  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.
