In: Annales de l’Institut Henri Poincaré, Probabilités et Statistiques, 2020, vol. 56, no. 4, p. 2281–2300
We consider a self-avoiding walk model (SAW) on the faces of the square lattice Z2. This walk can traverse the same face twice, but crosses any edge at most once. The weight of a walk is a product of local weights: each square visited by the walk yields a weight that depends on the way the walk passes through it. The local weights are parametrised by angles θ∈[π3,2π3] and satisfy the...
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In: Bernoulli, 2020, vol. 26, no. 3, p. 1665–1705
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In: Annals of Probability, 2020, vol. 48, no. 4, p. 1644–1692
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In: Pacific Journal of Mathematics, 2019, vol. 298, no. 2, p. 267–284
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In: Electronic Journal of Probability, 2018, vol. 23, p. -
We show that the canonical random-cluster measure associated to isoradial graphs is critical for all q≥1. Additionally, we prove that the phase transition of the model is of the same type on all isoradial graphs: continuous for 1≤q≤4 and discontinuous for q>4. For 1≤q≤4, the arm exponents (assuming their existence) are shown to be the same for all isoradial graphs. In particular,...
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In: Electronic Journal of Probability, 2018, vol. 23, p. -
We prove sharpness of the phase transition for the random-cluster model with q≥1 on graphs of the form S:=G×S, where G is a planar lattice with mild symmetry assumptions, and S a finite graph. That is, for any such graph and any q≥1, there exists some parameter pc=pc(S,q), below which the model exhibits exponential decay and above which there exists a.s. an infinite cluster. The result...
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In: Probability Surveys, 2018, vol. 15, p. 102–130
In this paper, we review a few known facts on the coordinate Bethe ansatz. We present a detailed construction of the Bethe ansatz vector ψ and energy Λ, which satisfy Vψ=Λψ, where V is the transfer matrix of the six-vertex model on a finite square lattice with periodic boundary conditions for weights a=b=1 and c>0. We also show that the same vector ψ satisfies Hψ=Eψ, where H is the...
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