The Annals of Applied Probability

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

Alan Hammond and Milind Hegde

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Cyclic time random walk random stirring model


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.

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