Phase transition for the interchange and quantum Heisenberg models on the Hamming graph

Abstract

We study a family of random permutation models on the Hamming graph $H(2,n)$ (i.e., the $2$-fold Cartesian product of complete graphs), containing the interchange process and the cycle-weighted interchange process with parameter $\theta >0$. This family contains the random walk representation of the quantum Heisenberg ferromagnet. We show that in these models the cycle structure of permutations undergoes a \textit{phase transition} -- when the number of transpositions defining the permutation is $\le c{n}^2$, for small enough $c>0$, all cycles are microscopic, while for more than $\ge C{n}^2$ transpositions, for large enough $C>0$, macroscopic cycles emerge with high probability. We provide bounds on values $C,c$ depending on the parameter $\theta$ of the model, in particular for the interchange process we pinpoint exactly the critical time of the phase transition. Our results imply also the existence of a phase transition in the quantum Heisenberg ferromagnet on $H(2,n)$, namely for low enough temperatures spontaneous magnetization occurs, while it is not the case for high temperatures. At the core of our approach is a novel application of the cyclic random walk, which might be of independent interest. By analyzing explorations of the cyclic random walk, we show that sufficiently long cycles of a random permutation are uniformly spread on the graph, which makes it possible to compare our models to the mean-field case, i.e., the interchange process on the complete graph, extending the approach used earlier by Schramm.

arXiv comment54 pages, 3 figures, journal version

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• The interchange process with reversals on the complete graph
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Cited by in corpus (3)

• The interchange process with reversals on the complete graph
• The mean-field quantum Heisenberg ferromagnet via representation theory
• Critical Parameters for Loop and Bernoulli Percolation