A. Bufetov and P. Nejjar Shock fluctuations in TASEP under a variety of time scalings arXiv:2003.12414 2020 https://arxiv.org/abs/2003.12414
Abstract: We consider the totally asymmetric simple exclusion process (TASEP) with two different initial conditions with shock discontinuities, made by block of fully packed particles. Initially a second class particle is at the left of a shock discontinuity. Using multicolored TASEP we derive an exact formulas for the distribution of the second class particle and colored height functions. These are given in terms of the height function at different positions of a single TASEP configuration. We study the limiting distributions of second class particles (and colored height functions). The result depends on how the width blocks of particles scale with the observation time; we study a variety of such scalings.
Abstract: We consider the asymmetric simple exclusion process on Z with a single second class particle initially at the origin. The first class particles form two rarefaction fans which come together at the origin, where the large time density jumps from 0 to 1. We are interested in X(t), the position of the second class particle at time t. We show that, under the KPZ 1/3 scaling, X(t) is asymptotically distributed as the difference of two independent, GUE-distributed random variables.The key part of the proof is to show that X(t) equals, up to a negligible term, the difference of a random number of holes and particles, with the randomness built up by ASEP itself. This provides a KPZ analogue to the 1994 result of Ferrari and Fontes, where this randomness comes from the initial data and leads to Gaussian limit laws.
2019
Patrik L. Ferrari and Peter Nejjar Statistics of TASEP with three merging characteristics preprint: arXiv:1910.14083 2019 https://arxiv.org/abs/1910.14083
Abstract: In this paper we consider the totally asymmetric simple exclusion process, with non-random initial condition having three regions of constant densities of particles. From left to right, the densities of the three regions are increasing. Consequently, there are three characteristics which meet, i.e., two shocks merge. We study the particle fluctuations at this merging point and show that they are given by a product of three (properly scaled) GOE Tracy-Widom distribution functions. We work directly in TASEP without relying on the connection to last passage percolation.
2017
P.L. Ferrari, P. Ghosal and P. Nejjar Limit law of a second class particle in TASEP with non-random initial condition preprint: arXiv:1710.02323 2017 https://arxiv.org/abs/1710.02323
Abstract: We consider the totally asymmetric simple exclusion process (TASEP) with non-random initial condition having density $\rho$ on $\mathbb{Z}_-$ and $\lambda$ on $\mathbb{Z}_+$, and a second class particle initially at the origin. For $\rho<\lambda$, there is a shock and the second class particle moves with speed $1-\lambda-\rho$. For large time $t$, we show that the position of the second class particle fluctuates on a $t^{1/3}$ scale and determine its limiting law. We also obtain the limiting distribution of the number of steps made by the second class particle until time $t$.
2016
P.L. Ferrari and P. Nejjar Fluctuations of the competition interface in presence of shocks arXiv:1603.07498 2016 http://arxiv.org/abs/1603.07498
Abstract: We consider last passage percolation (LPP) models with exponentially distributed random variables, which are linked to the totally asymmetric simple exclusion process (TASEP). The competition interface for LPP was introduced and studied by Ferrari and Pimentel in [Ann. Probab. 33 (2005), 1235-1254] for cases where the corresponding exclusion process had a rarefaction fan. Here we consider situations with a shock and determine the law of the fluctuations of the competition interface around its deterministic law of large number position. We also study the multipoint distribution of the LPP around the shock, extending our one-point result of [Probab. Theory Relat. Fields 61 (2015), 61-109].
2015
Patrik L. Ferrari and Peter Nejjar Anomalous shock fluctuations in TASEP and last passage percolation models Probab. Theory Related Fields, 161(1): 61-109 2015 http://dx.doi.org/10.1007/s00440-013-0544-6
Abstract: We consider the totally asymmetric simple exclusion process with initial conditions and/or jump rates such that shocks are generated. If the initial condition is deterministic, then the shock at time \(t\) will have a width of order \(t^{1/3}\). We determine the law of particle positions in the large time limit around the shock in a few models. In particular, we cover the case where at both sides of the shock the process of the particle positions is asymptotically described by the Airy\(_1\) process. The limiting distribution is a product of two distribution functions, which is a consequence of the fact that at the shock two characteristics merge and of the slow decorrelation along the characteristics. We show that the result generalizes to generic last passage percolation models.
Patrik L. Ferrari and Peter Nejjar Shock fluctuations in flat TASEP under critical scaling J. Stat. Phys., 160(4): 985-1004 2015 http://arxiv.org/abs/1408.4850
Abstract: We consider TASEP with two types of particles starting at every second site. Particles to the left of the origin have jump rate $1$, while particles to the right have jump rate $\alpha$. When $\alpha<1$ there is a formation of a shock where the density jumps to $(1-\alpha)/2$. For $\alpha<1$ fixed, the statistics of the associated height functions around the shock is asymptotically (as time $t\to\infty$) a maximum of two independent random variables as shown in [arXiv:1306.3336]. In this paper we consider the critical scaling when $1-\alpha=a t^{-1/3}$, where $t\gg 1$ is the observation time. In that case the decoupling does not occur anymore. We determine the limiting distributions of the shock and numerically study its convergence as a function of $a$. We see that the convergence to product $F_{\rm GOE}^2$ occurs quite rapidly as $a$ increases. The critical scaling is analogue to the one used in the last passage percolation to obtain the BBP transition processes.
0
P.L. Ferrari and P. Nejjar The second class particle process at shocks preprint: arXiv:2309.09570 0 https://arxiv.org/abs/2309.09570
Abstract: We consider the totally asymmetric simple exclusion process (TASEP) starting with a shock discontinuity at the origin, with asymptotic densities λ to the left of the origin and Ï to the right of it and λ<Ï. We find an exact identity for the distribution of a second class particle starting at the origin. Then we determine the limiting joint distributions of the second class particle. Bypassing the last passage percolation model, we work directly in TASEP, allowing us to extend previous one-point distribution results via a more direct and shorter ansatz.
P. Nejjar KPZ statistics of second class particles in ASEP via mixing preprint:arXiv:1911.09426 0 https://arxiv.org/abs/1911.09426
Abstract: We consider the asymmetric simple exclusion process on Z with a single second class particle initially at the origin. The first class particles form two rarefaction fans which come together at the origin, where the large time density jumps from 0 to 1. We are interested in X(t), the position of the second class particle at time t. We show that, under the KPZ 1/3 scaling, X(t) is asymptotically distributed as the difference of two independent, GUE-distributed random variables.