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Laura DeMarco, Northwestern University
Axel Saenz, University of Virginia
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 Johansson 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 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 a 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.
Laura DeMarco, Northwestern University
Laura DeMarco, Northwestern University
Rylee Lyman, Tufts University
Mapping class groups, \(GL(n,\mathbb{Z})\), and \(Out(F_n)\), the outer automorphism group of a free group are among some of the most well-studied infinite discrete groups. One facet they have in common is that, although finitely presented, they are "big" groups, in the sense that their elements exhibit a rich and wide array of dynamical behavior. The Nielsenâ€“Thurston normal form, Jordan normal form and relative train track representative, respectively, all attempt to expose and present this information in an organized way to aid reasoning about this behavior.
The group of outer automorphisms of a finite free product of finite groups is closely related to \(Out(F_n)\), but is comparatively understudied. In this talk we will introduce these groups, related geometric structures they act on, and review some of the known results. We would like to argue that these groups are also "big": to this end we have shown how to extend work of Bestvina, Feighn and Handel to construct relative train track representatives for outer automorphisms of free products.
There are no conferences next week.