Randomized box-ball systems, limit shape of rigged configurations and Thermodynamic Bethe ansatz

TL;DR

This paper studies the probabilistic behavior of generalized box-ball systems using thermodynamic Bethe ansatz, revealing the limit shape of associated rigged configurations and connecting it to algebraic structures.

Contribution

It introduces a probability distribution on states of generalized box-ball systems and determines their limit shape through thermodynamic Bethe ansatz analysis.

Findings

Limit shape of rigged configurations identified

Connection to deformed characters of KR modules established

Stationary local energy matches the limit shape analysis

Abstract

We introduce a probability distribution on the set of states in a generalized box-ball system associated with Kirillov-Reshetikhin (KR) crystals of type $A^{(1)}_n$. Their conserved quantities induce $n$-tuple of random Young diagrams in the rigged configurations. We determine their limit shape as the system gets large by analyzing the Fermionic formula by thermodynamic Bethe ansatz. The result is expressed as a logarithmic derivative of a deformed character of the KR modules and agrees with the stationary local energy of the associated Markov process of carriers.