Abstract
We consider the FK-Ising model in two dimensions at criticality. We obtain bounds on crossing probabilities of arbitrary topological rectangles, uniform with respect to the boundary conditions, generalizing results of [DHN11] and [CS12]. Our result relies on new discrete complex analysis techniques, introduced in [Che12]. We detail some applications, in particular the computation of so-called universal exponents, the proof of quasi-multiplicativity properties of arm probabilities, and bounds on crossing probabilities for the classical Ising model.
Highlights
The Ising model is one of the simplest and most fundamental models in equilibrium statistical mechanics. It was proposed as a model for ferromagnetism by Lenz in 1920 [Len20], and studied by Ising [Isi25], in an attempt to provide a microscopic explanation for the thermodynamical behavior of magnets
We develop tools that improve the connection between the discrete Ising model and the continuous objects describing its scaling limit
Recall that the Ising model is a random assignment of ±1 spins to the vertices of a graph G, where the probability of a spin configurationx∈G is proportional to exp (−βH (σ))
Summary
The Ising model is one of the simplest and most fundamental models in equilibrium statistical mechanics. Recall that the Ising model is a random assignment of ±1 spins to the vertices of a graph G, where the probability of a spin configuration (σx)x∈G is proportional to exp (−βH (σ)). Q > 0, the FK(p, q) model on a graph G is a measure on random subgraphs of G containing all its vertices: the probability of a configuration ω ⊂ G is proportional to p 1−p e(ω) qk(ω), where e (ω) is the number of edges of ω and k (ω) the number of clusters of ω (connected components of vertices). Edwards-Sokal coupling [ES88]: if one samples an FK-Ising configuration on G, assigns a ±1 spin to each cluster by an independent fair coin toss, and gives to each vertex of. FK-Ising interfaces at criticality were proved to converge to SLE(16/3) in [CDCH+14]
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