We consider integrals on unitary groups $U_d$ of the form $$\int_{U_d}U_{i_1j_1}... U_{i_qj_q}U^*_{j'_{1}i'_{1}} ... U^*_{j'_{q'}i'_{q'}}dU$$
We give an explicit formula in terms of characters of symmetric groups and Schur functions, which allows us to rederive an asymptotic expansion as $d\to\infty$. Using this we rederive and strenghthen a result of asymptotic freeness due to Voiculescu.
We then study large $d$ asymptotics of matrix model integrals and of the logarithm of Itzykson-Zuber integrals and show that they converge towards a limit when considered as power series. In particular we give an explicit formula for $$\lim_{d\to\infty}\frac{\partial^n}{\partial z^n}d^{-2} \log\int_{U_d} e^{zd Tr (XUYU^*)}dU|_{z=0}$$ assuming that the normalized traces $d^{-1} Tr(X^k)$ and $d^{-1} Tr (Y^k)$ converge in the large $d$ limit. We consider as well a different scaling and relate its asymptotics to Voiculescu's R-transform.
We investigate spectral properties of a Hermitised random matrix product which, contrary to previous product ensembles, allows for eigenvalues on the full real line. We prove that the eigenvalues form a bi-orthogonal ensemble, which reduces asymptotically to the Hermite Muttalib-Borodin ensemble. Explicit expressions for the bi-orthogonal functions as well as the correlation kernel are provided. Scaling the latter near the origin gives a limiting kernel involving Meijer G-functions, and the functional form of the global density is calculated. As a part of this study, we introduce a new matrix …
We apply the renormalized coupling constants and Virasoro constraints to derive the Itzykson-Zuber Ansatz on the form of the free energy in topological 2D gravity. We also treat the topological 1D gravity and the Hermitian one-matrix models in the same fashion. Some uniform behaviors are discovered in this approach.
This article is devoted to the asymptotic expansion of the generalized Harish Chandra-Itzykson-Zuber matrix integral for non-unitary symmetries characterized by a parameter beta(as usual beta =1,2 and 4 correspond to the orthogonal, unitary and symplectic group integrals). A WKB-expansion for f is derived from the heat kernel differential equation, for general values of k and beta. From an expansion in terms of zonal polynomials, one obtain an expansion in powers of the tau's for beta=1, and generalizations are considered for general beta. A duality relation, and a transformation of products o…
We reconsider the large $N$ asymptotics of Harish-Chandra-Itzykson-Zuber integrals. We provide, using Dyson's Brownian motion and the method of instantons, an alternative, transparent derivation of the Matytsin formalism for the unitary case. Our method is easily generalized to the orthogonal and symplectic ensembles. We obtain an explicit solution of Matytsin's equations in the case of Wigner matrices, as well as a general expansion method in the dilute limit, when the spectrum of eigenvalues spreads over very wide regions.
The HarishChandra-Itzykson-Zuber integral over the unitary group U(k) (beta=2) is present in numerous problems involving Hermitian random matrices. It is well known that the result is semi-classically exact. This simple result does not extend to other symmetry groups, such as the symplectic or orthogonal groups. In this article the analysis of this integral is extended first to the symplectic group Sp(k) (beta=4). There the semi-classical approximation has to be corrected by a WKB expansion. It turns out that this expansion stops after a finite number of terms ; in other words the WKB approxim…
In this paper we return to the classical work by Brézin, Itzykson, Parisi and Zuber, in which, among other things, the authors explicitly calculated the coefficients of the topological expansion in the cubic random matrix model in genus 0. Our main goal will be to rigorously prove the results of Brézin, Itzykson, Parisi and Zuber and to obtain an explicit formula for the coefficients of the topological expansion in genus 1. We will also prove some formulae and asymptotic results for the coefficients of the topological expansion in higher genera.
Concise review of the basic properties of unitary matrix integrals. They are studied with the help of the three matrix models: the ordinary unitary model, Brezin-Gross-Witten model and the Harish-Charndra-Itzykson-Zuber model. Especial attention is paid to the tricky sides of the story, from De Wit-t'Hooft anomaly in unitary integrals to the problem of correlators with Itzykson-Zuber measure. Of technical tools emphasized is the method of character expansions. The subject of unitary integrals remains highly under-investigated and a lot of new results are expected in this field when it attracts…
We consider solvable matrix models. We generalize Harish-Chandra-Itzykson-Zuber and certain other integrals (Gross-Witten integral and integrals over complex matrices) using the notion of tau function of matrix argument. In this case one can reduce the matrix integral to the integral over eigenvalues, which in turn is certain tau function. The resulting tau function may be analyzed either by the method of orthogonal polynomials or by the Schur functions expansion method.
