147 research outputs found
Random equations in aerodynamics
Literature was reviewed to identify aerodynamic models which might be treated by probablistic methods. The numerical solution of some integral equations that arise in aerodynamical problems were investigated. On the basis of the numerical studies a qualitative theory of random integral equations was developed to provide information on the behavior of the solutions of these equations (in particular, boundary and asymptotic behavior, and stability) and their statistical properties without actually obtaining explicit solutions of the equations
Condensation of the roots of real random polynomials on the real axis
We introduce a family of real random polynomials of degree n whose
coefficients a_k are symmetric independent Gaussian variables with variance
= e^{-k^\alpha}, indexed by a real \alpha \geq 0. We compute exactly
the mean number of real roots for large n. As \alpha is varied, one finds
three different phases. First, for 0 \leq \alpha \sim
(\frac{2}{\pi}) \log{n}. For 1 < \alpha < 2, there is an intermediate phase
where grows algebraically with a continuously varying exponent,
\sim \frac{2}{\pi} \sqrt{\frac{\alpha-1}{\alpha}} n^{\alpha/2}. And finally for
\alpha > 2, one finds a third phase where \sim n. This family of real
random polynomials thus exhibits a condensation of their roots on the real line
in the sense that, for large n, a finite fraction of their roots /n are
real. This condensation occurs via a localization of the real roots around the
values \pm \exp{[\frac{\alpha}{2}(k+{1/2})^{\alpha-1} ]}, 1 \ll k \leq n.Comment: 13 pages, 2 figure
QED cascades induced by circularly polarized laser fields
The results of Monte-Carlo simulations of electron-positron-photon cascades
initiated by slow electrons in circularly polarized fields of ultra-high
strength are presented and discussed. Our results confirm previous qualitative
estimations [A.M. Fedotov, et al., PRL 105, 080402 (2010)] of the formation of
cascades. This sort of cascades has revealed the new property of the
restoration of energy and dynamical quantum parameter due to the acceleration
of electrons and positrons by the field and may become a dominating feature of
laser-matter interactions at ultra-high intensities. Our approach incorporates
radiation friction acting on individual electrons and positrons.Comment: 13 pages, 10 figure
Distribution of roots of random real generalized polynomials
The average density of zeros for monic generalized polynomials,
, with real holomorphic and
real Gaussian coefficients is expressed in terms of correlation functions of
the values of the polynomial and its derivative. We obtain compact expressions
for both the regular component (generated by the complex roots) and the
singular one (real roots) of the average density of roots. The density of the
regular component goes to zero in the vicinity of the real axis like
. We present the low and high disorder asymptotic
behaviors. Then we particularize to the large limit of the average density
of complex roots of monic algebraic polynomials of the form with real independent, identically distributed
Gaussian coefficients having zero mean and dispersion . The average density tends to a simple, {\em universal}
function of and in the domain where nearly all the roots are located for
large .Comment: 17 pages, Revtex. To appear in J. Stat. Phys. Uuencoded gz-compresed
tarfile (.66MB) containing 8 Postscript figures is available by e-mail from
[email protected]
Intermittency in Branching Processes
We study the intermittency properties of two branching processes, one with a
uniform and another with a singular splitting kernel. The asymptotic
intermittency indices, as well as the leading corrections to the asymptotic
linear regime are explicitly computed in an analytic framework. Both models are
found to possess a monofractal spectrum with . Relations with
previous results are discussed.Comment: 20 pages, UCLA93/TEP/2
Correlations between zeros of a random polynomial
We obtain exact analytical expressions for correlations between real zeros of
the Kac random polynomial. We show that the zeros in the interval are
asymptotically independent of the zeros outside of this interval, and that the
straightened zeros have the same limit translation invariant correlations. Then
we calculate the correlations between the straightened zeros of the SO(2)
random polynomial.Comment: 31 pages, 2 figures; a revised version of the J. Stat. Phys. pape
Multifractal Multiplicity Distribution in Bunching-Parameter Analysis
A new multiplicity distribution with multifractal properties which can be
used in high-energy physics and quantum optics is proposed. It may be
considered as a generalization of the negative-binomial distribution. We find
the structure of the generating function for such distribution and discuss its
properties.Comment: LaTex, 12 pages, cite.st
Finite-size scaling of directed percolation in the steady state
Recently, considerable progress has been made in understanding finite-size
scaling in equilibrium systems. Here, we study finite-size scaling in
non-equilibrium systems at the instance of directed percolation (DP), which has
become the paradigm of non-equilibrium phase transitions into absorbing states,
above, at and below the upper critical dimension. We investigate the
finite-size scaling behavior of DP analytically and numerically by considering
its steady state generated by a homogeneous constant external source on a
d-dimensional hypercube of finite edge length L with periodic boundary
conditions near the bulk critical point. In particular, we study the order
parameter and its higher moments using renormalized field theory. We derive
finite-size scaling forms of the moments in a one-loop calculation. Moreover,
we introduce and calculate a ratio of the order parameter moments that plays a
similar role in the analysis of finite size scaling in absorbing nonequilibrium
processes as the famous Binder cumulant in equilibrium systems and that, in
particular, provides a new signature of the DP universality class. To
complement our analytical work, we perform Monte Carlo simulations which
confirm our analytical results.Comment: 21 pages, 6 figure
Two-Body Random Ensembles: From Nuclear Spectra to Random Polynomials
The two-body random ensemble (TBRE) for a many-body bosonic theory is mapped
to a problem of random polynomials on the unit interval. In this way one can
understand the predominance of 0+ ground states, and analytic expressions can
be derived for distributions of lowest eigenvalues, energy gaps, density of
states and so forth. Recently studied nuclear spectroscopic properties are
addressed.Comment: 8 pages, 4 figures. To appear in Physical Review Letter
Small ball probability, Inverse theorems, and applications
Let be a real random variable with mean zero and variance one and
be a multi-set in . The random sum
where are iid copies of
is of fundamental importance in probability and its applications.
We discuss the small ball problem, the aim of which is to estimate the
maximum probability that belongs to a ball with given small radius,
following the discovery made by Littlewood-Offord and Erdos almost 70 years
ago. We will mainly focus on recent developments that characterize the
structure of those sets where the small ball probability is relatively
large. Applications of these results include full solutions or significant
progresses of many open problems in different areas.Comment: 47 page
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