20.1 Glauber Dynamics for the 2-d Ising Model

Last time we showed that the mixing time is e √ n) for sufficiently large (but finite) β (though not all the way down to βc). Now we will show that the mixing time is O(n log n) for sufficiently small (but finite) β (again, not all the way up to βc). Getting both of these results to go all the way to βc is rather challenging and beyond the scope of this course. We also mention the following interesting conjecture: Conjecture: For the Ising model with the + (or −) boundary condition, the mixing time is poly(n) for all β > 0. [Intuition: The obstacle to rapid mixing for β > βc is the bottleneck between the plus-phase and the minus-phase; but one of the phases disappears with such a boundary condition.] We now turn to the result claimed above, that the mixing time is O(n log n) for sufficiently small β. Let d(X,Y ) denote the number of disagreements between the configurations X,Y . We use path coupling, meaning that we need consider only pairs X,Y that have one disagreement. Consider Xt, Yt with disagreement only at i0. Define the coupling in which Xt and Yt always pick the same site i and update it “optimally” (i.e., so as to maximize the probability of agreement).