Scholars@Duke
Richard Timothy Durrett

Richard Timothy Durrett

James B. Duke Distinguished Professor Emeritus of Mathematics
Mathematics
rtd@math.duke.edu
Dept of Math, Box 90320, Durham, NC 27708-0320
120 Science Drive, Durham, NC 27708-0320

Selected Publications

A stochastic spatial model for the sterile insect control strategy

Journal Article Stochastic Processes and their Applications · March 1, 2023 In the system we study, 1’s and 0’s represent occupied and vacant sites in the contact process with births at rate λ and deaths at rate 1. −1’s are sterile individuals that do not reproduce but appear spontaneously on vacant sites at rate α and die at rate ... Full text Cite

Corrigendum to: The contact process on periodic trees (Electronic Communications in Probability)

Journal Article Electronic Communications in Probability · January 1, 2023 In [1] we considered periodic trees in which the number of children in successive generations is (n, a1, …, ak ) with maxi ai ≤ Cn1−δ and (log ai)/ log n → bi as n → ∞. Our proof contained an error. In this note we correct the proof. The theorem has change ... Full text Cite

Competitive exclusion in a model with seasonality: Three species cannot coexist in an ecosystem with two seasons.

Journal Article Theoretical population biology · December 2022 Chan, Durrett, and Lanchier introduced a multitype contact process with temporal heterogeneity involving two species competing for space on the d-dimensional integer lattice. Time is divided into two seasons. They proved that there is an open set of the pa ... Full text Cite

Selective sweeps in SARS-CoV-2 variant competition.

Journal Article Proceedings of the National Academy of Sciences of the United States of America · November 2022 The main mathematical result in this paper is that change of variables in the ordinary differential equation (ODE) for the competition of two infections in a Susceptible-Infected-Removed (SIR) model shows that the fraction of cases due to the new variant s ... Full text Cite

Susceptible–infected epidemics on evolving graphs

Journal Article Electronic Journal of Probability · January 1, 2022 The evoSIR model is a modification of the usual SIR process on a graph G in which S −I connections are broken at rate ρ and the S connects to a randomly chosen vertex. The evoSI model is the same as evoSIR but recovery is impossible. In [14] the critical v ... Full text Cite

Motion by mean curvature in interacting particle systems

Journal Article Probability Theory and Related Fields · November 1, 2021 There are a number of situations in which rescaled interacting particle systems have been shown to converge to a reaction diffusion equation (RDE) with a bistable reaction term, see e.g., Cox et al. (Astérisque 349:1–127, 2013), Durrett (Ann Appl Prob 19:4 ... Full text Cite

Signatures of neutral evolution in exponentially growing tumors: A theoretical perspective.

Journal Article PLoS computational biology · February 2021 Recent work of Sottoriva, Graham, and collaborators have led to the controversial claim that exponentially growing tumors have a site frequency spectrum that follows the 1/f law consistent with neutral evolution. This conclusion has been criticized based o ... Full text Cite

The q-voter model on the torus

Journal Article Electronic Journal of Probability · January 1, 2021 In the q-voter model, the voter at x changes its opinion at rate fqx, where fx is the fraction of neighbors with the opposite opinion. Mean-field calculations suggest that there should be coexistence between opinions if q < 1 and clustering if q > 1. This ... Full text Cite

Controlling the spread of COVID-19 on college campuses.

Journal Article Mathematical biosciences and engineering : MBE · December 2020 This research was done during the DOMath program at Duke University from May 18 to July 10, 2020. At the time, Duke and other universities across the country were wrestling with the question of how to safely welcome students back to campus in the Fall. Bec ... Full text Cite

Poisson percolation on the oriented square lattice

Journal Article Stochastic Processes and their Applications · February 1, 2020 In Poisson percolation each edge becomes open after an independent exponentially distributed time with rate that decreases in the distance from the origin. As a sequel to our work on the square lattice, we describe the limiting shape of the component conta ... Full text Cite

The contact process on random graphs and Galton Watson trees

Journal Article Alea (Rio de Janeiro) · January 1, 2020 The key to our investigation is an improved (and in a sense sharp) understanding of the survival time of the contact process on star graphs. Using these results, we show that for the contact process on Galton-Watson trees, when the offspring distribution ( ... Full text Cite

The symbiotic contact process

Journal Article Electronic Journal of Probability · January 1, 2020 We consider a contact process on Zd with two species that interact in a symbiotic manner. Each site can either be vacant or occupied by individuals of species A and/or B. Multiple occupancy by the same species at a single site is prohibited. The name symbi ... Full text Cite

Coexistence in chase-escape

Journal Article Electronic Communications in Probability · January 1, 2020 We study a competitive stochastic growth model called chase-escape in which red particles spread to adjacent uncolored sites and blue particles only to adjacent red sites. Red particles are killed when blue occupies the same site. If blue has rate-1 passag ... Full text Cite

The contact process on periodic trees

Journal Article Electronic Communications in Probability · January 1, 2020 A little over 25 years ago Pemantle [6] pioneered the study of the contact process on trees, and showed that on homogeneous trees the critical values ⅄1 and ⅄2 for global and local survival were different. He also considered trees with periodic degree sequ ... Full text Cite

Probability: Theory and Examples

Book · April 5, 2019 This lively introduction to measure-theoretic probability theory covers laws of large numbers, central limit theorems, random walks, martingales, Markov chains, ergodic theorems, and Brownian motion. Concentrating on results that are the most useful for ap ... Full text Cite

Extrapolating weak selection in evolutionary games.

Journal Article Journal of mathematical biology · January 2019 This work is inspired by a 2013 paper from Arne Traulsen's lab at the Max Plank Institute for Evolutionary Biology (Wu et al. in PLoS Comput Biol 9:e1003381, 2013). They studied evolutionary games when the mutation rate is so small that each mutation goes ... Full text Cite

The Zealot voter model

Journal Article Annals of Applied Probability · January 1, 2019 Inspired by the spread of discontent as in the 2016 presidential election, we consider a voter model in which 0's are ordinary voters and 1's are zealots. Thinking of a social network, but desiring the simplicity of an infinite object that can have a nontr ... Full text Cite

Poisson percolation on the square lattice

Journal Article Alea · January 1, 2019 Suppose that on the square lattice the edge with midpoint x becomes open at rate ∥x∥σ-1 . Let ρ(x, t) be the probability that the corresponding edge is open at time t and let n(p; t) be the distance at which edges are open with probability p at time t. We ... Full text Cite

A simple evolutionary game arising from the study of the role of igf-II in pancreatic cancer

Journal Article Annals of Applied Probability · October 1, 2018 We study an evolutionary game in which a producer at x gives birth at rate 1 to an offspring sent to a randomly chosen point in x + Nc, while a cheater at x gives birth at rate λ > 1 times the fraction of producers in x + Nd and sends its offspring to a ra ... Full text Cite

Ordinary Differential Equation Models for Adoptive Immunotherapy.

