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arXiv · 2512.13969

Representation theory and cycle statistics for random walks on the symmetric group

Dominic Arcona
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Abstract

We use representation theory of $S_n$ to analyze the mixing of cycle type statistics $a_j(\sigma) = \{\text{# of $j$-cycles of $\sigma$}\}$ for any fixed $j$ in permutations $\sigma_t$ resulting from the $t$-step random transposition walk on $S_n$. We also derive analogous results for the star transposition walk. Our approach uses the method of moments; a key ingredient is a new formula for the coefficients in the irreducible character decomposition of the $S_n$-class function $(a_j)^r(\sigma)=\{(\text{# of $j$-cycles of $\sigma$})^r\}$ for any positive integers $r,j$ when $n\geq 2rj$.

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Dominic Arcona. 2025-12-16. Representation theory and cycle statistics for random walks on the symmetric group. https://arxiv.org/abs/2512.13969

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