Past

Given a family of independent events in a probability space, the probability

that none of the events occurs is of course the product of the probabilities

that the individual events do not occur. If there is some dependence between the

events, however, then bounding the probability that none occurs is a much less

trivial matter. The Lov

/notices/events/abstracts/stochastic-analysis/mt05/m

In Clarendon Lab

A self-interacting random walk is a random process evolving in an environment depending on its past behaviour.

The notion of Edge-Reinforced Random Walk (ERRW) was introduced in 1986 by Coppersmith and Diaconis [2] on a discrete graph, with the probability of a move along an edge being proportional to the number of visits to this edge. In the same spirit, Pemantle introduced in 1988 [5] the Vertex-Reinforced Random Walk (VRRW), the probability of move to an adjacent vertex being then proportional to the number of visits to this vertex (and not to the edge leading to the vertex). The Self-Interacting Diffusion (SID) is a continuous counterpart to these notions.

Although introduced by similar definitions, these processes show some significantly different behaviours, leading in their understanding to various methods. While the study of ERRW essentially requires some probabilistic tools, corresponding to some local properties, the comprehension of VRRW and SID needs a joint understanding of on one hand a dynamical system governing the general evolution, and on the other hand some probabilistic phenomena, acting as perturbations, and sometimes changing the nature of this dynamical system.

The purpose of our talk is to present our recent results on the subject [1,3,4,6].

Bibliography

[1] M. Bena

Numerous physical systems are justifiably modelled as Markov processes. However,

in practical applications the (usually implicit) assumptions concerning accurate

measurement of the system are often a fair departure from what is possible in

reality. In general, this lack of exact information is liable to render the

In Dept of Statistics

Recently random planar structures, such as planar graphs and outerplanar graphs, have received much attention. Typical questions one would ask about them are the following: how many of them are there, can we sample a random instance uniformly at random, and what properties does a random planar structure have ? To answer these questions we decompose the planar structures along their connectivity. For the asymptotic enumeration we interpret the decomposition in terms of generating funtions and derive the asymptotic number, using singularity analysis. For the exact enumeration and the uniform generation we use the so-called recursive method: We derive recursive counting formulas along the decomposition, which yields a deterministic polynomial time algorithm to sample a planar structure that is uniformly distributed. In this talk we show how to apply these methods to several labeled planar structures, e.g., planar graphs, cubic planar graphs, and outerplanar graphs.

In Dept of Statistics

/notices/events/abstracts/applied-analysis/tt05/Heywood.pdf