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

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 allow...

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Veröffentlicht in:Communications on pure and applied mathematics 2011-09, Vol.64 (9), p.1165-1198
Hauptverfasser: Duminil-Copin, Hugo, Hongler, Clément, Nolin, Pierre
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Hongler, Clément
Nolin, Pierre
description 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.
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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
title Connection probabilities and RSW-type bounds for the two-dimensional FK Ising model
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