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Probability and Combinatorics

Tuesday, September 24, 2019 - 3:00pm

Axel Saenz-Rodriguez



Temple University

Wachman Hall 617

The totally asymmetric simple exclusion process (TASEP) is a Markov process that is the prototypical model for transport phenomena in non-equilibrium statistical mechanics. It was first introduced by Spitzer in 1970, and in the last 20 years, it has gained a strong resurgence in the emerging field of "Integrable Probability" due to exact formulas from Johanson in 2000 and Tracy and Widom in 2007 (among other related formulas and results). In particular, these formulas led to great insights regarding fluctuations related to the Tracy-Widom distribution and scalings to the Kardar-Parisi-Zhang (KPZ) stochastic differential equation.

In this joint work with Leonid Petrov (University of Virginia), we introduce a new and simple Markov Process that maps the distribution of the TASEP at time $t>0$, given step initial time data, to the distribution of the TASEP at some earlier time $t-s>0$. This process "back in time" is closely related to the Hammersley Process introduced by JM Hammersley in 1972, which later found a resurgence in the longest increasing subsequence problem in the work of Aldous and Diaconis in 1995. Hence, we call our process the Backwards Hammersley-type Process (BHP). As an fun application of our results, we have a new proof of the limit shape for the TASEP. The central objects in our constructions and proofs are the Schur point processes and the Yang-Baxter equation for the sl_2 quantum affine Lie algebra. In this talk, we will discuss the background in more detail and will explain the main ideas behind the constructions and proof.