687 research outputs found
Global fluctuations and Gumbel statistics
We explain how the statistics of global observables in correlated systems can
be related to extreme value problems and to Gumbel statistics. This
relationship then naturally leads to the emergence of the generalized Gumbel
distribution G_a(x), with a real index a, in the study of global fluctuations.
To illustrate these findings, we introduce an exactly solvable nonequilibrium
model describing an energy flux on a lattice, with local dissipation, in which
the fluctuations of the global energy are precisely described by the
generalized Gumbel distribution.Comment: 4 pages, 3 figures; final version with minor change
Distribution of extremes in the fluctuations of two-dimensional equilibrium interfaces
We investigate the statistics of the maximal fluctuation of two-dimensional
Gaussian interfaces. Its relation to the entropic repulsion between rigid walls
and a confined interface is used to derive the average maximal fluctuation and the asymptotic behavior of the whole
distribution for finite with and the interface size and
tension, respectively. The standardized form of does not depend on
or , but shows a good agreement with Gumbel's first asymptote distribution
with a particular non-integer parameter. The effects of the correlations among
individual fluctuations on the extreme value statistics are discussed in our
findings.Comment: 4 pages, 4 figures, final version in PR
Probing the tails of the ground state energy distribution for the directed polymer in a random medium of dimension via a Monte-Carlo procedure in the disorder
In order to probe with high precision the tails of the ground-state energy
distribution of disordered spin systems, K\"orner, Katzgraber and Hartmann
\cite{Ko_Ka_Ha} have recently proposed an importance-sampling Monte-Carlo
Markov chain in the disorder. In this paper, we combine their Monte-Carlo
procedure in the disorder with exact transfer matrix calculations in each
sample to measure the negative tail of ground state energy distribution
for the directed polymer in a random medium of dimension .
In , we check the validity of the algorithm by a direct comparison with
the exact result, namely the Tracy-Widom distribution. In dimensions and
, we measure the negative tail up to ten standard deviations, which
correspond to probabilities of order . Our results are
in agreement with Zhang's argument, stating that the negative tail exponent
of the asymptotic behavior
as is directly related to the fluctuation exponent
(which governs the fluctuations
of the ground state energy for polymers of length ) via the simple
formula . Along the paper, we comment on the
similarities and differences with spin-glasses.Comment: 13 pages, 16 figure
Fluctuating Fronts as Correlated Extreme Value Problems: An Example of Gaussian Statistics
In this paper, we view fluctuating fronts made of particles on a
one-dimensional lattice as an extreme value problem. The idea is to denote the
configuration for a single front realization at time by the set of
co-ordinates of the
constituent particles, where is the total number of particles in that
realization at time . When are arranged in the ascending order
of magnitudes, the instantaneous front position can be denoted by the location
of the rightmost particle, i.e., by the extremal value
. Due to interparticle
interactions, at two different times for a single front
realization are naturally not independent of each other, and thus the
probability distribution [based on an ensemble of such front
realizations] describes extreme value statistics for a set of correlated random
variables. In view of the fact that exact results for correlated extreme value
statistics are rather rare, here we show that for a fermionic front model in a
reaction-diffusion system, is Gaussian. In a bosonic front model
however, we observe small deviations from the Gaussian.Comment: 6 pages, 3 figures, miniscule changes on the previous version, to
appear in Phys. Rev.
