The Cauchy two-matrix model

TL;DR

This paper introduces a new two-matrix model based on the Cauchy kernel, expressing correlation functions via biorthogonal polynomials and Riemann-Hilbert problems, and explores its universality and geometric properties.

Contribution

It presents a novel two-matrix model with Cauchy coupling, linking it to biorthogonal polynomials, Riemann-Hilbert problems, and algebraic curves, expanding the analytical framework of matrix models.

Findings

Correlation functions expressed via biorthogonal polynomials

Model related to a trigonal algebraic curve

Connection to universality and steepest descent analysis

Abstract

We introduce a new class of two(multi)-matrix models of positive Hermitean matrices coupled in a chain; the coupling is related to the Cauchy kernel and differs from the exponential coupling more commonly used in similar models. The correlation functions are expressed entirely in terms of certain biorthogonal polynomials and solutions of appropriate Riemann-Hilbert problems, thus paving the way to a steepest descent analysis and universality results. The interpretation of the formal expansion of the partition function in terms of multicolored ribbon-graphs is provided and a connection to the O(1) model. A steepest descent analysis of the partition function reveals that the model is related to a trigonal curve (three-sheeted covering of the plane) much in the same way as the Hermitean matrix model is related to a hyperelliptic curve.