Mathematics

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.

We consider the Fredrickson and Andersen one spin facilitated model (FA1f)on Z^d. Each site with rate one refreshes its occupation variable to a filled or to an empty state with probability p or q=1-p respectively, provided that at least one of its nearest neighbours is empty. We study the non-equilibrium dynamics started from an initial distribution $\nu$ different from the stationary product p-Bernoulli measure $\mu$, which has enough zeros. We then prove local convergence to equilibrium when the vacancy density q is above a proper threshold. The convergence is exponential (d=1) or stretched exponential (d>1). Joint work with N. Cancrini, F. Martinelli, C. Roberto and C. Toninelli.

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

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.