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

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