  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.
