Consider critical bond percolation on a large $2 n \times 2 n$ box on the square lattice. It is well-known that the size (i.e. number of vertices) of the largest open cluster is, with high probability, of order $n^2 \pi(n)$, where $\pi(n)$ denotes…

The understanding of site percolation on the triangular lattice progressed greatly in the last decade. Smirnov proved conformal invariance of critical percolation, thus paving the way for the construction of its scaling limit. Recently, the scaling limit of near critical…

Constrained diffusions in convex polyhedral cones with a general oblique reflection field, and with a diffusion coefficient scaled by a small parameter $\varepsilon> 0$, are considered. Using an interior Dirichlet heat kernel lower bound estimate for second order elliptic operators in bounded…

Let $A(t)$ be an $n\times p$ matrix with independent standard complex Brownian entries and set $M(t)=A(t)^*A(t)$. This is a process version of the Laguerre ensemble and as such we shall refer to it as the Laguerre process. The purpose of…

Consider a two-person zero-sum game played on a random $n$ by $n$ matrix where the entries are iid normal random variables. Let $Z$ be the number of rows in the support of the optimal strategy for player I given the…

We examine the percolation model on $\mathbb{Z}^d$ by an approach involving lattice animals and their surface-area-to-volume ratio. For $\beta \in [0,2(d-1))$, let $f(\beta)$ be the asymptotic exponential rate in the number of edges of the number of lattice animals containing…

In this paper we study three self-similar, long-range dependence, Gaussian processes. The first one, with covariance $$ \int^{s\wedge t}_0 u^a [(t-u)^b+(s-u)^b]du, $$ parameters $a>-1$, $-1 < b\leq 1$, $|b|\leq 1+a$, corresponds to fractional Brownian motion for $a=0$, $-1 < b…

We show that for all $p>p_c(\mathbb{Z}^d)$ percolation parameters, the probability that the cluster of the origin is finite but has at least $t$ vertices at distance one from the infinite cluster is exponentially small in $t$. We use this to…

Let $X=(X_t)_{t \ge 0}$ be a stable Levy process of index $\alpha \in (1,2)$ with the Levy measure $\nu(dx) = (c/x^{1+\alpha}) I_{(0,\infty)}(x) dx$ for $c>0$, let $x>0$ be given and fixed, and let $\tau_x = \inf\{ t>0 : X_t=x \}$…

For a coordinate symmetric random vector $(Y_1,\ldots,Y_n)={\bf Y} \in \mathbb{R}^n$, that is, one satisfying $(Y_1,\ldots,Y_n)=_d(e_1Y_1,\ldots,e_nY_n)$ for all $(e_1,\ldots,e_n) \in \{-1,1\}^n$, for which $P(Y_i=0)=0$ for all $i=1,2,\ldots,n$, the following Berry Esseen bound to the cumulative standard normal $\Phi$ for the standardized…