Phase Transition in the Aldous-Shields Model of Growing Trees

We study analytically the late time statistics of the number of particles in a growing tree model introduced by Aldous and Shields. In this model, a cluster grows in continuous time on a binary Cayley tree, starting from the root, by absorbing new particles at the empty perimeter sites at a rate proportional to c^{-l} where c is a positive parameter and l is the distance of the perimeter site from the root. For c=1, this model corresponds to random binary search trees and for c=2 it corresponds to digital search trees in computer science. By introducing a backward Fokker-Planck approach, we calculate the mean and the variance of the number of particles at large times and show that the variance undergoes a `phase transition' at a critical value c=sqrt{2}. While for c>sqrt{2} the variance is proportional to the mean and the distribution is normal, for c<sqrt{2} the variance is anomalously large and the distribution is non-Gaussian due to the appearance of extreme fluctuations. The model is generalized to one where growth occurs on a tree with $m$ branches and, in this more general case, we show that the critical point occurs at c=sqrt{m}.Comment: Latex 17 pages, 6 figure

Stochastic Analysis

The meeting took place on May 30-June 3, 2011, with over 55 people in attendance. Each day had 6 to 7 talks of varying length (some talks were 30 minutes long), except for Thursday: the traditional hike was moved to Thursday due to the weather (and weather on thursday was indeed fine). The talks reviewed directions in which progress in the general field of stochastic analysis occurred since the last meeting of this theme in Oberwolfach three years ago. Several themes were covered in some depth, in addition to a broad overview of recent developments. Among these themes a prominent role was played by random matrices, random surfaces/planar maps and their scaling limits, the KPZ universality class, and the interplay between SLE (Schramm-Loewner equation) and the GFF (Gaussian free field)

HIV-TRACE (Transmission Cluster Engine):A tool for large scale molecular epidemiology of HIV-1 and other rapidly evolving pathogens

In modern applications of molecular epidemiology, genetic sequence data are routinely used to identify clusters of transmission in rapidly evolving pathogens, most notably HIV-1. Traditional 'shoe-leather' epidemiology infers transmission clusters by tracing chains of partners sharing epidemiological connections (e.g., sexual contact). Here, we present a computational tool for identifying a molecular transmission analog of such clusters: HIV-TRACE (TRAnsmission Cluster Engine). HIV-TRACE implements an approach inspired by traditional epidemiology, by identifying chains of partners whose viral genetic relatedness imply direct or indirect epidemiological connections. Molecular transmission clusters are constructed using codon-aware pairwise alignment to a reference sequence followed by pairwise genetic distance estimation among all sequences. This approach is computationally tractable and is capable of identifying HIV-1 transmission clusters in large surveillance databases comprising tens or hundreds of thousands of sequences in near real time, that is, on the order of minutes to hours. HIV-TRACE is available at www.hivtrace.org and from www.github.com/veg/hivtrace, along with the accompanying result visualization module from www.github.com/veg/hivtrace-viz. Importantly, the approach underlying HIV-TRACE is not limited to the study of HIV-1 and can be applied to study outbreaks and epidemics of other rapidly evolving pathogens

Into the abyss : a study of the mise en abyme

As no single English study of the mise en abyme with its examples in our late-modern world has been undertaken, this thesis concerns the mise en abyme in English literature. In approximately the last third of the twentieth century, the concept has increasingly been associated with ‘postmodernism’ and the essential groundlessness of all claims to general or universal truth. In this thesis, I argue that the mise en abyme has become such a broad staple of character and narrative study that its meaning is diffuse in the extreme. First celebrated in the 1980s and 1990s, by several literary thinkers as a figure capturing the spirit of postmodernism, the eventual symptomatic dissipation of the mise en abyme in literary studies resulted from critical suggestions that the mise en abyme was after all, perhaps, bogus. It subsequently became associated with aesthetic phenomena far beyond its initial characterisation by André Gide in 1893. I argue that it has now become a trope of things wider than Gide’s initial allusion and has become a metaphor for abyssal - and abysmal - things. This thesis seeks to consider the history of the mise en abyme and to offer a contemporary account of what it might mean: it does this by uncovering the latent rhetorical figures which preceded the name ‘mise en abyme’. Formal readings of the play within the play in Hamlet and the gothic story read in The Fall of the House of Usher are both starting points to relink Gide’s idea to its, more common, metaphorical applications. Thus, metaphors of the abyss, the dark, the occulted, the uncanny and, most precisely, the ‘sinister’ are examined in this dissertation. The thesis first evaluates the theoretical inheritance of Gide’s work and then, in the second part, applies, through close reading, the meaning of Gide’s idea to recent, and representative literary examples. The thrust of the argument is that the reason many definitions, and applications, of the mise en abyme are such a source of problems, is because the mise en abyme, as an English literary phenomenon supporting the broad thesis of postmodern Gothic aesthetics, is concerned with representing abyssal metaphors. A clear delimitation of the mise en abyme is difficult whenever connotations of the abyss, the dark, the occult and the sinister are overlooked. So, this dissertation gives a circumspect view of what is designated as mise en abyme, and argues that, in late-modernity, its meaning is closest to the rhetorical figures named ekphrasis, metalepsis, and epanalepsis. This study concludes that, realistically, there is probably no such thing as the mise en abyme and instead, there are only rhetorical figures and metaphors of the sinister and of the abyss

