134 research outputs found
KPZ equation in one dimension and line ensembles
For suitably discretized versions of the Kardar-Parisi-Zhang equation in one
space dimension exact scaling functions are available, amongst them the
stationary two-point function. We explain one central piece from the technology
through which such results are obtained, namely the method of line ensembles
with purely entropic repulsion.Comment: Proceedings STATPHYS22, Bangalore, 200
Directed diffusion of reconstituting dimers
We discuss dynamical aspects of an asymmetric version of assisted diffusion
of hard core particles on a ring studied by G. I. Menon {\it et al.} in J. Stat
Phys. {\bf 86}, 1237 (1997). The asymmetry brings in phenomena like kinematic
waves and effects of the Kardar-Parisi-Zhang nonlinearity, which combine with
the feature of strongly broken ergodicity, a characteristic of the model. A
central role is played by a single nonlocal invariant, the irreducible string,
whose interplay with the driven motion of reconstituting dimers, arising from
the assisted hopping, determines the asymptotic dynamics and scaling regimes.
These are investigated both analytically and numerically through
sector-dependent mappings to the asymmetric simple exclusion process.Comment: 10 pages, 6 figures. Slight corrections, one added reference. To
appear in J. Phys. Cond. Matt. (2007). Special issue on chemical kinetic
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
Statistics of low energy excitations for the directed polymer in a random medium ()
We consider a directed polymer of length in a random medium of space
dimension . The statistics of low energy excitations as a function of
their size is numerically evaluated. These excitations can be divided into
bulk and boundary excitations, with respective densities
and . We find that both densities follow the scaling
behavior , where is the exponent governing the
energy fluctuations at zero temperature (with the well-known exact value
in one dimension). In the limit , both scaling
functions and behave as , leading to the droplet power law
in the regime . Beyond their common singularity near , the two scaling functions
are very different : whereas decays
monotonically for , the function first decays for
, then grows for , and finally presents a power law
singularity near . The density
of excitations of length accordingly decays as
where
. We obtain , and , suggesting the possible relation
.Comment: 15 pages, 25 figure
Exact solution for the stationary Kardar-Parisi-Zhang equation
We obtain the first exact solution for the stationary one-dimensional
Kardar-Parisi-Zhang equation. A formula for the distribution of the height is
given in terms of a Fredholm determinant, which is valid for any finite time
. The expression is explicit and compact enough so that it can be evaluated
numerically. Furthermore, by extending the same scheme, we find an exact
formula for the stationary two-point correlation function.Comment: 9 pages, 3 figure
Slow relaxation and aging kinetics for the driven lattice gas
We numerically investigate the long-time behavior of the density-density
auto-correlation function in driven lattice gases with particle exclusion and
periodic boundary conditions in one, two, and three dimensions using precise
Monte Carlo simulations. In the one-dimensional asymmetric exclusion process on
a ring with half the lattice sites occupied, we find that correlations induce
extremely slow relaxation to the asymptotic power law decay. We compare the
crossover functions obtained from our simulations with various analytic results
in the literature, and analyze the characteristic oscillations that occur in
finite systems away from half-filling. As expected, in three dimensions
correlations are weak and consequently the mean-field description is adequate.
We also investigate the relaxation towards the nonequilibrium steady state in
the two-time density-density auto-correlations, starting from strongly
correlated initial conditions. We obtain simple aging scaling behavior in one,
two, and three dimensions, with the expected power laws.Comment: 12 pages, 18 figures; to appear in Phys. Rev. E (2011
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
Exact probability function for bulk density and current in the asymmetric exclusion process
We examine the asymmetric simple exclusion process with open boundaries, a
paradigm of driven diffusive systems, having a nonequilibrium steady state
transition. We provide a full derivation and expanded discussion and digression
on results previously reported briefly in M. Depken and R. Stinchcombe, Phys.
Rev. Lett. {\bf 93}, 040602, (2004). In particular we derive an exact form for
the joint probability function for the bulk density and current, both for
finite systems, and also in the thermodynamic limit. The resulting distribution
is non-Gaussian, and while the fluctuations in the current are continuous at
the continuous phase transitions, the density fluctuations are discontinuous.
The derivations are done by using the standard operator algebraic techniques,
and by introducing a modified version of the original operator algebra. As a
byproduct of these considerations we also arrive at a novel and very simple way
of calculating the normalization constant appearing in the standard treatment
with the operator algebra. Like the partition function in equilibrium systems,
this normalization constant is shown to completely characterize the
fluctuations, albeit in a very different manner.Comment: 10 pages, 4 figure
Riemann-Hilbert approach to multi-time processes; the Airy and the Pearcey case
We prove that matrix Fredholm determinants related to multi-time processes
can be expressed in terms of determinants of integrable kernels \`a la
Its-Izergin-Korepin-Slavnov (IIKS) and hence related to suitable
Riemann-Hilbert problems, thus extending the known results for the single-time
case. We focus on the Airy and Pearcey processes. As an example of applications
we re-deduce a third order PDE, found by Adler and van Moerbeke, for the
two-time Airy process.Comment: 18 pages, 1 figur
Current Distribution and random matrix ensembles for an integrable asymmetric fragmentation process
We calculate the time-evolution of a discrete-time fragmentation process in
which clusters of particles break up and reassemble and move stochastically
with size-dependent rates. In the continuous-time limit the process turns into
the totally asymmetric simple exclusion process (only pieces of size 1 break
off a given cluster). We express the exact solution of master equation for the
process in terms of a determinant which can be derived using the Bethe ansatz.
From this determinant we compute the distribution of the current across an
arbitrary bond which after appropriate scaling is given by the distribution of
the largest eigenvalue of the Gaussian unitary ensemble of random matrices.
This result confirms universality of the scaling form of the current
distribution in the KPZ universality class and suggests that there is a link
between integrable particle systems and random matrix ensembles.Comment: 11 page
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