Quasi-polynomial mixing of the 2D stochastic Ising model with “plus” boundary up to criticality

Eyal Lubetzky ; Fabio Martinelli ; Allan Sly ; Fabio L. Toninelli

Journal of the European Mathematical Society, Tome 15 (2013) no. 2, pp. 339-386.

Résumé

We considerably improve upon the recent result of [37] 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 [37] 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(cLε))=0. In particular, for the all-plus boundary conditions and β large enough TMIX​≤exp(cLε). Here we show that the same conclusions hold for all β larger than the critical value βc​ and with exp(cLε) replaced by LclogL (i.e. quasi-polynomial mixing). The key point is a modification of the inductive scheme of [37] 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 complement the Brownian bridge picture established e.g. in [20,22,23].

Détail

DOI : 10.4171/jems/363

Classification : 60-XX, 82-XX, 00-XX
Keywords: Ising model, mixing time, phase coexistence, Glauber dynamics