Speaker
Description
We discuss properties of limiting distribution functions that arise in the periodic totally asymmetric simple exclusion process (pTASEP), and which are believed to be universal limiting distributions in the KPZ universality class for periodic models. For the periodic TASEP with periodic step initial condition, we prove that its one-point limiting distribution interpolates between a Gaussian and the GUE Tracy-Widom distribution, and also find different formulations that are in direct analogy with the latter. One of these representations lead to a purely discrete Riemann-Hilbert problem with infinitely many poles, and using it we connect the pTASEP with a system of coupled mKdV equations, and also with coupled heat equations. In addition, we also find a connection with the KP equation, extending a recent work of Quastel and Remenik. Towards the end of the walk, we also plan to briefly discuss some work in progress, showing how multi-point distributions for both TASEP and pTASEP are connected with matrix versions of the coupled mKdV and heat systems, and also with matrix versions of the Painlev´e II and KP equations. This is based on joint work with Jinho Baik (University of Michigan) and Zhipeng Liu (University of Kansas), and work in progress with Jinho Baik and
Andrei Prokhorov (University of Michigan).
