TY - JOUR

T1 - Word Measures on Symmetric Groups

AU - Hanany, Liam

AU - Puder, Doron

N1 - Publisher Copyright:
© The Author(s) 2022. Published by Oxford University Press. All rights reserved.

PY - 2023/6/1

Y1 - 2023/6/1

N2 - Fix a word w in a free group F on r generators. A w-random permutation in the symmetric group SN is obtained by sampling r independent uniformly random permutations σ1, . . . , σr ∈ SN and evaluating w (σ1, . . . , σr). In [39, 40], it was shown that the average number of fixed points in a w-random permutation is 1 + θ (N1−π(w)), where π (w) is the smallest rank of a subgroup H ≤ F containing w as a non-primitive element. We show that π (w) plays a role in estimates of all stable characters of symmetric groups. In particular, we show that for all t ≥ 2, the average number of t-cycles is 1/t + O(N−π(w)). As an application, we prove that for every s, every ε > 0 and every large enough r, Schreier graphs with r random generators depicting the action of SN on s-tuples, have 2nd eigenvalue at most 2√2r − 1 + ε asymptotically almost surely. An important ingredient in this work is a systematic study of not necessarily connected Stallings core graphs.

AB - Fix a word w in a free group F on r generators. A w-random permutation in the symmetric group SN is obtained by sampling r independent uniformly random permutations σ1, . . . , σr ∈ SN and evaluating w (σ1, . . . , σr). In [39, 40], it was shown that the average number of fixed points in a w-random permutation is 1 + θ (N1−π(w)), where π (w) is the smallest rank of a subgroup H ≤ F containing w as a non-primitive element. We show that π (w) plays a role in estimates of all stable characters of symmetric groups. In particular, we show that for all t ≥ 2, the average number of t-cycles is 1/t + O(N−π(w)). As an application, we prove that for every s, every ε > 0 and every large enough r, Schreier graphs with r random generators depicting the action of SN on s-tuples, have 2nd eigenvalue at most 2√2r − 1 + ε asymptotically almost surely. An important ingredient in this work is a systematic study of not necessarily connected Stallings core graphs.

UR - http://www.scopus.com/inward/record.url?scp=85161978232&partnerID=8YFLogxK

U2 - 10.1093/imrn/rnac084

DO - 10.1093/imrn/rnac084

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AN - SCOPUS:85161978232

SN - 1073-7928

VL - 2023

SP - 9221

EP - 9297

JO - International Mathematics Research Notices

JF - International Mathematics Research Notices

IS - 11

ER -