Discrete integrable systems and Pitman's transformation

TL;DR

This paper surveys the connection between Pitman's transformation and various classical integrable systems, highlighting how this link facilitates the analysis of dynamics from infinite configurations and invariant measures.

Contribution

It elucidates the relationship between Pitman's transformation and integrable systems like the box-ball, KdV, and Toda lattice, advancing understanding of their dynamics from infinite configurations.

Findings

Connection enables initiation of dynamics from infinite configurations

Progress reported on invariant measures for i.i.d. configurations

Unified perspective on discrete integrable systems and Pitman's transformation

Abstract

We survey recent work that relates Pitman's transformation to a variety of classical integrable systems, including the box-ball system, the ultra-discrete and discrete KdV equations, and the ultra-discrete and discrete Toda lattice equations. It is explained how this connection enables the dynamics of the integrable systems to be initiated from infinite configurations, which is important in the study of invariant measures. In the special case of spatially independent and identically distributed configurations, progress on the latter topic is also reported.