  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.
