## The Annals of Applied Probability

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

#### 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 Björnberg and Ueltschi [Ann. Appl. Probab. 28 (2018) 2063–2082], 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 [J. Phys. A 48 Article ID 345002] of the model on $\mathbb{Z}^{d}$ have shown behaviour with strong similarities to what is proven for trees.

#### Article information

Source
Ann. Appl. Probab., Volume 29, Number 4 (2019), 2067-2088.

Dates
Revised: October 2018
First available in Project Euclid: 23 July 2019

https://projecteuclid.org/euclid.aoap/1563869037

Digital Object Identifier
doi:10.1214/18-AAP1442

Mathematical Reviews number (MathSciNet)
MR3983335

Zentralblatt MATH identifier
07120703

#### Citation

Hammond, Alan; Hegde, Milind. Critical point for infinite cycles in a random loop model on trees. Ann. Appl. Probab. 29 (2019), no. 4, 2067--2088. doi:10.1214/18-AAP1442. https://projecteuclid.org/euclid.aoap/1563869037

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