Abstract
We study the one-dimensional discrete time totally asymmetric simple exclusion process with parallel update rules on a spatially periodic domain. A multi-point space-time joint distribution formula is obtained for general initial conditions. The formula involves contour integrals of Fredholm determinants with kernels acting on certain discrete spaces. For a class of initial conditions satisfying certain technical assumptions, we are able to derive large-time, large-period limit of the joint distribution, under the relaxation time scale t=O(L^{3/2}) when the height fluctuations are critically affected by the finite geometry. The assumptions are verified for the step and flat initial conditions. As a corollary we obtain the multi-point distribution of discrete time TASEP on the whole integer lattice {mathbb {Z}} by taking the period L large enough so that the finite-time distribution is not affected by the boundary. The large time limit for the multi-time distribution of discrete time TASEP on {mathbb {Z}} is then obtained for the step initial condition.
Highlights
Models in the one-dimensional Kardar–Parisi–Zhang (KPZ) universality class are expected to have the same limiting height fluctuations under the t : t2/3 : t1/3 KPZ scaling for temporal correlations, spatial correlations and height fluctuations
For periodic models the one-point 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(L3/2) when the height fluctuations are critically affected by the finite geometry
All limiting distribution formulas were obtained as scaling limits of finite-time joint distributions of a single model in the KPZ universality class, namely the continuous time periodic totally asymmetric simple exclusion process (TASEP on a ring)
Summary
Models in the one-dimensional Kardar–Parisi–Zhang (KPZ) universality class are expected to have the same limiting height fluctuations under the t : t2/3 : t1/3 KPZ scaling for temporal correlations, spatial correlations and height fluctuations. Describing the universal limiting fluctuating field and proving the convergence of concrete models to the universal limiting field have been the main goals in the field and attracted active research over the last two decades. The convergence to the limiting fluctuating field (at one-point or multi-point level) have been obtained for a large class of models (mostly exactly solvable), either for the whole space models [2,8,9,10,11,13,15,16,20,21], or for half space models [1,6,7]
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