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

Hugo Duminil-Copin; Clément Hongler; Pierre Nolin

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arXiv · 0912.4253

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

Hugo Duminil-Copin·Clément Hongler·Pierre Nolin

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Abstract

We prove Russo-Seymour-Welsh-type uniform bounds on crossing probabilities for the FK Ising model at criticality, independent of the boundary conditions. Our proof relies mainly on Smirnov's fermionic observable for the FK Ising model, which allows us to get precise estimates on boundary connection probabilities. It remains purely discrete, in particular we do not make use of any continuum limit, and it can be used to derive directly several noteworthy results - some new and some not - among which the fact that there is no spontaneous magnetization at criticality, tightness properties for the interfaces, and the existence of several critical exponents, in particular the half-plane one-arm exponent.

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Hugo Duminil-Copin, Clément Hongler, Pierre Nolin. 2009-12-21. Connection probabilities and RSW-type bounds for the FK Ising model. https://arxiv.org/abs/0912.4253

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