Critical point for infinite cycles in a random loop model on trees

TL;DR

This paper proves the existence of a sharp phase transition for infinite cycles in a random loop model on trees, establishing that infinite cycles occur for all sufficiently large time parameters, thus completing the phase diagram.

Contribution

It demonstrates that infinite cycles exist for all time parameters above a certain constant, confirming a sharp phase transition on trees.

Findings

Infinite cycles occur for all T greater than a constant

Established a sharp phase transition in the model

Confirmed similarities with behavior on bZ^d

Abstract

We study a spatial model of random permutations on trees with a time parameter $T>0$, a special case of which is the random stirring process. The model on trees was first analysed by Bj\"ornberg and Ueltschi[BU16], who established the existence of infinite cycles for $T$ slightly above a putatively identified critical value but left open behaviour at arbitrarily high values of $T$. We show the existence of infinite cycles for all $T$ greater than a constant, thus classifying behaviour for all values of $T$ and establishing the existence of a sharp phase transition. Numerical studies [BBBU15] of the model on $\mathbb{Z}^d$ have shown behaviour with strong similarities to what is proven for trees.