31,920 research outputs found
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
Anisotropic KPZ growth in 2+1 dimensions: fluctuations and covariance structure
In [arXiv:0804.3035] we studied an interacting particle system which can be
also interpreted as a stochastic growth model. This model belongs to the
anisotropic KPZ class in 2+1 dimensions. In this paper we present the results
that are relevant from the perspective of stochastic growth models, in
particular: (a) the surface fluctuations are asymptotically Gaussian on a
sqrt(ln(t)) scale and (b) the correlation structure of the surface is
asymptotically given by the massless field.Comment: 13 pages, 4 figure
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
Large time asymptotics of growth models on space-like paths I: PushASEP
We consider a new interacting particle system on the one-dimensional lattice
that interpolates between TASEP and Toom's model: A particle cannot jump to the
right if the neighboring site is occupied, and when jumping to the left it
simply pushes all the neighbors that block its way.
We prove that for flat and step initial conditions, the large time
fluctuations of the height function of the associated growth model along any
space-like path are described by the Airy_1 and Airy_2 processes. This includes
fluctuations of the height profile for a fixed time and fluctuations of a
tagged particle's trajectory as special cases.Comment: 48 pages, 4 figures, LaTeX; Final versio
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
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