Large cycles in random permutations related to the Heisenberg model

J. E. Björnberg

Published 2015-05-25Version 1

We study the weighted version of the interchange process where a permutation receives weight $\theta^{\#\mathrm{cycles}}$. For $\theta=2$ this is T\'oth's representation of the quantum Heisenberg ferromagnet on the complete graph. We prove, for $\theta>1$, that large cycles appear at `low temperature'.