2010/12/06 by Eyal Lubetzky, Fabio Martinelli, Lubetzky, Eyal +5
Mathematics · Physics and Astronomy · #60K35 #82C20 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR) #math-ph #math.MP #math.PR #msc:60K35 #msc:82C20
paper · pdf · doi:10.48550/arxiv.1012.1271
45 pages, 14 figures
arxiv created 2010/12/06 · arxiv updated 2015/03/17
We considerably improve upon the recent result of Martinelli and Toninelli on the mixing time of Glauber dynamics for the 2D Ising model in a box of side L at low temperature and with random boundary conditions whose distribution P stochastically dominates the extremal plus phase. An important special case is when P is concentrated on the homogeneous all-plus configuration, where the mixing time Tmix is conjectured to be polynomial in L. In [MT] it was shown that for a large enough inverse-temperature β and any ε>0 there exists c=c(β,ε) such that limL→∞P(Tmix≥ exp(c Lε))=0. In particular, for the all-plus boundary conditions and β large enough Tmix ≤ exp(c Lε). Here we show that the same conclusions hold for all β larger than the critical value βc and with exp(c Lε) replaced by Lc log L (i.e. quasi-polynomial mixing). The key point is a modification of the inductive scheme of [MT] together with refined equilibrium estimates that hold up to criticality, obtained via duality and random-line representation tools for the Ising model. In particular, we establish new precise bounds on the law of Peierls contours which quantitatively sharpen the Brownian bridge picture established e.g. in [Greenberg-Ioffe (2005)],[Higuchi (1979)],[Hryniv (1998)].