text: Presented on October 2, 2017 at 11:00 a.m. in the Klaus Advanced Computing Building, room 1116E.
  @cite_24: 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.
  @cite_44: 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.
  @cite_7: 
summary: The dynamics for critical 2D Potts FK models under boundary conditions takes after Glauber dynamics for the low temperature Ising model under plus boundary conditions. Improving on the original work of Martinelli @cite_24 , a delicate multi-scale analysis due to Martinelli and Toninelli @cite_44 , based on censoring inequalities (see ), yielded an upper bound of @math for the Ising model with plus boundary conditions. This was followed by an @math bound in @cite_7 via this approach, extended to all @math . Our proof of is based on this method.