We give a new proof of the fact that, near a turning point of the frozen boundary, the vertical tiles in a uniformly random lozenge tiling of a large sawtooth domain are distributed like the eigenvalues of a GUE random matrix. Our argument uses none of the standard tools of integrable probability. In their place, it uses a combinatorial interpretation of the Harish-Chandra/Itzykson-Zuber integral as a generating function for desymmetrized Hurwitz numbers.
In this paper, we prove a longstanding conjecture on the asymptotic behavior of a pair of oscillatory matrix integrals: the Harish-Chandra/Itzykson-Zuber (HCIZ) integral, and the Brezin-Gross-Witten (BGW) integral. The main result gives a complete asymptotic expansion of these integrals for small complex parameters. The coefficients of these asymptotic expansions are generating functions for monotone Hurwitz numbers sorted by genus.
We revisit the work of the first named author and using simpler algebraic arguments we calculate integrals of polynomial functions with respect to the Haar measure on the unitary group U(d). The previous result provided exact formulas only for 2d bigger than the degree of the integrated polynomial and we show that these formulas remain valid for all values of d. Also, we consider the integrals of polynomial functions on the orthogonal group O(d) and the symplectic group Sp(d). We obtain an exact character expansion and the asymptotic behavior for large d. Thus we can show the asymptotic freene…
The expression of the large $N$ Harish Chandra--Itzykson--Zuber (HCIZ) integral in terms of the moments of the two matrices is investigated using an auxiliary unitary two-matrix model, the associated biorthogonal polynomials and integrable hierarchy. We find that the large $N$ HCIZ integral is governed by the dispersionless Toda lattice hierarchy and derive its string equation. We use this to obtain various exact results on its expansion in powers of the moments.
The averages of ratios of characteristic polynomials det(lambda - X) of N x N random matrices X, are investigated in the large N limit for the GUE, GOE and GSE ensemble. The density of states and the two-point correlation function are derived from these ratios. The method relies on an extension of the Harish-Chandra-Itzykson-Zuber integrals to the GOE ensemble and to supergroups, which are explicitly evaluated as solutions of heat kernel differential equations. An external matrix source, linearly coupled to the random matrices, may also be added to the Gaussian distribution, and allows for a d…
We find new representations for Itzykson-Zuber like angular integrals for arbitrary beta, in particular for the orthogonal group O(n), the unitary group U(n) and the symplectic group Sp(2n). We rewrite the Haar measure integral, as a flat Lebesge measure integral, and we deduce some recursion formula on n. The same methods gives also the Shatashvili's type moments. Finally we prove that, in agreement with Brezin and Hikami's observation, the angular integrals are linear combinations of exponentials whose coefficients are polynomials in the reduced variables (x_i-x_j)(y_i-y_j).
