Limiting one-point distribution of periodic TASEP

Mixing times for the TASEP on the circle

“…Recall that T 0,v denotes the last passage time between the origin and v ∈ N 2 . Then there exist constants (c i ) i∈ [7] such that for all x > 0 and v…”

“…Estimates on point-to-line and line-to-line last passage times. We now study the different events B i for i ∈ [7] and give the following estimate on the last passage times.…”

“…In the following, we give bounds on probabilities of the different events (B i (θ)) i∈ [7] separately. We start with an estimate on the probability of the event B 7 (θ).…”

“…More recently, in a series of articles, Baik and Liu investigate the one-point and multi-point distribution of TASEP on the circle of length N , also called periodic TASEP, and obtain the limit function for various initial conditions, including step, flat, and uniformly random initial conditions [3,4,5,6,49]. For a positive density of particles k ≤ N/2, it turns out that a phase transition occurs at a time of order N 3/2 , called the relaxation time scale, where the behavior changes from a Tracy-Widom law to Gaussian fluctuations; see also [7] for a recent relation to the KP equation. Note that this phase transition is in accordance with our results on the mixing time, where we allow for any possible starting configuration.…”

Fluctuation Exponents of the KPZ equation on a large torus

“…For TASEP on the torus, more precise results have been derived in [3,4,5,7,6]. The rescaled random fluctuations are described explicitly in the relaxation regime, which has been further shown to interpolate the Gaussian statistics and the Tracy-Widom type statistics.…”