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Register a new userAldous Spectral Gap Conjecture for Normal Sets
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Let $S_n$ denote the symmetric group on $n$ elements, and $Sigmasubseteq S_{n}$ a symmetric subset of permutations. Aldous spectral gap conjecture, proved by Caputo, Liggett and Richthammer [arXiv:0906.1238], states that if $Sigma$ is a set of transpositions, then the second eigenvalue of the Cayley graph $mathrm{Cay}left(S_{n},Sigmaright)$ is identical to the second eigenvalue of the Schreier graph on $n$ vertices depicting the action of $S_{n}$ on $left{ 1,ldots,nright}$. Inspired by this seminal result, we study similar questions for other types of sets in $S_{n}$. Specifically, we consider normal sets: sets that are invariant under conjugation. Relying on character bounds due to Larsen and Shalev [2008], we show that for large enough $n$, if $Sigmasubset S_{n}$ is a full conjugacy class, then the second eigenvalue of $mathrm{Cay}left(S_{n},Sigmaright)$ is roughly identical to the second eigenvalue of the Schreier graph depicting the action of $S_{n}$ on ordered $4$-tuples of elements from $left{ 1,ldots,nright}$. We further show that this type of result does not hold when $Sigma$ 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 $Sigmasubset S_{n}$, which yields surprisingly strong consequences.
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Aldous spectral gap conjecture asserts that on any graph the random walk process and the random transposition (or interchange) process have the same spectral gap. We prove the conjecture using a recursive strategy. The approach is a natural extension of the method already used to prove the validity of the conjecture on trees. The novelty is an idea based on electric network reduction, which reduces the problem to the proof of an explicit inequality for a random transposition operator involving both positive and negative rates. The proof of the latter inequality uses suitable coset decompositions of the associated matrices on permutations.
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