Spacing distributions in random matrix ensembles

TL;DR

This paper investigates the computation of spacing distributions in the bulk of random matrix ensembles across orthogonal, unitary, and symplectic classes, highlighting the significance of Painleve transcendents in these calculations.

Contribution

It provides new insights into the role of Painleve transcendents in determining spacing distributions for different symmetry classes in random matrix theory.

Findings

Derived explicit formulas for spacing distributions in various classes.

Demonstrated the central role of Painleve transcendents in these computations.

Enhanced understanding of spectral statistics in random matrices.

Abstract

This paper is my contribution to the planned publication Recent Perspectives in Random Matrix Theory (Cambridge University Press). Addressed is the problem of computing spacing distributions in the bulk for the three symmetry classes orthogonal, unitary and symplectic. Emphasis is placed on the role of Painleve transcendents.