Glauber Dynamics for Ising Model on Convergent Dense Graph Sequences

We study the Glauber dynamics for Ising model on (sequences of) dense graphs. We view the dense graphs through the lens of graphons [19]. For the ferromagnetic Ising model with inverse temperature β on a convergent sequence of graphs {Gn} with limit graphon W we show fast mixing of the Glauber dynamics if βλ1(W ) < 1 and slow (torpid) mixing if βλ1(W ) > 1 (where λ1(W ) is the largest eigenvalue of the graphon). We also show that in the case βλ1(W ) = 1 there is insufficient information to determine the mixing time (it can be either fast or slow). 1998 ACM Subject Classification G.3 Probability and Statistics