The partition function of the two-matrix model as an isomonodromic tau-function

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The partition function of the two-matrix model as an

isomonodromic tau-function

Bertola Marco, Olivier Marchal

To cite this version:

Bertola Marco, Olivier Marchal. The partition function of the two-matrix model as an

isomon-odromic tau-function. Journal of Mathematical Physics, American Institute of Physics (AIP), 2009,

50, pp.013529. �10.1063/1.3054865�. �hal-00863565�

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The partition function of the two-matrix model as an

isomonodromic

␶

_function

M. Bertola3,2,a兲and O. Marchal1,2,b兲

1

Institut de Physique Théorique, CEA, IPhT, F-91191 Gif-sur-Yvette, France and CNRS, URA 2306, F-91191 Gif-sur-Yvette, France

2

Centre de Recherches Mathématiques, Université de Montréal, C. P. 6128, succ. centre ville, Montréal, Québec H3C 3J7, Canada

3

Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve W., Montréal, Québec H3G 1M8, Canada

共Received 9 September 2008; accepted 1 December 2008; published online 15 January 2009兲 We consider the Itzykson–Zuber–Eynard–Mehta two-matrix model and prove that the partition function is an isomonodromic␶ function in a sense that generalizes that of Jimbo et al.关 “Monodromy preserving deformation of linear ordinary dif-ferential equations with rational coefficients,” Physica D 2, 306共1981兲兴. In order to achieve the generalization we need to define a notion of␶ function for isomono-dromic systems where the adregularity of the leading coefficient is not a necessary requirement. © 2009 American Institute of Physics.关DOI:10.1063/1.3054865兴

I. INTRODUCTION

Random matrix models have been studied for years and have generated important results in many fields of both theoretical physics and mathematics.1,16,18The two-matrix model

d␮共M1, M2兲 = e−Tr共V1共M1兲+V2共M2兲−M1M2兲dM1dM2 共1.1兲

was used to model two-dimensional quantum gravity14and was investigated from a more math-ematical point of view in Refs.25,17,4,5,7,12, and8; the partition function of the model

Z_N共V₁_,V₂兲 =

冕冕

d␮共M1, M2兲共1.2兲

has important properties in the large N limit for the enumeration of discrete maps on surfaces15of arbitrary genus and it is also known to be a␶function for the 2-Toda hierarchy. In the case of the Witten conjecture, proved by Kontsevich23with the use of matrix integrals not too dissimilar from the above one, the enumerative properties of the␶function imply some nonlinear 共hierarchy of兲 partial differential equations共PDEs兲关the Korteweg-de Vries 共KdV兲 hierarchy for the mentioned example兴. On a similar level, one expects some hierarchy of PDEs for the case of the two-matrix model and possibly some Painlevé property共namely, the absence of movable essential singulari-ties兲. The Painlevé property is characteristic of␶functions for isomonodromic families of ordinary differential equations共ODEs兲 that depend on parameters; hence, a way of establishing such prop-erty for the partition function ZN is that of identifying it with an instance of isomonodromic ␶

function.20,21

This is precisely the purpose of this article; we capitalize on previous work that showed how to relate the matrix model to certain biorthogonal polynomials26,17,22,16and how these appear in a natural fashion as the solution of certain isomonodromic family.9

a兲_{Electronic mail: bertola@crm.umontreal.ca.} b兲_{Electronic mail: olivier.marchal@polytechnique.org.}

50, 013529-1

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The paper extends to the case of the two-matrix model the work contained in Refs.9,12, and

10; it uses, however, a different approach closer to a recent one.6

In Refs. 9,12,10, and19the partition function of the one-matrix model共and certain shifted Töplitz determinants兲 were identified as isomonodromic ␶ functions by using spectral residue formulas in terms of the spectral curve of the differential equation. Such spectral curve has interesting properties inasmuch as—in the one-matrix case—the spectral invariants can be related to the expectation values of the matrix model. Recently the spectral curve of the two-matrix model8 has been written explicitly in terms of expectation values of the two-matrix model and hence one could use their result and follow a similar path for the proof as the one followed in Ref.

12. Whichever one of the two approaches one chooses, a main obstacle is that the definition of isomonodromic␶ function20,21 relies on a genericity assumption for the ODE which fails in the case at hand, thus requiring a generalization in the definition共see, however, Ref.11for a different generalization兲.

According to this logic, one of the purposes of this paper is to extend the notion of␶function introduced by Jimbo et al.20to the two-matrix Itzykson–Zuber model. This task is accomplished in a rather general framework in Sec. III.

We then show that the partition function has a very precise relationship with the␶function so introduced, allowing us to共essentially兲 identify it as an isomonodromic␶function共Theorem 3.4兲. The proof is organized as follows. In Sec. II we recall a Riemann–Hilbert formulation of the problem due to Ref.24. In Sec. II B we transform it into a Riemann-Hilbert problem共RHP兲 with constant jumps, thus guaranteeing the existence of an ODE. As the asymptotic at infinity does not enter into the class of the Jimbo–Miwa theorem for isomonodromic␶function, we generalize their result to our setting in Sec. III. Using Schlesinger transformations, namely, the shift in the size of the matrix model from N to N + 1, we eventually show that the partition function equals the new defined isomonodromic␶function in Sec. III B.

