Existence of a phase transition of the interchange process on the Hamming graph

2016/05/11 by Batı Şengül, Bati Sengul, Piotr Miłoś +3
Mathematics · Physics and Astronomy · #FOS: Mathematics #FOS: Physical sciences #Markov Chains and Monte Carlo Methods #Mathematical Physics (math-ph) #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #math-ph #math.MP #math.PR

paper · pdf · doi:10.48550/arxiv.1605.03548

22 pages, 2 figures

arxiv created 2016/05/11 · openalex publication_date 2016/05/11 · arxiv updated 2016/05/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The interchange process on a finite graph is obtained by placing a particle on each vertex of the graph, then at rate 1, selecting an edge uniformly at random and swapping the two particles at either end of this edge. In this paper we develop new techniques to show the existence of a phase transition of the interchange process on the 2-dimensional Hamming graph. We show that in the subcritical phase, all of the cycles of the process have length O(log n), whereas in the supercritical phase a positive density of vertices lie in cycles of length at least n^2-ε for any ε>0.

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