70 research outputs found
Exact scaling functions for one-dimensional stationary KPZ growth
We determine the stationary two-point correlation function of the
one-dimensional KPZ equation through the scaling limit of a solvable
microscopic model, the polynuclear growth model. The equivalence to a directed
polymer problem with specific boundary conditions allows one to express the
corresponding scaling function in terms of the solution to a Riemann-Hilbert
problem related to the Painleve II equation. We solve these equations
numerically with very high precision and compare our, up to numerical rounding
exact, result with the prediction of Colaiori and Moore [1] obtained from the
mode coupling approximation.Comment: 24 pages, 6 figures, replaced with revised versio
Exact joint density-current probability function for the asymmetric exclusion process
We study the asymmetric exclusion process with open boundaries and derive the
exact form of the joint probability function for the occupation number and the
current through the system. We further consider the thermodynamic limit,
showing that the resulting distribution is non-Gaussian and that the density
fluctuations have a discontinuity at the continuous phase transition, while the
current fluctuations are continuous. The derivations are performed by using the
standard operator algebraic approach, and by the introduction of new operators
satisfying a modified version of the original algebra.Comment: 4 pages, 3 figure
Current large deviation function for the open asymmetric simple exclusion process
We consider the one dimensional asymmetric exclusion process with particle
injection and extraction at two boundaries. The model is known to exhibit four
distinct phases in its stationary state. We analyze the current statistics at
the first site in the low and high density phases. In the limit of infinite
system size, we conjecture an exact expression for the current large deviation
function.Comment: 4 pages, 3 figure
Free Energy of the Two-Matrix Model/dToda Tau-Function
We provide an integral formula for the free energy of the two-matrix model
with polynomial potentials of arbitrary degree (or formal power series). This
is known to coincide with the tau-function of the dispersionless
two--dimensional Toda hierarchy. The formula generalizes the case studied by
Kostov, Krichever, Mineev-Weinstein, Wiegmann, Zabrodin and separately
Takhtajan in the case of conformal maps of Jordan curves. Finally we generalize
the formula found in genus zero to the case of spectral curves of arbitrary
genus with certain fixed data.Comment: Ver 2: 18 pages added important formulas for higher genus spectral
curves, few typos removed (and few added). Ver 3: 19 pages (minor changes).
Typos removed, added appendix and improved exposition Ver 4: 19 pages, minor
corrections. Version submitted Ver 4; corrections prompted by referee and
accepted in Nuclear Phys.
Determinant solution for the Totally Asymmetric Exclusion Process with parallel update
We consider the totally asymmetric exclusion process in discrete time with
the parallel update. Constructing an appropriate transformation of the
evolution operator, we reduce the problem to that solvable by the Bethe ansatz.
The non-stationary solution of the master equation for the infinite 1D lattice
is obtained in a determinant form. Using a modified combinatorial treatment of
the Bethe ansatz, we give an alternative derivation of the resulting
determinant expression.Comment: 34 pages, 5 figures, final versio
Statistics of extremal intensities for Gaussian interfaces
The extremal Fourier intensities are studied for stationary
Edwards-Wilkinson-type, Gaussian, interfaces with power-law dispersion. We
calculate the probability distribution of the maximal intensity and find that,
generically, it does not coincide with the distribution of the integrated power
spectrum (i.e. roughness of the surface), nor does it obey any of the known
extreme statistics limit distributions. The Fisher-Tippett-Gumbel limit
distribution is, however, recovered in three cases: (i) in the non-dispersive
(white noise) limit, (ii) for high dimensions, and (iii) when only
short-wavelength modes are kept. In the last two cases the limit distribution
emerges in novel scenarios.Comment: 15 pages, including 7 ps figure
- âŠ