Related papers: Brownian disks and the Brownian snake

We consider an embedding of planar maps into an equilateral triangle $\Delta$ which we call the Cardy embedding. The embedding is a discrete approximation of a conformal map based on percolation observables that are used in Smirnov's proof…

In this article it is shown that the Brownian motion on the continuum random tree is the scaling limit of the simple random walks on any family of discrete $n$-vertex ordered graph trees whose search-depth functions converge to the Brownian…

We prove that, after suitable rescaling, the simple random walk on the trace of a large critical branching random walk converges to the Brownian motion on the integrated super-Brownian excursion.

Distance correlation is a new class of multivariate dependence coefficients applicable to random vectors of arbitrary and not necessarily equal dimension. Distance covariance and distance correlation are analogous to product-moment…

The integrated super-Brownian excursion (ISE) is the occupation measure of the spatial component of the head of the Brownian snake with lifetime process the normalized Brownian excursion. It is a random probability measure on $\mathbb{R}$,…

We consider an n-dimensional Brownian Motion trapped inside a bounded convex set by normally-reflecting boundaries. It is well-known that this process is uniformly ergodic. However, the rates of this ergodicity are not well-understood,…

We prove that the free Boltzmann quadrangulation with simple boundary and fixed perimeter, equipped with its graph metric, natural area measure, and the path which traces its boundary converges in the scaling limit to the free Boltzmann…

We consider the estimation of the drift and the level sets of the stationary distri- bution of a Brownian motion with drift, reflected in the boundary of a compact set $S\subset R^d$ , departing from the observation of a trajectory of this…

We establish an invariance principle connecting boundary random walks on $\mathbb N$ with Feller's Brownian motions on $[0,\infty)$. A Feller's Brownian motion is a Feller process on $[0,\infty)$ whose excursions away from the boundary $0$…

We give an explicit construction of the Brownian sphere biased by the distance between two distinguished points, which is based on the Miermont bijection for quadrangulations. We then describe various conditionings of this object, which are…

We study the volume of rigid loop-$O(n)$ quadrangulations with a boundary of length $2p$ in the non-generic critical regime. We prove that, as the half-perimeter $p$ goes to infinity, the volume scales in distribution to an explicit random…

We study properties of the random metric space called the Brownian map. For every h>0, we consider the connected components of the complement of the open ball of radius h centered at the root, and we let N(h,r) be the number of those…

We construct a new family of random permutons, called skew Brownian permuton, which describes the limits of several models of random constrained permutations. This family is parametrized by two real parameters. For a specific choice of the…

We study interacting systems of linear Brownian motions whose drift vector at every time point is determined by the relative ranks of the coordinate processes at that time. Our main objective has been to study the long range behavior of the…

We study the problem of estimating a compact set $S\subset \mathbb{R}^d$ from a trajectory of a reflected Brownian motion in $S$ with reflections on the boundary of $S$. We establish consistency and rates of convergence for various…

We establish the scaling limit of a class of boundary random walks to the full spectrum of Brownian-type processes on the half-line. By solving the associated martingale problem and employing weak convergence techniques, we prove that under…

Contraction properties of transport maps between probability measures play an important role in the theory of functional inequalities. The actual construction of such maps, however, is a non-trivial task and, so far, relies mostly on the…

We prove that random triangulations of types I, II, and III with a simple boundary under the critical Boltzmann weight converge in the scaling limit to the Brownian disk. The proof uses a bijection due to Poulalhon and Schaeffer between…

We study the limiting shape of the connected components of the vacant set of two-dimensional Brownian random interlacements: we prove that the connected component around $x$ is close in distribution to a rescaled \emph{Brownian amoeba} in…

We consider a continuous-time random walk in the quarter plane for which the transition intensities are constant on each of the four faces $(0,\infty)^2$, $F_1=\{0\}\times(0,\infty)$, $F_2=(0,\infty)\times\{0\}$ and $\{(0,0)\}$. We show…