35 research outputs found
Super-Rough Glassy Phase of the Random Field XY Model in Two Dimensions
We study both analytically, using the renormalization group (RG) to two loop
order, and numerically, using an exact polynomial algorithm, the
disorder-induced glass phase of the two-dimensional XY model with quenched
random symmetry-breaking fields and without vortices. In the super-rough glassy
phase, i.e. below the critical temperature , the disorder and thermally
averaged correlation function of the phase field , behaves, for , as where and is a microscopic length scale. We
derive the RG equations up to cubic order in and predict
the universal amplitude . The
universality of results from nontrivial cancellations between
nonuniversal constants of RG equations. Using an exact polynomial algorithm on
an equivalent dimer version of the model we compute numerically and
obtain a remarkable agreement with our analytical prediction, up to .Comment: 5 pages, 3 figure
Conservative interacting particles system with anomalous rate of ergodicity
We analyze certain conservative interacting particle system and establish
ergodicity of the system for a family of invariant measures. Furthermore, we
show that convergence rate to equilibrium is exponential. This result is of
interest because it presents counterexample to the standard assumption of
physicists that conservative system implies polynomial rate of convergence.Comment: 16 pages; In the previous version there was a mistake in the proof of
uniqueness of weak Leray solution. Uniqueness had been claimed in a space of
solutions which was too large (see remark 2.6 for more details). Now the
mistake is corrected by introducing a new class of moderate solutions (see
definition 2.10) where we have both existence and uniquenes
Kinetic hierarchy and propagation of chaos in biological swarm models
We consider two models of biological swarm behavior. In these models, pairs
of particles interact to adjust their velocities one to each other. In the
first process, called 'BDG', they join their average velocity up to some noise.
In the second process, called 'CL', one of the two particles tries to join the
other one's velocity. This paper establishes the master equations and BBGKY
hierarchies of these two processes. It investigates the infinite particle limit
of the hierarchies at large time-scale. It shows that the resulting kinetic
hierarchy for the CL process does not satisfy propagation of chaos. Numerical
simulations indicate that the BDG process has similar behavior to the CL
process
Exclusion processes with degenerate rates: convergence to equilibrium and tagged particle
Stochastic lattice gases with degenerate rates, namely conservative particle
systems where the exchange rates vanish for some configurations, have been
introduced as simplified models for glassy dynamics. We introduce two
particular models and consider them in a finite volume of size in
contact with particle reservoirs at the boundary. We prove that, as for
non--degenerate rates, the inverse of the spectral gap and the logarithmic
Sobolev constant grow as . It is also shown how one can obtain, via a
scaling limit from the logarithmic Sobolev inequality, the exponential decay of
a macroscopic entropy associated to a degenerate parabolic differential
equation (porous media equation). We analyze finally the tagged particle
displacement for the stationary process in infinite volume. In dimension larger
than two we prove that, in the diffusive scaling limit, it converges to a
Brownian motion with non--degenerate diffusion coefficient.Comment: 25 pages, 3 figure
Celebrating Cercignani's conjecture for the Boltzmann equation
Cercignani's conjecture assumes a linear inequality between the entropy and
entropy production functionals for Boltzmann's nonlinear integral operator in
rarefied gas dynamics. Related to the field of logarithmic Sobolev inequalities
and spectral gap inequalities, this issue has been at the core of the renewal
of the mathematical theory of convergence to thermodynamical equilibrium for
rarefied gases over the past decade. In this review paper, we survey the
various positive and negative results which were obtained since the conjecture
was proposed in the 1980s.Comment: This paper is dedicated to the memory of the late Carlo Cercignani,
powerful mind and great scientist, one of the founders of the modern theory
of the Boltzmann equation. 24 pages. V2: correction of some typos and one
ref. adde
The Newcomb-Benford Law in Its Relation to Some Common Distributions
An often reported, but nevertheless persistently striking observation, formalized as the Newcomb-Benford law (NBL), is that the frequencies with which the leading digits of numbers occur in a large variety of data are far away from being uniform. Most spectacular seems to be the fact that in many data the leading digit 1 occurs in nearly one third of all cases. Explanations for this uneven distribution of the leading digits were, among others, scale- and base-invariance. Little attention, however, found the interrelation between the distribution of the significant digits and the distribution of the observed variable. It is shown here by simulation that long right-tailed distributions of a random variable are compatible with the NBL, and that for distributions of the ratio of two random variables the fit generally improves. Distributions not putting most mass on small values of the random variable (e.g. symmetric distributions) fail to fit. Hence, the validity of the NBL needs the predominance of small values and, when thinking of real-world data, a majority of small entities. Analyses of data on stock prices, the areas and numbers of inhabitants of countries, and the starting page numbers of papers from a bibliography sustain this conclusion. In all, these findings may help to understand the mechanisms behind the NBL and the conditions needed for its validity. That this law is not only of scientific interest per se, but that, in addition, it has also substantial implications can be seen from those fields where it was suggested to be put into practice. These fields reach from the detection of irregularities in data (e.g. economic fraud) to optimizing the architecture of computers regarding number representation, storage, and round-off errors
A UNIFYING PROBABILISTIC INTERPRETATION OF BENFORDâS LAW
We propose a probabilistic interpretation of Benfordâs law, which
predicts the probability distribution of all digits in everyday-life numbers. Heuri-
stically, our point of view consists in considering an everyday-life number as a
continuous random variable taking value in an interval [0,A], whose maximum
A is itself an everyday-life number. This approach can be linked to the chara-
cterization of Benfordâs law by scale-invariance, as well as to the convergence
of a product of independent random variables to Benfordâs law. It also allows
to generalize Flehingerâs result about the convergence of iterations of Cesaro-
averages to Benfordâs la
A note on domino shuffling
We present a variation of James Propp's generalized domino shuffling, which provides an efficient way to obtain perfect matchings of weighted Aztec diamonds. Our modification is specially tailored to deal with cases when some of the weights are zero. This allows us to tile efficiently a large class of planar graphs, by embedding them in a large enough Aztec diamond. We also give a sufficient condition on the size of the latter diamond for the algorithm to succeed