4,895 research outputs found
On the binding of polarons in a mean-field quantum crystal
We consider a multi-polaron model obtained by coupling the many-body
Schr\"odinger equation for N interacting electrons with the energy functional
of a mean-field crystal with a localized defect, obtaining a highly non linear
many-body problem. The physical picture is that the electrons constitute a
charge defect in an otherwise perfect periodic crystal. A remarkable feature of
such a system is the possibility to form a bound state of electrons via their
interaction with the polarizable background. We prove first that a single
polaron always binds, i.e. the energy functional has a minimizer for N=1. Then
we discuss the case of multi-polarons containing two electrons or more. We show
that their existence is guaranteed when certain quantized binding inequalities
of HVZ type are satisfied.Comment: 28 pages, a mistake in the former version has been correcte
Application of Sequential Quasi-Monte Carlo to Autonomous Positioning
Sequential Monte Carlo algorithms (also known as particle filters) are
popular methods to approximate filtering (and related) distributions of
state-space models. However, they converge at the slow rate, which
may be an issue in real-time data-intensive scenarios. We give a brief outline
of SQMC (Sequential Quasi-Monte Carlo), a variant of SMC based on
low-discrepancy point sets proposed by Gerber and Chopin (2015), which
converges at a faster rate, and we illustrate the greater performance of SQMC
on autonomous positioning problems.Comment: 5 pages, 4 figure
Negative association, ordering and convergence of resampling methods
We study convergence and convergence rates for resampling schemes. Our first
main result is a general consistency theorem based on the notion of negative
association, which is applied to establish the almost-sure weak convergence of
measures output from Kitagawa's (1996) stratified resampling method. Carpenter
et al's (1999) systematic resampling method is similar in structure but can
fail to converge depending on the order of the input samples. We introduce a
new resampling algorithm based on a stochastic rounding technique of Srinivasan
(2001), which shares some attractive properties of systematic resampling, but
which exhibits negative association and therefore converges irrespective of the
order of the input samples. We confirm a conjecture made by Kitagawa (1996)
that ordering input samples by their states in yields a faster
rate of convergence; we establish that when particles are ordered using the
Hilbert curve in , the variance of the resampling error is
under mild conditions, where
is the number of particles. We use these results to establish asymptotic
properties of particle algorithms based on resampling schemes that differ from
multinomial resampling.Comment: 54 pages, including 30 pages of supplementary materials (a typo in
Algorithm 1 has been corrected
Birth and death processes with neutral mutations
In this paper, we review recent results of ours concerning branching
processes with general lifetimes and neutral mutations, under the infinitely
many alleles model, where mutations can occur either at birth of individuals or
at a constant rate during their lives.
In both models, we study the allelic partition of the population at time t.
We give closed formulae for the expected frequency spectrum at t and prove
pathwise convergence to an explicit limit, as t goes to infinity, of the
relative numbers of types younger than some given age and carried by a given
number of individuals (small families). We also provide convergences in
distribution of the sizes or ages of the largest families and of the oldest
families.
In the case of exponential lifetimes, population dynamics are given by linear
birth and death processes, and we can most of the time provide general
formulations of our results unifying both models.Comment: 20 pages, 2 figure
Gibbs measures based on 1D (an)harmonic oscillators as mean-field limits
We prove that Gibbs measures based on 1D defocusing nonlinear Schr{\"o}dinger
functionals with sub-harmonic trapping can be obtained as the mean-field/large
temperature limit of the corresponding grand-canonical ensemble for many
bosons. The limit measure is supported on Sobolev spaces of negative regularity
and the corresponding density matrices are not trace-class. The general proof
strategy is that of a previous paper of ours, but we have to complement it with
Hilbert-Schmidt estimates on reduced density matrices.Comment: Minor changes and precision
The mean-field approximation and the non-linear Schr\"odinger functional for trapped Bose gases
We study the ground state of a trapped Bose gas, starting from the full
many-body Schr{\"o}dinger Hamiltonian, and derive the nonlinear Schr{\"o}dinger
energy functional in the limit of large particle number, when the interaction
potential converges slowly to a Dirac delta function. Our method is based on
quantitative estimates on the discrepancy between the full many-body energy and
its mean-field approximation using Hartree states. These are proved using
finite dimensional localization and a quantitative version of the quantum de
Finetti theorem. Our approach covers the case of attractive interactions in the
regime of stability. In particular, our main new result is a derivation of the
2D attractive nonlinear Schr{\"o}dinger ground state
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