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 $\mathbb{R}$ yields a faster rate of convergence; we establish that when particles are ordered using the Hilbert curve in $\mathbb{R}^d$, the variance of the resampling error is ${\scriptscriptstyle\mathcal{O}}(N^{-(1+1/d)})$ under mild conditions, where $N$ 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

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

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

Classical field theory limit of many-body quantum Gibbs states in 2D and 3D

We provide the first rigorous derivation of Gibbs measures in two and three space dimensions, starting from many-body quantum systems in thermal equilibrium. More precisely, we prove that the grand-canonical Gibbs state of a large bosonic quantum system converges to the Gibbs measure of a nonlinear Schr{\"o}dinger-type classical field theory in terms of partition functions and reduced density matrices. The Gibbs measure thus describes the behavior of the infinite Bose gas at criticality, that is, close to the phase transition to a Bose-Einstein condensate. The Gibbs measure is concentrated on singular distributions and has to be appropriately renormalized, while the quantum system is well defined without any renormalization. By tuning a single real parameter (the chemical potential), we obtain a counter-term for the diverging repulsive interactions which provides the desired Wick renormalization of the limit classical theory. The proof relies on a new estimate on the entropy relative to quasi-free states and a novel method to control quantum variances.Comment: This revised version covers both the 2D and 3D cases and it replaces an older 2018 work that was only handling the 2D case. For the latter, please refer to versions 1-2 of this preprin

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 $1/\sqrt{N}$ 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

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

External shocks, internal shots: The geography of civil conflicts

This paper uses detailed information on the latitude and longitude of conflict events within a set of Sub-Saharan African countries to study the impact of external income shocks on the likelihood of violence. We consider a number of external demand shocks faced by the country or the regions within countries - changes in the world demand of agricultural commodities, financial crises in the partner countries or changes in foreign trade policy - and combine these with information reflecting the natural level of trade openness of the location. We find that (i) within-country, the incidence, intensity and onset of conflicts are generally negatively and significantly correlated with income shocks within locations; (ii) this relationship is significantly weaker for the most remote locations, i.e those located away from the main seaports, (iii) at country-level, we cannot detect any significant effect of these shock on conflict incidence or onset; but (iv) large and longlasting shocks seem to affect the location of conflict outbreaks. In general, our results suggest that external income shocks are important determinants of the intensity and geography of conflicts within countries. However, conflicts tend to start in remote locations which are naturally less affected by foreign shocks, which might explain why these seem to have little effect on conflict onset at the country-level