  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.
