Connection probabilities and RSW-type bounds for the two-dimensional FK Ising model

• Published in: Communications on pure and applied mathematics 2011-09, Vol.64 (9), p.1165-1198
• Main Authors: Duminil-Copin, Hugo, Hongler, Clément, Nolin, Pierre
• Format: Article
• Language: English
• Subjects: Applied mathematics; Boundary conditions; Exact sciences and technology; General mathematics; General, history and biography; Global analysis, analysis on manifolds; Mathematical analysis; Mathematics; Numerical analysis; Numerical analysis. Scientific computation; Numerical linear algebra; Partial differential equations; Sciences and techniques of general use; Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds
• ISSN: 0010-3640; 1097-0312
• DOI: 10.1002/cpa.20370

We prove Russo‐Seymour‐Welsh‐type uniform bounds on crossing probabilities for the FK Ising (FK percolation with cluster weight q = 2) model at criticality, independent of the boundary conditions. Our proof relies mainly on Smirnov's fermionic observable for the FK Ising model [24], which allows us to get precise estimates on boundary connection probabilities. We stay in a discrete setting; in particular, we do not make use of any continuum limit, and our result can be used to derive directly several noteworthy properties—including some new ones—among which are the fact that there is no infinite cluster at criticality, tightness properties for the interfaces, and the existence of several critical exponents, in particular the half‐plane, one‐arm exponent. Such crossing bounds are also instrumental for important applications such as constructing the scaling limit of the Ising spin field [6] and deriving polynomial bounds for the mixing time of the Glauber dynamics at criticality [17]. © 2011 Wiley Periodicals, Inc.