Research Projects

Two central areas in modern probability theory are the study of particles evolving in a random media and the analysis of the large-scale geometry of random sets. In this project, we shall explore these two areas under a common theme. The two common ingredients are a shot-noise potential induced by a Poisson point process and a diffusion or a system of diffusions. Each of the Poisson points transmit a (random) signal and the shot noise field is the net signal at each point; we may allow for additional randomness in the reception of signals. Under the first area, we study the behaviour of an independent particle/ string evolving according to a stochastic differential equation in the Poisson shot-noise field. This is supposed to model a random polymer evolving in a medium of random obstacles. There have been many papers by Donsker, Varadhan, Sznitman and others which deal with the simpler case of a single particle evolving according to a Brownian motion in a Poisson shot-noise field. Along with Siva Athreya and Carl Mueller, the first P.I. recently introduced this model and analyzed the large time behaviour of the "averaged partition function" of this model. In this project we will continue working on some of the open questions in this area. Some of the main questions are the behaviour of the quenched partition function and the motion of the string under the averaged and quenched measures. Under the second area, we consider the large scale connectivity properties of excursion sets of evolving shot-noise fields. The large scale geometry of these excursion sets is a fundamental question at the intersection of stochastic geometry and statistical physics with many diverse applications. These have been many recent papers on this area by Fields medalist Hugo Duminil-Copin, Ahlberg, Tassion, Peccati and the second P.I., among others. We will mostly consider a simple diffusion (i.e. Brownian motion) but allow for very general shot-noise fields. One of the main questions to determine whether there are exceptional times for percolation in these evolving models, when the behaviour differs from that of the static model. Enroute, we will try to answer the fundamental question of behaviour at "criticality" of certain percolation models.