Presented on October 2, 2017 at 11:00 a.m. in the Klaus Advanced Computing Building, room 1116E.
The critical slowdown picture of Glauber dynamics for the 2D Ising model is by now fairly well understood. For @math , the dynamics on an @math torus has @math via the work of Martinelli and Olivieri @cite_42 @cite_45 and Martinelli, Olivieri and Schonmann @cite_40 , showing that, in this regime, there is a uniform bound on the inverse gap (in fact under arbitrary boundary conditions; see [ .7] Martinelli97 ). That this dynamics has @math at any @math for some @math was shown by Chayes, Chayes and Schonmann @cite_18 , and thereafter with the sharp @math by Cesi @cite_27 . Finally, a polynomial upper bound on @math at @math was given in the aforementioned paper @cite_25 ; establishing the correct (believed to be universal and approximately @math ; cf. @cite_25 and its references) remains a challenging open problem.
@cite_18 @cite_42 @cite_27 @cite_40 @cite_45 @cite_25
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We consider a Glauber dynamics reversible with respect to the two-dimensional Ising model in a finite square of sideL, in the absence of an external field and at large inverse temperature β. We first consider the gap in the spectrum of the generator of the dynamics in two different cases: with plus and open boundary conditions. We prove that, when the symmetry under global spin flip is broken by the boundary conditions, the gap is much larger than the case in which the symmetry is present. For this latter we compute exactly the asymptotics of −(1 βL) log(gap) asL→∞ and show that it coincides with the surface tension along one of the coordinate axes. As a consequence we are able to study quite precisely the large deviations in time of the magnetization and to obtain an upper bound on the spin-spin time correlation in the infinite-volume plus phase. Our results establish a connection between the dynamical large deviations and those of the equilibrium Gibbs measure studied by Shlosman in the framework of the rigorous description of the Wulff shape for the Ising model. Finally we show that, in the case of open boundary conditions, it is possible to rescale the time withL in such a way that, asL→∞, the finite-dimensional distributions of the time-rescaled magnetization converge to those of a symmetric continuous-time Markov chain on the two-state space −m*(β),m*(β) ,m*(β) being the spontaneous magnetization. Our methods rely upon a novel combination of techniques for bounding from below the gap of symmetric Markov chains on complicated graphs, developed by Jerrum and Sinclair in their Markov chain approach to hard computational problems, and the idea of introducing “block Glauber dynamics” instead of the standard single-site dynamics, in order to put in evidence more effectively the effect of the boundary conditions in the approach to equilibrium. We consider Glauber dynamics (starting from an extremal configuration) in a monotone spin system, and show that interjecting extra updates cannot increase the expected Hamming distance or the total variation distance to the stationary distribution. We deduce that for monotone Markov random fields, when block dynamics contracts a Hamming metric, single-site dynamics mixes in O(n log n) steps on an n-vertex graph. In particular, our result completes work of Kenyon, Mossel and Peres concerning Glauber dynamics for the Ising model on trees. Our approach also shows that on bipartite graphs, alternating updates systematically between odd and even vertices cannot improve the mixing time by more than a factor of log n compared to updates at uniform random locations on an n-vertex graph. Our result is especially effective in comparing block and single-site dynamics; it has already been used in works of Martinelli, Toninelli, Sinclair, Mossel, Sly, Ding, Lubetzky, and Peres in various combinations. We study discrete time Glauber dynamics for random configurations with local constraints (e.g. proper coloring, Ising and Potts models) on finite graphs with n vertices and of bounded degree. We show that the relaxation time (defined as the reciprocal of the spectral gap 1 - _2 for the dynamics on trees and on certain hyperbolic graphs is polynomial in n. For these hyperbolic graphs, this yields a general polynomial sampling algorithm for random configurations. We then show that if the relaxation time T2 satisfies T2 = O(n), then the correlation coefficient, and the mutual information, between any local function (which dependsonly on the configuration in a fixed window) and the boundary conditions, decays exponentially in the distance between the window and the boundary. For the Ising model on a regular tree, this condition is sharp. We consider the Glauber dynamics for the 2D Ising model in a box of side L, at inverse temperature β and random boundary conditions τ whose distribution P either stochastically dominates the extremal plus phase (hence the quotation marks in the title) or is stochastically dominated by the extremal minus phase. A particular case is when P is concentrated on the homogeneous configuration identically equal to + (equal to −). For β large enough we show that for any ( >0 ) there exists ( c=c( , ) ) such that the corresponding mixing time T mix satisfies ( lim _ L , P (T_ mix exp ( cL^ ) ) =0 ). In the non-random case τ ≡ + (or τ ≡ −), this implies that ( T_ mix exp ( cL^ ) ). The same bound holds when the boundary conditions are all + on three sides and all − on the remaining one. The result, although still very far from the expected Lifshitz behavior T mix = O(L 2), considerably improves upon the previous known estimates of the form ( T_ mix exp ( c L^ 12 + ) ). The techniques are based on induction over length scales, combined with a judicious use of the so-called “censoring inequality” of Y. Peres and P. Winkler, which in a sense allows us to guide the dynamics to its equilibrium measure.