Toda flow with unbounded initial data

TL;DR

This paper constructs Toda flows from unbounded initial data, including certain unbounded sequences and random matrix models, leading to invariant measures.

Contribution

It introduces a method to define Toda flows with unbounded initial conditions, expanding the class of initial data beyond bounded sequences.

Findings

Allows unbounded ergodic sequences as initial data

Includes ta-ensembles matrix models as initial conditions

Yields invariant measures for the flow from these initial data

Abstract

A Toda flow is constructed starting from a certain class of unbounded initial conditions including sequences growing with power order of less than 1. Unbounded ergodic sequences are allowed, and especially \b{eta}-ensembles matrix models in random matrix theory can be an initial data and they yiled invariant measures for the flow.