Related papers: Complex Curve of the Two Matrix Model and its Tau-…
Exact eigenvalue correlation functions are computed for large $N$ hermitian one-matrix models with eigenvalues distributed in two symmetric cuts. An asymptotic form for orthogonal polynomials for arbitrary polynomial potentials that support…
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…
Random matrix models based on an integral over supermatrices are proposed as a natural extension of bosonic matrix models. The subtle nature of superspace integration allows these models to have very different properties from the analogous…
The super-algebraic structure of a generalized version of the Jaynes-Cummings model is investigated. We find that a Z2 graded extension of the so(2,1) Lie algebra is the underlying symmetry of this model. It is isomorphic to the…
This is a short note on the relation of the Matrix model with the non-commutative geometry of the 11-dimensional supermembrane. We put forward the idea that M-theory is described by the 't Hooft topological expansion of the Matrix model in…
Following the works by Wiegmann-Zabrodin, Elbau-Felder, Hedenmalm-Makarov, and others, we consider the normal matrix model with an arbitrary potential function, and explain how the problem of finding the support domain for the asymptotic…
We study the question of the existence of a dual extremal function for a bounded matrix function on the unit circle in connection with the problem of approximation by analytic matrix functions. We characterize the class of matrix functions,…
This paper is a brief review of recent results on the concept of ``generalized $\tau$-function'', defined as a generating function of all the matrix elements in a given highest-weight representation of a universal enveloping algebra ${\cal…
Dual multiplicity graphs are those simple, undirected graphs that have a weighted Hermitian adjacency matrix with only two distinct eigenvalues. From the point of view of frame theory, their characterization can be restated as which graphs…
We derive the discrete linear systems associated to multi--matrix models, the corresponding discrete hierarchies and the appropriate coupling conditions. We also obtain the $W_{1+\infty}$ constraints on the partition function. We then apply…
We propose the notion of $E_{2}$-quasi-exact solvability and apply this idea to find explicit solutions to the eigenvalue problem for a non-Hermitian Hamiltonian system depending on two parameters. The model considered reduces to the…
We explain how the spectral curve can be extracted from the ${\cal W}$-representation of a matrix model. It emerges from the part of the ${\cal W}$-operator, which is linear in time-variables. A possibility of extracting the spectral curve…
Recently, a two-matrix-model with a new type of interaction [1] has been introduced and analyzed using bi-orthogonal polynomial techniques. Here we present the complete 1/N^2 expansion for the formal version of this model, following the…
We develop the ODE/IM correspondence for the higher-order Mathieu equation arising from the quantum Seiberg-Witten curve of the pure $SU(r+1)$ ${\cal N}=2$ supersymmetric Yang-Mills theory. From the subdominant solutions, we construct the…
In this paper we studied the asymptotic eigenvalue statistics of the 2 matrix model with a quartic monomial and a general even polynomial potential. We studied the correlation kernel for the eigenvalues of one of the matrices in asymptotic…
The 2-matrix models can be defined in a setting more general than polynomial potentials, namely, the semiclassical matrix model. In this case, the potentials are such that their derivatives are rational functions, and the integration paths…
We introduce a class of Hermitian metrics, that we call pluriclosed star split, generalising both the astheno-K\"ahler metrics of Jost and Yau and the $(n-2)$-Gauduchon metrics of Fu-Wang-Wu on complex manifolds. They have links with…
An iterative scheme is set up for solving the loop equation of the hermitian one-matrix model with a multi-cut structure. Explicit results are presented for genus one for an arbitrary but finite number of cuts. Due to the complicated form…
Using recently developed methods of character expansions we solve exactly in the large N limit a new two-matrix model of hermitean matrices A and B with the action S={1\over 2}(\tr A^2+\tr B^2)-{\alpha\over 4}(\tr A^4+\tr B^4) -{\beta\over…
In this paper we investigate the tau-functions for the stationary sector of Gromov-Witten theory of the complex projective line and its version, relative to one point. In particular, we construct the integral representation for the points…