The probability that the cluster of the origin in critical site percolation on the triangular grid has diameter larger than $R$ is proved to decay like $R$ to the power $5/48$ as $R$ goes to infinity.

We give sharp rates of convergence for a natural Markov chain on the space of phylogenetic trees and dually for the natural random walk on the set of perfect matchings in the complete graph on $2n$ vertices. Roughly, the results…

We prove a self-normalized large deviation principle for sums of Banach space valued functions of a Markov chain. Self-normalization applies to situations for which a full large deviation principle is not available. We follow the lead of Dembo and Shao…

In this paper we give two examples of evanescent Markov chains which exhibit unusual behaviour on conditioning to survive for large times. In the first example we show that the conditioned processes converge vaguely in the discrete topology to a…

For nonlinear functions $f$ of a random vector $Y$, $E[f(Y)]$ and $f(E[Y])$ usually differ. Consequently the mathematical expectation of $Y$ is not intrinsic: when we change coordinate systems, it is not invariant.This article is about a fundamental and hitherto neglected…

A Large Deviation Principle (LDP) is proved for the family $(1/n)\sum_1^n f(x_i^n) Z_i$ where $(1/n)\sum_1^n \delta_{x_i^n}$ converges weakly to a probability measure on $R$ and $(Z_i)_{i\in N}$ are $R^d$-valued independent and identically distributed random variables having some exponential moments, i.e.,…

We study the $d$-dimensional branching random walk for $d>2$. This process has extremal equilibria for every intensity. We are interested in the large space scale and large space-time scale behavior of the equilibrium state. We show that the fluctuations of…

Consider a uniformly chosen element $X_n$ of the $n$-fold wreath product $\Gamma_n = G \wr G \wr \cdots \wr G$, where $G$ is a finite permutation group acting transitively on some set of size $s$. The eigenvalues of $X_n$ in…

We consider Funaki's model of a random string taking values in $\mathbf{R}^d$. It is specified by the following stochastic PDE, \[ \frac{\partial u(x)}{\partial t}=\frac{\partial^2 u(x)}{\partial x^2} +\dot{W}. \] where $\dot{W}=\dot{W}(x,t)$ is two-parameter white noise, also taking values in $\mathbf{R}^d$. We…

We study the behavior of shocks in the asymmetric simple exclusion process on $Z$ whose initial distribution is a product measure with a finite number of shocks. We prove that if the particle hopping rates of this process are in…