Journal Article Bulletin of mathematical biology · May 2018 Modified T cells that have been engineered to recognize the CD19 surface marker have recently been shown to be very successful at treating acute lymphocytic leukemias. Here, we explore four previous approaches that have used ordinary differential equations ... Full text Cite

Latent voter model on locally tree-like random graphs

Journal Article Stochastic Processes and their Applications · May 1, 2018 In the latent voter model, individuals who have just changed their choice have a latent period, which is exponential with rate λ, during which they will not change their opinion. We study this model on random graphs generated by a configuration model with ... Full text Open Access Cite

Block size in Geometric(p)-biased permutations

Journal Article Electronic Communications in Probability · 2018 Full text Link to item Cite

Asymptotic behavior of the brownian frog model

Journal Article Electronic Journal of Probability · January 1, 2018 We introduce an extension of the frog model to Euclidean space and prove properties for the spread of active particles. Fix r>0 and place a particle at each point x of a unit intensity Poisson point process P⊆ℝd−B(0,r). Around each point in P, put a ball o ... Full text Open Access Cite

Diffusion limit for the partner model at the critical value

Journal Article Electronic Journal of Probability · January 1, 2018 population with random formation and dissolution of partnerships, and with disease transmission only occuring within partnerships. Foxall, Edwards, and van den Driessche [7] found the critical value and studied the subcritical and supercritical regimes. Re ... Full text Cite

Stochastic Calculus: A Practical Introduction

Book · January 1, 2018 This compact yet thorough text zeros in on the parts of the theory that are particularly relevant to applications. It begins with a description of Brownian motion and the associated stochastic calculus, including their relationship to partial differential ... Full text Cite

Persistence and reversal of plasmid-mediated antibiotic resistance.

Journal Article Nature communications · November 2017 In the absence of antibiotic-mediated selection, sensitive bacteria are expected to displace their resistant counterparts if resistance genes are costly. However, many resistance genes persist for long periods in the absence of antibiotics. Horizontal gene ... Full text Cite

Temporal profiles of avalanches on networks.

Journal Article Nature communications · October 2017 An avalanche or cascade occurs when one event causes one or more subsequent events, which in turn may cause further events in a chain reaction. Avalanching dynamics are studied in many disciplines, with a recent focus on average avalanche shapes, i.e., the ... Full text Cite

Role of stem-cell divisions in cancer risk.

Journal Article Nature · August 2017 Full text Cite

Phase transitions for a planar quadratic contact process

Journal Article Advances in Applied Mathematics · June 1, 2017 We study a two dimensional version of Neuhauser's long range sexual reproduction model and prove results that give bounds on the critical values λf for the process to survive from a finite set and λe for the existence of a nontrivial stationary distributio ... Full text Cite

Spatial evolutionary games with weak selection.

Journal Article Proceedings of the National Academy of Sciences of the United States of America · June 2017 Recently, a rigorous mathematical theory has been developed for spatial games with weak selection, i.e., when the payoff differences between strategies are small. The key to the analysis is that when space and time are suitably rescaled, the spatial model ... Full text Cite

Genealogies in expanding populations

Journal Article Annals of Applied Probability · December 1, 2016 The goal of this paper is to prove rigorous results for the behavior of genealogies in a one-dimensional long range biased voter model introduced by Hallatschek and Nelson [Theor. Pop. Biol. 73 (2008) 158-170]. The first step, which is easily accomplished ... Full text Cite

Outcomes of Active Surveillance for Ductal Carcinoma in Situ: A Computational Risk Analysis.

Journal Article J Natl Cancer Inst · May 2016 BACKGROUND: Ductal carcinoma in situ (DCIS) is a noninvasive breast lesion with uncertain risk for invasive progression. Usual care (UC) for DCIS consists of treatment upon diagnosis, thus potentially overtreating patients with low propensity for progressi ... Full text Link to item Cite

Spatial Moran models, II: cancer initiation in spatially structured tissue.

Journal Article Journal of mathematical biology · April 2016 We study the accumulation and spread of advantageous mutations in a spatial stochastic model of cancer initiation on a lattice. The parameters of this general model can be tuned to study a variety of cancer types and genetic progression pathways. This inve ... Full text Cite

Evolutionary games on the torus with weak selection

Journal Article Stochastic Processes and their Applications · December 7, 2015 We study evolutionary games on the torus with N points in dimensions d≥3. The matrices have the form Ḡ=1+wG, where 1 is a matrix that consists of all 1’s, and w is small. As in Cox Durrett and Perkins (2011) we rescale time and space and take a limit as N→ ... Full text Cite

Coexistence of grass, saplings and trees in the Staver-Levin forest model

Journal Article Annals of Applied Probability · December 1, 2015 In this paper, we consider two attractive stochastic spatial models in which each site can be in state 0, 1 or 2: Krone's model in which 0 = vacant, 1 = juvenile and 2 = a mature individual capable of giving birth, and the Staver-Levin forest model in whic ... Full text Cite

Estimating Tumor Growth Rates In Vivo.

Journal Article Bulletin of mathematical biology · October 2015 In this paper, we develop methods for inferring tumor growth rates from the observation of tumor volumes at two time points. We fit power law, exponential, Gompertz, and Spratt’s generalized logistic model to five data sets. Though the data sets are small ... Full text Cite

Spatial networks evolving to reduce length

Journal Article Journal of Complex Networks · September 1, 2015 Motivated by results of Henry et al. (2011, PNAS, 108, 8605-8610), we propose a general scheme for evolving spatial networks to reduce their total edge lengths. We study the properties of the equilibria of two networks from this class, which interpolate be ... Full text Cite

HPV clearance and the neglected role of stochasticity.

Journal Article PLoS Comput Biol · March 2015 Clearance of anogenital and oropharyngeal HPV infections is attributed primarily to a successful adaptive immune response. To date, little attention has been paid to the potential role of stochastic cell dynamics in the time it takes to clear an HPV infect ... Full text Open Access Link to item Cite

Spatial moran models I. Stochastic tunneling in the neutral case

Journal Article Annals of Applied Probability · February 1, 2015 We consider a multistage cancer model in which cells are arranged in a d-dimensional integer lattice. Starting with all wild-type cells, we prove results about the distribution of the first time when two neutral mutations have accumulated in some cell in d ... Full text Cite

Two evolving social network models

Journal Article Alea · January 1, 2015 In our first model, individuals have opinions in [0, 1]d. Connections are broken at rate proportional to their length ℓ, an end point is chosen at random, a new connection to a random individual is proposed. In version (i) the new edge is always accepted. ... Cite

Spatial evolutionary games with small selection coefficients

Journal Article Electronic Journal of Probability · December 28, 2014 Here we will use results of Cox, Durrett, and Perkins [56] for voter model perturbations to study spatial evolutionary games on ℤd, d ≥ 3 when the interaction kernel is finite range, symmetric, and has covariance matrix σ2I. The games ... Full text Cite

Fingering in stochastic growth models

Journal Article Experimental Mathematics · October 2, 2014 Motivated by the widespread use of hybrid-discrete cellular automata in modeling cancer, we study two simple growth models on the two-dimensional lattice that incorporate a nutrient, assumed to be oxygen. In the first model, the oxygen concentration u(x, t ... Full text Cite

Exact solution for a metapopulation version of Schelling's model.

Journal Article Proceedings of the National Academy of Sciences of the United States of America · September 2014 In 1971, Schelling introduced a model in which families move if they have too many neighbors of the opposite type. In this paper, we will consider a metapopulation version of the model in which a city is divided into N neighborhoods, each of which has L ho ... Full text Cite

Chemical evolutionary games.

