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 consider the two-dimensional stochastic Ising model in finite square Λ with free boundary conditions, at inverse temperature β>β0 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 Λ, 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 derive uniform surface order large deviation estimates for the block magnetization in finite volume Ising (or Potts) models with plus or free (or a combination of both) boundary conditions in the phase coexistence regime ford≧3. The results are valid up to a limit of slab-thresholds, conjectured to agree with the critical temperature. Our arguments are based on the renormalization of the random cluster model withq≧1 andd≧3, and on corresponding large deviation estimates for the occurrence in a box of a largest cluster with density close to the percolation probability. The results are new even for the case of independent percolation (q=1). As a byproduct of our methods, we obtain further results in the FK model concerning semicontinuity (inp andq) of the percolation probability, the second largest cluster in a box and the tail of the finite cluster size distribution. 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 hard–core lattice gas (independent sets). We prove that the spectral gap of the Swendsen-Wang process for the Potts model on graphs with bounded degree is bounded from below by some constant times the spectral gap of any single-spin dynamics. This implies rapid mixing for the two-dimensional Potts model at all temperatures above the critical one, as well as rapid mixing at the critical temperature for the Ising model. After this we introduce a modified version of the Swendsen-Wang algorithm for planar graphs and prove rapid mixing for the two-dimensional Potts models at all non-critical temperatures. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 42, 520–535, 2013