Related papers: Combinatorics of dispersionless integrable systems…

We compute averages of products and ratios of characteristic polynomials associated with Orthogonal, Unitary, and Symplectic Ensembles of Random Matrix Theory. The pfaffian/determinantal formulas for these averages are obtained, and the…

We study S-matrix correlations for random matrix ensembles with a Hamiltonian which is the sum of a given deterministic part and of a random matrix with a Gaussian probability distribution. Using Efetov's supersymmetry formalism, we show…

The $\mathcal{A}$-tracial algebras are algebras endowed with multi-linear forms, compatible with the product, and indexed by partitions. Using the notion of $\mathcal{A}$-cumulants, we define and study the $\mathcal{A}$-freeness property…

We construct a replica field theory for a random matrix model with logarithmic confinement [K.A.Muttalib et.al., Phys. Rev. Lett. 71, 471 (1993)]. The corresponding replica partition function is calculated exactly for any size of matrix…

A spectral average which generalises the local spacing distribution of the eigenvalues of random $ N\times N $ hermitian matrices in the bulk of their spectrum as $ N\to\infty $ is known to be a $\tau$-function of the fifth Painlev\'e…

We consider the two-matrix model with potentials whose derivative are arbitrary rational function of fixed pole structure and the support of the spectra of the matrices are union of intervals (hard-edges). We derive an explicit formula for…

The generating functions for the gauge theory observables are often represented in terms of the unitary matrix integrals. In this work, the perturbative and non-perturbative aspects of the generic multi-critical unitary matrix models are…

We consider N x N matrices with complex entries that are perturbed by a complex Gaussian matrix with small variance. We prove that if the unperturbed matrix satisfies certain local laws then the bulk correlation functions are universal in…

We consider random stochastic matrices $M$ with elements given by $M_{ij}=|U_{ij}|^2$, with $U$ being uniformly distributed on one of the classical compact Lie groups or associated symmetric spaces. We observe numerically that, for large…

An exact map was established by Lacroix-A-Chez-Toine, Majumdar, and Schehr in [44] between the $N$ complex eigenvalues of complex non-Hermitian random matrices from the Ginibre ensemble, and the positions of $N$ non-interacting Fermions in…

We prove that a certain sequence of tau functions of the Garnier system satisfies Toda equation. We construct a class of algebraic solutions of the system by the use of Toda equation; then show that the associated tau functions are…

We study multi-matrix models which may be viewed as integrals of products of tau functions which depend on the eigenvalues of products of random matrices. In the present paper we consider tau functions of the hierarchy the two-component KP…

Okamoto has obtained a sequence of $\tau$-functions for the \PVI system expressed as a double Wronskian determinant based on a solution of the Gauss hypergeometric equation. Starting with integral solutions of the Gauss hypergeometric…

An element [\Phi] of the Grassmannian of n-dimensional subspaces of the Hardy space H^2, extended over the field C(x_1,..., x_n), may be associated to any polynomial basis {\phi} for C(x). The Pl\"ucker coordinates…

In this paper we study the Birkhoff coordinates (Cartesian action angle coordinates) of the Toda lattice with periodic boundary condition in the limit where the number $N$ of the particles tends to infinity. We prove that the transformation…

A new universality of Lyapunov spectra {\lambda_i} is shown for Hamiltonian systems. The universality appears in middle energy regime and is different from another universality which can be reproduced by random matrices in the following two…

Exact results from random matrix theory are used to systematically analyse the relationship between microscopic Dirac spectra and finite-volume partition functions. Results are presented for the unitary ensemble, and the chiral analogs of…

The factorization problem of the multi-component 2D Toda hierarchy is used to analyze the dispersionless limit of this hierarchy. A dispersive version of the Whitham hierarchy defined in terms of scalar Lax and Orlov--Schulman operators is…

Let $\mathfrak{D}$ be the space consists of pairs $(f,g)$, where $f$ is a univalent function on the unit disc with $f(0)=0$, $g$ is a univalent function on the exterior of the unit disc with $g(\infty)=\infty$ and $f'(0)g'(\infty)=1$. In…

We study the mixed Hessian of the dispersionless Toda $\tau$-function for the one-harmonic $s$-fold symmetric conformal map $f(w)=rw+aw^{1-s}$. This Hessian is the susceptibility matrix generated by the inverse conformal map. Our spectral…