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.
Citation
Alan Hammond. Milind Hegde. "Critical point for infinite cycles in a random loop model on trees." Ann. Appl. Probab. 29 (4) 2067 - 2088, August 2019. https://doi.org/10.1214/18-AAP1442
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