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…

The large deviation principle is established for the Fleming-Viot process with neutral mutation when the process starts from a point on the boundary. Since the diffusion coefficient is degenerate on the boundary, the boundary behavior of the process is investigated…

A distribution (function) $F$ on $[0,1]$ with $F(t)$ less or equal $t$ for all $t$ is called subdiagonal. The extreme subdiagonal distributions are identified as those whose distribution functions are almost surely the identity, or equivalently for which $F \circ…

We give another proof of the following result from a joint paper with Bálint Tóth: A Brownian motion reflected on an independent time-reversed Brownian motion is a Brownian motion.

We present a new method for locating the nodal line of the second eigenfunction for the Neumann problem in a planar domain.

We study the optimal control problem for $\mathbb{R}^d$-valued absolutely continuous stochastic processes with given marginal distributions at every time. When $d=1$, we show the existence and the uniqueness of a minimizer which is a function of a time and an…

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…