Journal Article Theoretical population biology · May 2014 Inspired by the use of hybrid cellular automata in modeling cancer, we introduce a generalization of evolutionary games in which cells produce and absorb chemicals, and the chemical concentrations dictate the death rates of cells and their fitnesses. Our l ... Full text Cite

The contact process with fast voting

Journal Article Electronic Journal of Probability · March 3, 2014 Consider a combination of the contact process and the voter model in which deaths occur at rate 1 per site, and across each edge between nearest neighbors births occur at rate λ and voting events occur at rate θ. We are interested in the asymptotics as θ→∞ ... Full text Cite

Multiopinion coevolving voter model with infinitely many phase transitions.

Journal Article Physical review. E, Statistical, nonlinear, and soft matter physics · December 2013 We consider an idealized model in which individuals' changing opinions and their social network coevolve, with disagreements between neighbors in the network resolved either through one imitating the opinion of the other or by reassignment of the discordan ... Full text Cite

Phase transitions in the quadratic contact process on complex networks

Journal Article Physical Review E - Statistical, Nonlinear, and Soft Matter Physics · June 27, 2013 The quadratic contact process (QCP) is a natural extension of the well-studied linear contact process where infected (1) individuals infect susceptible (0) neighbors at rate λ and infected individuals recover (1ï·0) at rate 1. In the QCP, a combination of ... Full text Cite

Applications of stochastic processes in biology and medicine

Journal Article International Journal of Stochastic Analysis · April 29, 2013 Full text Cite

Cancer modeling: A personal perspective

Journal Article Notices of the American Mathematical Society · March 1, 2013 Full text Cite

A branching process model of ovarian cancer.

Journal Article J Theor Biol · December 7, 2012 Ovarian cancer is usually diagnosed at an advanced stage, rendering the possibility of cure unlikely. To date, no cost-effective screening test has proven effective for reducing mortality. To estimate the window of opportunity for ovarian cancer screening, ... Full text Link to item Cite

Evolution of dispersal distance.

Journal Article Journal of mathematical biology · March 2012 The problem of how often to disperse in a randomly fluctuating environment has long been investigated, primarily using patch models with uniform dispersal. Here, we consider the problem of choice of seed size for plants in a stable environment when there i ... Full text Cite

Graph fission in an evolving voter model.

Journal Article Proceedings of the National Academy of Sciences of the United States of America · March 2012 We consider a simplified model of a social network in which individuals have one of two opinions (called 0 and 1) and their opinions and the network connections coevolve. Edges are picked at random. If the two connected individuals hold different opinions ... Full text Cite

Asymptotic behavior of Aldous' gossip process

Journal Article Annals of Applied Probability · December 1, 2011 Aldous [(2007) Preprint] defined a gossip process in which space is a discrete N × N torus, and the state of the process at time t is the set of individuals who know the information. Information spreads from a site to its nearest neighbors at rate 1/4 each ... Full text Cite

Persistence of activity in threshold contact processes, an "Annealed approximation" of random Boolean networks

Journal Article Random Structures and Algorithms · September 1, 2011 We consider a model for gene regulatory networks that is a modification of Kauffmann's J Theor Biol 22 (1969), 437-467 random Boolean networks. There are three parameters: $n = {\rm the}$ number of nodes, $r = {\rm the}$ number of inputs to each node, and ... Full text Cite

Intratumor heterogeneity in evolutionary models of tumor progression.

Journal Article Genetics · June 2011 With rare exceptions, human tumors arise from single cells that have accumulated the necessary number and types of heritable alterations. Each such cell leads to dysregulated growth and eventually the formation of a tumor. Despite their monoclonal origin, ... Full text Cite

Traveling waves of selective sweeps

Journal Article Annals of Applied Probability · April 1, 2011 The goal of cancer genome sequencing projects is to determine the genetic alterations that cause common cancers. Many malignancies arise during the clonal expansion of a benign tumor which motivates the study of recurrent selective sweeps in an exponential ... Full text Cite

Persistence of activity in random Boolean networks

Journal Article Random Structures and Algorithms · 2011 Cite

Brunet-derrida particle systems, free boundary problems and wiener-hopf equations

Journal Article Annals of Probability · January 1, 2011 We consider a branching-selection system in ℝ with N particles which give birth independently at rate 1 and where after each birth the leftmost particle is erased, keeping the number of particles constant. We show that, as N →∞, the empirical measure proce ... Full text Cite

Evolutionary dynamics of tumor progression with random fitness values.

Journal Article Theoretical population biology · August 2010 Most human tumors result from the accumulation of multiple genetic and epigenetic alterations in a single cell. Mutations that confer a fitness advantage to the cell are known as driver mutations and are causally related to tumorigenesis. Other mutations, ... Full text Cite

Evolution in predator-prey systems

Journal Article Stochastic Processes and their Applications · July 1, 2010 We study the adaptive dynamics of predatorprey systems modeled by a dynamical system in which the traits of predators and prey are allowed to evolve by small mutations. When only the prey are allowed to evolve, and the size of the mutational change tends t ... Full text Cite

Some features of the spread of epidemics and information on a random graph.

Journal Article Proceedings of the National Academy of Sciences of the United States of America · March 2010 Random graphs are useful models of social and technological networks. To date, most of the research in this area has concerned geometric properties of the graphs. Here we focus on processes taking place on the network. In particular we are interested in ho ... Full text Cite

Evolution of resistance and progression to disease during clonal expansion of cancer.

Journal Article Theoretical population biology · February 2010 Inspired by previous work of Iwasa et al. (2006) and Haeno et al. (2007), we consider an exponentially growing population of cancerous cells that will evolve resistance to treatment after one mutation or display a disease phenotype after two or more mutati ... Full text Cite

Contact processes on random graphs with power law degree distributions have critical value 0

Journal Article Annals of Probability · November 1, 2009 If we consider the contact process with infection rate λ on a random graph on n vertices with power law degree distributions, mean field calculations suggest that the critical value λc of the infection rate is positive if the power α>3. Physicists seem to ... Full text Cite

Coexistence for a multitype contact process with seasons

Journal Article Annals of Applied Probability · October 1, 2009 We introduce a multitype contact process with temporal heterogeneity involving two species competing for space on the d-dimensional integer lattice. Time is divided into seasons called alternately season 1 and season 2. We prove that there is an open set o ... Full text Cite

Chaos in a spatial epidemic model

Journal Article Annals of Applied Probability · August 1, 2009 We investigate an interacting particle system inspired by the gypsy moth, whose populations grow until they become sufficiently dense so that an epidemic reduces them to a low level. We consider this process on a random 3-regular graph and on the d-dimensi ... Full text Cite

A COSII genetic map of the pepper genome provides a detailed picture of synteny with tomato and new insights into recent chromosome evolution in the genus Capsicum.

