The universal Airy_1 and Airy_2 processes in the Totally Asymmetric Simple Exclusion Process

In the totally asymmetric simple exclusion process (TASEP) two processes arise in the large time limit: the Airy_1 and Airy_2 processes. The Airy_2 process is an universal limit process occurring also in other models: in a stochastic growth model on 1+1-dimensions, 2d last passage percolation, equilibrium crystals, and in random matrix diffusion. The Airy_1 and Airy_2 processes are defined and discussed in the context of the TASEP. We also explain a geometric representation of the TASEP from which the connection to growth models and directed last passage percolation is immediate.Comment: 13 pages, 4 figures, proceeding for the conference in honor of Percy Deift's 60th birthda

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

Tracy-Widom asymptotics for q-TASEP

We consider the q-TASEP that is a q-deformation of the totally asymmetric simple exclusion process (TASEP) on Z for q in [0,1) where the jump rates depend on the gap to the next particle. For step initial condition, we prove that the current fluctuation of q-TASEP at time t are of order t^{1/3} and asymptotically distributed as the GUE Tracy-Widom distribution, which confirms the KPZ scaling theory conjecture.Comment: 24 pages, 5 figure

On Time Correlations for KPZ Growth in One Dimension

Time correlations for KPZ growth in 1+1 dimensions are reconsidered. We discuss flat, curved, and stationary initial conditions and are interested in the covariance of the height as a function of time at a fixed point on the substrate. In each case the power laws of the covariance for short and long times are obtained. They are derived from a variational problem involving two independent Airy processes. For stationary initial conditions we derive an exact formula for the stationary covariance with two approaches: (1) the variational problem and (2) deriving the covariance of the time-integrated current at the origin for the corresponding driven lattice gas. In the stationary case we also derive the l arge time behavior for the covariance of the height gradients