97 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
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
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
Random Growth Models
The link between a particular class of growth processes and random matrices
was established in the now famous 1999 article of Baik, Deift, and Johansson on
the length of the longest increasing subsequence of a random permutation.
During the past ten years, this connection has been worked out in detail and
led to an improved understanding of the large scale properties of
one-dimensional growth models. The reader will find a commented list of
references at the end. Our objective is to provide an introduction highlighting
random matrices. From the outset it should be emphasized that this connection
is fragile. Only certain aspects, and only for specific models, the growth
process can be reexpressed in terms of partition functions also appearing in
random matrix theory.Comment: Review paper; 24 pages, 4 figures; Minor correction
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