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.