II. A RIEMANN–HILBERT FORMULATION OF THE TWO-MATRIX MODEL

According to a seminal work26,17 and following the notations and definitions introduced in Refs. 4 and 5, we consider paired sequences of monic polynomials 兵␲m共x兲,␴m共y兲其m=0,. . .,⬁ 共m

= deg␲m= deg␴m兲 that are biorthogonal in the sense that

冕冕

_⑂dxdy␲m共x兲␴n共y兲e−V1共x兲−V2共y兲+xy= hm␦mn, hm⫽ 0. 共2.1兲

The functions V1共x兲 and V2共y兲 appearing here are referred to as potentials, terminology drawn

from random matrix theory, in which such quantities play a fundamental role.

Henceforth, the second potential V2共y兲 will be chosen as a polynomial of degree d2+ 1,

V₂共y兲 =

兺

J=1 d₂+1 v_J Jy J_, _v d₂+1⫽ 0. 共2.2兲

For the purposes of most of the considerations to follow, the first potential V1共x兲 may have very

general analyticity properties as long as the manipulations make sense, but for definiteness and clarity, we choose it to be polynomial as well:

V1共x兲 =

兺

K=1 d₁+1 uK Kx K , ud₁+1⫽ 0. 共2.3兲

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冕冕

_⑂dxdy ª

兺

j

兺

k ⑂jk

冕

⌫_j dx

冕

⌫ˆ_k dy, ⑂ij苸 C, 共2.4兲

where the contours 兵⌫ˆk其k=1,. . .,d₂ will be chosen as follows. In the y plane, define d2+ 1 “wedge

sectors” 兵Sˆk其k=0,. . .,d₂ such that Sˆk is bounded by the pairs of rays rkª兵y 兩arg y =␪+ 2k␲/共d2+ 1兲其

and rk−1ª兵y 兩arg y =␪+ 2共k−1兲␲/共d2+ 1兲其, where␪ª arg vd₂+1. Then ⌫ˆk is any smooth oriented

contour within the sector Sˆkstarting from ⬁ asymptotic to the ray rk关or any ray within the sector

that is at an angle ⬍␲/ 2共2d2+ 1兲 to it, which is equivalent for purposes of integration兴 and

returning to ⬁ asymptotically along rk−1 关or at an angle ⬍␲/ 2共2d2+ 1兲 to it兴. These will be

referred to as the “wedge contours.” We also define a set of smooth oriented contours兵⌫ˇk其k=1,. . .,d₂

that have intersection matrix ⌫ˇj艚⌫ˆk=␦jk with the ⌫ˆk such that ⌫ˇk starts from ⬁ in sector Sˆ0,

asymptotic to the ray rˇ0ª兵y 兩arg共y兲=␪−␲/共d2+ 1兲其 and returns to ⬁ in sector Sˆkasymptotically

along the ray rˇkª兵y 兩arg共y兲=␪+ 2共k−1/2兲/共d2+ 1兲其. These will be called the “antiwedge”

con-tours.共See Fig.1.兲 The choice of these contours is determined by the requirement that all moment

integrals of the form

冕

_⌫ˆ k yje−V2共y兲+xy_dy_,

冕

⌫ˇ_k ykeV2共y兲−xy_dy_, _k_{= 1, . . . ,d} 2, j苸 N, 共2.5兲

be uniformly convergent in x苸C. In the case when the other potential V1共x兲 is also a polynomial,

of degree d1+ 1, the contours兵⌫k其k=1,. . .,d₁in the x plane may be defined similarly.

The partition function is defined here to be the multiple integral

Z_Nª 1 N!

冕冕

⑂N

兿

_j₌₁ N dxjdyj⌬共X兲⌬共Y兲

兿

j=1 N e−V1共xj兲−V2共yj兲+xjyj_, 共2.6兲

where ⌬共X兲 and ⌬共Y兲 denote the usual Vandermonde determinants and the factor 1/N! is chosen for convenience.

Γ

0 1 2 3 4 5 6 7

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Such multiple integral can also be represented as the following determinant:

Z_N= det关␮ij兴0ⱕi,jⱕN−1, ␮ijª

冕

⑂

xiyje−V1共x兲−V2共y兲+xy_dxdy_. 共2.7兲 The denomination of the partition function comes from the fact26,17,9that when⑂coincides with R_{⫻ R then Z}_N _coincides 共up to a normalization for the volume of the unitary group兲 with the following matrix integral:

冕冕

dM1dM2e−tr共V1共M1兲+V2共M2兲−M1M2兲, 共2.8兲

extended over the space of Hermitian matrices M1and M2of size N ⫻ N, namely, the

normaliza-tion factor for the measure d␮共M1, M2兲 introduced in共1.1兲.

A. Riemann–Hilbert characterization for the orthogonal polynomials

A Riemann–Hilbert characterization of the biorthogonal polynomials is a crucial step toward implementing a steepest-descent analysis. In our context it is also crucial in order to tie the random matrix side to the theory of isomonodromic deformations.

