Scientific

Particles attempt to follow a simple dynamic (random walk, constant flow, etc) in some space (interval, line, cycle, arbitrary graph). Add a simple interaction between particles, and the behaviour can change completely. The resulting dynamical systems are far more complex than the ingredients suggest. These processes (interchange process, TASEP, sorting networks, etc) have diverse to many topics: growth processes, queuing theory, representation theory, algebraic combinatorics. I will discuss recent progress on and open problems arising from several models of interacting particle systems.

In this talk, we extend the mutlicolor P/'olya urn schemes to countably infinitely many colors. We index the colors by \mathbb{Z}. Throughout the talk, we discuss mainly replacement matrices arising out of random walks. We show that the proportion of colors with suitable centering and scaling show central tendencies. Also the centering and scaling are fairly general. This behavior is in sharp contrast with the finite color case, where the asypmtotic behavior of the proportion of colors are determined by the qualitative properties (transience or recurrence) of the Markov chain underlying the replacement matrix. We also extend the infinite color case to fairly general graphs on \mathbb{R}^{d} and show that the proportion of colors show central tendencies similar to that in the case for \mathbb{Z}. Even the centering and scaling remains same.

The self-avoiding walk is a fundamental model in probability, combinatorics and statistical mechanics, for which many of the basic mathematical problems remain unsolved. Recent and ongoing progress for the four-dimensional self-avoiding walk has been based on a renormalization group analysis. This analysis takes as its starting point an exact representation of the self-avoiding walk problem as an equivalent problem for a perturbation of a Gaussian integral involving anti-commuting variables (fermions). This lecture will give a self-contained introduction to fermionic Gaussian integrals and will explain how they can be used to represent self-avoiding walks.

The talk is mainly based on the paper: D.C. Brydges, J.Z. Imbrie, G. Slade. Functional integral representations for self-avoiding walk. Probability Surveys, 6:34--61, (2009)

Particles attempt to follow a simple dynamic (random walk, constant flow, etc) in some space (interval, line, cycle, arbitrary graph). Add a simple interaction between particles, and the behaviour can change completely. The resulting dynamical systems are far more complex than the ingredients suggest. These processes (interchange process, TASEP, sorting networks, etc) have diverse to many topics: growth processes, queuing theory, representation theory, algebraic combinatorics. I will discuss recent progress on and open problems arising from several models of interacting particle systems.

In this talk I present a new model for seed banks, where individuals may obtain their type from ancestors which have lived in the near as well as the very far past. We discern three parameter regimes of the seed bank age distribution, which lead to substantially different behaviour in terms of genetic variability, in particular with respect to fixation of types and time to the most recent common ancestor. The classical Wright-Fisher model, as well as a seed bank model with bounded age distribution considered by Kaj, Krone and Lascoux (2001) are particular cases of the model. The mathematical methods are based not only on Markov chains, but also on renewal theory as well as on a Gibbsian approach introduced by Hammond and Sheffield (2011) in a different context. This talk is based in a joint work with Jochen Blath, Noemi Kurt, Dario Span`o.