Large cycles in random permutations related to the Heisenberg model

TL;DR

This paper investigates the emergence of large cycles in a weighted permutation process linked to the quantum Heisenberg ferromagnet, showing large cycles form at low temperatures for certain parameters.

Contribution

It establishes a connection between large cycle formation and low temperature regimes in a weighted permutation model related to quantum spin systems.

Findings

Large cycles appear at low temperature for >1.

The model extends Tth's representation of the quantum Heisenberg ferromagnet.

The study provides conditions for phase transition in cycle sizes.

Abstract

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'.