Abstract
We consider TASEP in continuous time with non-random initial conditions and arbitrary fixed density of particles $\rho \in (0,1)$. We show GOE Tracy-Widom universality of the one-point fluctuations of the associated height function. The result phrased in last passage percolation language is the universality for the point-to-line problem where the line has an arbitrary slope.
Highlights
We consider the totally asymmetric simple exclusion process (TASEP) in continuous time on Z
TASEP is a model in the Kardar-Parisi-Zhang (KPZ) universality class and one expects that for some model-dependent constants, c1, c2, t
Has in the t → ∞ limit a non-trivial distribution function, say D. It is well-known that for KPZ models the distribution D depends on classes of initial conditions [8
Summary
We consider the totally asymmetric simple exclusion process (TASEP) in continuous time on Z. The proof of our result is in his core probabilistic, where the only input from exactly solvable cases is the convergence to the Airy process for the so-called step initial condition and bounds on the tails of its onepoint distribution. The key idea is to bound the increment of the process by the ones of two stationary initial conditions, with densities slightly higher/lower than ρ, which are chosen such that the inequality holds on a set of high probability This probability is given in terms of some exit point probabilities. The main missing ingredient for an extension to PASEP is the convergence to the Airy process for step initial condition This is an open problem, but it looks easier than the analysis of PASEP with general densities ρ through exact formulas (compare with the formulas for ρ = 1/2 of [34, 35]).
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