相关论文: Matrix Models vs. Matrix Integrals
We construct ensembles of random integrable matrices with any prescribed number of nontrivial integrals and formulate integrable matrix theory (IMT) -- a counterpart of random matrix theory (RMT) for quantum integrable models. A type-M…
We study the hermitean and normal two matrix models in planar approximation for an arbitrary number of eigenvalue supports. Its planar graph interpretation is given. The study reveals a general structure of the underlying analytic complex…
A direct relation between the enumeration of ordinary maps and that of fully simple maps first appeared in the work of the first and last authors. The relation is via monotone Hurwitz numbers and was originally proved using Weingarten…
I review some recent works on the Hermitean one-matrix and d-dimensional gauge-invariant matrix models. Special attention is paid to solving the models at large-N by the loop equations. For the one-matrix model the main result concerns…
Dirichlet integrals and the associated Dirichlet statistical densities are widely used in various areas. Generalizations of Dirichlet integrals and Dirichlet models to matrix-variate cases, when the matrices are real symmetric positive…
The paper contains some new results and a review of recent achievements, concerning the multisupport solutions to matrix models. In the leading order of the 't Hooft expansion for matrix integral, these solutions are described by…
The theory of matrix models is reviewed from the point of view of its relation to integrable hierarchies. Determinantal formulas, relation to conformal field models and the theory of Generalized Kontsevich model are discussed in some…
The Hermitian, complex and fermionic two-matrix models with infinite set of variables are constructed. We show that these two-matrix models can be realized by the $W$-representations. In terms of the $W$-representations, we derive the…
Matrices are typically considered over fields or rings. Motivated by applications in parametric differential equations and data-driven modeling, we suggest to study matrices with entries from a Hilbert space and present an elementary theory…
We present a hermitian matrix chain representation of the general solution of the Hirota bilinear difference equation of three variables. In the large N limit this matrix model provides some explicit particular solutions of continuous…
We derive the loop equations for the one Hermitian matrix model in any dimension. These are a consequence of the Schwinger-Dyson equations of the model. Moreover we show that in leading order of large $N$ the loop equations form a closed…
We study integrals over Hermitian supermatrices of arbitrary size $p+q$, that are parametrized by an external field $X$ and a source $Y$, of respective size $m+n$ and $p+q$. We show that these integrals exhibit a simple topological…
This thesis deals with the geometric and integrable aspects associated with random matrix models. Its purpose is to provide various applications of random matrix theory, from algebraic geometry to partial differential equations of…
Matrix integrals used in random matrix theory for the study of eigenvalues of Hermitian ensembles have been shown to provide $\tau$-functions for several hierarchies of integrable equations. In this article, we extend this relation by…
We consider certain examples of applications of the general methods, based on geometry and integrability of matrix models, described in hep-th/0601212. In particular, the nonlinear differential equations, satisfied by quasiclassical…
We review some surprising links which have been discovered in the last few years between the theory of certain ordinary differential equations, and particular integrable lattice models and quantum field theories in two dimensions. An…
Small M-theories unify various models of a given family in the same way as the M-theory unifies a variety of superstring models. We consider this idea in application to the family of eigenvalue matrix models: their M-theory unifies various…
The product of a Hermitian matrix and a positive semidefinite matrix has only real eigenvalues. We present bounds for sums of eigenvalues of such a product.
Partition functions of eigenvalue matrix models possess a number of very different descriptions: as matrix integrals, as solutions to linear and non-linear equations, as tau-functions of integrable hierarchies and as special-geometry…
This talk gives an introduction into the subject of Seiberg-Witten curves and their relation to integrable systems. We discuss some motivations and origins of this relation and consider explicit construction of various families of…