Multipoint distribution of periodic TASEP

Abstract

The height fluctuations of the models in the KPZ class are expected to converge to a universal process. The spatial process at equal time is known to converge to the Airy process or its variations. However, the temporal process, or more generally the two-dimensional space-time fluctuation field, is less well understood. We consider this question for the periodic TASEP (totally asymmetric simple exclusion process). For a particular initial condition, we evaluate the multi-time and multi-location distribution explicitly in terms of a multiple integral involving a Fredholm determinant. We then evaluate the large time limit in the so-called relaxation time scale.

Riemann surfaces for KPZ with periodic boundaries

“…. < t n and corresponding positions x j was obtained by Baik and Liu for sharp wedge initial condition in [42]. After some rewriting in section 5.3.1, we obtain…”

“…Finally, the results of this paper are based on complicated asymptotics of Bethe ansatz formulas for TASEP in the limit where the number of lattice sites L and the number of particles N go to infinity with fixed density N/L [39,40,41,42]. A natural question is whether TASEP with finite L, N can already be described in terms of (finite genus) Riemann surfaces, so that the infinite genus Riemann surface R would emerge in a more transparent fashion in the large L, N limit.…”

“…In section 4, we study several meromorphic functions defined on these Riemann surfaces and needed for KPZ fluctuations. Finally, we explain in section 5 how our main results from section 2 are related to earlier exact formulas [39,40,41,42]. Some technical calculations are presented in appendix.…”

“…In this section, we give exact expressions for KPZ fluctuations with periodic boundary conditions equivalent to those obtained in [39,40,41,42], but written in a more unified way by interpreting various terms as natural objects living on Riemann surfaces evaluated on distinct sheets. Connections to stationary large deviations and interpretations in terms of particle-hole excitations and KdV solitons are discussed in some detail toward the end of the section.…”

“…. , h(x n , t n ) < u n ) for sharp wedge initial condition [42]. All these exact expressions have a somewhat complicated structure involving combinations of square roots and halfinteger polylogarithms.…”