50 research outputs found
Bethe anzats derivation of the Tracy-Widom distribution for one-dimensional directed polymers
The distribution function of the free energy fluctuations in one-dimensional
directed polymers with -correlated random potential is studied by
mapping the replicated problem to a many body quantum boson system with
attractive interactions. Performing the summation over the entire spectrum of
excited states the problem is reduced to the Fredholm determinant with the Airy
kernel which is known to yield the Tracy-Widom distributionComment: 5 page
Replica Bethe ansatz derivation of the Tracy-Widom distribution of the free energy fluctuations in one-dimensional directed polymers
The distribution function of the free energy fluctuations in one-dimensional
directed polymers with -correlated random potential is studied by
mapping the replicated problem to the -particle quantum boson system with
attractive interactions. We find the full set of eigenfunctions and eigenvalues
of this many-body system and perform the summation over the entire spectrum of
excited states. It is shown that in the thermodynamic limit the problem is
reduced to the Fredholm determinant with the Airy kernel yielding the universal
Tracy-Widom distribution, which is known to describe the statistical properties
of the Gaussian unitary ensemble as well as many other statistical systems.Comment: 23 page
Spatial correlations of the 1D KPZ surface on a flat substrate
We study the spatial correlations of the one-dimensional KPZ surface for the
flat initial condition. It is shown that the multi-point joint distribution for
the height is given by a Fredholm determinant, with its kernel in the scaling
limit explicitly obtained. This may also describe the dynamics of the largest
eigenvalue in the GOE Dyson's Brownian motion model. Our analysis is based on a
reformulation of the determinantal Green's function for the totally ASEP in
terms of a vicious walk problem.Comment: 11 pages, 2 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
From interacting particle systems to random matrices
In this contribution we consider stochastic growth models in the
Kardar-Parisi-Zhang universality class in 1+1 dimension. We discuss the large
time distribution and processes and their dependence on the class on initial
condition. This means that the scaling exponents do not uniquely determine the
large time surface statistics, but one has to further divide into subclasses.
Some of the fluctuation laws were first discovered in random matrix models.
Moreover, the limit process for curved limit shape turned out to show up in a
dynamical version of hermitian random matrices, but this analogy does not
extend to the case of symmetric matrices. Therefore the connections between
growth models and random matrices is only partial.Comment: 18 pages, 8 figures; Contribution to StatPhys24 special issue; minor
corrections in scaling of section 2.
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
The effect of detachment and attachment to a kink motion in the asymmetric simple exclusion process
We study the dynamics of a kink in a one-lane asymmetric simple exclusion
process with detachment and attachment of the particle at arbitrary sites. For
a system with one site of detachment and attachment we find that the kink is
trapped by the site, and the probability distribution of the kink position is
described by the overdumped Fokker-Planck equation with a V-shaped potential.
Our results can be applied to the motion of a kink in arbitrary number of sites
where detachment and attachment take place. When detachment and attachment take
place at every site, we confirm that the kink motion obeys the diffusion in a
harmonic potential. We compare our results with the Monte Carlo simulation, and
check the quantitative validity of our theoretical prediction of the diffusion
constant and the potential form.Comment: 10 pages, 5 figure
Exact solution of a partially asymmetric exclusion model using a deformed oscillator algebra
We study the partially asymmetric exclusion process with open boundaries. We
generalise the matrix approach previously used to solve the special case of
total asymmetry and derive exact expressions for the partition sum and currents
valid for all values of the asymmetry parameter q. Due to the relationship
between the matrix algebra and the q-deformed quantum harmonic oscillator
algebra we find that q-Hermite polynomials, along with their orthogonality
properties and generating functions, are of great utility. We employ two
distinct sets of q-Hermite polynomials, one for q1. It
turns out that these correspond to two distinct regimes: the previously studied
case of forward bias (q1) where the
boundaries support a current opposite in direction to the bulk bias. For the
forward bias case we confirm the previously proposed phase diagram whereas the
case of reverse bias produces a new phase in which the current decreases
exponentially with system size.Comment: 27 pages LaTeX2e, 3 figures, includes new references and further
comparison with related work. To appear in J. Phys.
Superdiffusivity of the 1D lattice Kardar-Parisi-Zhang equation
The continuum Kardar-Parisi-Zhang equation in one dimension is lattice
discretized in such a way that the drift part is divergence free. This allows
to determine explicitly the stationary measures. We map the lattice KPZ
equation to a bosonic field theory which has a cubic anti-hermitian
nonlinearity. Thereby it is established that the stationary two-point function
spreads superdiffusively.Comment: 21 page