133 research outputs found
Ground-States of Two Directed Polymers
Joint ground states of two directed polymers in a random medium are
investigated. Using exact min-cost flow optimization the true two-line
ground-state is compared with the single line ground state plus its first
excited state. It is found that these two-line configurations are (for almost
all disorder configurations) distinct implying that the true two-line
ground-state is non-separable, even with 'worst-possible' initial conditions.
The effective interaction energy between the two lines scales with the system
size with the scaling exponents 0.39 and 0.21 in 2D and 3D, respectively.Comment: 19 pages RevTeX, figures include
On Growth, Disorder, and Field Theory
This article reviews recent developments in statistical field theory far from
equilibrium. It focuses on the Kardar-Parisi-Zhang equation of stochastic
surface growth and its mathematical relatives, namely the stochastic Burgers
equation in fluid mechanics and directed polymers in a medium with quenched
disorder. At strong stochastic driving -- or at strong disorder, respectively
-- these systems develop nonperturbative scale-invariance. Presumably exact
values of the scaling exponents follow from a self-consistent asymptotic
theory. This theory is based on the concept of an operator product expansion
formed by the local scaling fields. The key difference to standard Lagrangian
field theory is the appearance of a dangerous irrelevant coupling constant
generating dynamical anomalies in the continuum limit.Comment: review article, 50 pages (latex), 10 figures (eps), minor
modification of original versio
Quantized Scaling of Growing Surfaces
The Kardar-Parisi-Zhang universality class of stochastic surface growth is
studied by exact field-theoretic methods. From previous numerical results, a
few qualitative assumptions are inferred. In particular, height correlations
should satisfy an operator product expansion and, unlike the correlations in a
turbulent fluid, exhibit no multiscaling. These properties impose a
quantization condition on the roughness exponent and the dynamic
exponent . Hence the exact values for two-dimensional
and for three-dimensional surfaces are derived.Comment: 4 pages, revtex, no figure
Vicinal Surfaces and the Calogero-Sutherland Model
A miscut (vicinal) crystal surface can be regarded as an array of meandering
but non-crossing steps. Interactions between the steps are shown to induce a
faceting transition of the surface between a homogeneous Luttinger liquid state
and a low-temperature regime consisting of local step clusters in coexistence
with ideal facets. This morphological transition is governed by a hitherto
neglected critical line of the well-known Calogero-Sutherland model. Its exact
solution yields expressions for measurable quantities that compare favorably
with recent experiments on Si surfaces.Comment: 4 pages, revtex, 2 figures (.eps
Directed polymers in high dimensions
We study directed polymers subject to a quenched random potential in d
transversal dimensions. This system is closely related to the
Kardar-Parisi-Zhang equation of nonlinear stochastic growth. By a careful
analysis of the perturbation theory we show that physical quantities develop
singular behavior for d to 4. For example, the universal finite size amplitude
of the free energy at the roughening transition is proportional to (4-d)^(1/2).
This shows that the dimension d=4 plays a special role for this system and
points towards d=4 as the upper critical dimension of the Kardar-Parisi-Zhang
problem.Comment: 37 pages REVTEX including 4 PostScript figure
Renormalization group study of one-dimensional systems with roughening transitions
A recently introduced real space renormalization group technique, developed
for the analysis of processes in the Kardar-Parisi-Zhang universality class, is
generalized and tested by applying it to a different family of surface growth
processes.
In particular, we consider a growth model exhibiting a rich phenomenology
even in one dimension. It has four different phases and a directed percolation
related roughening transition. The renormalization method reproduces extremely
well all the phase diagram, the roughness exponents in all the phases and the
separatrix among them. This proves the versatility of the method and elucidates
interesting physical mechanisms.Comment: Submitted to Phys. Rev.
Critical Exponents of the KPZ Equation via Multi-Surface Coding Numerical Simulations
We study the KPZ equation (in D = 2, 3 and 4 spatial dimensions) by using a
RSOS discretization of the surface. We measure the critical exponents very
precisely, and we show that the rational guess is not appropriate, and that 4D
is not the upper critical dimension. We are also able to determine very
precisely the exponent of the sub-leading scaling corrections, that turns out
to be close to 1 in all cases. We introduce and use a {\em multi-surface
coding} technique, that allow a gain of order 30 over usual numerical
simulations.Comment: 10 pages, 8 eps figures (2 figures added). Published versio
Significance analysis and statistical mechanics: an application to clustering
This paper addresses the statistical significance of structures in random
data: Given a set of vectors and a measure of mutual similarity, how likely
does a subset of these vectors form a cluster with enhanced similarity among
its elements? The computation of this cluster p-value for randomly distributed
vectors is mapped onto a well-defined problem of statistical mechanics. We
solve this problem analytically, establishing a connection between the physics
of quenched disorder and multiple testing statistics in clustering and related
problems. In an application to gene expression data, we find a remarkable link
between the statistical significance of a cluster and the functional
relationships between its genes.Comment: to appear in Phys. Rev. Let
Design and Analysis of Schemes for Adapting Migration Intervals in Parallel Evolutionary Algorithms
Ring Migration Topology Helps Bypassing Local Optima
Running several evolutionary algorithms in parallel and occasionally
exchanging good solutions is referred to as island models. The idea is that the
independence of the different islands leads to diversity, thus possibly
exploring the search space better. Many theoretical analyses so far have found
a complete (or sufficiently quickly expanding) topology as underlying migration
graph most efficient for optimization, even though a quick dissemination of
individuals leads to a loss of diversity. We suggest a simple fitness function
FORK with two local optima parametrized by and a scheme for
composite fitness functions. We show that, while the (1+1) EA gets stuck in a
bad local optimum and incurs a run time of fitness evaluations
on FORK, island models with a complete topology can achieve a run time of
by making use of rare migrations in order to explore the
search space more effectively. Finally, the ring topology, making use of rare
migrations and a large diameter, can achieve a run time of
, the black box complexity of FORK. This shows that the
ring topology can be preferable over the complete topology in order to maintain
diversity.Comment: 12 page
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