4,013 research outputs found
On the capacity functional of the infinite cluster of a Boolean model
Consider a Boolean model in with balls of random, bounded radii with
distribution , centered at the points of a Poisson process of intensity
. The capacity functional of the infinite cluster is given by
\theta_L(t) = \BP\{ Z_\infty\cap L\ne\emptyset\}, defined for each compact
.
We prove for any fixed and that is infinitely
differentiable in , except at the critical value ; we give a
Margulis-Russo type formula for the derivatives. More generally, allowing the
distribution to vary and viewing as a function of the measure
, we show that it is infinitely differentiable in all directions with
respect to the measure in the supercritical region of the cone of positive
measures on a bounded interval.
We also prove that grows at least linearly at the critical
value. This implies that the critical exponent known as is at most 1
(if it exists) for this model. Along the way, we extend a result of H.~Tanemura
(1993), on regularity of the supercritical Boolean model in with
fixed-radius balls, to the case with bounded random radii.Comment: 23 pages, 24 references, 1 figure in Annals of Applied Probability,
201
Correlations Estimates in the Lattice Anderson Model
We give a new proof of correlation estimates for arbitrary moments of the
resolvent of random Schr\"odinger operators on the lattice that generalizes and
extends the correlation estimate of Minami for the second moment. We apply this
moment bound to obtain a new -level Wegner-type estimate that measures
eigenvalue correlations through an upper bound on the probability that a local
Hamiltonian has at least eigenvalues in a given energy interval. Another
consequence of the correlation estimates is that the results on the Poisson
statistics of energy level spacing and the simplicity of the eigenvalues in the
strong localization regime hold for a wide class of translation-invariant,
selfadjoint, lattice operators with decaying off-diagonal terms and random
potentials.Comment: 16 page
Self-induced decoherence in dense neutrino gases
Dense neutrino gases exhibit collective oscillations where "self-maintained
coherence" is a characteristic feature, i.e., neutrinos of different energies
oscillate with the same frequency. In a non-isotropic gas, however, the flux
term of the neutrino-neutrino interaction has the opposite effect of causing
kinematical decoherence of neutrinos propagating in different directions, an
effect that is at the origin of the "multi-angle behavior" of neutrinos
streaming off a supernova core. We cast the equations of motion in a form where
the role of the flux term is manifest. We study in detail the symmetric case of
equal neutrino and antineutrino densities where the evolution consists of
collective pair conversions ("bipolar oscillations"). A gas of this sort is
unstable in that an infinitesimal anisotropy is enough to trigger a run-away
towards flavor equipartition. The "self-maintained coherence" of a perfectly
isotropic gas gives way to "self-induced decoherence."Comment: Revtex, 16 pages, 12 figure
Moments and central limit theorems for some multivariate Poisson functionals
This paper deals with Poisson processes on an arbitrary measurable space.
Using a direct approach, we derive formulae for moments and cumulants of a
vector of multiple Wiener-It\^o integrals with respect to the compensated
Poisson process. Second, a multivariate central limit theorem is shown for a
vector whose components admit a finite chaos expansion of the type of a Poisson
U-statistic. The approach is based on recent results of Peccati et al.\
combining Malliavin calculus and Stein's method, and also yields Berry-Esseen
type bounds. As applications, moment formulae and central limit theorems for
general geometric functionals of intersection processes associated with a
stationary Poisson process of -dimensional flats in are discussed
Martingale representation for Poisson processes with applications to minimal variance hedging
AbstractWe consider a Poisson process η on a measurable space equipped with a strict partial ordering, assumed to be total almost everywhere with respect to the intensity measure λ of η. We give a Clark–Ocone type formula providing an explicit representation of square integrable martingales (defined with respect to the natural filtration associated with η), which was previously known only in the special case, when λ is the product of Lebesgue measure on R+ and a σ-finite measure on another space X. Our proof is new and based on only a few basic properties of Poisson processes and stochastic integrals. We also consider the more general case of an independent random measure in the sense of Itô of pure jump type and show that the Clark–Ocone type representation leads to an explicit version of the Kunita–Watanabe decomposition of square integrable martingales. We also find the explicit minimal variance hedge in a quite general financial market driven by an independent random measure
Symmetry-breaking thermally induced collapse of dipolar Bose-Einstein condensates
We investigate a Bose-Einstein condensate with additional long-range dipolar
interaction in a cylindrically symmetric trap within a variational framework.
Compared to the ground state of this system, little attention has as yet been
payed to its unstable excited states. For thermal excitations, however, the
latter is of great interest, because it forms the "activated complex" that
mediates the collapse of the condensate. For a certain value of the s-wave
scatting length our investigations reveal a bifurcation in the transition
state, leading to the emergence of two additional and symmetry-breaking excited
states. Because these are of lower energy than their symmetric counterpart, we
predict the occurrence of a symmetry-breaking thermally induced collapse of
dipolar condensates. We show that its occurrence crucially depends on the trap
geometry and calculate the thermal decay rates of the system within leading
order transition state theory with the help of a uniform rate formula near the
rank-2 saddle which allows to smoothly pass the bifurcation.Comment: 6 pages, 3 figure
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