Recursions and ODEs for the correlators in integrable systems and random matrices

TL;DR

This paper develops systematic methods to derive ODEs and recursion relations for correlators in integrable systems, with applications to random matrix theory and Schlesinger systems, enhancing understanding of their mathematical structure.

Contribution

It introduces a systematic approach to obtain ODEs and recursion relations for correlators in integrable systems, applicable to random matrices and Fuchsian systems.

Findings

Derived ODEs for correlators in integrable systems.

Established recursion relations with polynomial coefficients.

Applied methods to random matrix theory and Schlesinger systems.

Abstract

An integrable system is often formulated as a flat connection, satisfying a Lax equation. It is given in terms of compatible systems having a common solution called the ``wave function" $\Psi$ living in a Lie group $G$, which satisfies some differential equations with rational coefficients. From this wave function, it is usual to define a sequence of ``correlators" $W_n$, that play an important role in many applications in mathematical physics. Here, we show how to systematically obtain ordinary differential equations (ODE) and recursion relations with polynomial coefficients for the correlators. An application is random matrix theory, where the wave functions are the expectation value of the characteristic polynomial, they form a family of orthogonal polynomials, and are known to satisfy an integrable system. The correlators are then the correlation functions of resolvents or of eigenvalue densities. We give the ODE and recursion on the matrix size that they satisfy. In addition, we discuss generic Fuchsian systems, namely, Schlesinger systems.