Sponsored by the Rutgers University Department of Mathematics and the
Center for Discrete Mathematics and Theoretical Computer Science (DIMACS)
Title: Second class particles in exclusion processes and Young tableaux
Speaker: Dan Romik, University of California, Davis
Date: Thursday, December 6, 2012 5:00pm
Location: Hill Center, Room 705, Rutgers University, Busch Campus, Piscataway, NJ
In probability theory, the Totally Asymmetric Simple Exclusion Process (known as TASEP) is an important random process of particles interacting on a one-dimensional lattice. It can be thought of as a simple model for a traffic jam: each particle tries to move one step forward at random times, succeeding if there is no particle blocking its way. One can also add a new type of particle known as a "second class particle", which can move forward if there is an empty space in front of it, but will be pushed back if an ordinary particle tries to move into its position from behind. A natural question is to ask which way the second class particle will go if started at the front of an "infinite traffic jam" with an infinite line of ordinary particles behind it and only empty space in front of it.
Young tableaux are important combinatorial objects related to the representation theory of the symmetric groups and to longest increasing subsequences in permutations. A natural operation on Young tableaux, known as "jeu de taquin", plays a key role in this theory.
In the talk I will describe recent joint work with Piotr Sniady, in which we discovered that understanding the behavior of the second class particle in the TASEP is actually equivalent to understanding the jeu de taquin operation on infinite Young tableaux. We then used this connection to prove that in a variant of the TASEP in which the probabilistic dynamics are driven by a natural measure on infinite Young tableaux (known as Plancherel measure) the second class particle moves asymptotically with a limiting (but random) speed; equivalently, in the language of Young tableaux, the "jeu de taquin path" is asymptotically a straight line with a random direction.