Fermionic construction of tau functions and random processes

Tau functions expressed as fermionic expectation values [E. Date, M. Jimbo, M. Kashiwara, T. Miwa, Transformation groups for soliton equations, in: M. Jimbo, T. Miwa (Eds.), Nonlinear Integrable Systems—Classical Theory and Quantum Theory, World Scientific, 1983, pp. 39–120] are shown to provide a natural and straightforward description of a number of random processes and statistical models involving hard core configurations of identical particles on the integer lattice, like a discrete version simple exclusion processes (ASEP), nonintersecting random walkers, lattice Coulomb gas models and others, as well as providing a powerful tool for combinatorial calculations involving paths between pairs of partitions. We study the decay of the initial step function within the discrete ASEP (d-ASEP) model as an example. For constant hopping rates we obtain Vershik–Kerov type of asymptotic configuration of particles.