Transition to Shocks in TASEP and Decoupling of Last Passage Times

Abstract

We consider the totally asymmetric simple exclusion process in a critical scaling parametrized by a ≥ 0, which creates a shock in the particle density of order aT −1/3 , T the observation time. When at a = 0 one has step initial data, we provide bounds on the limiting law of particle positions for a > 0, which in particular imply that in the double limit lim a→∞ lim T →∞ one recovers the product limit law and the degeneration of the correlation length observed at shocks of order 1. This result can be phrased in terms of a general last passage percolation model. We also obtain bounds on the decoupling of twopoint functions of several Airy processes.

Limit law of a second class particle in TASEP with non-random initial condition

“…For shocks created by deterministic initial data, [17] obtained the limit law of a first class particle located at the shock. Later, multipoint distributions of several particles at the shock where obtained [18], and the transition to shock fluctuations was studied [28]. For the case λ < ρ, what remained open is the limit law of the second class particle X t , and this is what we obtain in this paper.…”

Statistics of TASEP with Three Merging Characteristics

“…Fluctuations of a single shock for non-random initial conditions have been analyzed the first time in [14] and further investigated in [15,26], including the fluctuations of a second class particle [13]. For large time t, the limiting distribution of particle positions around the shock is given by a product of two distribution functions.…”

“…In the previous papers [13][14][15]26] the results are obtained by using the mapping to the last passage percolation (LPP) model and then analyzing an equivalent problem in terms of LPP quantities. Merging of two shocks corresponds to a LPP from a domain given by the union of three line segments with different slopes.…”

Time-time Covariance for Last Passage Percolation with Generic Initial Profile

“…To prove Theorem 2.5 we are going to use a comparison with the stationary model of density slightly higher or lower than 1/2. The comparison idea was first used in [15] and then generalized in [41], with applications in [24,27,39,40]. For that purpose, we need to introduce the notion of exit point, which is the location where the maximizer of the LPP exits its boundary terms.…”