Aldous's spectral gap conjecture for normal sets

Ori Parzanchevski, Doron Puder

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Abstract

Let Sn denote the symmetric group on n elements, and let Σ ⊆ Sn be a symmetric subset of permutations. Aldous's spectral gap conjecture, proved by Caputo, Liggett, and Richthammer [J. Amer. Math. Soc. 23 (2010), no. 3, 831-851], states that if Σ is a set of transpositions, then the second eigenvalue of the Cayley graph Cay(Sn, Σ) is identical to the second eigenvalue of the Schreier graph on n vertices depicting the action of Sn on {1, . . ., n}. Inspired by this seminal result, we study similar questions for other types of sets in Sn. Specifically, we consider normal sets: sets that are invariant under conjugation. Relying on character bounds due to Larsen and Shalev [Invent. Math. 174 (2008), no. 3, 645-687], we show that for large enough n, if Σ ⊂ Sn is a full conjugacy class, then the second eigenvalue of Cay(Sn, Σ) is roughly identical to the second eigenvalue of the Schreier graph depicting the action of Sn on ordered 4-tuples of elements from {1, . . ., n}. We further show that this type of result does not hold when Σ is an arbitrary normal set, but a slightly weaker one does hold. We state a conjecture in the same spirit regarding an arbitrary symmetric set Σ ⊂ Sn, which yields surprisingly strong consequences.

Original languageAmerican English
Pages (from-to)7067-7086
Number of pages20
JournalTransactions of the American Mathematical Society
Volume373
Issue number10
DOIs
StatePublished - Oct 2020

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