919 research outputs found
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
Renormalization group study of the two-dimensional random transverse-field Ising model
The infinite disorder fixed point of the random transverse-field Ising model
is expected to control the critical behavior of a large class of random quantum
and stochastic systems having an order parameter with discrete symmetry. Here
we study the model on the square lattice with a very efficient numerical
implementation of the strong disorder renormalization group method, which makes
us possible to treat finite samples of linear size up to . We have
calculated sample dependent pseudo-critical points and studied their
distribution, which is found to be characterized by the same shift and width
exponent: . For different types of disorder the infinite disorder
fixed point is shown to be characterized by the same set of critical exponents,
for which we have obtained improved estimates: and
. We have also studied the scaling behavior of the magnetization
in the vicinity of the critical point as well as dynamical scaling in the
ordered and disordered Griffiths phases
Disorder induced rounding of the phase transition in the large q-state Potts model
The phase transition in the q-state Potts model with homogeneous
ferromagnetic couplings is strongly first order for large q, while is rounded
in the presence of quenched disorder. Here we study this phenomenon on
different two-dimensional lattices by using the fact that the partition
function of the model is dominated by a single diagram of the high-temperature
expansion, which is calculated by an efficient combinatorial optimization
algorithm. For a given finite sample with discrete randomness the free energy
is a pice-wise linear function of the temperature, which is rounded after
averaging, however the discontinuity of the internal energy at the transition
point (i.e. the latent heat) stays finite even in the thermodynamic limit. For
a continuous disorder, instead, the latent heat vanishes. At the phase
transition point the dominant diagram percolates and the total magnetic moment
is related to the size of the percolating cluster. Its fractal dimension is
found d_f=(5+\sqrt{5})/4 and it is independent of the type of the lattice and
the form of disorder. We argue that the critical behavior is exclusively
determined by disorder and the corresponding fixed point is the isotropic
version of the so called infinite randomness fixed point, which is realized in
random quantum spin chains. From this mapping we conjecture the values of the
critical exponents as \beta=2-d_f, \beta_s=1/2 and \nu=1.Comment: 12 pages, 12 figures, version as publishe
The partially asymmetric zero range process with quenched disorder
We consider the one-dimensional partially asymmetric zero range process where
the hopping rates as well as the easy direction of hopping are random
variables. For this type of disorder there is a condensation phenomena in the
thermodynamic limit: the particles typically occupy one single site and the
fraction of particles outside the condensate is vanishing. We use extreme value
statistics and an asymptotically exact strong disorder renormalization group
method to explore the properties of the steady state. In a finite system of
sites the current vanishes as , where the dynamical exponent,
, is exactly calculated. For the transport is realized by active particles, which move with a constant velocity, whereas for
the transport is due to the anomalous diffusion of a single Brownian
particle. Inactive particles are localized at a second special site and their
number in rare realizations is macroscopic. The average density profile of
inactive particles has a width of, , in terms of the
asymmetry parameter, . In addition to this, we have investigated the
approach to the steady state of the system through a coarsening process and
found that the size of the condensate grows as for large
times. For the unbiased model is formally infinite and the coarsening is
logarithmically slow.Comment: 12 pages, 9 figure
Generalised extreme value statistics and sum of correlated variables
We show that generalised extreme value statistics -the statistics of the k-th
largest value among a large set of random variables- can be mapped onto a
problem of random sums. This allows us to identify classes of non-identical and
(generally) correlated random variables with a sum distributed according to one
of the three (k-dependent) asymptotic distributions of extreme value
statistics, namely the Gumbel, Frechet and Weibull distributions. These
classes, as well as the limit distributions, are naturally extended to real
values of k, thus providing a clear interpretation to the onset of Gumbel
distributions with non-integer index k in the statistics of global observables.
This is one of the very few known generalisations of the central limit theorem
to non-independent random variables. Finally, in the context of a simple
physical model, we relate the index k to the ratio of the correlation length to
the system size, which remains finite in strongly correlated systems.Comment: To appear in J.Phys.
Records in a changing world
In the context of this paper, a record is an entry in a sequence of random
variables (RV's) that is larger or smaller than all previous entries. After a
brief review of the classic theory of records, which is largely restricted to
sequences of independent and identically distributed (i.i.d.) RV's, new results
for sequences of independent RV's with distributions that broaden or sharpen
with time are presented. In particular, we show that when the width of the
distribution grows as a power law in time , the mean number of records is
asymptotically of order for distributions with a power law tail (the
\textit{Fr\'echet class} of extremal value statistics), of order
for distributions of exponential type (\textit{Gumbel class}), and of order
for distributions of bounded support (\textit{Weibull class}),
where the exponent describes the behaviour of the distribution at the
upper (or lower) boundary. Simulations are presented which indicate that, in
contrast to the i.i.d. case, the sequence of record breaking events is
correlated in such a way that the variance of the number of records is
asymptotically smaller than the mean.Comment: 12 pages, 2 figure
On the Role of Global Warming on the Statistics of Record-Breaking Temperatures
We theoretically study long-term trends in the statistics of record-breaking
daily temperatures and validate these predictions using Monte Carlo simulations
and data from the city of Philadelphia, for which 126 years of daily
temperature data is available. Using extreme statistics, we derive the number
and the magnitude of record temperature events, based on the observed Gaussian
daily temperatures distribution in Philadelphia, as a function of the number of
elapsed years from the start of the data. We further consider the case of
global warming, where the mean temperature systematically increases with time.
We argue that the current warming rate is insufficient to measurably influence
the frequency of record temperature events over the time range of the
observations, a conclusion that is supported by numerical simulations and the
Philadelphia temperature data.Comment: 11 pages, 6 figures, 2-column revtex4 format. For submission to
Journal of Climate. Revised version has some new results and some errors
corrected. Reformatted for Journal of Climate. Second revision has an added
reference. In the third revision one sentence that explains the simulations
is reworded for clarity. New revision 10/3/06 has considerable additions and
new results. Revision on 11/8/06 contains a number of minor corrections and
is the version that will appear in Phys. Rev.
Extreme fluctuations in noisy task-completion landscapes on scale-free networks
We study the statistics and scaling of extreme fluctuations in noisy
task-completion landscapes, such as those emerging in synchronized
distributed-computing networks, or generic causally-constrained queuing
networks, with scale-free topology. In these networks the average size of the
fluctuations becomes finite (synchronized state) and the extreme fluctuations
typically diverge only logarithmically in the large system-size limit ensuring
synchronization in a practical sense. Provided that local fluctuations in the
network are short-tailed, the statistics of the extremes are governed by the
Gumbel distribution. We present large-scale simulation results using the exact
algorithmic rules, supported by mean-field arguments based on a coarse-grained
description.Comment: 16 pages, 6 figures, revte
Local Leaders in Random Networks
We consider local leaders in random uncorrelated networks, i.e. nodes whose
degree is higher or equal than the degree of all of their neighbors. An
analytical expression is found for the probability of a node of degree to
be a local leader. This quantity is shown to exhibit a transition from a
situation where high degree nodes are local leaders to a situation where they
are not when the tail of the degree distribution behaves like the power-law
with . Theoretical results are verified by
computer simulations and the importance of finite-size effects is discussed.Comment: 4 pages, 2 figure
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