## Electronic Journal of Probability

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

#### 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
Accepted: 19 April 2018
First available in Project Euclid: 22 June 2019

https://projecteuclid.org/euclid.ejp/1561169148

Digital Object Identifier
doi:10.1214/18-EJP171

Mathematical Reviews number (MathSciNet)
MR3978214

Zentralblatt MATH identifier
07089002

#### 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

#### References

• [1] M. Ajtai, J. Komlós, and E. Szemerédi. Largest random component of a $k$-cube. Combint., 2(1):1–7, 1982.
• [2] O. Angel. Random infinite permutations and the cyclic time random walk. In Discrete random walks (Paris, 2003), Discrete Math. Theor. Comput. Sci. Proc., AC, pages 9–16 (electronic). Assoc. Discrete Math. Theor. Comput. Sci., Nancy, 2003.
• [3] C. Benassi, J. Fröhlich, and D. Ueltschi. Decay of Correlations in 2D Quantum Systems with Continuous Symmetry. Ann. H. Poinc., 18(9):2831–2847, 2017.
• [4] N. Berestycki. Emergence of giant cycles and slowdown transition in random transpositions and $k$-cycles. Electron. J. Probab., 16:no. 5, 152–173, 2011.
• [5] N. Berestycki and G. Kozma. Cycle structure of the interchange process and representation theory. Bull. Soc. Math. France, 143(2):265–280, 2015.
• [6] J. Björnberg. Large cycles in random permutations related to the Heisenberg model. Electron. C. Probab., 20:no. 55, 1–11, 2015.
• [7] Rick Durrett. Random graph dynamics. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge, 2010.
• [8] A. Gladkich and R. Peled. On the Cycle Structure of Mallows Permutations. Ann. Probab., 46(2), 1114-1169, 2018.
• [9] A. Hammond. Infinite cycles in the random stirring model on trees. Bull. Inst. Math. Acad. Sin. (N.S.), 8(1):85–104, 2013.
• [10] A. Hammond. Sharp phase transition in the random stirring model on trees. Probab. Th. Rel. Fields, 161(3-4):429–448, 2015.
• [11] T.E. Harris. Nearest-neighbor Markov interaction processes on multidimensional lattices. Advances in Mathematics, 9(1):66–89, aug 1972.
• [12] R. Kotecký, P. Milos, and D. Ueltschi. The random interchange process on the hypercube. Electron. C. Probab., 21:no. 4, 1–9, 2016.
• [13] O. Schramm. Compositions of random transpositions. Israel J. Math., 147:221–243, 2005.
• [14] B. Tóth. Improved lower bound on the thermodynamic pressure of the spin $1/2$ Heisenberg ferromagnet. Lett. Math. Phys., 28(1):75–84, 1993.