  Introduced in 1963, Glauber dynamics is one of the most practiced and
extensively studied methods for sampling the Ising model on lattices. It is
well known that at high temperatures, the time it takes this chain to mix in
$L^1$ on a system of size $n$ is $O(\log n)$. Whether in this regime there is
cutoff, i.e. a sharp transition in the $L^1$-convergence to equilibrium, is a
fundamental open problem: If so, as conjectured by Peres, it would imply that
mixing occurs abruptly at $(c+o(1))\log n$ for some fixed $c>0$, thus providing
a rigorous stopping rule for this MCMC sampler. However, obtaining the precise
asymptotics of the mixing and proving cutoff can be extremely challenging even
for fairly simple Markov chains. Already for the one-dimensional Ising model,
showing cutoff is a longstanding open problem.
  We settle the above by establishing cutoff and its location at the high
temperature regime of the Ising model on the lattice with periodic boundary
conditions. Our results hold for any dimension and at any temperature where
there is strong spatial mixing: For $\Z^2$ this carries all the way to the
critical temperature. Specifically, for fixed $d\geq 1$, the continuous-time
Glauber dynamics for the Ising model on $(\Z/n\Z)^d$ with periodic boundary
conditions has cutoff at $(d/2\lambda_\infty)\log n$, where $\lambda_\infty$ is
the spectral gap of the dynamics on the infinite-volume lattice. To our
knowledge, this is the first time where cutoff is shown for a Markov chain
where even understanding its stationary distribution is limited.
  The proof hinges on a new technique for translating $L^1$ to $L^2$ mixing
which enables the application of log-Sobolev inequalities. The technique is
general and carries to other monotone and anti-monotone spin-systems.
