T. E. Harris' contributions to interacting particle systems and percolation

Interacting particle systems and percolation have been among the most active areas of probability theory over the past half century. Ted Harris played an important role in the early development of both fields. This paper is a bird's eye view of his work in these fields, and of its impact on later research in probability theory and mathematical physics.Comment: Published in at http://dx.doi.org/10.1214/10-AOP593 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Metagenomic Analysis of the Structure and Function of the Human Gut Microbiota in Crohn's Disease

Inflammatory bowel diseases (IBD), such as Crohn’s disease, are chronic, immunologically mediated disorders that have severe medical consequences. The current hypothesis is that these diseases are due to an overly aggressive immune response to a subset of commensal enteric bacteria. Studies to date on IBD have suggested that the disorder may be caused by a combination of bacteria and host susceptibility; however the etiologies of these diseases remain an enigma. In this application, we propose to develop and demonstrate the ability to profile Crohn’s disease at an unprecedented molecular level by elucidation of specific biomarkers (bacterial strains, genes, or proteins) that correlate to disease symptoms. To achieve this goal, we will employ a multidisciplinary approach based on metagenomic and metaproteomic molecular tools to elucidate the composition of the commensal microbiota in monozygotic twins that are either healthy or exhibit Crohn’s disease (for concordant, both are diseased; for discordant, one is healthy and one is diseased). The central hypotheses of this proposal are (1) that specific members and/or functional activities of the gastrointestinal (GI) microbiota differ in patients with Crohn’s disease as compared to healthy individuals, and (2) that it will be possible to elucidate microbial signatures which correlate with the occurrence and progression of this disease by integration of data obtained from 16S rRNA based molecular fingerprinting, metagenomics, and metaproteomics approaches. To address these hypotheses, three specific aims are proposed: 1) Obtain data on community gene content (metagenome) in a subset of healthy twins and twins with Crohn’s Disease to assess potential differences in the metabolic capabilities of the gut microbiota associated with CD, 2) Obtain data on community protein content (metaproteome) in a subset of healthy twins and twins with Crohn’s Disease to assess the state of expressed proteins associated with CD, 3) Apply various statistical clustering and classification methods to correlate/associate microbial community composition, gene and protein content with patient metadata, including metabolite profiles and clinical phenotype. The ultimate goal of these efforts is to identify novel biomarkers for non-invasive diagnostics of CD and to eventually identify drug targets (i.e. bacterial strains) for cure or suppression of disease symptoms

A reconnaissance space sensing investigation of crustal structure for a strip from the eastern Sierra Nevada to the Colorado Plateau

The author has identified the following significant results. Research progress in an investigation using ERTS-1 MSS imagery to study regional tectonics and related natural resources is summarized. Field reconnaissance guided by analysis of ERTS-1 imagery has resulted in development of a tectonic model relating strike-slip faulting to crustal extension in the southern Basin Range Province. The tectonics of the northern Death Valley-Furnace Creek Fault Zone and spacially associated volcanism and mercury mineralization were also investigated. Field work in the southern Sierra Nevada has confirmed the existence of faults and diabase dike swarms aligned along several major lineaments first recognized in ERTS-1 imagery. Various image enhancement and analysis techniques employed in the study of ERTS-1 data are summarized

Dispersion processes

We study a synchronous dispersion process in which $M$ particles are initially placed at a distinguished origin vertex of a graph $G$. At each time step, at each vertex $v$ occupied by more than one particle at the beginning of this step, each of these particles moves to a neighbour of $v$ chosen independently and uniformly at random. The dispersion process ends once the particles have all stopped moving, i.e. at the first step at which each vertex is occupied by at most one particle. For the complete graph $K_n$ and star graph $S_n$, we show that for any constant $\delta>1$, with high probability, if $M \le n/2(1-\delta)$, then the process finishes in $O(\log n)$ steps, whereas if $M \ge n/2(1+\delta)$, then the process needs $e^{\Omega(n)}$ steps to complete (if ever). We also show that an analogous lazy variant of the process exhibits the same behaviour but for higher thresholds, allowing faster dispersion of more particles. For paths, trees, grids, hypercubes and Cayley graphs of large enough sizes (in terms of $M$) we give bounds on the time to finish and the maximum distance traveled from the origin as a function of the number of particles $M$

Exclusion processes in higher dimensions: Stationary measures and convergence

There has been significant progress recently in our understanding of the stationary measures of the exclusion process on $Z$. The corresponding situation in higher dimensions remains largely a mystery. In this paper we give necessary and sufficient conditions for a product measure to be stationary for the exclusion process on an arbitrary set, and apply this result to find examples on $Z^d$ and on homogeneous trees in which product measures are stationary even when they are neither homogeneous nor reversible. We then begin the task of narrowing down the possibilities for existence of other stationary measures for the process on $Z^d$. In particular, we study stationary measures that are invariant under translations in all directions orthogonal to a fixed nonzero vector. We then prove a number of convergence results as $t\to\infty$ for the measure of the exclusion process. Under appropriate initial conditions, we show convergence of such measures to the above stationary measures. We also employ hydrodynamics to provide further examples of convergence.Comment: Published at http://dx.doi.org/10.1214/009117905000000341 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Stochastic domination: the contact process, Ising models and FKG measures

We prove for the contact process on $Z^d$, and many other graphs, that the upper invariant measure dominates a homogeneous product measure with large density if the infection rate $\lambda$ is sufficiently large. As a consequence, this measure percolates if the corresponding product measure percolates. We raise the question of whether domination holds in the symmetric case for all infinite graphs of bounded degree. We study some asymmetric examples which we feel shed some light on this question. We next obtain necessary and sufficient conditions for domination of a product measure for ``downward'' FKG measures. As a consequence of this general result, we show that the plus and minus states for the Ising model on $Z^d$ dominate the same set of product measures. We show that this latter fact fails completely on the homogenous 3-ary tree. We also provide a different distinction between $Z^d$ and the homogenous 3-ary tree concerning stochastic domination and Ising models; while it is known that the plus states for different temperatures on $Z^d$ are never stochastically ordered, on the homogenous 3-ary tree, almost the complete opposite is the case. Next, we show that on $Z^d$, the set of product measures which the plus state for the Ising model dominates is strictly increasing in the temperature. Finally, we obtain a necessary and sufficient condition for a finite number of variables, which are both FKG and exchangeable, to dominate a given product measure.Comment: 27 page