Cutoff for the Ising model on the lattice
Introduced in 1963, Glauber dynamics is one of the most practiced and
extensively studied methods for sampling the Ising model on lattices. At
high temperatures, the time it takes this chain to mix in $L^1$ on a
system of size $n$ is $O(\log n)$. It was conjectured that in this regime
there is *cutoff*, i.e. a sharp transition in the $L^1$-convergence to
equilibrium. We settle the above by establishing cutoff and its location
in 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.
The proof hinges on a new technique for translating $L^1$-mixing to
$L^2$-mixing of projections of the chain combined with the application of
logarithmic-Sobolev inequalities. The technique is general and carries to
other monotone and anti-monotone spin-systems, e.g.\ gas hard-core, Potts,
anti-ferromagentic Ising, arbitrary boundary conditions and other graphs.
Joint work with Eyal Lubetzky.