Proof of Aldous' spectral gap conjecture

Authors:
Pietro Caputo, Thomas M. Liggett and Thomas Richthammer

Journal:
J. Amer. Math. Soc. **23** (2010), 831-851

MSC (2010):
Primary 60K35, 60J27, 05C50

DOI:
https://doi.org/10.1090/S0894-0347-10-00659-4

Published electronically:
January 26, 2010

MathSciNet review:
2629990

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Abstract: 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 with rows and columns indexed by permutations.

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Additional Information

**Pietro Caputo**

Affiliation:
Dipartimento di Matematica, Università di Roma Tre, Italy and Department of Mathematics, University of California, Los Angeles, California 90095

Email:
caputo@mat.uniroma3.it

**Thomas M. Liggett**

Affiliation:
Department of Mathematics, University of California, Los Angeles, California 90095

Email:
tml@math.ucla.edu

**Thomas Richthammer**

Affiliation:
Department of Mathematics, University of California, Los Angeles, California 90095

Email:
richthammer@math.ucla.edu

DOI:
https://doi.org/10.1090/S0894-0347-10-00659-4

Keywords:
Random walk,
weighted graph,
spectral gap,
interchange process,
symmetric exclusion process

Received by editor(s):
June 26, 2009

Published electronically:
January 26, 2010

Additional Notes:
The first author was partially supported by the Advanced Research Grant “PTRELSS” ADG-228032 of the European Research Council. He thanks Filippo Cesi for helpful discussions

Partial support from NSF Grant DMS-0301795 is acknowledged

Article copyright:
© Copyright 2010
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.