < 0 and some c onstant C 2 > 0 . It will b e shown in a separate pap er that this assumption is sufficient to guaran tee that the equilibrium densities exist, ha ve compact support, are regular and their supports do not include the origin. In order to enforce the normalization of ρ 1 , ρ 2 –as customary– we will introduce in the action tw o suitable Lagrange m ultipliers b S := b S [ γ 1 , γ 2 , ρ 1 , ρ 2 ] := S [ ρ 1 , ρ 2 ] + γ +  Z ρ 1 d x − 1  + γ −  Z ρ 2 d x − 1  (5-9) The v anishing of the first v ariation of the functional b S yields the equations V 1 ( z ) T − 2 Z R + ρ 1 ln | z − x | d x + Z R − ρ 2 ln | z − x | d x = γ + , z ∈ S upp ( ρ 1 ) V 2 ( z ) T − 2 Z R − ρ 2 ln | z − x | d x + Z R + ρ 1 ln | z − x | d x = γ − , z ∈ S upp ( ρ 2 ) (5-10) Differen tiating w.r.t. z yields 1 T V 0 1 ( z ) − 2 P .V . Z ρ 1 1 z − x d x + Z ρ 2 1 z − x d x = 0 , z ∈ S upp ( ρ 1 ) 1 T V 0 2 ( z ) − 2 P .V . Z ρ 2 1 z − x d x + Z ρ 1 1 z − x d x = 0 , z ∈ S upp ( ρ 2 ) (5-11) where P .V . R indicates the Cauch y principal v alue. These equations are b est written in terms of the resolv ents (or W eyl functions ) W i ( z ) := Z ρ i ( x )d x z − x , z ∈ C \ S upp ( ρ i ) . (5-12) 23 Indeed, with the help of the Sokhotskyi–Plemelj formula, the equations ab o ve tak e a simpler form  W 1 , + ( z ) + W 1 , − ( z ) = 1 T V 0 1 ( z ) + W 2 ( z ) , z ∈ S upp ( ρ 1 ) W 2 , + ( z ) + W 2 , − ( z ) = 1 T V 0 2 ( z ) + W 1 ( z ) , z ∈ S upp ( ρ 2 ) (5-13) W i, + − W i, − = − 2 iπ ρ i (5-14) Note that in the RHS of the system of equations, the resolven t of the other measure is ev aluated at a regular p oin t due to the fact that the tw o supp orts are disjoint. W e define, partly motiv ated by [ 21 ], the shifted resolv ents Y 1 := − W 1 + 2 V 0 1 + V 0 2 3 T , Y 2 := W 2 − V 0 1 + 2 V 0 2 3 T . (5-15) Observ e that, in view of Assumption ( 5.1 ) , { Y 1 , Y 2 } ha ve the same analytic structure as { W 1 , W 2 } , while the equations describing the jumps simplify to:  Y 1 , + + Y 1 , − = − Y 2 ( z ) , z ∈ S upp ( ρ 1 ) Y 2 , + + Y 2 , − = − Y 1 ( z ) , z ∈ S upp ( ρ 2 ) (5-16) Y 1 , + − Y 1 , − = 2 iπ ρ 1 , Y 2 , + − Y 2 , − = − 2 iπ ρ 2 (5-17) Using these equations one obtains b y direct insp ection that R ( x ) := Y 1 2 + Y 2 2 + Y 1 Y 2 (5-18) has no jumps on either supp orts. Multiplying eq. ( 5-18 ) on b oth sides b y Y 1 − Y 2 one obtains R ( z )( Y 1 − Y 2 ) = Y 1 3 − Y 2 3 (5-19) whic h can b e rewritten as Y 1 3 − R ( z ) Y 1 = Y 2 3 − R ( z ) Y 2 := D ( z ) . (5-20) The first expression may a priori hav e at most jumps across the supp ort of ρ 1 , while the second may ha ve jumps only across the supp ort of ρ 2 : since the t wo supp orts are disjoint w e conclude that D ( x ) is a regular function on the real axis. Remark 5.1 If we intr o duc e the function Y 0 := − Y 1 − Y 2 the jump r elations ( 5-16 , 5-17 ) c an b e r ewritten (after a few str aightforwar d manipulations) as Y 