Word Measures on Symmetric Groups
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arXiv
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| Format: | Preprint |
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2020
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| _version_ | 1866915767453220864 |
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| author | Hanany, Liam Puder, Doron |
| author_facet | Hanany, Liam Puder, Doron |
| contents | Fix a word $w$ in a free group $F$ on $r$ generators. A $w$-random permutation in the symmetric group $S_N$ is obtained by sampling $r$ independent uniformly random permutations $σ_{1},\ldots,σ_{r}\in S_{N}$ and evaluating $w\left(σ_{1},\ldots,σ_{r}\right)$. In [arXiv:1104.3991, arXiv:1202.3269] it was shown that the average number of fixed points in a $w$-random permutation is $1+θ\left(N^{1-π\left(w\right)}\right)$, where $π\left(w\right)$ is the smallest rank of a subgroup $H\le F$ containing $w$ as a non-primitive element. We show that $π\left(w\right)$ plays a role in estimates of all stable characters of symmetric groups. In particular, we show that for all $t\ge2$, the average number of $t$-cycles is $\frac{1}{t}+O\left(N^{-π\left(w\right)}\right)$. As an application, we prove that for every $s$, every $\varepsilon>0$ and every large enough $r$, Schreier graphs with $r$ random generators depicting the action of $S_{N}$ on $s$-tuples, have second eigenvalue at most $2\sqrt{2r-1}+\varepsilon$ asymptotically almost surely. An important ingredient in this work is a systematic study of not-necessarily connected Stallings core graphs. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2009_00897 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Word Measures on Symmetric Groups Hanany, Liam Puder, Doron Group Theory 20e05 (Primary) 20B30, 68r15, 05c48, 05c50, 05c81, 20C30 (Secondary) Fix a word $w$ in a free group $F$ on $r$ generators. A $w$-random permutation in the symmetric group $S_N$ is obtained by sampling $r$ independent uniformly random permutations $σ_{1},\ldots,σ_{r}\in S_{N}$ and evaluating $w\left(σ_{1},\ldots,σ_{r}\right)$. In [arXiv:1104.3991, arXiv:1202.3269] it was shown that the average number of fixed points in a $w$-random permutation is $1+θ\left(N^{1-π\left(w\right)}\right)$, where $π\left(w\right)$ is the smallest rank of a subgroup $H\le F$ containing $w$ as a non-primitive element. We show that $π\left(w\right)$ plays a role in estimates of all stable characters of symmetric groups. In particular, we show that for all $t\ge2$, the average number of $t$-cycles is $\frac{1}{t}+O\left(N^{-π\left(w\right)}\right)$. As an application, we prove that for every $s$, every $\varepsilon>0$ and every large enough $r$, Schreier graphs with $r$ random generators depicting the action of $S_{N}$ on $s$-tuples, have second eigenvalue at most $2\sqrt{2r-1}+\varepsilon$ asymptotically almost surely. An important ingredient in this work is a systematic study of not-necessarily connected Stallings core graphs. |
| title | Word Measures on Symmetric Groups |
| topic | Group Theory 20e05 (Primary) 20B30, 68r15, 05c48, 05c50, 05c81, 20C30 (Secondary) |
| url | https://arxiv.org/abs/2009.00897 |