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On the Aldous-Caputo Spectral Gap Conjecture for Hypergraphs

Part of: Representation theory of groups Graph theory Special processes Probability theory on algebraic and topological structures Markov processes Permutation groups

Published online by Cambridge University Press:  21 May 2025

GIL ALON ,
GADY KOZMA  and
DORON PUDER
GIL ALON
Affiliation:
Department of Mathematics and Computer Science, The Open University of Israel, 1 University Road, P. O. Box 808, Raanana, 4353701, Israel. e-mail: gilal@openu.ac.il
GADY KOZMA
Affiliation:
Faculty of Mathematics and Computer Science, Ziskind Building, Weizmann Institute, Rehovot, 7610001, Israel. e-mail: gady.kozma@weizmann.ac.il
DORON PUDER
Affiliation:
School of Mathematical Sciences, Schreiber building, Tel Aviv University, Tel Aviv, 6997801, Israel. e-mail: doronpuder@gmail.com
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Abstract

In their celebrated paper [CLR10], 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 $\mathbb{R}\left[{\mathrm{Sym}}_{n}\right]$ of the symmetric group. Here we generalise the Octopus Inequality and apply it to generalising the Caputo–Liggett–Richthammer Theorem to certain hypergraphs, proving some cases of a conjecture of Caputo.

MSC classification

Primary: 20C30: Representations of finite symmetric groups
Secondary: 05C81: Random walks on graphs 05C65: Hypergraphs 20B30: Symmetric groups 05C50: Graphs and linear algebra (matrices, eigenvalues, etc.) 60B15: Probability measures on groups or semigroups, Fourier transforms, factorization 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , First View , pp. 1 - 40
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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References

Aldous, D., Caputo, P., Durrett, R., Holroyd, A. E., Jung, P. U. and Puha, A. L.. The life and mathematical legacy of Thomas M. Liggett. Notices Amer. Math. Soc. 68(1) 2020.10.1090/noti2203CrossRefGoogle Scholar
Alon, G. and Kozma, G.. Ordering the representations of $S_n$ using the interchange process. Canad. Math. Bull. 56(1) (2013), 13–30.10.4153/CMB-2011-147-2CrossRefGoogle Scholar
Alon, G. and Kozma, G.. Comparing with octopi. Ann. Inst. H. Poincaré Probab. Statist. 56(4) (2020), 2672–2685.10.1214/20-AIHP1054CrossRefGoogle Scholar
Bristiel, A. and Caputo, P.. Entropy inequalities for random walks and permutations. Ann. Ins. Henri Poincaré (B) Probab. Stat. 60(1) (2024), 54–81.10.1214/22-AIHP1267CrossRefGoogle Scholar
Cesi, F.. A few remarks on the octopus inequality and Aldous’ spectral gap conjecture. Comm. Algebra. 44(1) (2016), 279–302.10.1080/00927872.2014.975349CrossRefGoogle Scholar
Chen, J. P.. The moving particle lemma for the exclusion process on a weighted graph. Electron. Commun. Probab. 22 (2017), 1–13.Google Scholar
Caputo, P., Liggett, T. and Richthammer, T.. Proof of Aldous’ spectral gap conjecture. J. Amer. Math. Soc. 23(3) (2010), 831–851.10.1090/S0894-0347-10-00659-4CrossRefGoogle Scholar
Diaconis, P. and Shahshahani, M.. Generating a random permutation with random transpositions. Z. Wahrscheinlichkeitstheorie verw. Gebiete 57(2) (1981), 159–179.10.1007/BF00535487CrossRefGoogle Scholar
Fulton, W. and Harris, J.. Representation theory: a first course, vol. 129. (Springer Science & Business Media, 2013).Google Scholar
Handjani, S. and Jungreis, D.. Rate of convergence for shuffling cards by transpositions. J. Theoret. Probab. 9 (1996), 983–993.Google Scholar
Marcus, A. W., Spielman, D. A. and Srivastava, N.. Ramanujan graphs and the solution of the Kadison–Singer problem. In International Congress of Mathematicans, ICM 2014, vol. 3 (2014), p. 363–386.Google Scholar
Piras, D.. Generalizations of Aldous’ Spectral Gap Conjecture. PhD. thesis. Tesi di Laurea, Universitá degli Studi Roma Tre (2010).Google Scholar
Parzanchevski, O. and Puder, D.. Aldous’s spectral gap conjecture for normal sets. Trans. Amer. Math. Soc. 373(10) (2020), 7067–7086.10.1090/tran/8155CrossRefGoogle Scholar