We first recall the approach given by Kuijlaars and McLaughin共referred to as KM in the rest of the article兲 in Ref.24suitably extended and adapted共in a rather trivial way兲 to the setting and notation of the present work. We quote—paraphrasing and with a minor generalization—their theorem without proof.

Theorem 2.1:共KM asymptotic24兲 The monic biorthogonal polynomial␲n共x兲 is the (1,1) entry

of the solution⌫共x兲 (if it exists) of the following Riemann–Hilbert problem for ⌫共x兲 . 共1兲 The matrix ⌫共x兲 is piecewise analytic in C\ ⊔⌫j.

共2兲 The (nontangential) boundary values of ⌫共x兲 satisfy the relations

⌫共x兲+= ⌫共x兲−

冤

1 wj,1 ¯ wj,d₂ 1 0 0 1

冥

, x苸 ⌫j, 共2.9兲 w_j,␯= wj,␯共x兲 ª e−V1共x兲

兺

k=1 d₂ ⑂_jk

冕

⌫ˆ_k y␯−1e−V2共y兲+xy_dy_. 共2.10兲 共3兲 As x→⬁, we have the following asymptotic expansion:

⌫共x兲 ⬃

冉

Id+ YN,1 x + O

冉

1 x2

冊冊

冢

xN 0 0 0 x−mN−1_Id r_N 0 0 0 x−mN_Id d₂−r_N

冣

, 共2.11兲

where we have defined the integers mN and rNas follows:

N= mNd2+ rN, mN,rN苸 N, 0 ⱕ rnⱕ d2− 1. 共2.12兲

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⌫N共x兲 ª ⌫共x兲 ª

冤

␲N共x兲 C0共␲N兲 ¯ Cd₂−1共␲N兲 pN−1共x兲 C0共pN−1兲 ¯ Cd₂−1共pN−1兲 ] ] pN−d₂共x兲 C0共pN−d₂兲 ¯ Cd₂−1共pN−d₂兲

冥

, 共2.13兲 C_i共f共z兲兲 ª 1 2␲i

冕冕

_⑂ f共x兲 x− zy i_e−V₁共x兲−V2共y兲+xy_dydx_, 共2.14兲 where the polynomials denoted above by pN−1, . . . , pN−d₂ are some polynomials of degree not

exceeding N − 1, whose detailed properties are largely irrelevant for our discussion; we refer to Ref.24for these details.

By a left multiplication of this solution by a suitable constant matrix共i.e., some suitable linear combination of the last rows兲, we can see that the matrix

⌫ˆNª

冤

␲n C0共␲n兲 ¯ Cd₂−1共␲n兲 ␲n−1 C0共␲n−1兲 ¯ Cd₂−1共␲n−1兲 ] ] ␲n−d₂ C0共␲n−d₂兲 ¯ Cd₂−1共␲n−d₂兲

冥

共2.15兲

and ⌫Nare related as

⌫ˆN共x兲 = UN⌫N共x兲, 共2.16兲

where UNis a constant matrix共depending on N and on the coefficients of the polynomials but not

on x兲. As an immediate consequence, ⌫ˆN solves the same RHP as ⌫ with the exception of the

normalization at infinity共2.11兲. The matrix UNwill not play a major role in the following proof,

but we mention it for completeness.

The RHP featured in Theorem 2.1 is not immediately suitable to make the connection to the theory of isomonodromic deformations as described in Refs.20and21; we recall that this is the theory that describes the deformations of an ODE in the complex plane which leave the Stokes matrices共i.e., the so-called extended monodromy data兲 invariant. The solution ⌫N共or ⌫ˆN兲 does not

solve any ODE as formulated because the jumps on the contours are nonconstant. If, however, we can relate ⌫N with some other RHP with constant jumps, then its solution can be immediately

shown to satisfy a polynomial ODE, which allows us to use the machinery of Refs.20 and21. This is the purpose of Sec. III A.

B. A RHP with constant jumps

In Ref.9the biorthogonal polynomials were characterized in terms of an ODE or—which is the same—of a RHP with constant jumps. In order to connect the two formulations, we will use some results contained in Ref. 13, and we start by defining some auxiliary quantities: for 1 ⱕ k ⱕ d2, define the d2sequences of functions兵␺m

共k兲_共x兲其 m苸N as follows: ␺m 共k兲_{共x兲 ª} 1 2␲i

冕

_⌫ˇ k ds

冕冕

⑂ dzdw␲m共z兲e −V₁共z兲 x− z V₂

_⬘

共s兲 − V₂

_⬘

共w兲 s− w e −V₂共w兲+V2共s兲+zw−xs_, _{1 ⱕ k ⱕ d} 2, 共2.17兲 and let ␺_m共0兲共x兲 ª␲m共x兲e−V1共x兲. 共2.18兲

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In terms of these define, for N ⱖ d2, the sequence of共d2+ 1兲⫻共d2+ 1兲 matrix valued functions ⌿N 共x兲, ⌿ˆ N共x兲 ª

冤

␺_N共0兲共x兲 ¯ ␺_N共d2兲共x兲 ] ] ␺_N_−d 2 共0兲 _{共x兲 ¯} _␺ N−d₂ 共d2兲 共x兲

冥

. 共2.19兲

For convenience共the notation with a hat describes only that we have removed the normalization matrix UN兲, we define also

⌿

N

ª U_N−1_⌿_ˆ

N

. 共2.20兲

The relationship with the matrices ⌫Nand ⌫ˆNintroduced in the previous section is detailed in the

following crucial theorem.

