CONFERENCES
Rare Events SimulationSøren Asmussen (Lund University - Sweden)Rare events come up in a variety of applied areas like insurance risk, reliability, telecommunications and statistical physics. For example in data transmission problems, one may be interested in the bit loss rate which is typically of magnitude $p=10^{-9}$ or smaller. To simulate just a single bit on its way through the system may be time consuming, and since to obtain a reasonable precision using straightforward simulation one needs to simulate a number replications (here bits) somewhat larger than $p^{-1}$, more sophisticated methods are called upon. The present talk gives a survey of such methods and open problems in the area. The most established method is importance sampling, simulating using a changed probability measure $Q$ rather than the given one $P$. The problem is the choice of $Q$. Taking $Q$ as the distribution $Q_A$ given the rare event $A$ gives variance 0 but is not feasible since the estimator involves the Radon-Nikodym derivative $dP/dQ_A=P(A)$ on $A$ which is not available (one simulates because $P(A)$ is unknown!). Instead, one would try to make $Q$ look as much like $Q_A$ as possible, and for this purpose large deviations techniques have become an established tool. In particular, exponential change of measure plays an important role when the underlying distributions are light-tailed. More recently, the large deviations approach has, however, turned out to have severe limitations, as will be demonstrated from problems involving processes with boundaries and distributions with heavy tails. Computational Methods in Statistical GeneticsDavid Balding (Imperial College - UK)A fundamental problem in statistical genetics is modelling the complex correlation structure of datasets that results from the fact that different individuals have different amounts of shared ancestry. Tree-based models are natural, but statistical inference under such models is challenging. I will review computational statistical methods in this application, including approaches based on Markov chain Monte Carlo and importance sampling, as well as more computationally efficient approximate methods. I will discuss applications in human population genetics, including population-based methods for disease gene mapping. Concentration Phenomena in Dynamical SystemsPierre Collet (CNRS, École Polytechnique - France)Concentration phenomena is a common fact in statistical mechanics connected to large deviation theory. It states that some observables do not deviate appreciably from their average in some adequate limit. Recently some new versions of this effect was discovered for independent random variables and more general observables leading to important statistical consequences. Extensions were proven for non independent random variables. We will discuss the result for expanding maps of the interval and present some statistical consequences. Convergence to the Brownian WebLuiz Renato Fontes (Universidade de São Paulo - Brazil)Arratia, and later Tóth and Werner, constructed random processes that formally correspond to coalescing one-dimensional Brownian motions starting from every space-time point. We extend their work by constructing and characterizing what we call the Brownian web as a random variable taking values in an appropriate (metric) space whose points are (compact) sets of paths. This leads to general convergence criteria and, in particular, to convergence in distribution of coalescing random walks in the scaling limit to the Brownian web. Entropic Repulsion Phenomena for Harmonic CrystalsGiambattista Giacomin (Université Paris 7 - France)I will present results aimed at understanding in detail the repulsion phenomena that arise when random interfaces are forced to cohabit with hard walls and/or with other interfaces. What is observed is the result of an entropy-energy competition that in the context of Gaussian effective models (harmonic crystals or lattice free fields) can be handled with precision. I will first focus on the results available in the most basic situation of one interface and one flat wall, stressing in particular the dependence of the phenomenon on the dimension of the space and the relevance of the results in investigating other interface phenomena like wetting. I will then analyse the repulsion effects in presence of a rough wall and for models of interfaces which are not allowed to intersect with each other. Finite Symbolic SequencesRicardo Lima (CNRS, Marseille - France)Complexity and entropy are two main tools in characterizing the properties of symbolic words. The analysis of their asymptotics has an old and reach story. Instead, we shall review some properties and constraints of these functions, when applied to finite lenght sequences. The recent interest on the subject is related with the use of symbolic dynamics in the study of dynamical systems as well as with the analysis of biological sequences. This is why we shall also try to describe some possible implications for such practical purposes. Limit Behavior of Algebraic AutomataServet Martínez (Universidad de Chile - Chile)We introduce algebraic automata and describe its dynamics. We discuss the mathematical tools involved in this study as well as the limit behavior of larger classes of automata. Cover Times and Thick Points for Random WalksYuval Peres (University of California, Berkeley - USA)In joint work with A. Dembo, J. Rosen and O. Zeitouni, we proved two conjectures about simple random walk in two dimensions: The first, due to Erdos and Taylor (1960), involves the number of visits to the most visited lattice site in the first $n$ steps of the walk. The second, due to Aldous (1989), concerns the number of steps it takes a simple random walk to cover all points of the $n$ by $n$ lattice torus. I will describe the analytical tool behind these proofs, a multiscale refinement of the "Second moment method", motivated by probability on trees. I will also present a different refinement, used in work with I. Benjamini, H. Kesten and O. Schramm to analyze the uniform spanning forest in dimensions greater than four. On the Role of Social Clusters in the Transmission of Infectious DiseasesRinaldo Schinazi (Université d'Aix Marseille - France)We introduce a spatial stochastic model for the spread of tuberculosis and HIV. We have three parameters: the size of the social cluster for each individual and the infection rates within and outside the social cluster. We show that when the infection rate from outside the cluster is low (this is presumably the case for tuberculosis and HIV) then an epidemic is possible only if the typical social cluster and the within infection rate are large enough. ContactContact us at: epb6@ime.usp.br. |