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2019 Existence of a phase transition of the interchange process on the Hamming graph
Piotr Miłoś, Batı Şengül
Electron. J. Probab. 24(none): 1-21 (2019). DOI: 10.1214/18-EJP171

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 lies in cycles of length at least $n^{2-\varepsilon }$ for any $\varepsilon >0$.

Citation

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Piotr Miłoś. Batı Şengül. "Existence of a phase transition of the interchange process on the Hamming graph." Electron. J. Probab. 24 1 - 21, 2019. https://doi.org/10.1214/18-EJP171

Information

Received: 19 May 2017; Accepted: 19 April 2018; Published: 2019
First available in Project Euclid: 22 June 2019

zbMATH: 07089002
MathSciNet: MR3978214
Digital Object Identifier: 10.1214/18-EJP171

Subjects:
Primary: 60G99, 81S99, 82B99

JOURNAL ARTICLE
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Vol.24 • 2019
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