The totally asymmetric simple exclusion process (TASEP) is a very elementary irreversible model featuring classical particles hopping between neighbouring sites of a one-dimensional lattice. TASEP has the very rare property of being exactly solvable: its stationary state displays nice combinatorial structures, and the generator of the dynamics can be diagonalized exactly using Bethe ansatz methods introduced for quantum integrable spin chains.



Despite its apparent simplicity, the dynamics of TASEP exhibits at large scales (when the lattice spacing goes to zero) universal features shared by a variety of non-equilibrium systems with many interacting degrees of freedom, such as growing interfaces or one-dimensional fluids. Indeed, TASEP belongs to the celebrated non-equilibrium universality class known as KPZ, which has become in the past few years a prominent topic in mathematical physics at the interface between probability theory and non-equilibrium statistical physics, with some nice experimental realizations. During the internship, the student will familiarize herself / himself with TASEP and KPZ universality, either from the Bethe ansatz and quantum integrability side, or from the connection to combinatorics and extreme value statistics of Brownian bridges, depending on the background and inclination of the student.