0 , ± ( z ) = Y 1 , ∓ ( z ) z ∈ S upp ( ρ 1 ) Y 0 , ± ( z ) = Y 2 , ∓ ( z ) z ∈ S upp ( ρ 2 ) . (5-21) This implies that we c an think of the Y j ( z ) ’s as the thr e e br anches (thr e e sheets ) of a Riemann surfac e r e alize d as a triple c over of the x –plane br anche d at the endp oints of the sp e ctr al b ands. The Riemann surfac e is depicte d in Fig. 2 . Thm. 5.1 wil l b e pr oving mor e formal ly this statement, r e alizing Y 0 , Y 1 , Y 2 as br anches of a cubic e quation (which the r e ader c an alr e ady guess fr om ( 5-20 )). 24 Y 1 Y 0 Y 2 Figure 2: A pictorial example of the three sheets of the Riemann surface of Y ( x ). In blue the supp ort of ρ 1 (righ t) and in green that of ρ 2 (left). Theorem 5.1 Under Assumption ( 5.1 ), the two shifte d r esolvents satisfy the same cubic e quation in the form E ( y , z ) := y 3 − R ( z ) y − D ( z ) = 0 . (5-22) The c o efficients R ( x ) , D ( x ) ar e r elate d to the e quilibrium me asur es as fol lows: R ( z ) = ( V 0 1 ) 2 + ( V 0 2 ) 2 + V 0 1 V 0 2 3 T 2 − R 1 ( z ) − R 2 ( z ) R i ( z ) := 1 T Z V 0 i ( z ) − V 0 i ( x ) z − x ρ i ( x )d x = 1 T ∇ i V 0 i (5-23) D ( x ) = U 0 U 1 U 2 + U 2 R 1 + U 1 R 2 + 1 T ∇ 12 V 0 1 − 1 T ∇ 21 V 0 2 (5-24) (5-25) wher e the functions U 0 , U 1 , U 2 and the op er ators ∇ 1 , ∇ 2 , ∇ 12 , ∇ 21 have b e en define d as U 1 := 2 V 0 1 + V 0 2 3 T , U 2 := − 2 V 0 2 + V 0 1 3 T , U 0 := V 0 2 − V 0 1 3 T (5-26) ∇ j f ( z ) := Z f ( z ) − f ( x ) z − x d ρ j ( x ) , ∇ 12 f ( z ) := Z Z f ( z ) − f ( x ) z − x d ρ 1 ( x )d ρ 2 ( y ) x − y , (5-27) ∇ 21 f ( z ) := Z Z f ( z ) − f ( x ) z − y d ρ 2 ( y )d ρ 1 ( x ) y − x (5-28) F urthermor e the supp ort of ρ 1 , ρ 2 must c onsist of finite union of finite disjoint interv als . 25 Pro of . The statement ab out the supp ort of the measures follows from the argument used in [ 12 ] that that we sk etc h here. By results of Chapter 13 of [ 36 ]), Assumption 5.1 implies that the supp orts of the equilibrium measures are compact. Moreov er, it follows that the branchpoints (and branc hcuts) of the functions Y 1 , 2 , 0 coincide with the supp orts of ρ 1 , ρ 2 . Since they solve a cubic algebraic equation, the branc hp oin ts are determined as zero es of the discriminant ∆ := 4 R 3 − 27 D 2 of ( 5-22 ). As suggested by the previous discussion, ev en without an explicit expression for R , D , it follo ws from Morera’s theorem that R, D are also real-analytic and so is ∆. Thus ∆ cannot ha ve infinitely many zeroes in a compact domain and so the num b er of endpoints of the supp orts is a-priori finite. The only part that is left to b e pro ven are the formulæ for R, D . Computing the jump of W j 2 from eqs. ( 5-14 ) one obtains ( W 1 ) 2 + − ( W 1 ) 2 − = − 2 iπ ρ 1  1 T V 0 1 + W 2  (5-29) and a similar equation for W 2 . This implies that W 1 2 ( z ) = Z ρ 1 ( x ) ( V 0 1 ( x ) /T + W 2 ( x )) z − x d x = = 1 T V 0 1 ( z ) W 1 ( z ) − Z ρ 1 ( x ) ( V 0 1 ( z ) − V 0 1 ( x )) T ( z − x ) d x | {z } =: R 1 ( z ) + Z Z ρ 1 ( x ) ρ 2 ( y )d x d y ( z − x )( x − y ) (5-30) Note that R 1 ( z ) is regular on the supp ort of ρ 1 since V 0 1 is. Hence we ha v e W 1 2 ( z ) = 1 T V 0 1 ( z ) W 1 ( z ) − R 1 ( z ) + =: W 12 z }| { Z Z ρ 1 ( x ) ρ 2 ( y )d x d y ( z − x )( x − y ) (5-31) W 2 2 ( z ) = 1 T V 0 2 ( z ) W 2 ( z ) − R 2 ( z ) + Z Z ρ 1 ( x ) ρ 2 ( y )d x d y ( z − y )( y − x ) | {z } =: W 21 . (5-32) Adding the tw o together and using the iden tity 1 ( z − x )( x − y ) + 1 ( z − y )( y − x ) = 1 ( z − x )( z − y ) (5-33) w e obtain W 1 2 + W 2 2 = 1 T V 0 1 W 1 + 1 T V 0 2 W 2 − R 1 − R 2 + W 1 W 2 , (5-34) whic h is precisely eq. ( 5-18 ) when rewritten with W 1 , W 2 replaced by their expressions in terms of the Y 1 , Y 2 v ariables resulting from ( 5-15 ). The second formula for D ( x ) can be obtained by noticing that D ( x ) = Y 0 Y 1 Y 2 . Explicitly , this reads (after rearranging the terms and using the definitions for the shifted resolven ts ( 5-15 ) and the U j ’s ( 5-26 ) D ( x ) = U 0 U 1 U 2 − 2 U 0 W 1 W 2 − W 2 1 W 2 + W 1 W 2 2 − U 2  W 2 1 − V 0 1 T W 1  − U 1  W 2 2 − V 0 2 T W 2  (5-35) 26 Substituting in the last tw o terms the expressions ( 5-31 , 5-32 ) we obtain D ( x ) = U 0 U 1 U 2 − 2 U 0 W 1 W 2 − W 2 1 W 2 + W 1 W 2 2 + U 2 R 1 + U 1 R 2 − U 2 W 12 − U 1 W 21 (5-36) Using W 1 W 2 = W 12 + W 21 ( 5-31 , 5-32 ) we obtain D ( x ) = U 0 U 1 U 2 − W 2 1 W 2 + W 1 W 2 2 + U 2 R 1 + U 1 R 2 + V 0 1 T W 12 − V 0 2 T W 21 (5-37) No w we can rewrite V 0 1 W 12 = ∇ 12 ( V 0 1 ) + Z Z V 0 1 ( x ) ρ 1 ( x ) ρ 2 ( y )d x d y ( z − x )( x − y ) (5-38) V 0 2 W 21 = ∇ 21 ( V 0 2 ) + Z Z V 0 2 ( y ) ρ 1 ( x ) ρ 2 ( y )d x d y ( z − y )( y − x ) (5-39) so that we ha ve obtained D ( x ) = U 0 U 1 U 2 + U 2 R 1 + U 1 R 2 + 1 T ∇ 12 V 0 1 − 1 T ∇ 21 V 0 2 + R (5-40) R := 1 T Z Z  V 0 1 ( x ) z − x + V 0 2 ( y ) z − y  ρ 1 ( x ) ρ 2 ( y )d x d y ( x − y ) − W 2 1 W 2 + W 1 W 2 2 (5-41) W e w ant to sho w that R ≡ 0: indeed it is clear that the only part of D ( x ) which may hav e jump- discon tinuities is R , but we kno w that D ( x ) has no such discon tinuities. Hence R has no discontin uities; on the other hand it is clear from its definition that R cannot ha ve an y other singularities and hence it is an en tire function. Insp ection shows that R ( z ) → 0 a s z → ∞ and hence R must b e identically v anishing by Liouville’s theorem. This concludes the pro of of the theorem. Q.E.D. The presence of a “spectral curv e” will be one of the crucial ingredients for the large–degree asymptotic analysis of the biorthogonal p olynomials using the Riemann–Hilb ert formulation giv en in [ 8 ] and the Deift–Zhou nonlinear steep est descent metho d. Indeed w e will show in a separate publication that the OPs are mo deled by (spinorial) Baker–Akhiezer vectors (similarly to [ 9 ]) that naturally live on the