Word Measures on Symmetric Groups
Liam Hanany, Doron Puder

TL;DR
This paper investigates the properties of word-induced permutations in symmetric groups, revealing how subgroup ranks influence permutation statistics and eigenvalues of associated Schreier graphs.
Contribution
It extends previous results by linking subgroup rank to all stable characters of symmetric groups and analyzes spectral properties of Schreier graphs with random generators.
Findings
Average number of t-cycles approaches 1/t with error term depending on subgroup rank
Second eigenvalue of Schreier graphs is asymptotically bounded by 2√(2r-1)+ε
Subgroup rank π(w) governs estimates of stable characters in symmetric groups
Abstract
Fix a word in a free group on generators. A -random permutation in the symmetric group is obtained by sampling independent uniformly random permutations and evaluating . In [arXiv:1104.3991, arXiv:1202.3269] it was shown that the average number of fixed points in a -random permutation is , where is the smallest rank of a subgroup containing as a non-primitive element. We show that plays a role in estimates of all stable characters of symmetric groups. In particular, we show that for all , the average number of -cycles is . As an application, we prove that for every , every and every large enough ,…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
