Related papers: Thick points for intersections of planar sample pa…

The original density is 1 for $t\in (0,1)$, $b$ is an integer base ($b\geq 2$%), and $p\in (0,1)$ is a parameter. The first construction stage divides the unit interval into $b$ subintervals and multiplies the density in each subinterval by…

We introduce a definition of thickness in $\mathbb{R}^d$ and obtain a lower bound for the Hausdorff dimension of the intersection of finitely or countably many thick compact sets using a variant of Schmidt's game. As an application we prove…

We consider Random Walk in Random Scenery, denoted $X_n$, where the random walk is symmetric on $Z^d$, with $d>4$, and the random field is made up of i.i.d random variables with a stretched exponential tail decay, with exponent $\alpha$…

We consider n non-intersecting Brownian motions with two fixed starting positions and two fixed ending positions in the large n limit. We show that in case of 'large separation' between the endpoints, the particles are asymptotically…

We consider a random geometric graph obtained by placing a Poisson point process of intensity 1 in the d-dimensional torus of side length n^(1/d) and connecting two points by an edge if their distance is at most r. We consider the case of…

We consider $p$ independent Brownian motions in $\R^d$. We assume that $p\geq 2$ and $p(d-2)<d$. Let $\ell_t$ denote the intersection measure of the $p$ paths by time $t$, i.e., the random measure on $\R^d$ that assigns to any measurable…

Motivated by an approximation problem from mathematical finance, we analyse the stability of the boundary crossing probability for the multivariate Brownian motion process, with respect to small changes of the boundary. Under broad…

We study simple random walk on the class of random planar maps which can be encoded by a two-dimensional random walk with i.i.d. increments or a two-dimensional Brownian motion via a "mating-of-trees" type bijection. This class includes the…

We consider the random walk on a simple point process on $\Bbb{R}^d$, $d\geq2$, whose jump rates decay exponentially in the $\alpha$-power of jump length. The case $\alpha =1$ corresponds to the phonon-induced variable-range hopping in…

Let C be two times continuously differentiable curve in R^2 with at least one point at which the curvature is non-zero. For any i,j > 0 with i+j =1, let Bad(i,j) denote the set of points (x,y) in R^2 for which max {||qx ||^{1/i},…

We obtain large deviations estimates for the self-intersection local times for a symmetric random walk in dimension 3. Also, we show that the main contribution to making the self-intersection large, in a time period of length $n$, comes…

We study a model of $n$ one-dimensional non-intersecting Brownian motions with two prescribed starting points at time $t=0$ and two prescribed ending points at time $t=1$ in a critical regime where the paths fill two tangent ellipses in the…

Let B_n be the number of self-intersections of a symmetric random walk with finite second moments in the integer planar lattice. We obtain moderate deviation estimates for B_n - E B_n and E B_n- B_n, which are given in terms of the best…

We define two families of Poissonian soups of bidirectional trajectories on $\mathbb{Z}^2$, which can be seen to adequately describe the local picture of the trace left by a random walk on the two-dimensional torus $(\mathbb{Z}/N…

We consider pairs of 3-dimensional Brownian paths, started at the origin and conditioned to have no intersections after time zero. We show that there exists a unique measure on pairs of paths that is invariant under this conditioning, while…

The Erd\"os-Taylor theorem [Acta Math. Acad. Sci. Hungar, 1960] states that if $\mathsf{L}_N$ is the local time at zero, up to time $2N$, of a two-dimensional simple, symmetric random walk, then $\tfrac{\pi}{\log N} \,\mathsf{L}_N$…

We study a configuration model on bipartite planar maps in which, given $n$ even integers, one samples a planar map with $n$ faces uniformly at random with these face degrees. We prove that when suitably rescaled, such maps always admit…

In the context of order statistics of discrete time random walks (RW), we investigate the statistics of the gap, $G_n$, and the number of time steps, $L_n$, between the two highest positions of a Markovian one-dimensional random walker,…

We prove limit theorems for random walks with $n$ steps in the $d$-dimensional Euclidean space as both $n$ and $d$ tend to infinity. One of our results states that the path of such a random walk, viewed as a compact subset of the…

There is a close connection between intersections of Brownian motion paths and percolation on trees. Recently, ideas from probability on trees were an important component of the multifractal analysis of Brownian occupation measure, in joint…