Theorem 2.2:共Factorization theorem兲 The following identities hold:

⌿ˆ N共x兲 = ⌫ˆN共x兲⌿0共x兲, ⌿N共x兲 = ⌫N共x兲⌿0共x兲, 共2.21兲 ⌿0共x兲 ª V共x兲W共x兲, where V ª

冉

e −V₁共x兲 ₀ 0 V₀,

冊

, W共x兲 ª

冉

1 0 0 W0共x兲

冊

共2.22兲 and V0and W0共x兲 are the d2⫻ d2 matrices with elements

共V0兲jk=

冤

v₂ v₃ ¯ v_d 2+1 v₃ v_d 2+1 v_d 2 vd2+1 v_d 2+1

冥

共2.23兲 =

再

v_j_+k if j + k ⱕ d₂+ 1 0 if j + k ⬎ d2+ 1,

冎

共2.24兲共W0共x兲兲jk=

冕

⌫ˇ_k yj−1eV2共y兲−xy_dy_, _{1 ⱕ j,} _k_{ⱕ d} 2. 共2.25兲

Proof: The proof is contained in Appendix A 共based on Refs. 13 and 3兲 and is a direct

verification based on matrix multiplication, noticing that the matrix V0 is nothing but the matrix

representation of 共V2

⬘

共y兲−V2

⬘

共s兲兲/共y −s兲 as a quadratic form in the bases 1, y , y2, . . . , yd2−1 and

1 , s , s2_{, . . . , s}d₂−1_.

䊐 The point of the matter now is that this new matrix ⌿N 共or ⌿兲 solves a Riemann–HilbertN

problem with constant jumps; this is due to the right multiplication of the factor ⌿0ª V共x兲W共x兲.

To this end, we state the following theorem.

Theorem 2.3: The matrix⌿Nis the unique solution of the following RHP.

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共x兲⌿N

−1

= DN共x兲 where DN共x兲 is a polynomial in x of degree d1.

共4兲 ⳵u_K⌿N共x兲⌿N −1 = UK,N共x兲 is polynomial in x of degree K . 共5兲 ⳵v J⌿N共x兲⌿N −1 = VJ,N共x兲 is polynomial in x . 共6兲 det共⌿N+1⌿N

−1_{兲⬅C with C a constant independent of x .}

(Sketch of) Proof: In order to prove point共1兲, we have to directly consider the definition of

⌿N

共noticing that ⌿N has the same jumps since they differ only by a left multiplier兲. Using

Sokhotski–Plemelji formulas and the results of Appendix B, one can quickly obtain the proof. Point共2兲 is a consequence of the factorization Theorem 2.2 and the asymptotic representation of ⌫N given in Theorem 2.1.

The uniqueness of the solution is proved by showing that det ⌿Nis constant in x; this follows

from the fact that the determinant has no jumps due to the unimodularity of the jump matrices, and the fact that det ⌿0 is constant and at infinity det ⌿N tends to a constant. Thus, by Liouville’s

theorem det ⌿Nmust be constant. At this point, the argument goes that any other solution ⌿ mustN

be invertible as well; the ratio ⌿N⌿N−1has no jump, thus being entire, and tends to the identity at

infinity. Once more, Liouville’s theorem assures that then the two matrices are identically equal. The points共3兲–共6兲 in the above theorem can be found in Refs.8and5, and the proofs will not be repeated here in detail. For example, point 共3兲 can be proven by noticing that since ⌿N has

constant jumps, the expression ⌿N

⬘

⌿N

−1

has no jumps and is hence entire. The fact that it is in fact a polynomial of degree d1 follows from the asymptotic behavior at infinity. In a similar fashion,

one can prove the other two points. 䊐

The main upshot of points共3兲–共5兲 is that ⌿Nis the solution of an isomonodromic deformation

system: namely, we may think of the equations as deformation equations which leave the 共gener-alized兲 monodromy of the ODE in 共3兲 invariant, as defined in Refs.20and21. We can thus apply the ideas of the said papers to our situation after the suitable generalizations.

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In the next section, we shall define a proper notion of isomonodromic␶function: it should be pointed out that the definition of Refs.20and21cannot be applied as such because—as shown in Ref.5—the ODE that the matrix ⌿N共or ⌿兲 solves has a highly degenerate leading coefficient atN

the singularity at infinity.

In the list, the crucial ingredients are the differential equations 共in x or relatively to the parameters uKand vJ兲. First, the fact that DN共x兲 is a polynomial comes from explicit computation

共see Ref.8, for example兲. The result concerning the determinant of RN共x兲 can also be found in Ref.

