Related papers: Multiple points on the boundaries of Brownian loop…
We construct a measure on the thick points of a Brownian loop soup in a bounded domain D of the plane with given intensity $\theta>0$, which is formally obtained by exponentiating the square root of its occupation field. The measure is…
We provide a decomposition of the trace of the Brownian motion into a simple path and an independent Brownian soup of loops that intersect the simple path. More precisely, we prove that any subsequential scaling limit of the loop erased…
It is well-known (see Dvoretzky, Erd{\H o}s and Kakutani [8] and Le Gall [12]) that a planar Brownian motion $(B_t)_{t\ge 0}$ has points of infinite multiplicity, and these points form a dense set on the range. Our main result is the…
We define and study a family of generalized non-intersection exponents for planar Brownian motions that is indexed by subsets of the complex plane: For each $A\subset\CC$, we define an exponent $\xi(A)$ that describes the decay of certain…
This article deals with limit theorems for certain loop variables for loop soups whose intensity approaches infinity. We first consider random walk loop soups on finite graphs and obtain a central limit theorem when the loop variable is the…
The Brownian map is a random geodesic metric space arising as the scaling limit of random planar maps. We strengthen the so-called confluence of geodesics phenomenon observed at the root of the map, and with this, reveal several properties…
We provide a new construction of Brownian disks in terms of forests of continuous random trees equipped with nonnegative labels corresponding to distances from a distinguished point uniformly distributed on the boundary of the disk. This…
We study vertex-like operators built from the Brownian loop soup in the limit as the loop soup intensity tends to infinity. More precisely, following Camia, Gandolfi and Kleban (Nuclear Physics B 902, 2016), we take a Brownian loop soup in…
Given a point and an expanding map on the unit interval, we consider the set of points for which the forward orbit under this map is bounded away from the given point. For maps like multiplication by an integer modulo 1, such sets have full…
We interpret the Hilbert entropy of a convex projective structure on a closed higher-genus surface as the Hausdorff dimension of the non-differentiability points of the limit set in the full flag space $\mathcal F(\mathbb R^3)$.…
We study Conformal Loop Ensemble (CLE$_{\kappa}$) in doubly connected domains: annuli, the punctured disc, and the punctured plane. We restrict attention to CLE$_{\kappa}$ for which the loops are simple, i.e. $\kappa\in (8/3,4]$. In the…
Brownian particles that are replicated and annihilated at equal rate have strongly correlated positions, forming a few compact clusters separated by large gaps. We characterize the distribution of the particles at a given time, using a…
Consider percolation on $T\times \mathbb{Z}^d$, the product of a regular tree of degree $k\geq 3$ with the hypercubic lattice $\mathbb{Z}^d$. It is known that this graph has $0<p_c<p_u<1$, so that there are non-trivial regimes in which…
In a series of recent preprints, we have proven that with probability one the Hausdorff dimension on the outer boundary of planar Brownian motion is 4/3, confirming a conjecture by Mandelbrot. It is also shown that the Hausdorff dimension…
We consider the random walk loop soup on the discrete half-plane corresponding to a central charge c in (0, 1]. We look at the clusters of discrete loops and show that the scaling limit of the outer boundaries of outermost clusters is the…
We prove in the setting of $Q$--Ahlfors regular PI--spaces the following result: if a domain has uniformly large boundary when measured with respect to the $s$--dimensional Hausdorff content, then its visible boundary has large…
We show that the scaling limit of the random walk loop soup on suitable planar graphs is the Brownian loop soup, under a topology on multisets of unrooted, unparameterized, and macroscopic loops. The result holds assuming only convergence…
If $X(t,x)$ is the density of one-dimensional super-Brownian motion, we prove that $\text{dim}(\partial\{x:X(t,x)>0\})=2-2\lambda_0\in(0,1)$ a.s. on $\{X_t\neq 0\}$, where $-\lambda_0\in(-1,-1/2)$ is the lead eigenvalue of a killed…
We derive the exact value of intersection exponents between planar Brownian motions or random walks, confirming predictions from theoretical physics by Duplantier and Kwon. Let B and B' be independent Brownian motions (or simple random…
We consider the loop soup at intensity ${1\over 2}$ conditioned on having local time $0$ on a set of vertices with positive occupation field in their vicinities. We give a relation between this loop soup and the usual loop soup conditioned…