Contest based on a directed polymer in a random medium
We introduce a simple one-parameter game derived from a model describing the
properties of a directed polymer in a random medium. At his turn, each of the
two players picks a move among two alternatives in order to maximize his final
score, and minimize opponent's return. For a game of length , we find that
the probability distribution of the final score develops a traveling wave
form, , with the wave profile unusually
decaying as a double exponential for large positive and negative . In
addition, as the only parameter in the game is varied, we find a transition
where one player is able to get his maximum theoretical score. By extending
this model, we suggest that the front velocity is selected by the nonlinear
marginal stability mechanism arising in some traveling wave problems for which
the profile decays exponentially, and for which standard traveling wave theory
applies
Classical diffusion of N interacting particles in one dimension: General results and asymptotic laws
I consider the coupled one-dimensional diffusion of a cluster of N classical
particles with contact repulsion. General expressions are given for the
probability distributions, allowing to obtain the transport coefficients. In
the limit of large N, and within a gaussian approximation, the diffusion
constant is found to behave as N^{-1} for the central particle and as (\ln
N)^{-1} for the edge ones. Absolute correlations between the edge particles
increase as (\ln N)^{2}. The asymptotic one-body distribution is obtained and
discussed in relation of the statistics of extreme events.Comment: 6 pages, 2 eps figure
Diffusion of Tagged Particle in an Exclusion Process
We study the diffusion of tagged hard core interacting particles under the
influence of an external force field. Using the Jepsen line we map this many
particle problem onto a single particle one. We obtain general equations for
the distribution and the mean square displacement of the tagged
center particle valid for rather general external force fields and initial
conditions. A wide range of physical behaviors emerge which are very different
than the classical single file sub-diffusion $ \sim t^{1/2}$ found
for uniformly distributed particles in an infinite space and in the absence of
force fields. For symmetric initial conditions and potential fields we find
$ = {{\cal R} (1 - {\cal R})\over 2 N {\it r} ^2} $ where $2 N$ is
the (large) number of particles in the system, ${\cal R}$ is a single particle
reflection coefficient obtained from the single particle Green function and
initial conditions, and $r$ its derivative. We show that this equation is
related to the mathematical theory of order statistics and it can be used to
find even when the motion between collision events is not Brownian
(e.g. it might be ballistic, or anomalous diffusion). As an example we derive
the Percus relation for non Gaussian diffusion
Modeling temporal fluctuations in avalanching systems
We demonstrate how to model the toppling activity in avalanching systems by
stochastic differential equations (SDEs). The theory is developed as a
generalization of the classical mean field approach to sandpile dynamics by
formulating it as a generalization of Itoh's SDE. This equation contains a
fractional Gaussian noise term representing the branching of an avalanche into
small active clusters, and a drift term reflecting the tendency for small
avalanches to grow and large avalanches to be constricted by the finite system
size. If one defines avalanching to take place when the toppling activity
exceeds a certain threshold the stochastic model allows us to compute the
avalanche exponents in the continum limit as functions of the Hurst exponent of
the noise. The results are found to agree well with numerical simulations in
the Bak-Tang-Wiesenfeld and Zhang sandpile models. The stochastic model also
provides a method for computing the probability density functions of the
fluctuations in the toppling activity itself. We show that the sandpiles do not
belong to the class of phenomena giving rise to universal non-Gaussian
probability density functions for the global activity. Moreover, we demonstrate
essential differences between the fluctuations of total kinetic energy in a
two-dimensional turbulence simulation and the toppling activity in sandpiles.Comment: 14 pages, 11 figure
Edwards-Wilkinson surface over a spherical substrate: noise in the height fluctuations
We study the steady state fluctuations of an Edwards-Wilkinson type surface
with the substrate taken to be a sphere. We show that the height fluctuations
on circles at a given latitude has the effective action of a perfect Gaussian
noise, just as in the case of fixed radius circles on an infinite planar
substrate. The effective surface tension, which is the overall coefficient of
the action, does not depend on the latitude angle of the circles.Comment: 6 page
Crowding at the Front of the Marathon Packs
We study the crowding of near-extreme events in the time gaps between
successive finishers in major international marathons. Naively, one might
expect these gaps to become progressively larger for better-placing finishers.
While such an increase does indeed occur from the middle of the finishing pack
down to approximately 20th place, the gaps saturate for the first 10-20
finishers. We give a probabilistic account of this feature. However, the data
suggests that the gaps have a weak maximum around the 10th place, a feature
that seems to have a sociological origin.Comment: 5 pages, 2 figures; version 2: published manuscript with various
changes in response to referee comments and some additional improvement
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