Scaling limits of random walks and their related parameters on critical random trees and graphs

In this thesis we study random walks in random environments, a major area in Probability theory. Within this broad topic, we are mainly focused in studying scaling limits of random walks on random graphs at criticality, that is precisely when we witness the emergence of a giant component that has size proportional to the number of vertices of the graph. Critical random graphs of interest include critical Galton-Watson trees and maximal components that belong to the Erd}os- R_enyi universality class. The first part of the thesis expands upon using analytic and geometric properties of those random graphs to establish distributional convergence of certain graph parameters, such as the blanket time. Our contribution refines the previous existing results on the order of the mean blanket time. The study of this problem can be seen as a stepping stone to deal with the more delicate problem of establishing convergence in distribution of the rescaled cover times of the discrete-time walks in each of the applications of our main result. Relying on powerful resistance techniques developed in recent years, another part of the thesis investigates random walks in random enviroments on tree-like spaces and their scaling limits in a certain regime, that is when the potential of the random walk in random environment converges. Results include novel scaling continuum limits of a biased random walk on large critical branching random walk and a self-reinforced discrete process on size-conditioned critical Galton-Watson trees. In both cases the diffusions that are not on natural scale are identified as Brownian motions on a continuum random fractal tree with its natural metric replaced by a distorted resistance metric

Big Data for Small Parks: Examining Regional Vegetation Patterns to Assess the Current Condition and Vulnerability of Eastern National Parks to Climate Change

The United States National Park Service mission is to preserve natural and cultural resources unimpaired for future generations. Given climate change, the paradigm of restoring natural resources to their pre-European settlement condition is no longer appropriate or achievable management. Instead, we must promote resilience and plan for adaptation. This approach poses many challenges, including knowledge gaps about the current condition of park ecosystems including wetlands, and lack of information about the matrix surrounding parks, which will strongly influence park ecosystem response to climate change. My dissertation research focused on filling these knowledge gaps to provide much needed information to managers in northeastern national parks (NP). We constructed multimetric indicators (MMIs) of wetland condition for vegetation, soil, water chemistry, and algae to assess wetland condition in Acadia NP, compared patterns of structure and tree diversity in park and matrix forests, and assessed migration potential of eastern tree species through dispersal simulations and spatial analyses of tree regeneration. Using the MMIs, we found Acadia NP wetlands to be in good condition overall, and identified degraded wetlands to prioritize for restoration. Our study of 50 eastern NPs found parks to have consistently older forest structure, such as higher density of large trees and greater coarse woody debris volume, than matrix forests. Our follow-up study in 39 eastern NPs documented consistently higher tree diversity in parks than matrix forests. These results suggest that park forests may respond differently and potentially be more resilient to climate change than matrix forests. However, our assessments of tree migration capacity documented significant dispersal barriers north of many southern oak (Quercus spp.), hickory (Carya spp.) and pine (Pinus spp.) species predicted to gain suitable habitat in the northeastern US. In roughly the same area, we documented widespread regeneration debt of these same southern tree species, with invasive plant species, deer overabundance, and anthropogenic land cover the likely drivers. Taken together, these results indicate that while parks may be somewhat resilient in the short-term, without intervention, longer-term adaptive capacity of northeastern forests to climate change will be severely impacted by migration barriers and regeneration debts in the mid-Atlantic region

Locating Identity and Ethnicity in Cornish Civil Society: Penzance, a Case Study

The thesis consists of two volumes, the first being the main text and the second the appendices.Recently there has been considerable interest in Cornish ethnicity reflected both by a rise in the numbers in Cornwall who identify as Cornish and by academic research. Cornish studies have constructed a regional narrative embracing Celticity and an economy based on primary industries, particularly mining, from which has evolved a distinctive culture. This study adopting an ethnographic approach, extends Cornish studies by considering a number of elements which have not previously been addressed. These include investigating how identity may be played out in a particular place to see whether there may be differences in how ethnicity is performed within Cornwall, looking at how it may be practiced collectively in the context of civil society and examining the relationship between ethnicity and place identity. Three settings within Penzance have been selected to represent some of the issues prevalent in twenty first century Cornwall. They include a study of festivals celebrating ethnicity and place identity, an investigation of how kinship and ethnicity are the basis for social cohesion on a social housing estate and an analysis of a dispute over harbour re-development reflecting tensions between regeneration and conservation. Investigating the civil society associated with each of these settings has identified a number of discourses which influence place images, are the focus for debate and reflect different ways in which ethnicity is articulated and performed. Influences on Cornish identity have been exposed which have not been previously explored by Cornish Studies including the relationship between civil society and the state, the importance of place mythology and the impact of inward migration. The study concludes that collective identities, ethnicity and place images are constantly in flux driven by discourses debated within the micro-politics of civil society and that the overarching narratives of Cornishness contain tensions and cleavages which help explain the fractured nature of much of public life in Cornwall