A simple renormalization flow for FK-percolation models

We present a setup that enables to define in a concrete way a renormalization flow for the FK-percolation models from statistical physics (that are closely related to Ising and Potts models). In this setting that is applicable in any dimension of space, one can interpret perturbations of the critical (conjectural) scaling limits in terms of stationary distributions for rather simple Markov processes on spaces of abstract discrete weighted graphs.Comment: 12 pages, to appear in the Jean-Michel Bismut 65th anniversary volum

SLEs as boundaries of clusters of Brownian loops

In this research announcement, we show that SLE curves can in fact be viewed as boundaries of certain simple Poissonian percolation clusters: Recall that the Brownian loop-soup (introduced in the paper arxiv:math.PR/0304419 with Greg Lawler) with intensity c defines a Poissonian collection of (simple if one focuses only on the outer boundary) loops in a domain. This random family of (possibly intersecting) loops is conformally invariant (and there are almost surely infinitely many small loops in any sample). We show that there exists a critical value a in (0,1] such that if one colors all the interiors of the loops, the obtained clusters are bounded when ca, one single cluster fills the domain. We prove that for small c, the outer boundaries of the clusters are SLE-type curves where $\kappa \le 4$ and $c$ related by the usual relation $c=(3\kappa-8)(6-\kappa)/2\kappa$ (i.e. c corresponds to the central charge of the model). Conjecturally, the critical value a is equal to one and corresponds to SLE4 loops, so that this should give for any c in (0,1] a construction of a natural countable family of random disjoint SLE$_\kappa$ loops (i.e. $\kappa$ should span $(8/3,4]$), that behaves ``nicely'' under perturbation of the domain. A precise relation between chordal SLE and the loop-soup goes as follows: Consider the sample of a certain restriction measure (i.e. a certain union of Brownian excursions) in a domain, attach to it all the above-described clusters that it intersects. The outer boundary of the obtained set is exactly an SLE$_\kappa$, if the restriction measure exponent is equal to the highest-weight of the corresponding representation with central charge c.Comment: Research anouncement, to appear in C. R. Acad. Sci. Pari

Random soups, carpets and fractal dimensions

We study some properties of a class of random connected planar fractal sets induced by a Poissonian scale-invariant and translation-invariant point process. Using the second-moment method, we show that their Hausdorff dimensions are deterministic and equal to their expectation dimension. We also estimate their low-intensity limiting behavior. This applies in particular to the "conformal loop ensembles" defined via Poissonian clouds of Brownian loops for which the expectation dimension has been computed by Schramm, Sheffield and Wilson.Comment: To appear in J. London Math. So