8where one has det共⌿N+1⌿N

−1_兲=det共a

N共x兲兲=Cste. The properties concerning the differential

equa-tions relatively to parameters can be found in Ref. 8 too. Under all these assumptions, we will show that the proof of Jimbo et al. can be adapted and that we can define a suitable␶function in the same way Jimbo et al. did.

III. DEFINITION OF THE ␶ FUNCTION

In this section, we will place ourselves in a more general context than the one described above; we will show that, under a few assumptions, one can define a good notion of␶function. More generally, we will denote with ta the isomonodromic parameters or “isomonodromic

times”共in our case they are the uKand the vJ兲 and a subscript a or b is understood as a derivation

relative to ta or tb. For a function f of the isomonodromic times, we will denote by the usual

symbol its differential,

df=

兺

a

⳵t_afdta=

兺

a

fadta. 共3.1兲

Our setup falls in the following framework that it is useful to ascertain from the specifics of the case at hand. Suppose we are given a matrix whose behavior at infinity is

⌿共x兲 ⬃ Y共x兲⌶共x兲, Y共x兲 ª

冉

1 +Y1 x + Y2 x2 + ¯

冊

x S , 共3.2兲

where ⌶共x兲=⌶共x;t兲 is some explicit expression 共the “bare” isomonodromic solution兲 and S is a matrix independent of the isomonodromic times. Note that this asymptotic falls exactly in the one we get for ⌿N共x兲. This implies that if we define the one-form-valued matrix H共x;t兲 by

H共x;t兲 = d⌶共x;t兲⌶共x;t兲−1 共3.3兲

then H共x兲=兺Hadta 共we suppress explicit mention of the t dependence henceforth兲 is some

solu-tion of the zero-curvature equasolu-tions:

⳵_aH_b₋⳵_bH_a₌关H_a_,H_b兴. 共3.4兲

This result is a direct computation using Ha=共⳵a⌶兲⌶−1 and ⳵b共⌶−1兲=−⌶−1共⳵b⌶兲⌶−1. We will assume共which is the case in our setting兲 that all H_aare polynomials in x. We will also use that the dressed deformations ⍀a given by ⌿a= ⍀a⌿ are polynomials, which is the case in our setup,

thanks to properties共4兲 and 共5兲 of Theorem 2.3. Moreover, according to the asymptotic, they are given by

⍀a=共YHaY−1兲pol. 共3.5兲

In this very general共and generic兲 setting, we can formulate the definition of a “␶ function” as follows.

Definition 3.1: The ␶differential is the one-form defined by

␻ª

兺

␻adtaª

兺

a

res tr共Y−1_Y

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The main point of the matter is that—without any further details—we can now prove that the

␶differential is in fact closed and hence locally defines a function.

Theorem 3.1: The␶differential is a closed differential and locally defines a␶function as

dlog␶=␻. 共3.7兲

Proof: We need to prove the closure of the differential. We first recall the main relations

between the bare and dressed deformations,

⳵_aY= ⍀aY− YHa, YHaY−1= ⍀a− Ra, Raª⳵aYY−1. 共3.8兲

We note that—by construction—⍀a=共YHaY−1兲polis a polynomial while Ra= O共x−1兲 irrespective

of the form of S. We compute the cross derivatives directly,

共3.9兲 where, in the last step, we have used that YHbY−1= ⍀b− Rband that the contribution coming from

⍀bvanishes since it is a polynomial. Rewriting the same with a ↔ b and subtracting, we obtain

共3.10兲 Note that, up to this point, we only used the zero-curvature equations for the connection ⵜ =兺共⳵_a− Ha兲dta and the fact that Ha are polynomials in x. We thus need to prove that the last

quantity in共3.10兲vanishes: this follows from the following computation, which uses once more the fact that Haand ⍀aare all polynomials. Indeed, we have res Tr共Ha

⬘

Hb兲=0 and hence 关using

共3.8兲兴

共3.11兲 Using integration by parts 共and cyclicity of the trace兲 on the first term here above, we obtain precisely the last quantity in共3.10兲. The theorem is proved. 䊐

A. Application to our problem

We now apply the general definition above to our setting, with the identifications ⌿ = ⌿N,

Y= ⌫N 共as a formal power series at ⬁兲, and ⌶=⌿0. We will write YN instead of ⌫N in the

expressions below to emphasize that we consider its asymptotic expansion at ⬁. In order to apply the generalized Definition 3.1 to our setting, we identify

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• the times ta used in Sec. III with the coefficients vjand uKof the two potentials关共2.3兲and

共2.2兲兴, and

• the “bare isomonodromic solution” ⌶共x;t兲 of Sec. III with the bare solution ⌿0共x;uជ, vជ兲

appearing in Theorem 2.3.

This reduces the definition of the␶function to the one below.

Definition 3.2: The ␶function is defined by the following PDE: d共log␶N兲 = Res

x→⬁Tr共YN −1

Y_N

⬘

d共⌿0兲⌿0

−1_兲, _共3.12兲

where YNis the formal asymptotic expansion of ⌫Nat infinity,

Y_N= Y˜N

冢

xN 0 0 0 x−mN−1_Id r_N 0 0 0 x−m_N_Id d₂−r_N

冣

. 共3.13兲

Remark 3.1: The matrix S of the previous section [Eq.(3.2)] in our case becomes S= SN=

冢

N 0 0

0 共− mN− 1兲Idr_N 0

0 0 − mNIdd₂−r_N

冣

共3.14兲 as follows by inspection of Eq.(3.13).

