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Limit law of a second class particle in TASEP with non-random initial condition. (English. French summary) Zbl 1472.82023

In this paper a totally asymmetric simple exclusion process (TASEP) with non-random initial condition and density \(\lambda\) on \(\mathbb{Z}_-\) and \(\rho\) on \(\mathbb{Z}_+\) as one of the simplest non-reversible interacting particle systems on \(\mathbb{Z}\) lattice is considered. An initial and further particle configurations are assumed and described by the occupation variables \(\{\eta_j\}\). Particles (first-class particles) can jump (they are independent) one step to the right only if their right neighboring site is empty. The particles cannot overtake each other and a labeling to them is associated. The position of particle \(k\) at time \(t\) is denoted by \(x_k(t)\) with the right-to-left ordering. In this paper the second-class particles are considered: when a first-class particle tries to jump on a site occupied by a second-class particle, the jump is not suppressed and the two particles interchanges their positions. The applications of second-class particles are very often when the interacting system generates shocks as the discontinuities in the particle density.
The main result of paper is given by Theorem 1.1 which is in the form of the limiting distribution and uses two ingredients: 1) the asymptotic independence of the last passage times from two disjoint initial set of points of a last passage percolation (LPP) model; 2) a tightness-type result on the two LPP problems (by Proposition 3.2 and Corollary 3.4) that extends to general the densities of the Pimentel method.
The paper is divided into two sections where Section 2 shows the connection between TASEP and LPP and the proof of Theorem 1.1, which is mainly based on preliminary results on the control of LPP at different points.

MSC:

82C22 Interacting particle systems in time-dependent statistical mechanics
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60B20 Random matrices (probabilistic aspects)
82C43 Time-dependent percolation in statistical mechanics
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