Macroscopic cycles for the interchange and quantum Heisenberg models on random regular graphs

TL;DR

This paper demonstrates that for large parameters, the interchange process and quantum Heisenberg model exhibit macroscopic clusters on random regular graphs, extending known results beyond complete and two-block graphs.

Contribution

It proves the existence of macroscopic clusters for these models on random regular graphs, a significant generalization from previously studied specific graph structures.

Findings

Macroscopic clusters form in the models on random regular graphs.

Results extend known phenomena from complete and two-block graphs.

Applicable for large enough parameters.

Abstract

The interchange process is a random permutation model that was introduced as a way to study the quantum Heisenberg model. For this model, progress had been made on some specific graphs: trees, the hypercube, the Hamming graph, the complete graph and the two block graph. Here we show that for large enough parameters, both the interchange process and the quantum Heisenberg model have macroscopic clusters on random d-regular graphs. Such a result was only known for the complete graph and the two blocks graph.