46 research outputs found
Fluctuations of a one-dimensional polynuclear growth model in a half space
We consider the multi-point equal time height fluctuations of a
one-dimensional polynuclear growth model in a half space. For special values of
the nucleation rate at the origin, the multi-layer version of the model is
reduced to a determinantal process, for which the asymptotics can be analyzed.
In the scaling limit, the fluctuations near the origin are shown to be
equivalent to those of the largest eigenvalue of the orthogonal/symplectic to
unitary transition ensemble at soft edge in random matrix theory.Comment: 51 pages, 8 figure
Polynuclear growth model, GOE and random matrix with deterministic source
We present a random matrix interpretation of the distribution functions which
have appeared in the study of the one-dimensional polynuclear growth (PNG)
model with external sources. It is shown that the distribution, GOE, which
is defined as the square of the GOE Tracy-Widom distribution, can be obtained
as the scaled largest eigenvalue distribution of a special case of a random
matrix model with a deterministic source, which have been studied in a
different context previously. Compared to the original interpretation of the
GOE as ``the square of GOE'', ours has an advantage that it can also
describe the transition from the GUE Tracy-Widom distribution to the GOE.
We further demonstrate that our random matrix interpretation can be obtained
naturally by noting the similarity of the topology between a certain
non-colliding Brownian motion model and the multi-layer PNG model with an
external source. This provides us with a multi-matrix model interpretation of
the multi-point height distributions of the PNG model with an external source.Comment: 27pages, 4 figure
Large time asymptotics of growth models on space-like paths II: PNG and parallel TASEP
We consider the polynuclear growth (PNG) model in 1+1 dimension with flat
initial condition and no extra constraints. The joint distributions of surface
height at finitely many points at a fixed time moment are given as marginals of
a signed determinantal point process. The long time scaling limit of the
surface height is shown to coincide with the Airy_1 process. This result holds
more generally for the observation points located along any space-like path in
the space-time plane. We also obtain the corresponding results for the discrete
time TASEP (totally asymmetric simple exclusion process) with parallel update.Comment: 39 pages,6 figure
Current moments of 1D ASEP by duality
We consider the exponential moments of integrated currents of 1D asymmetric
simple exclusion process using the duality found by Sch\"utz. For the ASEP on
the infinite lattice we show that the th moment is reduced to the problem of
the ASEP with less than or equal to particles.Comment: 13 pages, no figur
Dynamics of a tagged particle in the asymmetric exclusion process with the step initial condition
The one-dimensional totally asymmetric simple exclusion process (TASEP) is
considered. We study the time evolution property of a tagged particle in TASEP
with the step-type initial condition. Calculated is the multi-time joint
distribution function of its position. Using the relation of the dynamics of
TASEP to the Schur process, we show that the function is represented as the
Fredholm determinant. We also study the scaling limit. The universality of the
largest eigenvalue in the random matrix theory is realized in the limit. When
the hopping rates of all particles are the same, it is found that the joint
distribution function converges to that of the Airy process after the time at
which the particle begins to move. On the other hand, when there are several
particles with small hopping rate in front of a tagged particle, the limiting
process changes at a certain time from the Airy process to the process of the
largest eigenvalue in the Hermitian multi-matrix model with external sources.Comment: 48 pages, 8 figure
Fluctuations of the one-dimensional asymmetric exclusion process using random matrix techniques
The studies of fluctuations of the one-dimensional Kardar-Parisi-Zhang
universality class using the techniques from random matrix theory are reviewed
from the point of view of the asymmetric simple exclusion process. We explain
the basics of random matrix techniques, the connections to the polynuclear
growth models and a method using the Green's function.Comment: 41 pages, 10 figures, minor corrections, references adde
Two ways to solve ASEP
The purpose of this article is to describe the two approaches to compute
exact formulas (which are amenable to asymptotic analysis) for the probability
distribution of the current of particles past a given site in the asymmetric
simple exclusion process (ASEP) with step initial data. The first approach is
via a variant of the coordinate Bethe ansatz and was developed in work of Tracy
and Widom in 2008-2009, while the second approach is via a rigorous version of
the replica trick and was developed in work of Borodin, Sasamoto and the author
in 2012.Comment: 10 pages, Chapter in "Topics in percolative and disordered systems
Generalized Green Functions and current correlations in the TASEP
We study correlation functions of the totally asymmetric simple exclusion
process (TASEP) in discrete time with backward sequential update. We prove a
determinantal formula for the generalized Green function which describes
transitions between positions of particles at different individual time
moments. In particular, the generalized Green function defines a probability
measure at staircase lines on the space-time plane. The marginals of this
measure are the TASEP correlation functions in the space-time region not
covered by the standard Green function approach. As an example, we calculate
the current correlation function that is the joint probability distribution of
times taken by selected particles to travel given distance. An asymptotic
analysis shows that current fluctuations converge to the process.Comment: 46 pages, 3 figure