The multifractal spectrum of Brownian intersection local times

Let \ell be the projected intersection local time of two independent Brownian paths in R^d for d=2,3. We determine the lower tail of the random variable \ell(U), where U is the unit ball. The answer is given in terms of intersection exponents, which are explicitly known in the case of planar Brownian motion. We use this result to obtain the multifractal spectrum, or spectrum of thin points, for the intersection local times.Comment: Published at http://dx.doi.org/10.1214/009117905000000116 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Robustness of scale-free spatial networks

A growing family of random graphs is called robust if it retains a giant component after percolation with arbitrary positive retention probability. We study robustness for graphs, in which new vertices are given a spatial position on the $d$-dimensional torus and are connected to existing vertices with a probability favouring short spatial distances and high degrees. In this model of a scale-free network with clustering we can independently tune the power law exponent $\tau$ of the degree distribution and the rate $\delta d$ at which the connection probability decreases with the distance of two vertices. We show that the network is robust if $\tau<2+1/\delta$, but fails to be robust if $\tau>3$. In the case of one-dimensional space we also show that the network is not robust if $\tau<2+1/(\delta-1)$. This implies that robustness of a scale-free network depends not only on its power-law exponent but also on its clustering features. Other than the classical models of scale-free networks our model is not locally tree-like, and hence we need to develop novel methods for its study, including, for example, a surprising application of the BK-inequality.Comment: 34 pages, 4 figure

A conditioning principle for Galton-Watson trees

We show that an infinite Galton-Watson tree, conditioned on its martingale limit being smaller than \eps, converges as \eps\downarrow 0 in law to the regular $\mu$-ary tree, where $\mu$ is the essential minimum of the offspring distribution. This gives an example of entropic repulsion where the limit has no entropy.Comment: This is now superseded by a new paper, arXiv:1204.3080, written jointly with Nina Gantert. The new paper contains much stronger results (e.g. the two-point concentration of the level at which the Galton-Watson tree ceases to be minimal) based on a significantly more delicate analysis, making the present paper redundan

Galton-Watson trees with vanishing martingale limit

We show that an infinite Galton-Watson tree, conditioned on its martingale limit being smaller than \eps, agrees up to generation $K$ with a regular $\mu$-ary tree, where $\mu$ is the essential minimum of the offspring distribution and the random variable $K$ is strongly concentrated near an explicit deterministic function growing like a multiple of \log(1/\eps). More precisely, we show that if $\mu\ge 2$ then with high probability as \eps \downarrow 0, $K$ takes exactly one or two values. This shows in particular that the conditioned trees converge to the regular $\mu$-ary tree, providing an example of entropic repulsion where the limit has vanishing entropy.Comment: This supersedes an earlier paper, arXiv:1006.2315, written by a subset of the authors. Compared with the earlier version, the main result (the two-point concentration of the level at which the Galton-Watson tree ceases to be minimal) is much stronger and requires significantly more delicate analysi

Bayesian model choice in cumulative link ordinal regression models

The use of the proportional odds (PO) model for ordinal regression is ubiquitous in the literature. If the assumption of parallel lines does not hold for the data, then an alternative is to specify a non-proportional odds (NPO) model, where the regression parameters are allowed to vary depending on the level of the response. However, it is often difficult to fit these models, and challenges regarding model choice and fitting are further compounded if there are a large number of explanatory variables. We make two contributions towards tackling these issues: firstly, we develop a Bayesian method for fitting these models, that ensures the stochastic ordering conditions hold for an arbitrary finite range of the explanatory variables, allowing NPO models to be fitted to any observed data set. Secondly, we use reversible-jump Markov chain Monte Carlo to allow the model to choose between PO and NPO structures for each explanatory variable, and show how variable selection can be incorporated. These methods can be adapted for any monotonic increasing link functions. We illustrate the utility of these approaches on novel data from a longitudinal study of individual-level risk factors affecting body condition score in a dog population in Zenzele, South Africa.TJM is supported by Biotechnology and Biological Sciences Research Council grant number BB/I012192/1. MM is supported by a grant from the International Fund for Animal Welfare (IFAW) and the World Society for the Protection of Animals (WSPA), with additional support from the Charles Slater Fund and the Jowett Fund. JW is supported by the Alborada Trust and the RAPIDD program of the Science and Technology Directorate, Department of Homeland Security and the Fogarty International Centre.This is the final version of the article. It was first available from International Society for Bayesian Analysis via http://dx.doi.org/10.1214/14-BA88

Evolution of collision numbers for a chaotic gas dynamics

We put forward a conjecture of recurrence for a gas of hard spheres that collide elastically in a finite volume. The dynamics consists of a sequence of instantaneous binary collisions. We study how the numbers of collisions of different pairs of particles grow as functions of time. We observe that these numbers can be represented as a time-integral of a function on the phase space. Assuming the results of the ergodic theory apply, we describe the evolution of the numbers by an effective Langevin dynamics. We use the facts that hold for these dynamics with probability one, in order to establish properties of a single trajectory of the system. We find that for any triplet of particles there will be an infinite sequence of moments of time, when the numbers of collisions of all three different pairs of the triplet will be equal. Moreover, any value of difference of collision numbers of pairs in the triplet will repeat indefinitely. On the other hand, for larger number of pairs there is but a finite number of repetitions. Thus the ergodic theory produces a limitation on the dynamics.Comment: 4 pages, published versio

The demography of free-roaming dog populations and applications to disease and population control

Understanding the demography of domestic dog populations is essential for effective disease control, particularly of canine-mediated rabies. Demographic data are also needed to plan effective population management. However, no study has comprehensively evaluated the contribution of demographic processes (i.e. births, deaths and movement) to variations in dog population size or density, or determined the factors that regulate these processes, including human factors. We report the results of a 3-year cohort study of domestic dogs, which is the first to generate detailed data on the temporal variation of these demographic characteristics. The study was undertaken in two communities in each of Bali, Indonesia and Johannesburg, South Africa, in rabies-endemic areas and where the majority of dogs were free-roaming. None of the four communities had been engaged in any dog population management interventions by local authorities or animal welfare organizations. All identified dogs in the four communities were monitored individually throughout the study. We observed either no population growth or a progressive decline in population size during the study period. There was no clear evidence that population size was regulated through environmental resource constraints. Rather, almost all of the identified dogs were owned and fed regularly by their owners, consistent with population size regulated by human demand. Finally, a substantial fraction of the dogs originated from outside the population, entirely through the translocation of dogs by people, rather than from local births. These findings demonstrate that previously reported growth of dog populations is not a general phenomenon and challenge the widely held view that free-roaming dogs are unowned and form closed populations. Synthesis and applications. These observations have broad implications for disease and population control. The accessibility of dogs for vaccination and evaluation through owners and the movement of dogs (some of them infected) by people will determine the viable options for disease control strategies. The impact of human factors on population dynamics will also influence the feasibility of annual vaccination campaigns to control rabies and population control through culling or sterilization. The complex relationship between dogs and people is critically important in the transmission and control of canine-mediated rabies. For effective management, human factors must be considered in the development of disease and population control programmes