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Sharp phase transition in the random stirring model on trees

We establish that the phase transition for infinite cycles in the random stirring model on an infinite regular tree of high degree is sharp. That is, we prove that there exists d_0 such that, for any d \geq d_0, the set of parameter values at which the random stirring model on the rooted regular tree with offspring degree d almost surely contains an infinite cycle consists of a semi-infinite interval. The critical point at the left-hand end of this interval is at least 1/d + 1/(2d^2) and at most 1/d + 2/(d^2). This version is a major revision, with a much shorter proof. Principal among the changes are a reworking of the argument in Section 4 of the old version, which was proposed by a referee, and the use of a simpler means of handling a boundary case, which eliminates the previous Section 6.