Integrals of tau functions

Abstract

We consider integrals of tau functions of Zakharov-Shabat systems whose higher times are related to the eigenvalues of products of random matrices. Apart of random matrices there is the set of $n$ pairs of given matrices which play the role of parameters. In terms of these matrices we introduce the notions of words, dressed words and dual words, these notions are related to the graphs and dual graphs drawn on a Riemann surface $\Sigma$. The integrals of tau functions over independent ensembles of random matrices can be computed in form of series over partitions of products of the Schur polynomials with a multiplier which depends on the Euler characteristic of the Riemann surface. This form allows to compare the integrals of tau functions with correlation functions of certain quantum models. We present a tau function whose integral is equal to the correlation function of the Wilson loops of the two-dimensional Yang-Mills model on $\Sigma$.

arXiv commentcorrection of typos and completing Section 2

References in corpus (27)

• The $n$-component KP hierarchy and representation theory
• Products of Rectangular Random Matrices: Singular Values and Progressive Scattering
• An algebro-geometric proof of Witten's conjecture
• Complete Set of Cut-and-Join Operators in Hurwitz-Kontsevich Theory
• Conformal Matrix Models as an Alternative to Conventional Multi-Matrix Models
• Generalized Kazakov-Migdal-Kontsevich Model: group theory aspects
• Character Expansion Methods for Matrix Models of Dually Weighted Graphs
• Recent exact and asymptotic results for products of independent random matrices
• On the structure of Normal Matrix Model
• Free fermions and tau-functions
• Laplacian Growth and Whitham Equations of Soliton Theory
• Matrix Models for Random Partitions
• Hypergeometric τ-functions, Hurwitz numbers and enumeration of paths
• The matrix model for dessins d'enfants
• Fermionic construction of partition functions for two-matrix models and perturbative Schur function expansions
• Schur function expansion for normal matrix model and associated discrete matrix models
• Differential equations for singular values of products of Ginibre random matrices
• Combinatorial solutions to integrable hierarchies
• Dressing method and the coupled KP hierarchy
• Solvable Matrix Models
• Non-commutative extensions of two-dimensional topological field theories and Hurwitz numbers for real algebraic curves
• Hard edge limit of the product of two strongly coupled random matrices
• Spectral correlation functions of the sum of two independent complex Wishart matrices with unequal covariances
• The Non-Orientable Map Asymptotics Constant $p_g$
• Pfaffian structures and certain solutions to BKP hierarchies II. Multiple integrals
• Cyclic Foam Topological Field Theories
• Matrix integrals and Hurwitz numbers

Cited by in corpus (1)

• On Products of Random Matrices