A system of grabbing particles related to Galton-Watson trees

We consider a system of particles with arms that are activated randomly to grab other particles as a toy model for polymerization. We assume that the following two rules are fulfilled: Once a particle has been grabbed then it cannot be grabbed again, and an arm cannot grab a particle that belongs to its own cluster. We are interested in the shape of a typical polymer in the situation when the initial number of monomers is large and the numbers of arms of monomers are given by i.i.d. random variables. Our main result is a limit theorem for the empirical distribution of polymers, where limit is expressed in terms of a Galton-Watson tree

A random tunnel number one 3-manifold does not fiber over the circle

We address the question: how common is it for a 3-manifold to fiber over the circle? One motivation for considering this is to give insight into the fairly inscrutable Virtual Fibration Conjecture. For the special class of 3-manifolds with tunnel number one, we provide compelling theoretical and experimental evidence that fibering is a very rare property. Indeed, in various precise senses it happens with probability 0. Our main theorem is that this is true for a measured lamination model of random tunnel number one 3-manifolds. The first ingredient is an algorithm of K Brown which can decide if a given tunnel number one 3-manifold fibers over the circle. Following the lead of Agol, Hass and W Thurston, we implement Brown's algorithm very efficiently by working in the context of train tracks/interval exchanges. To analyze the resulting algorithm, we generalize work of Kerckhoff to understand the dynamics of splitting sequences of complete genus 2 interval exchanges. Combining all of this with a "magic splitting sequence" and work of Mirzakhani proves the main theorem. The 3-manifold situation contrasts markedly with random 2-generator 1-relator groups; in particular, we show that such groups "fiber" with probability strictly between 0 and 1.Comment: This is the version published by Geometry & Topology on 15 December 200

Spatial Distribution of Leprosy in the Amazon Region of Brazil

To detect areas with increased case-detection rates, we used spatial scan statistics to identify 5 of 10 clusters of leprosy in the Amazon region of Brazil. Despite increasing economic development, population growth, and road infrastructure, leprosy is endemic to this region, which is a source of case exportation to other parts of Brazil

Evaluating spatial surveillance: detection of known outbreaks in real data

Since the anthrax attacks of October 2001 and the SARS outbreaks of recent years, there has been an increasing interest in developing surveillance systems to aid in the early detection of such illness. Systems have been established which do this is by monitoring primary health-care visits, pharmacy sales, absenteeism records, and other non-traditional sources of data. While many resources have been invested in establishing such systems, relatively little effort has as yet been expended in evaluating their performance. One way to evaluate a given surveillance system is to compare the signals it generates with known outbreaks identified in other systems. In public health practice, for example, public health departments investigate reports of illness and sometimes track hospital admissions. Comparison of new systems with extant systems cannot generate estimates of test characteristics such as sensitivity and specificity, since the actual number of positives and negatives cannot be known. However, the comparison can reveal whether a new or proposed system’s signals match outbreaks detected by the existing system. This could help support or reject the new system as an alternative or complement to the extant system. We propose three methods to test the null hypothesis that the new system does not signal true outbreaks more often than would be expected by chance. The methods dier in the restrictiveness of the assumptions required. Each test may detect weaknesses in the new system, depending on the distribution of outbreaks and can be used to construct confidence limits on the agreement between the new system’s signals and the outbreaks, given the distribution of the signals. They can be used to assess whether the new system works in that it detects the outbreaks better than chance would suggest and can also determine if the new systems’ signals are generated earlier than an extant system

Random division of an interval

The well-known relation between random division of an interval and the Poisson process is interpreted as a Laplace transformation. With the use of this interpretation a number of (in part known) results is derived very easily

The structure of typical clusters in large sparse random configurations

The initial purpose of this work is to provide a probabilistic explanation of a recent result on a version of Smoluchowski's coagulation equations in which the number of aggregations is limited. The latter models the deterministic evolution of concentrations of particles in a medium where particles coalesce pairwise as time passes and each particle can only perform a given number of aggregations. Under appropriate assumptions, the concentrations of particles converge as time tends to infinity to some measure which bears a striking resemblance with the distribution of the total population of a Galton-Watson process started from two ancestors. Roughly speaking, the configuration model is a stochastic construction which aims at producing a typical graph on a set of vertices with pre-described degrees. Specifically, one attaches to each vertex a certain number of stubs, and then join pairwise the stubs uniformly at random to create edges between vertices. In this work, we use the configuration model as the stochastic counterpart of Smoluchowski's coagulation equations with limited aggregations. We establish a hydrodynamical type limit theorem for the empirical measure of the shapes of clusters in the configuration model when the number of vertices tends to $\infty$. The limit is given in terms of the distribution of a Galton-Watson process started with two ancestors

Random Neighbor Theory of the Olami-Feder-Christensen Earthquake Model

We derive the exact equations of motion for the random neighbor version of the Olami-Feder-Christensen earthquake model in the infinite-size limit. We solve them numerically, and compare with simulations of the model for large numbers of sites. We find perfect agreement. But we do not find any scaling or phase transitions, except in the conservative limit. This is in contradiction to claims by Lise & Jensen (Phys. Rev. Lett. 76, 2326 (1996)) based on approximate solutions of the same model. It indicates again that scaling in the Olami-Feder-Christensen model is only due to partial synchronization driven by spatial inhomogeneities. Finally, we point out that our method can be used also for other SOC models, and treat in detail the random neighbor version of the Feder-Feder model.Comment: 18 pages, 6 ps-figures included; minor correction in sec.