Consider an infinite dimensional diffusion process process on $T^{{𝐙}^d}$, where $T$ is the circle, defined by the action of its generator $L$ on $C^2\left(T^{{𝐙}^d}\right)$ local functions as $Lf\left(\eta \right)={\sum }_{i\in {𝐙}^d}\left(\frac{1}{2}{a}_i\frac{{\partial }^{2}f}{\partial {\eta }_i^2}+b_i\frac{\partial f}{\partial {\eta }_i}\right)$. Assume that the coefficients, $a_i$ and $b_i$ are smooth, bounded, finite range with uniformly bounded second order partial derivatives, that $a_i$ is only a function of ${\eta }_i$ and that ${inf}_{i,\eta }{a}_i\left(\eta \right)\>0$. Suppose $\nu$ is an invariant product measure. Then, if $\nu$ is the Lebesgue measure or if $d=1,2$, it is the unique invariant measure. Furthermore, if $\nu$ is translation invariant, then...

Consider an infinite dimensional diffusion process process on TZd, where T is the circle, defined by the action of its generator L on C2(TZd) local functions as $Lf\left(\eta \right)={\sum }_{i\in {𝐙}^d}\left(\frac{1}{2}{a}_i\frac{{\partial }^{2}f}{\partial {\eta }_i^2}+b_i\frac{\partial f}{\partial {\eta }_i}\right)$. Assume that the coefficients, ai and bi are smooth, bounded, finite range with uniformly bounded second order partial derivatives, that ai is only a function of ${\eta }_i$ and that ${inf}_{i,\eta }{a}_i\left(\eta \right)>0$. Suppose ν is an invariant product measure. Then, if ν is the Lebesgue measure or if d=1,2, it is the unique invariant measure. Furthermore, if ν is translation...

A simple renewal process is a stochastic process $\left\{X_n\right\}$ taking values in $\{0,1\}$ where the lengths of the runs of $1$’s between successive zeros are independent and identically distributed. After observing $X_0,X_1,...X_n$ one would like to estimate the time remaining until the next occurrence of a zero, and the problem of universal estimators is to do so without prior knowledge of the distribution of the process. We give some universal estimates with rates for the expected time to renewal as well as for the conditional distribution...

We construct a class of conformally invariant measures on sets (or paths) and we study the critical exponents called intersection exponents associated to these measures. We show that these exponents exist and that they correspond to intersection exponents between planar Brownian motions. More precisely, using the definitions and results of our paper [27], we show that any set defined under such a conformal invariant measure behaves exactly as a pack (containing maybe a non-integer number) of Brownian...

There has been much success in describing the limiting spatial fluctuations of growth models in the Kardar–Parisi–Zhang (KPZ) universality class. A proper rescaling of time should introduce a non-trivial temporal dimension to these limiting fluctuations. In one-dimension, the KPZ class has the dynamical scaling exponent z = 3/2, that means one should find a universal space–time limiting process under the scaling of time as tT, space like t2/3X and fluctuations like t1/3 as t → ∞. In this paper we...

We consider the standard first passage percolation model in ℤd for d ≥ 2 and we study the maximal flow from the upper half part to the lower half part (respectively from the top to the bottom) of a cylinder whose basis is a hyperrectangle of sidelength proportional to n and whose height is h(n) for a certain height function h. We denote this maximal flow by τn (respectively φn). We emphasize the fact that the cylinder may be tilted. We look at the probability that these flows, rescaled by the surface...

The asymptotics of the probability that the self-intersection local time of a random walk on ${\mathbb{Z}}^d$ exceeds its expectation by a large amount is a fascinating subject because of its relation to some models from Statistical Mechanics, to large-deviation theory and variational analysis and because of the variety of the effects that can be observed. However, the proof of the upper bound is notoriously difficult and requires various sophisticated techniques. We survey some heuristics and some recently elaborated...