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

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 Bjornberg 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 Z(d) have shown behaviour with strong similarities to what is proven for trees.