Our first online meeting will take place on Monday the 22nd of February, from 4pm to 6pm (Central European Time). Our speakers will be Sylvain Prolhac and Guilherme Silva. The zoom link will be sent to the members of the project through the mailing list. If you are interested in participating and you are not a PIICQ participant, please register sending us an email!
First talk by Sylvain Prolhac at 4pm
Title: Riemann surfaces for TASEP and KPZ fluctuations in finite volume.
Abstract: The totally asymmetric simple exclusion process (TASEP) is an integrable Markov process described at large scales by KPZ universality. We show that Bethe ansatz for height fluctuations of TASEP with periodic boundary conditions may be formulated in terms of meromorphic differentials on a compact Riemann surface. In the KPZ scaling limit, the Riemann surface for TASEP converges to the infinite genus Riemann surface for half-integer polylogarithms. The talk will be mainly based on the paper https://arxiv.org/abs/2006.15096.
Second talk by Guilherme Silva at 5pm
Title: Integrable systems for multipoint distributions of classical and periodic TASEP.
Abstract: It is a classical result that the GUE Tracy-Widom distribution is expressed in terms of the Painlevé II equation. Recently, Quastel and Remenik extended this result to the one-time multi-location limiting distributions of the KPZ fixed point, connecting it to a one-time multi-location matrix KP equation.
In this talk, we plan to discuss connections of limiting distributions that arise in the classical and periodic TASEP with the KP and other integrable equations. For step initial condition, we show that the multi-time multi-location limiting distributions of both TASEP and periodic TASEP admit a Riemann-Hilbert representation which, in turn, yield that they satisfy certain non-commutative versions of nonlinear heat and nonlinear mKdV systems. Coupled together, these systems yield a characterization of these distributions in terms of a multi-time multi-location matrix KP equation, extending the result by Quastel and Remenik.
Our results also indicate that the limiting distributions of TASEP and periodic TASEP have distinct qualitative features, with the periodic TASEP giving rise to soliton solutions of the related integrable systems. Time permiting, we will also indicate some asymptotic results that we have so far obtained for some particular cases of these distributions.
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).