We consider the totally asymmetric simple exclusion process (TASEP) with two different initial conditions with shock discontinuities formed by blocks of fully packed particles. Initially a second class particle is at the left of a shock discontinuity. Using multicolored TASEP we derive 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.
We consider the totally asymmetric simple exclusion process with \emph{soft-shock} initial particle density, which is a step function increasing in the direction of flow and the step size chosen small to admit KPZ scaling. The initial configuration is deterministic and the dynamics create a shock. We prove that the fluctuations of a particle at the macroscopic position of the shock converge to the maximum of two independent GOE Tracy-Widom random variables, which establishes a conjecture of Ferrari and Nejjar. Furthermore, we show the joint fluctuations of particles near the shock are determin…
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 $α$. When $α<1$ there is a formation of a shock where the density jumps to $(1-α)/2$. For $α<1$ 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 $1-α=a t^{-1/3}$, where $t\gg 1$ is the observation time. In that case the decoupling does…
Interacting particle systems in the KPZ universality class on a ring of size $L$ with $O(L)$ number of particles are expected to change from KPZ dynamics to equilibrium dynamics at the so-called relaxation time scale $t=O(L^{3/2})$. In particular the system size is expected to have little effect to the particle fluctuations in the sub-relaxation time scale $t\ll L^{3/2}$. We prove that this is indeed the case for the totally asymmetric simple exclusion process (TASEP) with two types of initial conditions. For flat initial condition, we show that the particle fluctuations are given by the Airy$…
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.
We consider the totally asymmetric simple exclusion process (TASEP) with non-random initial condition having density $ρ$ on $\mathbb{Z}_-$ and $λ$ on $\mathbb{Z}_+$, and a second class particle initially at the origin. For $ρ<λ$, there is a shock and the second class particle moves with speed $1-λ-ρ$. 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 $t$.
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 f…
We consider a totally asymmetric simple exclusion on $\mathbb{Z}$ with the step initial condition, under the additional restriction that the first particle cannot cross a deterministally moving wall. We prove that such a wall may induce asymptotic fluctuation distributions of particle positions of the form $$ \mathbb{P}\Big(\sup_{τ\in \mathbb{R}}\{\textrm{Airy}_2(τ) -g(τ)\}\leq S\Big)$$ with arbitrary barrier functions $g$. This is the same class of distributions that arises as one-point asymptotic fluctuations of TASEPs with arbitrary initial conditions. Examples include Tracy-Widom GOE and G…
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 aroun…
The convergence of the totally asymmetric simple exclusion process to the solution of the Burgers equation is a classical result. In his seminal 1981 paper, Herman Rost proved the convergence of the density fields and local equilibrium when the limiting solution of the equation is a rarefaction fan. An important tool of his proof is the subadditive ergodic theorem. We prove his results by showing how second class particles transport the rarefaction-fan solution, as characteristics do for the Burgers equation, avoiding subadditivity. In the way we show laws of large numbers for tagged particles…
In this paper we show that a variety of interacting particle systems with multiple species can be viewed as random walks on Hecke algebras. This class of systems includes the asymmetric simple exclusion process (ASEP), M-exclusion TASEP, ASEP(q,j), stochastic vertex models, and many others. As an application, we study the asymptotic behavior of second class particles in some of these systems.
