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

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.