The partial derivatives of ln␶N with respect to the times uK and vJsplit into two sets which

have different forms:

⳵u_Klog␶N= − Res x→⬁Tr

冉

Y_N−1Y_N

_⬘

x K KE11

冊

, 共3.15兲 ⳵v

Jlog␶N= Res_x→_⬁Tr共YN

−1_Y

N

⬘

⳵v

J共⌿0兲⌿0

−1_兲. _共3.16兲

These two equations are the specializations of Definition 3.1 using that⌶共x;t兲=⌿0共x;uជ, vជ兲 and thus H共x兲=d⌶⌶−1_{take on two forms, depending on which of the times we are differentiating with.} The block form of the derivatives⳵_u

K⌿0⌿0

−1 _or_⳵

v

J⌿0⌿0

−1_{follows simply by the block form of}_⌿ 0 (in Theorem 2.2), noticing that the dependence on the uK is only in the (1,1) entry [共⌿0兲11

= e−V1共x兲 _{], while the dependence on the v}

Jis only in the d2⫻ d2 lower-right block.

One can notice that the situation we are looking at is a generalization of what happens in the one-matrix case. In the one-matrix model, the matrix S is zero and therefore YN are 共formal兲

Laurent series. The matrix ⌿0matrix is absent in that case since there is only one potential, and

thus one recovers the usual definition of isomonodromic␶function共see Ref.12兲. Note also that in

the derivation with respect to vJ, we have obtained the second equality using the block diagonal

structure of ⌿0 共first row/column does not play a role兲. It is remarkable that the two systems are

completely decoupled, i.e., that in the first one the matrix ⌿0共containing all the dependence in V2兲

disappears and that in the second one the matrix A0共containing the potential V1兲 also disappears.

Now that we have defined a proper generalization of isomonodromic␶functions corresponding to our setting; we still need to make the link with the partition function.

B. Discrete Schlesinger transformation: ␶ function quotient

In this section, we investigate the relationship between the␶function of Definition 3.2 and the partition function ZNof the matrix model. We anticipate that the two objects turn out to be the

same共up to a nonzero factor that will be explicitly computed, Theorem 3.4兲: the proof relies on two steps, the first of which we prepare in this section. These are

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• proving that they satisfy the same recurrence relation in N and • identifying the initial conditions for the recurrence relation.

We start by investigating the relationship between ␶N and␶N+1; this analysis is essentially

identical to the theory developed in Ref.21and used in and Ref. 6, but we report it here for the convenience of the reader.

From the fact that the ⌿Nhas constant jumps, we deduce that ⌿N+1⌿N

−1

is an entire function. Moreover, asymptotically it looks like

Y ˜ N. 共3.18兲

Thus, remembering that Y˜Nis a series x−1, Liouville’s theorem states that ⌿N+1⌿N

−1

is a polynomial of degree 1, and hence, for some constant matrices RN

0 and RN 1 we must have ⌿N+1⌿N −1 = RN共x兲 = RN 0 + xRN 1 . 共3.19兲

From the fact that det共RN兲 does not depend on x 共last property of Theorem 2.2兲, we know that

R_N−1共x兲 is a polynomial of degree at most 1 as well 共this is easy if one considers the expression of the inverse of a matrix using the comatrix兲.

Comparing the asymptotics of ⌿N+1 and RN共x兲⌿N term by term in the expansion in inverse

powers of x and after some elementary algebra, one obtains共Ref.20, Appendix A兲

RN共x兲 = E␣₀x+ RN,0 and RN

−1

共x兲 = E1x+ RN,0 −1

. 共3.20兲

Here, we have introduced the notation␣0= rN+ 1 which corresponds to the index of the column

where the coefficient x−1is to be found in the asymptotic of ⌿N+1⌿N

−1

. The notation Ejdenotes the

matrix with 1 in position j , j and 0 elsewhere. These notations are the standard notation used originally by Jimbo and Miwa in a Schlesinger transformation. The matrix共RN,0兲␣,␤is given by

␤=␣0 ␤= 1 ␤⫽␣0,1 ␣=␣0 −共YN,2兲␣₀,1+

兺

_␥⫽␣ 0共YN,1兲␣0,␥共YN,1兲␥,1 共YN,1兲␣₀,1 −共YN,1兲␣₀,1 −共YN,1兲␣₀,␤ ␣= 1 1 共YN,1兲␣₀,1 0 0 ␣⫽␣0,1 − 共YN,1兲␣,1 共YN,1兲␣₀,1 0 ␦␣,␤ 共3.21兲 and共RN,0 −1_兲 ␣,␤is given by

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␤=␣0 ␤= 1 ␤⫽␣0,1 ␣=␣0 0 共YN,1兲␣₀,1 0 ␣= 1 − 1 共YN,1兲␣₀,1 −−共YN,2兲␣0,1 共YN,1兲␣₀,1 +共YN,1兲1,1 − 共YN,1兲␣₀,␤ 共YN,1兲␣₀,1 ␣⫽␣0,1 0 共YN,1兲␣,1 ␦␣,␤. 共3.22兲

While the formulas above might seem complicated, we will use the two important observa-tions共that can be obtained easily with the definitions without the explicit formulas given above兲

E_␣ 0RN,0 −1 + RN,0E1= RN,0 −1_E ␣₀+ E1RN,0= 0, 共3.23兲 R_N−1共x兲R_N

⬘

共x兲 = RN,0 −1 E_␣

0 does not depend on x.