three- sheeted cov ering sp ecified b y ( 5-22 ). A A rectangular mixed 3 –matrix mo del with ghost fields The mo del ( 3-16 ) (for any v alue of h ) can b e obtained from the following matrix mo dels A.1 In teger h < N Consider the standard Leb esgue measure on the space M at (( N − h ) × N , C ) of complex ( N − h ) × N matrices, viewed as a linear space: if A ∈ M at (( N − h ) × N , C ) we will use a shorthand notation d A d A † := N − h Y i =1 N Y j =1 d < A ij d = A ij (A-1) 27 for the v olume element. Let, as ab o v e, M 1 , M 2 ∈ H + N and consider the following normalizable measure on the space H + N × H + N × M at ( N − h, N , C ) d b µ C ( M 1 , M 2 , A ) := d M 1 d M 2 d A d A † α ( M 1 ) β ( M 2 )e − T r A ( M 1 + M 2 ) A † (A-2) Since the measure is Gaussian in A one immediately sees that (up to inessen tial proportionality constants) Z M at ( N − h,N , C ) d b µ C ( M 1 , M 2 , A ) ∝ d µ h ( M 1 , M 2 ) = d M 1 d M 2 α ( M 1 ) β ( M 2 ) det( M 1 + M 2 ) N − h , h ∈ { 0 , 1 , . . . , N − 1 } (A-3) A.2 Half–in teger h < N One could also consider a similar mo del where M at ( N − h, N , C ) is replaced by M at (2 N − 2 h, N , R ) where h ∈ 1 2 Z and h < N obtaining then d A := 2 N − 2 h Y i =1 N Y j =1 d A ij d b µ R ( M 1 , M 2 , A ) := d M 1 d M 2 d Aα ( M 1 ) β ( M 2 )e − T r A ( M 1 + M 2 ) A t (A-4) Z M at ( N − h,N , R ) d b µ R ( M 1 , M 2 , A ) ∝ d µ ( M 1 , M 2 ) = = d M 1 d M 2 α ( M 1 ) β ( M 2 ) det( M 1 + M 2 ) N − h , (A-5) Of course, if h is actually an integer then this case reduces to the previous one. A.3 In teger and half in teger h > N F or the sake of completeness w e note that when h > N the determinant in the reduced measure ( 3-16 ) is a positive p ow er in the numerator; this can b e obtained from a Gaussian integral ov er Grassmann v ariables whic h can be obtained from “complex” an ticommuting v ariables (for h integer) or “real” (for h ∈ 1 2 Z ) muc h along the line of the previous (commuting v ariable) case. Clearly , the plethora of mo dels is endless and each v alue of h can b e studied with the aid of a different set of biorthogonal p olynomial. W e singled out the case h = 0, since it corresp onds to the Cauch y BOPs that hav e many features in common with the muc h b etter–known orthogonal p olynomials and y et seem to hav e a v ery rich but still tractable asymptotic theory . 28 B Relation to the O (1) –mo del The O ( n ) matrix model [ 18 ] is a m ultimatrix model for a (positive) Hermitean matrix M and n Hermitean matrices A j with distribution d M n Y j =1 d A j exp   − N T r   V ( M ) + M · n X j =1 A j 2     . (B-1) With some manipulations, the in tegration o ver the Gaussian v ariables A j can be performed and the result written in terms of the eigenv alues z j of the matrix M , yielding a (unnormalized) measure o ver the space of eigenv alues giv en by ∆( Z ) 2 Q i

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