Values of Brownian intersection exponents III: Two-sided exponents

This paper determines values of intersection exponents between packs of planar Brownian motions in the half-plane and in the plane that were not derived in our first two papers. For instance, it is proven that the exponent $\xi (3,3)$ describing the asymptotic decay of the probability of non-intersection between two packs of three independent planar Brownian motions each is $(73-2 \sqrt {73}) / 12$. More generally, the values of $\xi (w_1, >..., w_k)$ and \tx (w_1', ..., w_k') are determined for all $k \ge 2$, $w_1, w_2\ge 1$, $w_3, ...,w_k\in[0,\infty)$ and all $w_1',...,w_k'\in[0,\infty)$. The proof relies on the results derived in our first two papers and applies the same general methods. We first find the two-sided exponents for the stochastic Loewner evolution processes in a half-plane, from which the Brownian intersection exponents are determined via a universality argument

One-arm exponent for critical 2D percolation

The probability that the cluster of the origin in critical site percolation on the triangular grid has diameter larger than $R$ is proved to decay like $R^{-5/48}$ as $R\to\infty$

Conformal Loop Ensembles: The Markovian characterization and the loop-soup construction

For random collections of self-avoiding loops in two-dimensional domains, we define a simple and natural conformal restriction property that is conjecturally satisfied by the scaling limits of interfaces in models from statistical physics. This property is basically the combination of conformal invariance and the locality of the interaction in the model. Unlike the Markov property that Schramm used to characterize SLE curves (which involves conditioning on partially generated interfaces up to arbitrary stopping times), this property only involves conditioning on entire loops and thus appears at first glance to be weaker. Our first main result is that there exists exactly a one-dimensional family of random loop collections with this property---one for each k in (8/3,4]---and that the loops are forms of SLE(k). The proof proceeds in two steps. First, uniqueness is established by showing that every such loop ensemble can be generated by an "exploration" process based on SLE. Second, existence is obtained using the two-dimensional Brownian loop-soup, which is a Poissonian random collection of loops in a planar domain. When the intensity parameter c of the loop-soup is less than 1, we show that the outer boundaries of the loop clusters are disjoint simple loops (when c>1 there is almost surely only one cluster) that satisfy the conformal restriction axioms. We prove various results about loop-soups, cluster sizes, and the c=1 phase transition. Taken together, our results imply that the following families are equivalent: 1. The random loop ensembles traced by certain branching SLE(k) curves for k in (8/3, 4]. 2. The outer-cluster-boundary ensembles of Brownian loop-soups for c in (0, 1]. 3. The (only) random loop ensembles satisfying the conformal restriction axioms.Comment: This 91 page-long paper contains the previous versions (v2) of both papers arxiv:1006.2373 and arxiv:1006.2374 that correspond to Part I and Part II of the present paper. This merged longer paper is to appear in Annals of Mathematic

Conformal fields, restriction properties, degenerate representations and SLE

In this note, we show how to relate the Schramm-Loewner Evolution processes (SLE) to highest-weight representations of the Virasoro Algebra. The conformal restriction properties of SLE that have been recently studied in the paper arXiv:math.PR/0209343 by G. Lawler, O. Schramm and the second author play an instrumental role. In this setup, various considerations from conformal field theory can be interpreted and reformulated via SLE. This enables to make a concrete link between the two-dimensional discrete critical systems from statistical physics and conformal field theory.Comment: To appear in C.R.Acad. Sci. Paris, Ser. I Math. Minor modifications from the first versio