Periodic TASEP with general initial conditions

Abstract

We consider the one-dimensional totally asymmetric simple exclusion process with an arbitrary initial condition in a spatially periodic domain, and obtain explicit formulas for the multi-point distributions in the space-time plane. The formulas are given in terms of an integral involving a Fredholm determinant. We then evaluate the large-time, large-period limit in the relaxation time scale, which is the scale such that the system size starts to affect the height fluctuations. The limit is obtained assuming certain conditions on the initial condition, which we show that the step, flat, and step-flat initial conditions satisfy. Hence, we obtain the limit theorem for these three initial conditions in the periodic model, extending the previous work on the step initial condition. We also consider uniform random and uniform-step random initial conditions.

Multi-point distribution of discrete time periodic TASEP

“…(2) Under the relaxation time scale t = O(L 3/2 ) and the 1 : 2 : 3 KPZ scaling, we obtain large-time, large-period limits for the multi-point joint distributions under certain assumptions on the initial condition, which are verified for step and flat cases. These limiting formulas agree with those obtained in [4,5], thus providing an evidence that the height fluctuations for periodic models in the KPZ class are in fact universal.…”

“…Remark 2.3. Theorem 2.2 generalizes Theorem 3.1 of [5] (and also Theorem 4.6 of [4] for the special step initial condition) on the finite-time multi-point distribution of continuous time periodic TASEP. In fact their formulas can be obtained from our formula (2.6) by taking p = ǫ, t = ǫt and letting ǫ → 0.…”

“…See figure 1. We remark that the Bethe polynomials (and their roots) we are considering here is a one-parameter generalization of the one considered in [3,4,5] for continuous time periodic TASEP which corresponds to p = 0 degeneration of (2.15), Definition 2.9 (Left and right Bethe roots). For |z| < Ö c , we define the sets…”

“…For periodic models the onepoint marginals of limiting height fluctuations were obtained for classical step, flat and stationary initial conditions in [3,14], and independently in [19] under the so-called relaxation time scale t = O(L 3/2 ) when the height fluctuations are critically affected by the finite geometry. Further extensions including formulas for multi-point space-time joint distributions were subsequently obtained in [4] for step initial condition and [5] for more general initial conditions. These limiting formulas are expected to be universal for all related models on periodic domains.…”

“…Comparing to the previous work [4,5] and [15] on the multi-point distributions of continuous time TASEP, on either periodic domain or Z, our work consists of formulas with similar structures but involves an extra parameter p describing the hopping probability, which makes the algebraic properties a bit more complicated. In particular the polynomial whose roots are related to the kernels in the periodic formulas now depends on the extra parameter p. We point out that all the formulas for continuous time TASEP can be obtained from our formulas by rescaling the time and taking p → 0.…”