Presented on October 2, 2017 at 11:00 a.m. in the Klaus Advanced Computing Building, room 1116E.
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.
@cite_24 @cite_44 @cite_7
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We give the first comprehensive analysis of the effect of boundary conditions on the mixing time of the Glauber dynamics in the so-called Bethe approximation. Specifically, we show that the spectral gap and the log-Sobolev constant of the Glauber dynamics for the Ising model on an n-vertex regular tree with (+)-boundary are bounded below by a constant independent of n at all temperatures and all external fields. This implies that the mixing time is O(logn) (in contrast to the free boundary case, where it is not bounded by any fixed polynomial at low temperatures). In addition, our methods yield simpler proofs and stronger results for the spectral gap and log-Sobolev constant in the regime where the mixing time is insensitive to the boundary condition. Our techniques also apply to a much wider class of models, including those with hard-core constraints like the antiferromagnetic Potts model at zero temperature (proper colorings) and the hardcore lattice gas (independent sets). We consider the two-dimensional stochastic Ising model in finite square  with free boundary conditions, at inverse temperature >0 and zero external field. Using duality and recent results of Ioffe on the Wulff construction close to the critical temperature, we extend some of the results obtained by Martinelli in the low-temperature regime to any temperature below the critical one. In particular we show that the gap in the spectrum of the generator of the dynamics goes to zero in the thermodynamic limit as an exponential of the side length of , with a rate constant determined by the surface tension along one of the coordinate axes. We also extend to the same range of temperatures the result due to Shlosman on the equilibrium large deviations of the magnetization with free boundary conditions. We consider the Glauber dynamics for the 2D Ising model in a box of side L, at inverse temperature  and random boundary conditions  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  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   + (or   ), 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.