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Bounding the number of self-avoiding walks: Hammersley–Welsh with polygon insertion

Duminil-Copin, Hugo ; Ganguly, Shirshendu ; Hammond, Alan ; Manolescu, Ioan

In: Annals of Probability, 2020, vol. 48, no. 4, p. 1644–1692

On the Gibbs states of the noncritical Potts model on $$\mathbb Z ^2$$

Coquille, Loren ; Duminil-Copin, Hugo ; Ioffe, Dmitry ; Velenik, Yvan

In: Probability Theory and Related Fields, 2014, vol. 158, no. 1-2, p. 477-512

Universality for the random-cluster model on isoradial graphs

Duminil-Copin, Hugo ; Li, Jhih-Huang ; Manolescu, Ioan

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,...

The Bethe Ansatz for the six-vertex and XXZ models: An exposition

Duminil-Copin, Hugo ; Gagnebin, Maxime ; Harel, Matan ; Manolescu, Ioan ; Tassion, Vincent

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...