Journal Article TAG. Theoretical and applied genetics. Theoretische und angewandte Genetik · May 2009 We report herein the development of a pepper genetic linkage map which comprises 299 orthologous markers between the pepper and tomato genomes (including 263 conserved ortholog set II or COSII markers). The expected position of additional 288 COSII markers ... Full text Cite

A waiting time problem arising from the study of multi-stage carcinogenesis

Journal Article Annals of Applied Probability · April 1, 2009 We consider the population genetics problem: how long does it take before some member of the population has m specified mutations? The case m = 2 is relevant to onset of cancer due to the inactivation of both copies of a tumor suppressor gene. Models for l ... Full text Cite

SPecial invited paper coexistence in stochastic spatial models

Journal Article Annals of Applied Probability · April 1, 2009 In this paper I will review twenty years of work on the question: When is there coexistence in stochastic spatial models? The answer, announced in Durrett and Levin [Theor. Pop. Biol. 46 (1994) 363-394], and that we explain in this paper is that this can b ... Full text Cite

Degenerate diffusions arising from gene duplication models

Journal Article Annals of Applied Probability · February 1, 2009 We consider two processes that have been used to study gene duplication, Watterson's [Genetics 105 (1983) 745-766] double recessive null model and Lynch and Force's [Genetics 154 (2000) 459-473] subfunctionalization model. Though the state spaces of these ... Full text Cite

Reply to Michael Behe

Journal Article Genetics · February 1, 2009 Full text Cite

Coexistence in a particle system with seasons

Journal Article Ann. Appl. Probab. · 2009 Cite

Waiting for two mutations: with applications to regulatory sequence evolution and the limits of Darwinian evolution.

Journal Article Genetics · November 2008 Results of Nowak and collaborators concerning the onset of cancer due to the inactivation of tumor suppressor genes give the distribution of the time until some individual in a population has experienced two prespecified mutations and the time until this m ... Full text Cite

Coexistence in host-pathogen systems

Journal Article Stochastic Processes and their Applications · June 1, 2008 Lanchier and Neuhauser have initiated the study of host-symbiont systems but have concentrated on the case in which the birth rates for unassociated hosts are equal. Here we allow the birth rates to be different and identify cases in which a host with a sp ... Full text Cite

One-dimensional stepping stone models, sardine genetics and Brownian local time

Journal Article Annals of Applied Probability · February 1, 2008 Consider a one-dimensional stepping stone model with colonies of size M and per-generation migration probability v, or a voter model on ℤ in which interactions occur over a distance of order K. Sample one individual at the origin and one at L. We show that ... Full text Cite

Downcrossings and local time

Chapter · January 1, 2008 Let formula presented be the standard Brownian motion with all paths continuous. Let formula presented be the maximum process and formula presented be reflecting Brownian motion. If formula presented is the number of є to 0 times Y crosses down from e to 0 ... Full text Cite

Limiting behavior for the distance of a random walk

Journal Article Electronic Journal of Probability · January 1, 2008 In this paper we study some aspects of the behavior of random walks on large but finite graphs before they have reached their equilibrium distribution. This investigation is motivated by a result we proved recently for the random transposition random walk: ... Full text Cite

Population genetics of polymorphism and divergence under fluctuating selection.

Journal Article Genetics · January 2008 Current methods for detecting fluctuating selection require time series data on genotype frequencies. Here, we propose an alternative approach that makes use of DNA polymorphism data from a sample of individuals collected at a single point in time. Our met ... Full text Cite

Two phase transitions for the contact process on small worlds

Journal Article Stochastic Processes and their Applications · December 1, 2007 In our version of Watts and Strogatz's small world model, space is a d-dimensional torus in which each individual has in addition exactly one long-range neighbor chosen at random from the grid. This modification is natural if one thinks of a town where an ... Full text Cite

On the width of hybrid zones

Journal Article Stochastic Processes and their Applications · December 1, 2007 Hybrid zones occur when two species are found in close proximity and interbreeding occurs, but the species' characteristics remain distinct. These systems have been treated in the biology literature using partial differential equations models. Here we inve ... Full text Cite

Simple models of genomic variation in human SNP density.

Journal Article BMC genomics · June 2007 BackgroundDescriptive hierarchical Poisson models and population-genetic coalescent mixture models are used to describe the observed variation in single-nucleotide polymorphism (SNP) density from samples of size two across the human genome.Res ... Full text Cite

Stepping-stone spatial structure causes slow decay of linkage disequilibrium and shifts the site frequency spectrum.

Journal Article Genetics · June 2007 The symmetric island model with D demes and equal migration rates is often chosen for the investigation of the consequences of population subdivision. Here we show that a stepping-stone model has a more pronounced effect on the genealogy of a sample. For s ... Full text Cite

Dependence of paracentric inversion rate on tract length.

Journal Article BMC bioinformatics · April 2007 BackgroundWe develop a Bayesian method based on MCMC for estimating the relative rates of pericentric and paracentric inversions from marker data from two species. The method also allows estimation of the distribution of inversion tract lengths. Full text Cite

Wagner's canalization model.

Journal Article Theoretical population biology · March 2007 Wagner (1996, Does evolutionary plasticity evolve? Evolution 50, 1008-1023.) and Siegal and Bergman, 2002 and Azevedo et al., 2006 have studied a simple model of the evolution of a network of N genes, in order to explain the observed phenomenon that system ... Full text Cite

Waiting for regulatory sequences to appear

Journal Article Annals of Applied Probability · February 1, 2007 One possible explanation for the substantial organismal differences between humans and chimpanzees is that there have been changes in gene regulation. Given what is known about transcription factor binding sites, this motivates the following probability qu ... Full text Cite

A new coexistence result for competing contact processes

Journal Article Annals of Applied Probability · August 1, 2006 Neuhauser [Probab. Theory Related Fields 91 (1992) 467-506] considered the two-type contact process and showed that on ℤ 2 coexistence is not possible if the death rates are equal and the particles use the same dispersal neighborhood. Here, we show that it ... Full text Cite

Random graph dynamics

Journal Article Random Graph Dynamics · January 1, 2006 The theory of random graphs began in the late 1950s in several papers by Erdos and Renyi. In the late twentieth century, the notion of six degrees of separation, meaning that any two people on the planet can be connected by a short chain of people who know ... Full text Cite

A phase transition in the random transposition random walk

Journal Article Probability Theory and Related Fields · January 1, 2006 Our work is motivated by Bourque and Pevzner's (2002) simulation study of the effectiveness of the parsimony method in studying genome rearrangement, and leads to a surprising result about the random transposition walk on the group of permutations on n ele ... Full text Cite

Power laws for family sizes in a duplication model

Journal Article Annals of Probability · November 1, 2005 Qian, Luscombe and Gerstein [J. Molecular Biol. 313 (2001) 673-681] introduced a model of the diversification of protein folds in a genome that we may formulate as follows. Consider a multitype Yule process starting with one individual in which there are n ... Full text Cite

A coalescent model for the effect of advantageous mutations on the genealogy of a population

Journal Article Stochastic Processes and their Applications · October 1, 2005 When an advantageous mutation occurs in a population, the favorable allele may spread to the entire population in a short time, an event known as a selective sweep. As a result, when we sample n individuals from a population and trace their ancestral lines ... Full text Cite

Random partitions approximating the coalescence of lineages during a selective sweep

Journal Article Annals of Applied Probability · August 1, 2005 When a beneficial mutation occurs in a population, the new, favored allele may spread to the entire population. This process is known as a selective sweep. Suppose we sample n individuals at the end of a selective sweep. If we focus on a site on the chromo ... Full text Cite

Random Oxford graphs

Journal Article Stochastic Processes and their Applications · August 1, 2005 Inspired by a concept in comparative genomics, we investigate properties of randomly chosen members of G1(m, n, t), the set of bipartite graphs with m left vertices, n right vertices, t edges, and each vertex of degree at least one. We give asymptotic resu ... Full text Cite

Can stable social groups be maintained by homophilous imitation alone?

