On the Six-Vertex Model's Free Energy.

In this paper, we provide new proofs of the existence and the condensation of Bethe roots for the Bethe Ansatz equation associated with the six-vertex model with periodic boundary conditions and an arbitrary density of up arrows (per line) in the regime . As an application, we provide a short, fully rigorous computation of the free energy of the six-vertex model on the torus, as well as an asymptotic expansion of the six-vertex partition functions when the density of up arrows approaches 1/2. This latter result is at the base of a number of recent results, in particular the rigorous proof of continuity/discontinuity of the phase transition of the random-cluster model, the localization/delocalization behaviour of the six-vertex height function when and , and the rotational invariance of the six-vertex model and the Fortuin-Kasteleyn percolation.

Planar random-cluster model: fractal properties of the critical phase.

This paper is studying the critical regime of the planar random-cluster model on with cluster-weight . More precisely, we prove which are uniform in their boundary conditions and depend only on their extremal lengths. They imply in particular that any fractal boundary is touched by macroscopic clusters, uniformly in its roughness or the configuration on the boundary.

Uniform Lipschitz Functions on the Triangular Lattice Have Logarithmic Variations.

Uniform integer-valued Lipschitz functions on a domain of size  of the triangular lattice are shown to have variations of order  . The level lines of such functions form a loop (2) model on the edges of the hexagonal lattice with edge-weight one. An infinite-volume Gibbs measure for the loop (2) model is constructed as a thermodynamic limit and is shown to be unique.