Contact Interactions on a Lattice

Harris, T. E.

doi:10.1214/aop/1176996493

Cited by 933 publications

(807 citation statements)

References 9 publications

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“…The contact process is first introduced in [2] by Harris in 1974 and has been an important model for the development of the theory of interacting particle systems since then. For a detailed survey of the study of the contact process, see Chapter Six of [5] and Part one of [6].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Mean field limit for survival probability of the high-dimensional contact process

Xue

2017

Statistics & Probability Letters

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In this paper we are concerned with the contact process on the squared lattice. The contact process intuitively describes the spread of the infectious disease on a graph, where an infectious vertex becomes healthy at a constant rate while a healthy vertex is infected at rate proportional to the number of infectious neighbors. As the dimension of the lattice grows to infinity, we give a mean field limit for the survival probability of the process conditioned on the event that only the origin of the lattice is infected at t = 0. The binary contact path process is a main auxiliary tool for our proof.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Mean field limit for survival probability of the high-dimensional contact process

Xue

2017

Statistics & Probability Letters

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“…We finally turn our attention to a process that comprises symmetric decoagulation (that is, •• → •• and •• → •• taking place with equal probability in each direction) and spontaneous decay (• → •) occurring at independent rates. Introduced as a crude model of an epidemic [44], this contact process is known to exhibit a transition from a phase in which the absorbing state (empty lattice) is reached with certainty to a phase in which there is some probability that the epidemic remains active forever in the thermodynamic (infinite system size) limit [45]. This transition occurs as the ratio between the decoagulation and decay rates is increased beyond a critical value.…”

Section: Iv2 Reaction-diffusion Systems and Directed Percolationmentioning

confidence: 99%

The Lee-Yang theory of equilibrium and nonequilibrium phase transitions

2003

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We present a pedagogical account of the Lee-Yang theory of equilibrium phase transitions and review recent advances in applying this theory to nonequilibrium systems. Through both general considerations and explicit studies of specific models, we show that the Lee-Yang approach can be used to locate and classify phase transitions in nonequilibrium steady states. I IntroductionIn this work we seek a mathematical understanding of phase transitions in the steady state of stochastic many-body systems. Systems at equilibrium with their environment provide examples of such steady states, and the mechanisms underlying equilibrium phase transitions are long known and understood. Experimentally, one can distinguish between two types of transition: the first-order transition at which there is phase coexistence, e.g. between a high-density solid and a low-density fluid, and the continuous transition at which fluctuations and correlations grow to such an extent as to be macroscopically observable.From a thermodynamical perspective, one can understand first-order transitions by associating with each phase a free energy. For a given set of external parameters, the phase 'chosen' by the system is that with the lowest free energy, and so a phase transition occurs when the free energies of two (or more) phases are equal. The sudden changes in macroscopically measurable quantities that take place at first-order transitions are described mathematically as discontinuities in the first derivative of the free energy. Discontinuities in higher derivatives relate to continuous (higherorder) phase transitions.The tools of equilibrium statistical physics allow onein principle at least-to express the free energy solely in terms of microscopic interactions. More specifically, the free energy is given by the logarithm of a partition function, a quantity that normalises the steady-state probability distribution of microscopic configurations. Initially it was not universally accepted that this approach could faithfully describe phase transitions, in particular the first-order solidfluid transition [1]. In order to show that the statistical mechanical approach can reproduce the correct discontinuities in the free energy at a first-order transition, Lee and Yang [2] introduced a description of phase transitions concerning zeros of the partition function when generalised to the complex plane of an intensive thermodynamic quantity. Initially, the theory was couched in terms of zeros in the complex fugacity plane which is appropriate for fluids in the grand canonical (fluctuating particle number) ensemble. However, by mapping a lattice gas onto the Ising model [3], the theory was found to hold equally well for a magnet in a fixed magnetic field. Later [4-6] it became clear that the distribution of zeros in the complex temperature plane can reveal information about phase transitions in the canonical ensemble.In section II we present a brief, self-contained discussion of the Lee-Yang theory of equilibrium phase transitions which relates nonanalytic ...

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“…Such stochastic systems have been intensively studied since their introduction by Harris (1974); we refer the reader to the surveys of Durrett (1991) and of Liggett (2004). Contact processes are common in spatially explicit epidemiological models.…”

Section: The Basic Modelmentioning

confidence: 99%

Pair statistics clarify percolation properties of spatially explicit simulations

Achter

Webb

2006

Theoretical Population Biology

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Dispersal is a fundamental control on the spatial structure of a population. We investigate the precise mechanism by which a mixed strategy of short-and long-distance dispersal affects spatial patterning. Using techniques from pair approximation and percolation theory, we demonstrate that dispersal controls the extent to which a population is completely connected by modulating the proportion of neighboring sites which are simultaneously occupied. We show that near the percolation threshold this pair statistic, rather than other metrics proposed earlier, best explains clustering, and we suggest more general circumstances under which this may hold.Key words: dispersal, percolation, pair approximation, aggregation, spatial models, connectivity 2 Understanding the sources and consequences of spatial heterogeneity is a fundamental task in ecology (Levin, 1992). Both in terrestrial (Cain et al., 2000) and marine (Carlon and Olson, 1993) systems, dispersal is one of the basic forces which shapes the spatial structure of a population (Levin and Muller-Landau, 2000; Tilman and Kareiva, 1997). A poorly-dispersing species shows spatial autocorrelation, whereas long-distance dispersal contributes to the spatial mixing of a population (Reed et al., 2000). This spatial patterning has important consequences for phenomena such as species coexistence (Bolker and Pacala, 1999), vulnerability to and recovery from disturbance (Etter and Caswell, 1994;Hiebeler, 2004), and other aspects of ecosystem function (Pacala and Deutschman, 1995). Through such mechanisms, whether dispersal is highly localized, long-distance, or a mixture of the two can affect the spatiotemporal dynamics of a population.In this paper, we examine the effects of mixed dispersal strategies on the spatial structure of a population. To this end we consider a spatially explicit birth-death model, in which organisms have a mixed strategy of local and global dispersal. We will see that the exact strategy -that is, the relative frequency of local and global dispersal -has important consequences for the spatial structure of a population. Specifically, we show that whether a population is completely connected depends on the proportion of pairs of neighboring sites which are simultaneously occupied, and that dispersal influences patterns of aggregation by changing this pair statistic. (Earlier statements in the literature (Iwasa, 2000;Kubo et al., 1996) imply that clustering is determined by a conditional site occupation probability which is related to, but distinct from, the pair statistic.) Briefly, local density controls global connectedness.Much prior work on spatially-explicit, single-species systems has focused on the special case of either purely local (de Aguiar et al., 2004;Haraguchi and Sasaki, 2000) or purely global (Deredec and Courchamp, 2003;Tilman, 1994) dispersal. Other researchers have examined both cases simultaneously (Boots and Sasaki, 2000;Socolar et al., 2001), without considering intermediate or mixed strategies.A class of models which expli...

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Contact Interactions on a Lattice

Cited by 933 publications

References 9 publications

Mean field limit for survival probability of the high-dimensional contact process

Mean field limit for survival probability of the high-dimensional contact process

The Lee-Yang theory of equilibrium and nonequilibrium phase transitions

Pair statistics clarify percolation properties of spatially explicit simulations

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