Periodic TASEP: when integrable systems meet integrable probability (once again)

Abstract: It is well-known that the Tracy-Widom distributions admit representations involving solutions to particular integrable systems. Other marginals of the KPZ fixed point, such as the Airy2 process, also admit similar representations. And very recently, first by Quastel and Remenik and shortly afterwards by Le Doussal, statistics of the KPZ fixed point were found to be connected to the KP equation.

In this talk, we plan to overview some analogue connections, but now for distributions of the periodic TASEP (pTASEP), which are believed to be the universal analogue of the KPZ universality class for periodic setup. For the step periodic initial condition, we compare the limiting one-point distribution of the pTASEP with the GUE Tracy-Widom distribution, highlighting the key features that allow to connect both of them to coupled systems of mKdV and heat equations. We also discuss some asymptotic properties of this limiting distribution, showing that it interpolates between the GUE Tracy-Widom and a Gaussian. For pTASEP with general initial condition, we also explain how very few analytic aspects of its limiting one-point distribution give a connection with the KP equation, in analogous way to Quastel-Remenik’s mentioned result. This talk is based on joint work with Jinho Baik (University of Michigan) and Zhipeng Liu (University of Kansas). Time permitting, we also briefly discuss a work in progress with Jinho Baik and Andrei Prokhorov (University of Michigan), greatly extending the mentioned results to multipoint distributions.

mathematical physicscombinatoricsprobabilityrepresentation theorystatistics theory

Audience: researchers in the topic

Junior Integrable Probability Seminar

Series comments: The Junior Integrable Probability Seminar is a series of talks taking place online and involving young researchers working in Integrable Probability and related fields. Go to sites.google.com/view/junior-ips for zoom link and password for the talk a day before the seminar.