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

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 $A_n^{(1)}$. 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.

arXiv comment24 pages, minor corrections

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• Bi-infinite solutions for KdV- and Toda-type discrete integrable systems based on path encodings
• Generalized hydrodynamics in complete box-ball system for $U_q({\widehat{sl}}_n)$
• Scaling limit of soliton lengths in a multicolor box-ball system