Showing papers in "Probability Surveys in 2006"

Determinantal Processes and Independence

Abstract: We give a probabilistic introduction to determinantal and per- manental point processes. Determinantal processes arise in physics (fermions, eigenvalues of random matrices) and in combinatorics (nonintersecting paths, random spanning trees). They have the striking property that the number of points in a region D is a sum of independent Bernoulli random variables, with parameters which are eigenvalues of the relevant operator on L 2 (D). Moreover, any determinantal process can be represented as a mixture of determinantal projection processes. We give a simple explanation for these known facts, and establish analogous representations for permanental pro- cesses, with geometric variables replacing the Bernoulli variables. These representations lead to simple proofs of existence criteria and central limit theorems, and unify known results on the distribution of absolute values in certain processes with radially symmetric distributions.

On the constructions of the skew Brownian motion

Abstract: This article summarizes the various ways one may use to construct the Skew Brownian motion, and shows their connections. Recent applications of this process in modelling and numerical simulation motivates this survey. This article ends with a brief account of related results, extensions and applications of the Skew Brownian motion.

Recent advances in invariance principles for stationary sequences

Abstract: In this paper we survey some recent results on the central limit theorem and its weak invariance principle for stationary sequences. We also describe several maximal inequalities that are the main tool for obtaining the invariance principles, and also they have interest in themselves. The classes of dependent random variables considered will be martingale-like sequences, mixing sequences, linear processes, additive functionals of ergodic Markov chains.

Localization and delocalization of random interfaces

Abstract: The study of effective interface models has been quite active recently, with a particular emphasis on the effect of various external potentials (wall, pinning potential, ...) leading to localization/delocalization transitions. I review some of the results that have been obtained. In particular, I discuss pinning by a local potential, entropic repulsion and the (pre)wetting transition, both for models with continuous and discrete heights. This text is based on lecture notes for a mini-course given during the workshop "Topics in Random Interfaces and Directed Polymers" held in Leipzig, September 12-17 2005.

An essay on the general theory of stochastic processes

Abstract: This text is a survey on the general theory of stochastic processes, with a view towards random times and enlargements of filtrations. The first five chapters present standard materials, which where developed by the French probability school and which are usually written in French. The material presented in the last three chapters is less standard and takes into account some recent developments.

Level crossings and other level functionals of stationary Gaussian processes

Abstract: This paper presents a synthesis on the mathematical work done on level crossings of stationary Gaussian processes, with some extensions. The main results [(factorial) moments, representation into the Wiener Chaos, asymptotic results, rate of convergence, local time and number of crossings] are described, as well as the different approaches [normal comparison method, Rice method, Stein-Chen method, a general m-dependent method] used to obtain them; these methods are also very useful in the general context of Gaussian fields. Finally some extensions [time occupation functionals, number of maxima in an interval, process indexed by a bidimensional set] are proposed, illustrating the generality of the methods. A large inventory of papers and books on the subject ends the survey.

Uniqueness and Non-uniqueness in Percolation Theory

Abstract: This paper is an up-to-date introduction to the problem of uniqueness versus non-uniqueness of infinite clusters for percolation on ℤd and, more generally, on transitive graphs. For iid percolation on ℤd, uniqueness of the infinite cluster is a classical result, while on certain other transitive graphs uniqueness may fail. Key properties of the graphs in this context turn out to be amenability and nonamenability. The same problem is considered for certain dependent percolation models – most prominently the Fortuin–Kasteleyn random-cluster model – and in situations where the standard connectivity notion is replaced by entanglement or rigidity. So-called simultaneous uniqueness in couplings of percolation processes is also considered. Some of the main results are proved in detail, while for others the proofs are merely sketched, and for yet others they are omitted. Several open problems are discussed.

Markov chain comparison

Abstract: This is an expository paper, focussing on the following scenario. We have two Markov chains, M and M'. By some means, we have obtained a bound on the mixing time of M'. We wish to compare M with M' in order to derive a corresponding bound on the mixing time of M. We investigate the application of the comparison method of Diaconis and Saloff-Coste to this scenario, giving a number of theorems which characterize the applicability of the method. We focus particularly on the case in which the chains are not reversible. The purpose of the paper is to provide a catalogue of theorems which can be easily applied to bound mixing times.

The realization of positive random variables via absolutely continuous transformations of measure on Wiener space

Abstract: Let $\mu$ be a Gaussian measure on some measurable space $\{W = \{w\}, \mathcal{B}(W)\}$ and let $ u$ be a measure on the same space which is absolutely continuous with respect to $ u$. The paper surveys results on the problem of constructing a transformation $T$ on the $W$ space such that $Tw = w + u(w)$ where $u$ takes values in the Cameron-Martin space and the image of $\mu$ under $T$ is $\mu$. In addition we ask for the existence of transformations $T$ belonging to some particular classes.

The geometry of Brownian surfaces

Abstract: Motivated by Segal's axiom of conformal field theory, we do a survey on geometrical random fields. We do a history in order to arrive at a field theoretical analog of Klauder's quantization in Hamiltonoan quantum mechanic by using infinite dimensional Airault-Malliavin Brownian motion.