Journal Article Journal of Economic Behavior and Organization · July 1, 2005 A central problem in the biological and social sciences concerns the conditions required for emergence and maintenance of cooperation among unrelated individuals. Most models and experiments have been pursued in a game-theoretic context and involve reward ... Full text Cite

Bayesian and maximum likelihood estimation of genetic maps.

Journal Article Genetical research · April 2005 There has recently been increased interest in the use of Markov Chain Monte Carlo (MCMC)-based Bayesian methods for estimating genetic maps. The advantage of these methods is that they can deal accurately with missing data and genotyping errors. Here we pr ... Full text Cite

The stepping stone model. II: Genealogies and the infinite sites model

Journal Article Annals of Applied Probability · February 1, 2005 This paper extends earlier work by Cox and Durrett, who studied the coalescence times for two lineages in the stepping stone model on the two-dimensional torus. We show that the genealogy of a sample of size n is given by a time change of Kingman's coalesc ... Full text Cite

Competing super-brownian motions as limits of interacting particle systems

Journal Article Electronic Journal of Probability · January 1, 2005 We study two-type branching random walks in which the birth or death rate of each type can depend on the number of neighbors of the opposite type. This competing species model contains variants of Durrett’s predator-prey model and Durrett and Levin’s colic ... Full text Cite

Adaptive evolution drives the diversification of zinc-finger binding domains.

Journal Article Molecular biology and evolution · December 2004 The human genome is estimated to contain 700 zinc-finger genes, which perform many key functions, including regulating transcription. The dramatic increase in the number of these genes as we move from yeast to C. elegans to Drosophila and to humans, as wel ... Full text Cite

Microsatellite mutation models: insights from a comparison of humans and chimpanzees.

Journal Article Genetics · September 2004 Using genomic data from homologous microsatellite loci of pure AC repeats in humans and chimpanzees, several models of microsatellite evolution are tested and compared using likelihood-ratio tests and the Akaike information criterion. A proportional-rate, ... Full text Cite

Subfunctionalization: How often does it occur? How long does it take?

Journal Article Theoretical population biology · September 2004 The mechanisms responsible for the preservation of duplicate genes have been debated for more than 70 years. Recently, Lynch and Force have proposed a new explanation: subfunctionalization--after duplication the two gene copies specialize to perform comple ... Full text Cite

Approximating selective sweeps.

Journal Article Theoretical population biology · September 2004 The fixation of advantageous mutations in a population has the effect of reducing variation in the DNA sequence near that mutation. Kaplan et al. (1989) used a three-phase simulation model to study the effect of selective sweeps on genealogies. However, mo ... Full text Cite

Bayesian estimation of genomic distance.

Journal Article Genetics · January 2004 We present a Bayesian approach to the problem of inferring the number of inversions and translocations separating two species. The main reason for developing this method is that it will allow us to test hypotheses about the underlying mechanisms, such as t ... Full text Cite

Rigorous results for the NK model

Journal Article Annals of Probability · October 1, 2003 Motivated by the problem of the evolution of DNA sequences, Kauffman and Levin introduced a model in which fitnesses were assigned to strings of 0's and 1's of length N based on the values observed in a sliding window of length K + 1. When K ≥ 1, the lands ... Full text Cite

Shuffling Chromosomes

Journal Article Journal of Theoretical Probability · July 1, 2003 The gene order of chromosomes can be rearranged by chromosomal inversions that reverse the order of segments. Motivated by a comparative study of two Drosophila species, we investigate the number of reversals that are needed to scramble the gene order when ... Full text Cite

Dinucleotide repeats in the Drosophila and human genomes have complex, length-dependent mutation processes.

Journal Article Molecular biology and evolution · May 2003 We use methods of maximum likelihood estimation to fit several microsatellite mutation models to the observed length distribution of dinucletoide repeats in the Drosophila and human genomes. All simple models are rejected by this procedure. Two new models, ... Full text Cite

A surprising poisson process arising from a species competition model

Journal Article Stochastic Processes and their Applications · December 1, 2002 Motivated by the work of Tilman (Ecology 75 (1994) 2) and May and Nowak (J. Theoret. Biol. 170 (1994) 95) we consider a process in which points are inserted randomly into the unit interval and a new point kills each point to its left independently and with ... Full text Cite

The stepping stone model: New formulas expose old myths

Journal Article Annals of Applied Probability · November 1, 2002 We study the stepping stone model on the two-dimensional torus. We prove several new hitting time results for random walks from which we derive some simple approximation formulas for the homozygosity in the stepping stone model as a function of the separat ... Full text Cite

Competition and species packing in patchy environments.

Journal Article Theoretical population biology · May 2002 In models of competition in which space is treated as a continuum, and population size as continuous, there are no limits to the number of species that can coexist. For a finite number of sites, N, the results are different. The answer will, of course, dep ... Full text Cite

Once edge-reinforced random walk on a tree

Journal Article Probability Theory and Related Fields · April 1, 2002 We consider a nearest neighbor walk on a regular tree, with transition probabilities proportional to weights or conductances of the edges. Initially all edges have weight 1, and the weight of an edge is increased to c > 1 when the edge is traversed for the ... Full text Cite

Mutual invadability implies coexistence in spatial models

Journal Article Memoirs of the American Mathematical Society · January 1, 2002 In (1994) Durrett and Levin proposed that the equilibrium behavior of stochastic spatial models could be determined from properties of the solution of the mean field ordinary differential equation (ODE) that is obtained by pretending that all sites are alw ... Full text Cite

Discussion on the meeting on 'statistical modelling and analysis of genetic data'

Journal Article Journal of the Royal Statistical Society. Series B: Statistical Methodology · January 1, 2002 Full text Cite

Bayesian estimation of the number of inversions in the history of two chromosomes.

Journal Article Journal of computational biology : a journal of computational molecular cell biology · January 2002 We present a Bayesian approach to the problem of inferring the history of inversions separating homologous chromosomes from two different species. The method is based on Markov Chain Monte Carlo (MCMC) and takes full advantage of all the information from m ... Full text Cite

Mutual invadability implies coexistence in spatial models

Journal Article Memoirs of the American Mathematical Society · January 1, 2002 In (1994) Durrett and Levin proposed that the equilibrium behavior of stochastic spatial models could be determined from properties of the solution of the mean field ordinary differential equation (ODE) that is obtained by pretending that all sites are alw ... Cite

A simple formula useful for positional cloning.

Journal Article Genetics · January 2002 We derive a formula for the distribution of the length T of the recombination interval containing a target gene and using N gametes in a region where R kilobases correspond to 1 cM. The formula can be used to calculate the number of meiotic events required ... Full text Cite

Chutes and Ladders in Markov Chains

Journal Article Journal of Theoretical Probability · December 1, 2001 We investigate how the stationary distribution of a Markov chain changes when transitions from a single state are modified. In particular, adding a single directed edge to nearest neighbor random walk on a finite discrete torus in dimensions one, two, or t ... Full text Cite

Dynamics of microsatellite divergence under stepwise mutation and proportional slippage/point mutation models.

Journal Article Genetics · October 2001 Recently Kruglyak, Durrett, Schug, and Aquadro showed that microsatellite equilibrium distributions can result from a balance between polymerase slippage and point mutations. Here, we introduce an elaboration of their model that keeps track of all parts of ... Full text Cite

Exponential distance statistics to detect the effects of population subdivision.

