  On any locally-finite geometry, the stochastic Ising model is known to be
contractive when the inverse-temperature $\beta$ is small enough, via classical
results of Dobrushin and of Holley in the 1970's. By a general principle
proposed by Peres, the dynamics is then expected to exhibit cutoff. However, so
far cutoff for the Ising model has been confirmed mainly for lattices, heavily
relying on amenability and log Sobolev inequalities. Without these, cutoff was
unknown at any fixed $\beta>0$, no matter how small, even in basic examples
such as the Ising model on a binary tree or a random regular graph.
  We use the new framework of information percolation to show that, in any
geometry, there is cutoff for the Ising model at high enough temperatures.
Precisely, on any sequence of graphs with maximum degree $d$, the Ising model
has cutoff provided that $\beta<\kappa/d$ for some absolute constant $\kappa$
(a result which, up to the value of $\kappa$, is best possible). Moreover, the
cutoff location is established as the time at which the sum of squared
magnetizations drops to 1, and the cutoff window is $O(1)$, just as when
$\beta=0$.
  Finally, the mixing time from almost every initial state is not more than a
factor of $1+\epsilon_\beta$ faster then the worst one (with
$\epsilon_\beta\to0$ as $\beta\to 0$), whereas the uniform starting state is at
least $2-\epsilon_\beta$ times faster.