Recently the correlation functions of the so-called Itzykson-Zuber/Harish-Chandra integrals were computed (by one of the authors and collaborators) for all classical groups using an integration formula that relates integrals over compact groups with respect to the Haar measure and Gaussian integrals over a maximal nilpotent Lie subalgebra of their complexification. Since the integration formula a posteriori had the same form for the classical series, a conjecture was formulated that such a formula should hold for arbitrary semisimple Lie groups. We prove this conjecture using an abstract Lie-t…
We calculate a general spectral correlation function of products and ratios of characteristic polynomials for a $N\times N$ random matrix taken from the chiral Gaussian Unitary Ensemble (chGUE). Our derivation is based upon finding an Itzykson-Zuber type integral for matrices from the non-compact manifold ${\sf{Gl(n,{\mathcal{C}})/U(1)\times ...\times U(1)}}$ (matrix Macdonald function). The correlation function is shown to be always represented in a determinant form generalising the known expressions for only positive moments. Finally, we present the asymptotic formula for the correlation fun…
In this note, we study the asymptotic of spherical integrals, which are analytical extension in index of the normalized Schur polynomials for $β=2$ , and of Jack symmetric polynomials otherwise. Such integrals are the multiplicative counterparts of the Harish-Chandra-Itzykson-Zuber (HCIZ) integrals, whose asymptotic are given by the so-called $R$-transform when one of the matrix is of rank one. We argue by a saddle-point analysis that a similar result holds for all $β>0$ in the multiplicative case, where the asymptotic is governed by the logarithm of the $S$-transform. As a consequence of this…
We introduce the set of framed convex polyhedra with N faces as the symplectic quotient C^2N//SU(2). A framed polyhedron is then parametrized by N spinors living in C^2 satisfying suitable closure constraints and defines a usual convex polyhedron plus a phase for each face. We show that there is an action of the unitary group U(N) on this phase space, which changes the shape of faces and allows to map any polyhedron onto any other with the same total area. This realizes the isomorphism of the space of framed polyhedra with the Grassmannian space U(N)/SU(2)*U(N-2). We show how to write averages…
Pizzetti's formula explicitly shows the equivalence of the rotation invariant integration over a sphere and the action of rotation invariant differential operators. We generalize this idea to the integrals over real, complex, and quaternion Stiefel manifolds in a unifying way. In particular we propose a new way to calculate group integrals and try to uncover some algebraic structures which manifest themselves for some well-known cases like the Harish-Chandra integral. We apply a particular case of our formula to an Itzykson-Zuber integral for the coset SO(4)/[SO(2)xSO(2)]. This integral natura…
It is shown that the correlation functions of the random variables $\det(λ- X)$, in which $X$ is a real symmetric $ N\times N$ random matrix, exhibit universal local statistics in the large $N$ limit. The derivation relies on an exact dual representation of the problem: the $k$-point functions are expressed in terms of finite integrals over (quaternionic) $k\times k$ matrices. However the control of the Dyson limit, in which the distance of the various parameters $\la$'s is of the order of the mean spacing, requires an integration over the symplectic group. It is shown that a generalization of…
We study the partition function from random matrix theory using a well known connection to orthogonal polynomials, and a recently developed Riemann-Hilbert approach to the computation of detailed asymptotics for these orthogonal polynomials. We obtain the first proof of a complete large N expansion for the partition function, for a general class of probability measures on matrices, originally conjectured by Bessis, Itzykson, and Zuber. We prove that the coefficients in the asymptotic expansion are analytic functions of parameters in the original probability measure, and that they are generatin…
We present a new proof of Harish-Chandra's formula $$Π(h_1) Π(h_2) \int_G e^{\langle \mathrm{Ad}_g h_1, h_2 \rangle} dg = \frac{ [ \! [ Π, Π] \!] }{|W|} \sum_{w \in W} ε(w) e^{\langle w(h_1),h_2 \rangle},$$ where $G$ is a compact, connected, semisimple Lie group, $dg$ is normalized Haar measure, $h_1$ and $h_2$ lie in a Cartan subalgebra of the complexified Lie algebra, $Π$ is the discriminant, $\langle \cdot, \cdot \rangle$ is the Killing form, $[ \! [ \cdot, \cdot ] \!]$ is an inner product that extends the Killing form to polynomials, $W$ is a Weyl group, and $ε(w)$ is the sign of $w \in W$…
We reconsider the problem of calculating a general spectral correlation function containing an arbitrary number of products and ratios of characteristic polynomials for a N x N random matrix taken from the Gaussian Unitary Ensemble (GUE). Deviating from the standard "supersymmetry" approach, we integrate out Grassmann variables at the early stage and circumvent the use of the Hubbard-Stratonovich transformation in the "bosonic" sector. The method, suggested recently by one of us, is shown to be capable of calculation when reinforced with a generalization of the Itzykson-Zuber integral to a non…
We give an overview of the recursive characterisations of random matrix ensembles that are currently at the forefront of random matrix theory by way of studying two classes of ensembles using two different types of recursive schemes: Established theory on Selberg correlation integrals is used to derive linear differential equations on the eigenvalue densities and resolvents of the classical matrix ensembles, which lead to $1$-point recursions, understood to be analogues of the Harer-Zagier recursion, for the expansion coefficients of the associated $1$-point cumulants, while loop equation anal…
