Universal correlations in random matrices: quantum chaos, the $1/r^2$ integrable model, and quantum gravity

TL;DR

This paper explores the universal correlations in random matrix theory and their applications across quantum chaos, integrable models, and quantum gravity, highlighting the interconnectedness of these fields through RMT.

Contribution

It reviews the connections between RMT and various physical systems, and discusses the loop equations method for calculating correlation functions in different matrix ensembles.

Findings

Unified framework for correlations in quantum chaos, integrable models, and quantum gravity.

Application of loop equations to compute correlation functions in RMT.

Analysis of smoothed eigenvalue correlators in 2-matrix models.

Abstract

Random matrix theory (RMT) provides a common mathematical formulation of distinct physical questions in three different areas: quantum chaos, the 1-d integrable model with the $1/r^2$ interaction (the Calogero-Sutherland-Moser system), and 2-d quantum gravity. We review the connection of RMT with these areas. We also discuss the method of loop equations for determining correlation functions in RMT, and smoothed global eigenvalue correlators in the 2-matrix model for gaussian orthogonal, unitary and symplectic ensembles.