Journal Article Theoretical population biology · September 2001 Statistical tests are needed to determine whether spatial structure has had a significant effect on the genetic differentiation of subpopulations. Here we introduce a new family of statistics based on a sum of an exponential function of the distances betwe ... Full text Cite

On the quantity and quality of single nucleotide polymorphisms in the human genome

Journal Article Stochastic Processes and their Applications · May 1, 2001 Single nucleotide polymorphisms (SNPs) are useful markers for locating genes since they occur throughout the human genome and thousands can be scored at once using DNA microarrays. Here, we use branching processes and coalescent theory to show that if one ... Full text Cite

The equilibrium distribution for a generalized Sankoff-Ferretti model accurately predicts chromosome size distributions in a wide variety of species

Journal Article Journal of Applied Probability · January 1, 2001 Sankoff and Ferretti (1996) introduced several models of the evolution of chromosome size by reciprocal translocations, where for simplicity they ignored the existence of centromeres. However, when they compared the models to data on six organisms they fou ... Full text Cite

Quantification of homoplasy for nucleotide transitions and transversions and a reexamination of assumptions in weighted phylogenetic analysis.

Journal Article Systematic biology · December 2000 Nucleotide transitions are frequently down-weighted relative to transversions in phylogenetic analysis. This is based on the assumption that transitions, by virtue of their greater evolutionary rate, exhibit relatively more homoplasy and are therefore less ... Full text Cite

Distribution and abundance of microsatellites in the yeast genome can Be explained by a balance between slippage events and point mutations.

Journal Article Molecular biology and evolution · August 2000 We fit a Markov chain model of microsatellite evolution introduced by Kruglyak et al. to data on all di-, tri-, and tetranucleotide repeats in the yeast genome. Our results suggest that many features of the distribution of abundance and length of microsate ... Full text Cite

Lessons on pattern formation from planet WATOR.

Journal Article Journal of theoretical biology · July 2000 It is well known that if reacting species experience unequal diffusion rates, then dynamics that lead to a constant steady state in a "well-mixed" environment can in a spatial setting lead to interesting patterns. In this paper, we focus on complementary p ... Full text Cite

Selective mapping: a strategy for optimizing the construction of high-density linkage maps.

Journal Article Genetics · May 2000 Historically, linkage mapping populations have consisted of large, randomly selected samples of progeny from a given pedigree or cell lines from a panel of radiation hybrids. We demonstrate that, to construct a map with high genome-wide marker density, it ... Full text Cite

A spatially-structured stochastic model to simulate heterogenous transmission of viruses in fungal populations

Journal Article Ecological Modelling · March 30, 2000 A spatially explicit, interacting particle system model was developed to simulate the heterogeneous transmission of viruses in fungal populations. This model is based primarily on hypoviruses in the chestnut blight fungus, Cryphonectria parasitica, which d ... Full text Cite

Boundary Modified Contact Processes

Journal Article Journal of Theoretical Probability · January 1, 2000 We introduce a one dimensional contact process for which births to the right of the rightmost particle and to the left of the leftmost particle occur at rate λe (where e is for external). Other births occur at rate λi (where i is for internal). Deaths occu ... Full text Cite

Rescaled voter models converge to super-Brownian motion

Journal Article Annals of Probability · January 1, 2000 We show that a sequence of voter models, suitably rescaled in space and time, converges weakly to super-Brownian motion. The result includes both nearest neighbor and longer range voter models and complements a limit theorem of Mueller and Tribe in one dim ... Full text Cite

Spatial models for hybrid zones.

Journal Article Heredity · January 2000 We introduce a spatially explicit model of natural hybrid zones that allows us to consider how patterns of allele frequencies and linkage disequilibria change over time. We examine the influence of hybrid zone origins on patterns of variation at two loci, ... Full text Cite

Local frequency dependence and global coexistence.

Journal Article Theoretical population biology · June 1999 In sessile organisms such as plants, interactions occur locally so that important ecological aspects like frequency dependence are manifest within local neighborhoods. Using probabilistic cellular automata models, we investigated how local frequency-depend ... Full text Cite

A new stochastic model of microsatellite evolution

Journal Article Journal of Applied Probability · January 1, 1999 We introduce a continuous-time Markov chain model for the evolution of microsatellites, simple sequence repeats in DNA. We prove the existence of a unique stationary distribution for our model, and fit the model to data from approximately 106 base pairs of ... Full text Cite

A new stochastic model of microsatellite evolution

Journal Article Journal of Applied Probability · January 1, 1999 We introduce a continuous-time Markov chain model for the evolution of microsatellites, simple sequence repeats in DNA. We prove the existence of a unique stationary distribution for our model, and fit the model to data from approximately 106 base pairs of ... Full text Cite

The Equilibrium Behavior of Reversible Coagulation-Fragmentation Processes

Journal Article Journal of Theoretical Probability · January 1, 1999 The coagulation-fragmentation process models the stochastic evolution of a population of N particles distributed into groups of different sizes that coagulate and fragment at given rates. The process arises in a variety of contexts and has been intensively ... Full text Cite

Rescaled contact processes converge to super-Brownian motion in two or more dimensions

Journal Article Probability Theory and Related Fields · January 1, 1999 We show that in dimensions two or more a sequence of long range contact processes suitably rescaled in space and time converges to a super-Brownian motion with drift. As a consequence of this result we can improve the results of Bramson, Durrett, and Swind ... Full text Cite

Stochastic spatial models

Journal Article SIAM Review · January 1, 1999 In the models we will consider, space is represented by a grid of sites that can be in one of a finite number of states and that change at rates that depend on the states of a finite number of sites. Our main aim here is to explain an idea of Durrett and L ... Full text Cite

Equilibrium distributions of microsatellite repeat length resulting from a balance between slippage events and point mutations.

Journal Article Proceedings of the National Academy of Sciences of the United States of America · September 1998 We describe and test a Markov chain model of microsatellite evolution that can explain the different distributions of microsatellite lengths across different organisms and repeat motifs. Two key features of this model are the dependence of mutation rates o ... Full text Cite

Spatial aspects of interspecific competition.

Journal Article Theoretical population biology · February 1998 Using several variants of a stochastic spatial model introduced by Silvertown et al., we investigate the effect of spatial distribution of individuals on the outcome of competition. First, we prove rigorously that if one species has a competitive advantage ... Full text Cite

A spatial model for the abundance of species

Journal Article Annals of Probability · January 1, 1998 The voter model, with mutations occurring at a positive rate a, has a unique equilibrium distribution. We investigate the logarithms of the relative abundance of species for these distributions in d ≥ 2. We show that, as α → 0, the limiting distribution is ... Full text Cite

Allelopathy in spatially distributed populations

Journal Article Journal of Theoretical Biology · March 21, 1997 In a homogeneously mixing population of E. coli, colicin-producing and colicin-sensitive strategies both may be evolutionarily stable for certain parameter ranges, with the outcome of competition determined by initial conditions. In contrast, in a spatiall ... Full text Cite

Holomorphic diffusions and boundary behavior of harmonic functions

Journal Article Annals of Probability · January 1, 1997 We study a family of differential operators {Lα, α ≥ 0} in the unit ball D of Cn with n ≥ 2 that generalize the classical Laplacian, α = 0, and the conformal Laplacian, α = 1/2 (that is, the Laplace-Beltrami operator for Bergman metric in D). Using the dif ... Full text Cite

