Electronic Journal of Probability

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

Piotr Miłoś and Batı Şengül

Full-text: Open access

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 $\varepsilon >0$.

Article information

Source
Electron. J. Probab., Volume 24 (2019), paper no. 64, 21 pp.

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

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1561169148

Digital Object Identifier
doi:10.1214/18-EJP171

Mathematical Reviews number (MathSciNet)
MR3978214

Zentralblatt MATH identifier
07089002

Subjects
Primary: 60G99: None of the above, but in this section 81S99: None of the above, but in this section 82B99: None of the above, but in this section

Keywords
random permutation phase transition Hamming graph

Rights
Creative Commons Attribution 4.0 International License.

Citation

Miłoś, Piotr; Şengül, Batı. Existence of a phase transition of the interchange process on the Hamming graph. Electron. J. Probab. 24 (2019), paper no. 64, 21 pp. doi:10.1214/18-EJP171. https://projecteuclid.org/euclid.ejp/1561169148


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