Totally Asymmetric Simple Exclusion Process (TASEP) is the most well known interacting particle system in the so-called Kardar-Parisi-Zhang (KPZ) universality class. As the time goes to infinity, the height fluctuations and spatial correlation length have scaling exponents 1/3 and 2/3 respectively. Moreover, the limiting fluctuations are expected to be model independent and only dependent on the initial data.

In this talk, we discuss the TASEP on a periodic domain (periodic TASEP). Compared to the TASEP on the infinite line, which usually contains infinitely many particles, periodic TASEP has a finite number of particles. It was conjectured that periodic TASEP exhibits a crossover behavior between the Gaussian and KPZ universalities when the time is proportional to the 3/2 power of the system size. This time scale is called the relaxation time of the system. We will describe how one can use a method called coordinate Bethe ansatz to solve the periodic TASEP explicitly and further find the limiting fluctuations in the relaxation time scale.

This is joint with Jinho Baik (University of Michigan).