Long-Range Order for Critical Book-Ising and Book-Percolation.

In this paper, we investigate the behaviour of statistical physics models on a book with pages that are isomorphic to half-planes. We show that even for models undergoing a continuous phase transition on , the phase transition becomes discontinuous as soon as the number of pages is sufficiently large. In particular, we prove that the Ising model on a three pages book has a discontinuous phase transition (if one allows oneself to consider large coupling constants along the line on which pages are glued).

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.