Scaling Limit for the Space-Time Covariance of the Stationary Totally Asymmetric Simple Exclusion Process

Abstract

The totally asymmetric simple exclusion process (TASEP) on the one-dimensional lattice with the Bernoulli ρ measure as initial conditions, 0 < ρ < 1, is stationary in space and time. Let N t (j) be the number of particles which have crossed the bond from j to j + 1 during the time span [0, t]. For j = (1 − 2ρ)t + 2w(ρ(1 − ρ)) 1/3 t 2/3 we prove that the fluctuations of N t (j) for large t are of order t 1/3 and we determine the limiting distribution function F w (s), which is a generalization of the GUE Tracy-Widom distribution. The family F w (s) of distribution functions have been obtained before by Baik and Rains in the context of the PNG model with boundary sources, which requires the asymptotics of a Riemann-Hilbert problem. In our work we arrive at F w (s) through the asymptotics of a Fredholm determinant. F w (s) is simply related to the scaling function for the space-time covariance of the stationary TASEP, equivalently to the asymptotic transition probability of a single second class particle.

Anisotropic Growth of Random Surfaces in 2 + 1 Dimensions

“…Since this time x, y ∈ R 1 are always in the bulk, we just use the bound of Lemma 6.8 and get 25) where Ψ(x) is the digamma function, which has the Taylor expansion at infinity given by Ψ(x) = ln(x) − 1/(2x) + O(1/x 2 ). (4.26) Thus R t,1 ≤ const ln(1/ε 1 ), (4.27) with const t-independent, as long as z, t → ∞ as t → ∞.…”

Fluctuating Hydrodynamics Approach to Equilibrium Time Correlations for Anharmonic Chains

“…All three peaks have a width much less than ct. But then, in case γ = γ , the product f γ (x, t)f γ (x, t) 0 for large t. Hence for the memory kernel (4.17) we invoke a small overlap approximation as 18) which is to be inserted in Eq. (4.16).…”

Exact solutions for KPZ-type growth processes, random matrices, and equilibrium shapes of crystals

“…In the picture of the directed polymer, it chooses either one of the two boundaries and the fluctuations from the boundary portions are comparable in size to the ones coming from the bulk. Such fluctuation properties are studied for PNG in [39] and for the TASEP in [16].…”