Large deviations and one-sided scaling limit of randomized multicolor box-ball system

TL;DR

This paper establishes a large deviations principle for the invariant Young diagrams in a randomized multicolor box-ball system, showing almost sure convergence to equilibrium shapes and refining thermodynamic Bethe ansatz analysis for detailed scaling behavior.

Contribution

It introduces a large deviations framework for the invariant Young diagrams and refines the TBA analysis to describe detailed scaling forms in the randomized BBS.

Findings

Convergence of Young diagrams to equilibrium shape at exponential rate

Large deviations principle for row lengths of Young diagrams

Exact scaling forms of vacancy, row length, and column multiplicity

Abstract

The basic $\kappa$-color box-ball (BBS) system is an integrable cellular automaton on one dimensional lattice whose local states take $\{0,1,\cdots,\kappa \}$ with $0$ regarded as an empty box. The time evolution is defined by a combinatorial rule of quantum group theoretical origin, and the complete set of conserved quantities is given by a $\kappa$-tuple of Young diagrams. In the randomized BBS, a probability distribution on $\{0,1,\cdots,\kappa \}$ to independently fill the consecutive $n$ sites in the initial state induces a highly nontrivial probability measure on the $\kappa$-tuple of those invariant Young diagrams. In a recent work \cite{kuniba2018randomized}, their large $n$ `equilibrium shape' has been determined in terms of Schur polynomials by a Markov chain method and also by a very different approach of Thermodynamic Bethe Ansatz (TBA). In this paper, we establish a large deviations principle for the row lengths of the invariant Young diagrams. As a corollary, they are shown to converge almost surely to the equilibrium shape at an exponential rate. We also refine the TBA analysis and obtain the exact scaling form of the vacancy, the row length and the column multiplicity, which exhibit nontrivial factorization in a one-parameter specialization.