Word maps, random permutations and random graphs

CASSIDY, EWAN,GEORGE (2025) Word maps, random permutations and random graphs. Doctoral thesis, Durham University.

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Abstract

The aim of this thesis is to study word maps on the symmetric group, with applications in the study of spectral properties of random regular graphs.

We establish that, if $w\in F_{r}$ is not a proper power, then $\mathop{\mathbb{E}}_{\phi_{n}\in\hom(F_{r},S_{n})}\left[\chi\left(\phi_{n}(w)\right)\right]=O\left(\frac{1}{\dim\chi}\right)$ as $n\to\infty$ , where $\chi$ is any stable irreducible character of $S_{n}$ .

We use this to prove that random sequences of representations of $F_{r}$ that factor through non--trivial irreducible representations of $S_{n}$ converge strongly to the left regular representation $\lambda:F_{r}\to U\left(\ell^{2}\left(F_{r}\right)\right)$ , for any non--trivial irreducible representation of dimension $\leq Cn^{n^{\frac{1}{20}-\delta}}$ .

An immediate consequence is that a random $2r$ --regular Schreier graph depicting the action of $r$ random permutations on $n^{\frac{1}{20}-\delta}$ --tuples of distinct elements in $[n]$ has a near optimal spectral gap, with probability $\to1$ as $n\to\infty$ .

Item Type:	Thesis (Doctoral)
Award:	Doctor of Philosophy
Faculty and Department:	Faculty of Science > Mathematical Sciences, Department of
Thesis Date:	2025
Copyright:	Copyright of this thesis is held by the author
Deposited On:	16 Sep 2025 13:46

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Word maps, random permutations and random graphs

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