相关论文: Extremal distance and conformal radius of a CLE_4 …

We study some approximation problems on a strict subset of the circle by analytic functions of the Hardy space H2 of the unit disk (in C), whose modulus satisfy a pointwise constraint on the complentary part of the circle. Existence and…

We show that a uniformly acute triangulation of the plane is rigid under Luo's discrete conformal change, extending previous results on hexagonal triangulations. Our result is a discrete analogue of the conformal rigidity of the plane. We…

Extremal length is a conformal invariant that transfers naturally to the discrete setting, giving square tilings as a natural combinatorial analog of conformal mappings. Recent work by S. Hersonsky has explored generalizing these ideas to…

This paper studies, in dimensions greater than two, stationary diffusion processes in random environment which are small, isotropic perturbations of Brownian motion satisfying a finite range dependence. Such processes were first considered…

This paper describes joint work with Oded Schramm and Wendelin Werner establishing the values of the planar Brownian intersection exponents from which one derives the Hausdorff dimension of certain exceptional sets of planar Brownian…

We prove that properly rescaled large planar Eulerian triangulations converge to the Brownian map. This result requires more than a standard application of the methods that have been used to obtain the convergence of other families of…

For the supercritical Bernoulli bond percolation on $\mathbb{Z}^d$ ($d \geq 2$), we give a coupling between the random walk on the infinite cluster and its limit Brownian motion, such that the maximum distance between the paths during…

We show that, under certain natural assumptions, large random plane bipartite maps with a boundary converge after rescaling to a one-parameter family ($\mathrm{BD}_L$, $0 < L < \infty$) of random metric spaces homeomorphic to the closed…

The random walk loop soup is a Poissonian ensemble of lattice loops; it has been extensively studied because of its connections to the discrete Gaussian free field, but was originally introduced by Lawler and Trujillo Ferreras as a discrete…

As a first step toward a characterization of the limiting extremal process of branching Brownian motion, we proved in a recent work [Comm. Pure Appl. Math. 64 (2011) 1647-1676] that, in the limit of large time $t$, extremal particles…

We study the structure of extreme level sets of a standard one dimensional branching Brownian motion, namely the sets of particles whose height is within a fixed distance from the order of the global maximum. It is well known that such…

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.…

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…

In a previous paper, the authors proved a conjecture of Lalley and Sellke that the empirical (time-averaged) distribution function of the maximum of branching Brownian motion converges almost surely to a Gumbel distribution. The result is…

The conformal loop ensemble (CLE) has two phases: for $\kappa \in (8/3, 4]$, the loops are simple and do not touch each other or the boundary; for $\kappa \in (4,8)$, the loops are non-simple and may touch each other and the boundary. For…

In this paper, we establish limit theorems for the supremum of the support, denoted by $M_t$, of a supercritical super-Brownian motion $\{X_t, t\ge0\}$ on $\mathbb{R}$. We prove that there exists an $m(t)$ such that $(X_t-m(t), M_t-m(t))$…

We obtain exact formulae for three basic quantities in random conformal geometry that depend on the modulus of an annulus. The first is for the law of the modulus of the Brownian annulus describing the scaling limit of uniformly sampled…

Let each of n particles starting at the origin in Z^2 perform simple random walk until reaching a site with no other particles. Lawler, Bramson, and Griffeath proved that the resulting random set A(n) of n occupied sites is (with high…

We study co-adapted couplings of (canonical hypoelliptic) diffu-sions on the (subRiemannian) Heisenberg group, that we call (Heisenberg) Brow-nian motions and are the joint laws of a planar Brownian motion with its L{\'e}vy area. We show…

We review some of the results related to conformal restriction: the chordal case and the radial case. We describe Brownian intersection exponents, conformal restriction property and SLE, and study their properties.