The recurrence relation satisfied by the sequence兵␶N其 is derived in the next theorem.

Theorem 3.2: Up to multiplication by functions that do not depend on the isomonodromic

parameters (i.e., independent of the potentials V₁and V2兲, the following identity holds: ␶N+1

␶N

=共Y1兲1,␣₀. 共3.24兲

Proof: The proof follows Ref.21, but we report it here for convenience of the reader. Con-sider the following identity:

⌿N+1= YN+1⌿0= RNYN⌿0. 共3.25兲

This implies that

YN+1= RNYN. 共3.26兲

Taking the derivative with respect to x gives Y_N−1₊₁Y_N

_⬘

₊₁= YN −1_R N −1_R N

⬘

Y_N+ YN −1_Y N

⬘

. 共3.27兲 Therefore, we have

−1

is polynomial in x and RN

−1

R_N

⬘

does not depend on x. Therefore, we are left only with

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dlog␶N+1− d log␶N= − Res x→⬁Tr共RN −1 R_N

⬘

dYNYN −1 兲. 共3.31兲

A direct matrix computation using the explicit form of RN yields

dlog␶N+1− d log␶N= d log共共YN,1兲1,␣₀兲共3.32兲

and hence

␶N+1 ␶N

=共Y1兲1,␣₀. 共3.33兲

The last equality is to be understood up to a multiplicative constant not depending on the

param-eters uKand vJin␶. 䊐

In order to complete the first step, we need to express the entry共Y1兲1,␣₀ in terms of the ratio

of two consecutive partition functions. This is accomplished in the following section.

Theorem 3.3; For the matrix ⌫Nthe asymptotic expansion at infinity(2.11)is such that

共YN,1兲1,␣₀=共vd₂+1兲SNhN=共vd₂+1兲SN

Z_N₊₁

Z_N , 共3.34兲

where SNand␣0苸兵0,1, ... ,d2− 1其 are defined by the following relation:

N= d2SN+␣0− 1. 共3.35兲

Proof: In order to compute 共YN,1兲1,␣₀, it is sufficient to compute the leading term of the

expansion at ⬁ appearing in the first row of the matrix ⌫N. Recalling the expression 共2.13兲, we

start by the following direct computation using integration by parts:

冕冕

_␬dzdw␲N共z兲ziwk−1e−V1共z兲−V2共z兲+zw=

冕冕

␬ dzdw␲N共z兲e−V1共z兲wk−1e−V2共w兲 di dwi共e zw_兲 =共− 1兲i

冕冕

␬ dzdw␲N共z兲e −V₁共z兲+zw di dwi共w k−1_e−V₂共w兲_兲 =

冕冕

␬ dzdw␲N共z兲qd₂i+k−1共w兲e−V1共z兲−V2共z兲+zw, 共3.36兲

where qd₂i+k−1共w兲 is a polynomial of the indicated degree whose leading coefficient is vd₂+1

i

. The last right hand side is 0 if d2i+ k − 1 ⬍ N because of orthogonality. If d2i+ k − 1 = N, the integral

gives vd₂+1

i _h

N by the normality conditions concerning our biorthogonal set. This computation

allows us to expand the Cauchy transform of共⌫N兲1,␣₀near ⬁ as follows:

C共p_Nw_␣ 0共x兲兲 = 1 2␲i

冕冕

_␬dzdw ␲N共z兲 z− ww ␣0−1_e−V1共z兲−V2共z兲+zw = −

兺

i=0 S_N−1 1 2␲i

冕冕

_␬dzdw␲N共z兲 zi xi+1w ␣0−1_e−V1共z兲−V2共z兲+zw + 1 2␲i 1 xSN+1

冕冕

␬ dzdw␲N共z兲 x− zz S_N_w␣0−1_e−V1共z兲−V2共z兲+zw+ O共x−SN−2兲. 共3.37兲

By orthogonality the first sum vanishes term by term and the leading coefficient of the second term is vd₂+1

S_N

hN. 䊐

We are finally in the position of formulating and proving the main theorem of the paper.

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∀ N苸 N:ZN=共vd₂+1兲兺j=0

N−1_S

j␶

N, 共3.38兲

where we recall that Sj is given by the decomposition of j+ 1 in the Euclidian division by d2 : Sj= E关共j +1兲/d2兴 . A short computation of the power in vd₂+1 gives

∀ N苸 N:ZN=共vd₂+1兲d2共␣N共␣N−1兲/2兲+␣N共N−␣Nd2兲␶N, 共3.39兲

where␣N= E关N/d2兴.

