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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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)
Divisions Faculty of Science > Mathematical Sciences, Department of
Date Deposited 16 Sep 2025 12:46
Last Modified 16 Mar 2026 18:37

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