We study fluctuations of the magnetization (average of spins) in an Ising model on a sequence of "well-connected" approximately dn regular graphs on n vertices. We show that if dn √ n, then the fluctuations are universal, and the same as that of the Curie Weiss model, in the entire Ferromagnetic parameter regime. We then give a counterexample to show that dn √ n is actually tight, in the sense that the limiting distribution changes if dn ∼ √ n except in the high temperature regime. By refining our argument, we show that in the high temperature regime universality holds for dn n 1/3. As a by product of our proof technique, we prove rates of convergence, as well as exponential concentration for the sum of spins, and tight estimates for several statistics of interest.
This is based on joint work with Nabarun Deb at Columbia University.