Transition to Shocks in TASEP and Decoupling of Last Passage Times

We consider the totally asymmetric simple exclusion process in a critical scaling parametrized by $a\geq0$, which creates a shock in the particle density of order $aT^{-1/3},$ $T$ the observation time. When starting from step initial data, we provide bounds on the limiting law which in particular imply that in the double limit $\lim_{a \to \infty}\lim_{T \to \infty}$ one recovers the product limit law and the degeneration of the correlation length observed at shocks of order $1$. This result is shown to apply to a general last-passage percolation model. We also obtain bounds on the two-point functions of several $\mathrm{Airy}$ processes.Comment: A few typos have been corrected. Published in the Latin American Journal of Probability and Mathematical Statistics , Vol. 15, p. 1311-1334 (2018

Shock fluctuations in flat TASEP under critical scaling

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\cite{FN14}. 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 $F_{\rm GOE}^2$ 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\cite{BBP06}.Comment: 26 pages, 5 figures, LaTeX (minor improvements

Anomalous shock fluctuations in TASEP and last passage percolation models

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.Comment: 51 pages, 7 figures; Results for TASEP and LPP extended and better illustrate

Fluctuations of the competition interface in presence of shocks

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].Comment: 33 pages, 4 figures, LaTe

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

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 $t$.Comment: 30 pages, 4 figures, LaTeX; Minor improvement

Shock Fluctuations in KPZ Growth Models

The Kardar-Parisi-Zhang (KPZ) universality class is a class of stochastic growth models which has attracted much interest, especially since the discovery about 15 years ago that the Tracy-Widom distributions from random matrix theory arise in it. Since then, more and more subclasses of the KPZ class have been studied, and experimental evidence for the soundness of KPZ scalings and statistics has been given. The aims of this thesis are the following. First, we introduce the KPZ class and discuss its conjectured universal scaling properties, limiting distributions and processes.As examples of growth models belonging to the KPZ class where these aspects have been studied, we treat in particular the (totally) asymmetric simple exclusion process ((T)ASEP) and last passage percolation (LPP). We describe the Tracy-Widom distributions, and the Airy processes which appear in these models. As a first result, we obtain the limiting distribution of certain particle positions in TASEP with particular initial data. Second, we focus on the study of shocks. After introducing the main concepts, we prove the emergence of an independence structure, which appears on a general level in LPP. With this independence, we provide the limiting distributions of shock positions in concrete cases in TASEP and show that they are given by products of Tracy-Widom distributions. We also show that the correlation length in KPZ models, which in all settings considered so far was t to the power 2/3 (t being the observation time), degenerates at the shock to t to the power 1/3. Finally, we consider a critical scaling, which, depending on the choice of the parameter, interpolates between shocks, flat profiles, and rarefaction fans. We prove that the fluctuations of particle positions in this critical scaling are, in the large time limit, given by a new transition process. The correlation length is shown to be t to the power 2/3 again. We perform a numerical study which suggests that we recover the product structure of shocks by letting the scaling parameter tend to infinity

The effect of the spin-orbit coupling in the relativistic contribution of the atomic states overlapping

In this study, we made an explicit relativistic contribution to the overlapping of the atomic states. This contribution highlights the effect of the fine structure in this kind of overlapping. The weight of the relativistic term in the particular case of 3d transition metals is analyzed.In this study, we made an explicit relativistic contribution to the overlapping of the atomic states. This contribution highlights the effect of the fine structure in this kind of overlapping. The weight of the relativistic term in the particular case of 3d transition metals is analyzed