Proof: Recall that the␶function is only defined up to a multiplicative constant not depending on N nor on the coefficients ukand vj. With this in mind and combining Theorems 3.2 and 3.3, we

find that ␶N+1 ␶N = Theorem 3.2 共Y1兲1,␣₀ = Theorem 3.3 共vd₂+1兲SN Z_N₊₁ Z_N , 共3.40兲

where N = d2SN+␣0− 1. Hence, for every n0:

␶NZn₀= ZN␶n₀共vd₂+1兲兺j=n0

N−1_S

j_. 共3.41兲

One would like to take n0= 0 because it enables explicit computations. As we will prove now, there

is a way of extending naturally all the reasoning down to 0. Indeed, the RHP for ⌫N共Theorem 2.1兲

is perfectly well defined for N = 0 and has solution

⌫0=

冢

1 C0共1兲 C1共1兲 ¯ Cd₂−1共1兲 0 1 0 ¯ 0 ] ₀ ] 0 0 0 . . . 1

冣

. 共3.42兲

Consequently, we can take

␶NZ0=共vd₂+1兲兺j=0

N−1

S_j_Z

N␶0. 共3.43兲

Also note that Z0⬅1 共by definition兲.

We can compute␶0directly from Definition 3.2 because of the particularly simple and explicit

expression of ⌿0= ⌫0⌿0,

dln␶0= res Tr共Y0 −1

Y₀

⬘

d⌿0⌿−1兲. 共3.44兲

We claim that this expression is identically zero共and, hence, we can define␶0⬅1兲; indeed,

Y₀−1Y₀

⬘

=

冢

0 ⴱ ¯ ⴱ 0 0 ¯ 0 ] ] ₀ 0 0 ¯ 0

冣

共3.45兲 and d⌿0共x兲⌿0 −1_{共x兲 =}

冢

쐓 0 ¯ 0 0 쐓 ¯ 쐓 ] ] ] 0 쐓 ¯ 쐓

冣

共3.46兲

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The presence of the power in vd₂+1is due to a bad normalization of the partition function itself

共ZN兲 and can be easily canceled out by taking vd₂+1= 1 from the start共it is just a normalization of

the weight function兲. This finding is consistent with the work of Bergere and Eynard2

since all their results concerning the partition function and its derivatives with respect to parameters have special cases for ud₁+1 and vd₂+1. It also signals the fact that the RHP is badly defined when v_d

2+1= 0 because the contour integrals involved diverge and the whole setup breaks down. Indeed if vd₂+1= 0, this simply means that V2 is a polynomial of lower degree, and thus the RHP that we

should set up should be of smaller size from the outset.

IV. OUTLOOK

In this article, we have restricted ourselves to contours going from infinity to infinity. This allows us to use integration by parts without picking up any boundary term. A natural extension of this work could be to see what happens when contours end in the complex plane and especially study what happens when the end points move共models with hard edges兲. This generalization is important in the computation of the gap probabilities of the Dyson model,26which correspond to a random matrix model with Gaussian potentials but with the integration restricted to intervals of the real axis.

ACKNOWLEDGMENTS

We would like to thank John Harnad for proposing the problem and Seung Yeop Lee and Alexei Borodin for fruitful discussions. This work was done at the University of Montréal at the Department of Mathematics and Statistics and the Centre de Recherche Mathématique共CRM兲 and O.M. would like to thank both for their hospitality. This work was partly supported by the Enigma European Network, Grant No. MRT-CT-2004-5652, by the ANR project Géométrie et intégrabilité en physique mathématique, Grant No. ANR-05-BLAN-0029-01, by the Enrage European Net-work, Grant No. MRTN-CT-2004-005616, by the European Science Foundation through the Mis-gam program, by the French and Japanese governments through PAI Sakurav, and by the Quebec government with the FQRNT.

APPENDIX A: FACTORIZATION OF ⌿N

Starting from the definition of the last d2columns of ⌿N

共2.19兲, we observe that ␺_m共k兲共x兲 ª 1 2i␲

冕

冤

1 v₂ v₃ ¯ v_d 2+1 v₃ v_d 2+1 v_d 2 v_d 2+1 v_d 2+1

冥

冤

1 f₁ f₂ ¯ fd₂ f₁

_⬘

f₂

_⬘

¯ f_d 2

⬘

] ] f₁共d2−1兲 ¯ _f d₂ 共d2−1兲

冥

, 共B3兲

where the Wronskian sub-block in the second term is constructed by choosing d2 homologically

independent contour classes for the integrations ⌫ˇ ,

fk共x兲 ª

冕

⌫ˇ_k

e−xs+V2共s兲_ds_, _k_{= 1, . . . ,d}

2. 共B4兲

The dressing matrix ⌿0 exhibits a Stokes’ phenomenon共of Airy’s type兲 which is the inevitable

drawback of removing the x dependence from the jump matrix. We can now compute the jumps and see that it does not depend on x. For the kth column we have

j=1 d₂ ⑂_{ᐉ j}共⌫j 共y兲 # ⌫ˇk兲, 共B7兲 1

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