Representation Theory and Cycle Statistics for Random Walks on the Symmetric Group

Abstract

We use representation theory of $S_n$ to analyze the mixing of cycle type statistics $a_j(\sigma) = \{$# 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 $n\geq 2rj$.