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• W4386689546 abstract "Abstract Let $alpha $ and $beta $ be uniformly random permutations of orders $2$ and $3$, respectively, in $S_{N}$, and consider, say, the permutation $alpha beta alpha beta ^{-1}$. How many fixed points does this random permutation have on average? The current paper studies questions of this kind and relates them to surprising topological and algebraic invariants of elements in free products of groups. Formally, 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 $mathbb{E}big [textrm{fix}_{gamma }big (Nbig )big ]$, 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 $mathbb{E}big [textrm{fix}_{gamma }big (Nbig )big ]$ as $Nto infty $ is an integer, and is equal to the number of subgroups $Hle Gamma $ containing $gamma $ such that $Hcong mathbb{Z}$ or $Hcong C_{2}*C_{2}$. Equivalently, this is the number of subgroups $Hle Gamma $ containing $gamma $ and having (rational) Euler characteristic zero. We also prove there is an asymptotic expansion for $mathbb{E}big [textrm{fix}_{gamma }big (Nbig )big ]$ and determine the limit distribution of the number of fixed points as $Nto infty $. These results are then generalized to all statistics of cycles of fixed lengths." @default.
• W4386689546 created "2023-09-13" @default.
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• W4386689546 date "2023-09-11" @default.
• W4386689546 modified "2023-09-27" @default.
• W4386689546 title "Local Statistics of Random Permutations from Free Products" @default.
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• W4386689546 doi "https://doi.org/10.1093/imrn/rnad207" @default.
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