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54
Coalescing random walks and voter model consensus times on the torus in Zd
 THE ANNALS OF PROBABILITY
, 1989
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Stochastic spatial models
 SIAM Rev
, 1999
"... Abstract. 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 Levin (1994): the behavior of these ..."
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Cited by 55 (2 self)
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Abstract. 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 Levin (1994): the behavior of these models can be predicted from the properties of the mean field ODE, i.e., the equations for the densities of the various types that result from pretending that all sites are always independent. We will illustrate this picture through a discussion of eight families of examples from statistical mechanics, genetics, population biology, epidemiology, and ecology. Some of our findings are only conjectures based on simulation, but in a number of cases we are able to prove results for systems with “fast stirring ” by exploiting connections between the spatial model and an associated reaction diffusion equation.
Recent Progress in Coalescent Theory
"... Coalescent theory is the study of random processes where particles may join each other to form clusters as time evolves. These notes provide an introduction to some aspects of the mathematics of coalescent processes and their applications to theoretical population genetics and in other fields such ..."
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Cited by 48 (3 self)
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Coalescent theory is the study of random processes where particles may join each other to form clusters as time evolves. These notes provide an introduction to some aspects of the mathematics of coalescent processes and their applications to theoretical population genetics and in other fields such as spin glass models. The emphasis is on recent work concerning in particular the connection of these processes to continuum random trees and spatial models such as coalescing random walks.
Dynamics of Ising Spin Systems at Zero Temperature
"... We consider zerotemperature, stochastic Ising models t on Z d with nearestneighbor interactions and an initial spin configuration 0 chosen from a symmetric Bernoulli distribution (corresponding physically to a deep quench). Whether 1 exists  i.e., whether each spin flips only finitely ..."
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Cited by 33 (17 self)
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We consider zerotemperature, stochastic Ising models t on Z d with nearestneighbor interactions and an initial spin configuration 0 chosen from a symmetric Bernoulli distribution (corresponding physically to a deep quench). Whether 1 exists  i.e., whether each spin flips only finitely many times as t ! 1 (for almost every 0 and realization ! of the dynamics)  depends on the nature of the couplings, in particular the presence of continuous disorder. For the homogeneous (nondisordered) ferromagnet, we prove that 1 does not exist, at least for d 2 (although it does exist for some lattices other than Z d ). For continuous disorder, under mild conditions, we show that 1 does exist. We also analyze a dynamical order parameter that measures how much 1 depends on 0 and how much on !.
Rescaled voter models converge to superBrownian motion
 Ann. Probab
, 2000
"... We show that a sequence of voter models, suitably rescaled in space and time, converges weakly to superBrownian motion. The result includes both nearest neighbor and longer range voter models and complements a limit theorem of Mueller and Tribe in one dimension. ..."
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Cited by 33 (14 self)
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We show that a sequence of voter models, suitably rescaled in space and time, converges weakly to superBrownian motion. The result includes both nearest neighbor and longer range voter models and complements a limit theorem of Mueller and Tribe in one dimension.
A new model for evolution in a spatial continuum
"... o b a b i l i t y Vol. 15 (2010), Paper no. 7, pages 162–216. Journal URL ..."
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Cited by 20 (5 self)
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o b a b i l i t y Vol. 15 (2010), Paper no. 7, pages 162–216. Journal URL
Discrete Opinion Dynamics with Stubborn Agents
"... We study discrete opinion dynamics in a social network with ”stubborn agents” who influence others but do not change their opinions. We generalize the classical voter model by introducing nodes (stubborn agents) that have a fixed state. We show that the presence of stubborn agents with opposing opin ..."
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Cited by 17 (1 self)
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We study discrete opinion dynamics in a social network with ”stubborn agents” who influence others but do not change their opinions. We generalize the classical voter model by introducing nodes (stubborn agents) that have a fixed state. We show that the presence of stubborn agents with opposing opinions precludes convergence to consensus; instead, opinions converge in distribution with disagreement and fluctuations. In addition to the first moment of this distribution typically studied in the literature, we study the behavior of the second moment in terms of network properties and the opinions and locations of stubborn agents. We also study the problem of ”optimal placement of stubborn agents” where the location of a fixed number of stubborn agents is chosen to have the maximum impact on the longrun expected opinions of agents.
Clusters and recurrence in the twodimensional zerotemperature stochastic Ising model
, 2001
"... We analyze clustering and (local) recurrence of a standard Markov process model of spatial domain coarsening. The continuous time process, whose state space consists of assignments of +1 or −1 to each site in Z 2, is the zerotemperature limit of the stochastic homogeneous Ising ferromagnet (with Gl ..."
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Cited by 17 (8 self)
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(Show Context)
We analyze clustering and (local) recurrence of a standard Markov process model of spatial domain coarsening. The continuous time process, whose state space consists of assignments of +1 or −1 to each site in Z 2, is the zerotemperature limit of the stochastic homogeneous Ising ferromagnet (with Glauber dynamics): the initial state is chosen uniformly at random and then each site, at rate one, polls its 4 neighbors and makes sure it agrees with the majority, or tosses a fair coin in case of a tie. Among the main results (almost sure, with respect to both the process and initial state) are: clusters (maximal domains of constant sign) are finite for times t < ∞, but the cluster of a fixed site diverges (in diameter) as t → ∞; each of the two constant states is (positive) recurrent. We also present other results and conjectures concerning positive and null recurrence and the role of absorbing states.
Recurrence and Ergodicity of Interacting Particle Systems
, 1999
"... Many interacting particle systems with short range interactions are not ergodic, but converge weakly towards a mixture of their ergodic invariant measures. The question arises whether a.s. the process eventually stays close to one of these ergodic states, or if it changes between the attainable ergo ..."
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Cited by 17 (4 self)
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Many interacting particle systems with short range interactions are not ergodic, but converge weakly towards a mixture of their ergodic invariant measures. The question arises whether a.s. the process eventually stays close to one of these ergodic states, or if it changes between the attainable ergodic states infinitely often ("recurrence"). Under the assumption that there exists a convergencedetermining class of distributions that is (strongly) preserved under the dynamics, we show that the system is in fact recurrent in the above sense. We apply our method to several interacting particle systems, obtaining new or improved recurrence results. In addition, we answer a question raised by Ed Perkins concerning the change of the locally predominant type in a model of mutually catalytic branching. Running head: Recurrence and Ergodicity Keywords and phrases: Interacting particle systems, longtime behavior, clustering, recurrence, ergodicity, mutually catalytic branching branching 1991 ...
The stepping stone model, II: Genealogies and the infinite sites model, submitted
, 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 twodimensional torus. We show that the genealogy of a sample of size n is given by a time change of Kingman’s coalescent. With DNA sequence data in mind, we inves ..."
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Cited by 17 (3 self)
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This paper extends earlier work by Cox and Durrett, who studied the coalescence times for two lineages in the stepping stone model on the twodimensional torus. We show that the genealogy of a sample of size n is given by a time change of Kingman’s coalescent. With DNA sequence data in mind, we investigate mutation patterns under the infinite sites model, which assumes that each mutation occurs at a new site. Our results suggest that the spatial structure of the human population contributes to the haplotype structure and a slower than expected decay of genetic correlation with distance revealed by recent studies of the human genome. 1. Introduction. Sequencing