Scattering of the Toda system and the Gaussian $\beta$-ensemble

TL;DR

This paper links the classical Toda integrable system with the Gaussian β-ensemble, providing a new symplectic proof that certain matrix models follow this distribution, enhancing understanding of log-gases.

Contribution

It introduces a symplectic proof connecting the Toda flow on matrices to the Gaussian β-ensemble, offering new insights into matrix models for log-gases.

Findings

Dumitriu-Edelman tridiagonal model has spectrum following Gaussian β-ensemble

Classical Toda flow provides a framework for understanding matrix distributions

Scattering asymptotics relate to log-gases on the real line

Abstract

The classical Toda flow is a well-known integrable Hamiltonian system that diagonalizes matrices. By keeping track of the distribution of entries and precise scattering asymptotics, one can exhibit matrix models for log-gases on the real line. These types of scattering asymptotics date back to fundamental work of Moser. More precisely, using the classical Toda flow acting on symmetric real tridiagonal matrices, we give a "symplectic" proof of the fact that the Dumitriu-Edelman tridiagonal model has a spectrum following the Gaussian $\beta$-ensemble.