Coexistence results for some competition models

Journal Article Annals of Applied Probability · January 1, 1997 Barley yellow dwarf is a widespread disease that affects small grains and many grass species, as well as wheat, barley and oat. The disease is caused by an aphid transmitted virus. Rochow conducted a study near Ithaca, New York, which showed that a shift i ... Full text Cite

Super-Tree Random Measures

Journal Article Journal of Theoretical Probability · January 1, 1997 We use supercritical branching processes with random walk steps of geometrically decreasing size to construct random measures. Special cases of our construction give close relatives of the super-(spherically symmetric stable) processes. However, other case ... Full text Cite

Spatial models for species-area curves

Journal Article Journal of Theoretical Biology · March 14, 1996 Inspired by earlier work of Hubbell, we introduce a simple spatial model to explain observed species-area curves. As in the theory of MacArthur and Wilson, our curves result from a balance between migration and extinction. Our model predicts that the wide ... Full text Cite

Spatial models for species area curves

Journal Article Annals of Probability · January 1, 1996 The relationship between species number and area is an old problem in biology. We propose here an interacting particle system - the multitype voter model with mutation - as a mathematical model to study this problem. We analyze the species area curves of t ... Full text Cite

Hybrid zones and voter model interfaces

Journal Article Bernoulli · January 1, 1995 We study the dynamics of hybrid zones in the absence of selection. In dimensions d > 1 the width of the hybrid zone grows as √t but in one dimension the width converges to a non-degenerate limit. We believe that tight interfaces are common in one-dimension ... Full text Cite

Coexistence results for catalysts

Journal Article Probability Theory and Related Fields · December 1, 1994 In this paper we consider a modification of Ziff, Gulari and Barshad's (1986) model of oxidation of carbon monoxide on a catalyst surface in which the reactants are mobile on the catalyst surface. We find regions in the parameter space in which poisoning o ... Full text Cite

Stochastic spatial models: A user's guide to ecological applications

Journal Article Philosophical Transactions of the Royal Society B: Biological Sciences · January 1, 1994 Full text Cite

The importance of being discrete (and spatial)

Journal Article Theoretical Population Biology · January 1, 1994 We consider and compare four approaches to modeling the dynamics of spatially distributed systems: mean field approaches (described by ordinary differential equations) in which every individual is considered to have equal probability of interacting with ev ... Full text Cite

Asymptotic behavior of excitable cellular automata

Journal Article Experimental Mathematics · January 1, 1993 We study two families of excitable cellular automata known as the Greenberg-Hastings model and the cyclic cellular automaton. Each family consists of local deterministic oscillating lattice dynamics, with parallel discrete-time updating, parametrized by th ... Full text Cite

Asymptotic behavior of Brownian polymers

Journal Article Probability Theory and Related Fields · September 1, 1992 We consider a system that models the shape of a growing polymer. Our basic problem concerns the asymptotic behavior of Xt, the location of the end of the polymer at time t. We obtain bounds on Xt in the (physically uninteresting) case that d=1 and the inte ... Full text Cite

Multicolor particle systems with large threshold and range

Journal Article Journal of Theoretical Probability · January 1, 1992 In this paper we consider the Greenberg-Hastings and cyclic color models. These models exhibit (at least) three different types of behavior. Depending on the number of colors and the size of two parameters called the threshold and range, the Greenberg-Hast ... Full text Cite

Some rigorous results for the Greenberg-Hastings Model

Journal Article Journal of Theoretical Probability · October 1, 1991 In this paper, we obtain some rigorous results for a cellular automaton known as the Greenberg-Hastings Model. The state space is {0, 1, 2}Zd. The dynamics are deterministic and discrete time. A site which is 1 changes to 2, a site which is 2 changes to 0, ... Full text Cite

The critical contact process seen from the right edge

Journal Article Probability Theory and Related Fields · September 1, 1991 Durrett (1984) proved the existence of an invariant measure for the critical and supercritical contact process seen from the right edge. Galves and Presutti (1987) proved, in the supercritical case, that the invariant measure was unique, and convergence to ... Full text Cite

Complete convergence theorem for a competition model

Journal Article Probability Theory and Related Fields · March 1, 1991 In this paper we consider a hierarchical competition model. Durrett and Swindle have given sufficient conditions for the existence of a nontrivial stationary distribution. Here we show that under a slightly stronger condition, the complete convergence theo ... Full text Cite

Annihilating branching processes

Journal Article Stochastic Processes and their Applications · January 1, 1991 We consider Markov processes ηt ⊂ Zd in which (i) particles die at rate δ ≥ 0, (ii) births from x to a neighboring y occur at rate 1, and (iii) when a new particle lands on an occupied site the particles annihilate each other and a vacant site results. Whe ... Full text Cite

On weighted heights of random trees

Journal Article Journal of Theoretical Probability · January 1, 1991 Consider the family tree T of a branching process starting from a single progenitor and conditioned to have v=v(T) edges (total progeny). To each edge we associate a weight W(e). The weights are i.i.d. random variables and independent of T. The weighte ... Full text Cite

Are there bushes in a forest?

Journal Article Stochastic Processes and their Applications · January 1, 1991 In this paper we consider a process in which each site x ε{lunate} Zd can be occupied by grass, bushes or trees and ask the question: Are there equilibria in which bushes and trees are both present? The answer is sometimes yes and sometimes no. © 1991. ... Full text Cite

Exponential convergence for one dimensional contact processes

Journal Article Acta Mathematica Sinica · December 1, 1990 The complete convergence theorem implies that starting from any initial distribution the one dimensional contact process converges to a limit as t→∞. In this paper we give a necessary and sufficient condition on the initial distribution for the convergence ... Full text Cite

Large deviations for independent random walks

Journal Article Probability Theory and Related Fields · March 1, 1990 We consider a system of independent random walks on ℤ. Let ξn(x) be the number of particles at x at time n, and let Ln(x)=ξ0(x)+ ... +ξn(x) be the total occupation time of x by time n. In this paper we study the large deviations of Ln(0)-Ln(1). The behavio ... Full text Cite

Ergodicity of reversible reaction diffusion processes

Journal Article Probability Theory and Related Fields · March 1, 1990 Reaction-diffusion processes were introduced by Nicolis and Prigogine, and Haken. Existence theorems have been established for most models, but not much is known about ergodic properties. In this paper we study a class of models which have a reversible mea ... Full text Cite

Correlation lengths for oriented percolation

Journal Article Journal of Statistical Physics · June 1, 1989 Oriented percolation has two correlation lengths, one in the "space" and one in the "time" direction. In this paper we define these quantities for the two-dimensional model in terms of the exponential decay of suitably chosen quantities, and study the rela ... Full text Cite

Scaling inequalities for oriented percolation

Journal Article Journal of Statistical Physics · June 1, 1989 We look at seven critical exponents associated with two-dimensional oriented percolation. Scaling theory implies that these quantities satisfy four equalities. We prove five related inequalitites. © 1989 Plenum Publishing Corporation. ... Full text Cite

Large deviations for the contact process and two dimensional percolation

Journal Article Probability Theory and Related Fields · December 1, 1988 The following results are proved: 1) For the upper invariant measure of the basic one-dimensional supercritical contact process the density of 1's has the usual large deviation behavior: the probability of a large deviation decays exponentially with the nu ... Full text Cite

A simple proof of the stability criterion of Gray and Griffeath

Journal Article Probability Theory and Related Fields · December 1, 1988 Gray and Griffeath studied attractive nearest neighbor spin systems on the integers having "all 0's" and "all 1's" as traps. Using the contour method, they established a necessary and sufficient condition for the stability of the "all 1's" equilibrium unde ... Full text Cite

Random walk in random environment: A counterexample?

