A few remarks on the octopus inequality and Aldous' spectral gap\n conjecture

2013/10/23 by Filippo Cesi, Cesi, Filippo · 1 citation
Mathematics · #05C25 #05C50 #20C30 #60K35 #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Geometric and Algebraic Topology #Graph theory and applications #Probability (math.PR) #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.1310.6156

openalex publication_date 2013/10/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A conjecture by D. Aldous, which can be formulated as a statement about the\nfirst nontrivial eigenvalue of the Laplacian of certain Cayley graphs on the\nsymmetric group generated by transpositions, has been recently proven by\nCaputo, Liggett and Richthammer. Their proof is a subtle combination of two\ningredients: a nonlinear mapping in the group algebra of the symmetric groups\nwhich permits a proof by induction, and a quite hard estimate named the octopus\ninequality. In this paper we present a simpler and more transparent proof of\nthe octopus inequality, which emerges naturally when looking at the Aldous'\nconjecture from an algebraic perspective. We also show that the analogous of\nthe Aldous' conjecture, where the spectral gap is replaced by the Kazhdan\nconstant, does not hold in general.\n

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• The Aldous property for normal Cayley graphs on symmetric groups

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