Cutoff for the Ising model on the lattice

Understand

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.