Macroscopic cycles for the interchange process on random regular graphs

The interchange process can be seen as a random walk on permutations of a graph: at rate 1, independently on every edge, we exchange the value on both endpoints of the edge. This model 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.