Approach to Stationarity for KPZ Fluctuations in Finite Volume

Explore a 51-minute lecture from the Harvard CMSA program on Classical, quantum, and probabilistic integrable systems, delivered by Sylvain Prolhac from Université Paul Sabatier, Toulouse. Discover how the cumulants of the height for the KPZ fixed point in finite volume behave as affine functions of time at late stages, with the constant term being the final remnant of the initial state information. Learn about two computational approaches through the totally asymmetric exclusion process: an iterated matrix product representation leading to expressions involving extreme value statistics of Brownian paths, and Bethe ansatz providing explicit expressions for simple initial conditions. The comparison between these approaches yields conjectures for generating functions of Brownian paths.