Hamiltonian Structure of Equations Appearing in Random Matrices

TL;DR

This paper reveals that the level spacing distributions in Gaussian Unitary Ensemble random matrices are governed by Hamiltonian equations with a specific structure, connecting random matrix theory to integrable systems and isomonodromic deformations.

Contribution

It demonstrates the Hamiltonian structure of equations for level spacing distributions in GUE, linking them to isomonodromic deformations and classical R-matrix theory.

Findings

Level spacing distributions are described by Hamiltonian equations.

Connections established between random matrices and integrable systems.

Hamiltonian structure explained via classical R-matrix framework.

Abstract

The level spacing distributions in the Gaussian Unitary Ensemble, both in the ``bulk of the spectrum,'' given by the Fredholm determinant of the operator with the sine kernel ${\sin \pi(x-y) \over \pi(x-y)}$ and on the ``edge of the spectrum,'' given by the Airy kernel ${\rm{Ai}(x) \rm{Ai}'(y) - \rm{Ai}(y) \rm{Ai}'(x) \over (x-y)}$, are determined by compatible systems of nonautonomous Hamiltonian equations. These may be viewed as special cases of isomonodromic deformation equations for first order $ 2\times 2 $ matrix differential operators with regular singularities at finite points and irregular ones of Riemann index 1 or 2 at $\infty$. Their Hamiltonian structure is explained within the classical R-matrix framework as the equations induced by spectral invariants on the loop algebra ${\tilde{sl}(2)}$, restricted to a Poisson subspace of its dual space ${\tilde{sl}^*_R(2)}$, consisting of elements that are rational in the loop parameter.