On the AldousCaputo Spectral Gap Conjecture for Hypergraphs
Abstract
In their celebrated paper (arXiv:0906.1238), Caputo, Liggett and Richthammer proved Aldous' conjecture and showed that for an arbitrary finite graph, the spectral gap of the interchange process is equal to the spectral gap of the underlying random walk. A crucial ingredient in the proof was the Octopus Inequality  a certain inequality of operators in the group ring $\mathbb{R}[S_n]$ of the symmetric group. Here we generalize the Octopus Inequality and apply it to generalize the CaputoLiggettRichthammer Theorem to certain hypergraphs, proving some cases of a conjecture of Caputo.
 Publication:

arXiv eprints
 Pub Date:
 November 2023
 DOI:
 10.48550/arXiv.2311.02505
 arXiv:
 arXiv:2311.02505
 Bibcode:
 2023arXiv231102505A
 Keywords:

 Mathematics  Group Theory;
 Mathematics  Combinatorics;
 Mathematics  Probability;
 Mathematics  Representation Theory;
 20c30 (Primary) 05c81;
 05c65;
 20B30;
 05c50;
 60b15;
 60j10;
 60k35(Secondary)
 EPrint:
 32 pages + 9 pages of Appendix containing code in SageMath