Local Statistics of Random Permutations from Free Products

Doron Puder, Tomer Zimhoni

Published 2022-03-23Version 1

Let $\Gamma=G_1*\ldots*G_k$ be a free product of groups where each of $G_1,\ldots,G_k$ is either finite, finitely generated free, or an orientable hyperbolic surface group. For a fixed element $\gamma\in\Gamma$, a $\gamma$-random permutation in the symmetric group $S_N$ is the image of $\gamma$ through a uniformly random homomorphism $\Gamma\to S_{N}$. In this paper we study local statistics of $\gamma$-random permutations and their asymptotics as $N$ grows. We first consider $\mathbb{E}\left[\mathrm{fix}_{\gamma}\left(N\right)\right]$, the expected number of fixed points in a $\gamma$-random permutation in $S_{N}$. We show that unless $\gamma$ has finite order, the limit of $\mathbb{E}\left[\mathrm{fix}_{\gamma}\left(N\right)\right]$ as $N\to\infty$ is an integer, and is equal to the number of subgroups $H\le\Gamma$ containing $\gamma$ such that $H\cong\mathbb{Z}$ or $H\cong C_{2}*C_{2}$. Equivalently, this is the number of subgroups $H\le\Gamma$ containing $\gamma$ and having (rational) Euler characteristic zero. We also prove there is an asymptotic expansion for $\mathbb{E}\left[\mathrm{fix}_{\gamma}\left(N\right)\right]$ and determine the limit distribution of the number of fixed points as $N\to\infty$. These results are then generalized to all statistics of cycles of fixed lengths.