Journal Article Communications in Mathematical Physics · June 1, 1988 We describe a family of random walks in random environments which have exponentially decaying correlations, nearest neighbor transition probabilities which are bounded away from 0, and yet are subdiffusive in any dimension d<∞. © 1988 Springer-Verlag. ... Full text Cite

Crabgrass, measles, and gypsy moths: An introduction to interacting particle systems

Journal Article The Mathematical Intelligencer · June 1, 1988 Full text Cite

Connectivity properties of Mandelbrot's percolation process

Journal Article Probability Theory and Related Fields · March 1, 1988 In 1974, Mandelbrot introduced a process in [0, 1]2 which he called "canonical curdling" and later used in this book(s) on fractals to generate self-similar random sets with Hausdorff dimension D∈(0,2). In this paper we will study the connectivity or "perc ... Full text Cite

Crabgrass, measles, and gypsy moths: An introduction to modern probability

Journal Article Bulletin of the American Mathematical Society · January 1, 1988 Full text Cite

Limit theorems for the spread of epidemics and forest fires

Journal Article Stochastic Processes and their Applications · January 1, 1988 We prove that the "spatial epidemic with removal" grows linearly and has an asymptotic shape on the set of nonextinction. © 1988. ... Full text Cite

Inhomogeneous percolation problems and incipient infinite clusters

Journal Article Journal of Physics A: Mathematical and General · December 1, 1987 The authors consider inhomogeneous percolation models with density p c+f(x) and examine the forms of f(x) which produce incipient structures. Taking f(x) approximately= mod x mod - lambda and assuming the existence of a correlation length exponent v for th ... Full text Cite

Splitting intervals II: Limit laws for lengths

Journal Article Probability Theory and Related Fields · May 1, 1987 In the processes under consideration, a particle of size L splits with exponential rate Lα, 0<α<∞, and when it splits, it splits into two particles of size LV and L(1-V) where V is independent of the past with d.f. F on (0, 1). Let Ztbe the number of parti ... Full text Cite

Critical behavior of the two-dimensional first passage time

Journal Article Journal of Statistical Physics · December 1, 1986 We study the two-dimensional first passage problem in which bonds have zero and unit passage times with probability p and 1-p, respectively. We prove that as the zero-time bonds approach the percolation threshold pc, the first passage time exhibits the sam ... Full text Cite

Multidimensional random walks in random environments with subclassical limiting behavior

Journal Article Communications in Mathematical Physics · March 1, 1986 In this paper we will describe and analyze a class of multidimensional random walks in random environments which contain the one dimensional nearest neighbor situation as a special case and have the pleasant feature that quite a lot can be said about them. ... Full text Cite

Some general results concerning the critical exponents of percolation processes

Journal Article Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete · September 1, 1985 In this paper we will give some results concerning the critical exponents of percolation processes which are valid for "any" model. These results show that in several respects the behavior which occurs for percolation on the binary tree provides bounds on ... Full text Cite

Thermodynamic inequalities for percolation

Journal Article Communications in Mathematical Physics · June 1, 1985 In this paper we describe the percolation analogues of the Gibbs and Helmholtz potentials and use these quantities to prove some general inequalities concerning the critical exponents of percolation processes. © 1985 Springer-Verlag. ... Full text Cite

Fixed points of the smoothing transformation

Journal Article Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete · September 1, 1983 Let W 1,..., W N be N nonnegative random variables and let {Mathematical expression} be the class of all probability measures on [0, ∞). Define a transformation T on {Mathematical expression} by letting Tμ be the distribution of W 1X1+ ... + W N X N, where ... Full text Cite

Maxima of branching random walks

Journal Article Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete · June 1, 1983 In recent years several authors have obtained limit theorems for Ln, the location of the rightmost particle in a supercritical branching random walk but all of these results have been proved under the assumption that the offspring distribution has φ{symbol ... Full text Cite

Extension of domains with finite gauge

Journal Article Mathematische Annalen · March 1, 1983 Full text Cite

Oriented Percolation in Dimensions D ≥ 4: Bounds and Asymptotic Formulas

Journal Article Mathematical Proceedings of the Cambridge Philosophical Society · January 1, 1983 Let pc(d) be the critical probability for oriented percolation in Zdand let (d) be the time constant for the first passage process based on the exponential distribution. In this paper we show that as d →∞, dpc(d)→ 1 and dμ{d)→γ where y is a constant in [e- ... Full text Cite

Contact processes in several dimensions

Journal Article Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete · December 1, 1982 Full text Cite

An introduction to infinite particle systems

Journal Article Stochastic Processes and their Applications · January 1, 1981 In 1970, Spitzer wrote a paper called "Interaction of Markov processes" in which he introduced several classes of interacting particle systems. These processes and other related models, collectively referred to as infinite particle systems, have been the o ... Full text Cite

Conditioned limit theorems for random walks with negative drift

Journal Article Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete · January 1, 1980 In this paper we will solve a problem posed by Iglehart. In (1975) he conjectured that if Sn is a random walk with negative mean and finite variance then there is a constant α so that (S[n.]/αn1/2|N>n) converges weakly to a process which he called the Brow ... Full text Cite

A new parameter in brain stem evoked response: Component wave areas

Journal Article Laryngoscope · 1979 Using a newly developed brain stem evoked response (BSER) parameter in preliminary testing, the authors can individually identify the auditory thresholds of 500 Hz, 1000 Hz and 2000 Hz to within 15-25 db. The authors have developed an accurate method of br ... Cite

Maxima of branching random walks vs. independent random walks

Journal Article Stochastic Processes and their Applications · January 1, 1979 In recent years several authors have obtained limit theorems for the location of the right most particle in a supercritical branching random walk. In this paper we will consider analogous problems for an exponentially growing number of independent random w ... Full text Cite

The genealogy of critical branching processes

Journal Article Stochastic Processes and their Applications · January 1, 1978 In this paper we will obtain results concerning the distribution of generations and the degree of relationship of the individuals in a critical branching process {Z(t), t≥0} and we will apply these results to obtain a "central limit theorem" for critical b ... Full text Cite

Weak convergence with random indices

Journal Article Stochastic Processes and their Applications · January 1, 1977 Suppose {Xnn≥-0} are random variables such that for normalizing constants an>0, bn, n≥0 we have Yn(·)=(X[n, ·]-bn/an ⇒ Y(·) in D(0.∞) . Then an and bn must in specific ways and the process Y possesses a scaling property. If {Nn} are positive integer valued ... Full text Cite

Downcrossings and local time

Journal Article Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete · June 1, 1976 Full text Cite

WAITING TIMES WITHOUT MEMORY.

Journal Article J Appl Probab · January 1, 1976 A waiting time without memory, or age-independent residual life-time, is a positive-valued random variable T with the property that for any x, y greater than 0, given that T greater than x, the conditional probability of T greater than x plus y is the same ... Full text Cite

Evolving spatial networks to minimize length

Journal Article Journal of Complex Networks Full text Link to item Cite

Two evolving social network models

Journal Article Alea (Rio de Janeiro): Latin American journal of probability and mathematical statistics Open Access Cite