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Contact Processes

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Stochastic Interacting Systems: Contact, Voter and Exclusion Processes

Part of the book series: Grundlehren der mathematischen Wissenschaften ((GL,volume 324))

Abstract

The contact process is often thought of as a model for the spread of infection. The collection of individuals that may be infected at any given time is taken to be the set of vertices of a connected, undirected graph S. For such a graph, the degree of a vertex x is the number of vertices y that are connected to x by an edge. The main examples to be treated below are the d dimensional integer lattice Z d (in which the degree of each vertex is 2d), and the homogeneous tree T d in which every vertex has degree d +1. In general, we will assume that the degrees of the vertices are uniformly bounded. A path through S is a sequence of consecutive edges in the graph, and its length is the number of edges used. The distance between two vertices x, y ∈ S is the minimal length of a path from x to y, and is denoted by |y-x|.

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References

  • M. Aizenman and G. Grimmett, Strict monotonicity for critical points in percolation and ferromagnetic models, J. Statist. Phys. 63 (1991), 817–835.

    MathSciNet  Google Scholar 

  • E. D. Andjel, The contact process in high dimensions, Ann. Prob. 16 (1988), 1174–1183.

    MathSciNet  MATH  Google Scholar 

  • E. D. Andjel, Survival of multidimensional contact process in random environments, Bol. Soc. Bras. Mat. 23 (1992), 109–119.

    MathSciNet  MATH  Google Scholar 

  • E. D. Andjel, R. Schinazi and R. H. Schonmann, Edge processes of one-dimensional stochastic growth models, Ann. Inst. H. Poincaré Probab. Statist. 26 (1990), 489–506.

    MathSciNet  MATH  Google Scholar 

  • D. J. Barsky and C. C. Wu, Critical exponents for the contact process under the triangle condition, J. Statist. Phys. 91 (1998), 95–124.

    MathSciNet  MATH  Google Scholar 

  • V. Belitsky, P. A. Ferrari, N. Konno and T. M. Liggett, A strong correlation inequality for contact processes and oriented percolation, Stoch. Proc. Appl. 67 (1997), 213–225.

    MathSciNet  MATH  Google Scholar 

  • C. Bezuidenhout and L. Gray, Critical attractive spin systems, Ann. Probab. 22 (1994), 1160–1194.

    MathSciNet  MATH  Google Scholar 

  • C. Bezuidenhout and G. Grimmett, The critical contact process dies out, Ann. Probab. 18 (1990), 1462–1482.

    MathSciNet  MATH  Google Scholar 

  • C. Bezuidenhout and G. Grimmett, Exponential decay for subcritical contact and percolation processes, Ann. Probab. 19 (1991), 984–1009.

    MathSciNet  MATH  Google Scholar 

  • M. Bramson, R. Durrett and R. H. Schonmann, The contact process in a random environment, Ann. Probab. 19 (1991), 960–983.

    MathSciNet  MATH  Google Scholar 

  • M. Bramson, R. Durrett and G. Swindle, Statistical mechanics of crabgrass, Ann. Probab. 17(1989), 444–481.

    MathSciNet  MATH  Google Scholar 

  • L. Buttel, J. T. Cox and R. Durrett, Estimating the critical values of stochastic growth models, J. Appl. Probab. 30 (1993), 455–461.

    MathSciNet  Google Scholar 

  • M. Cassandro, A. Galves, E. Olivieri and M. E. Vares, Metastable behavior of stochastic dynamics: A pathwise approach, J. Statist. Phys. 35 (1984), 603–634.

    MathSciNet  MATH  Google Scholar 

  • J. W. Chen, Small density fluctuation for one-dimensional contact processes under nonequi-librium, Acta Math. Sci. 13 (1993), 399–405.

    MATH  Google Scholar 

  • J. W. Chen, The contact process on a finite system in higher dimensions, Chinese J. Contemp. Math. 15 (1994), 13–20.

    Google Scholar 

  • J. W. Chen, Smoothness and stability of one-dimensional contact processes, Acta Math.Sinica 38 (1995), 91–98.

    MathSciNet  MATH  Google Scholar 

  • J. W. Chen, R. Durrett and X. F. Liu, Exponential convergence for one dimensional contact processes, Acta Math. Sinica 6 (1990), 349–353.

    MathSciNet  MATH  Google Scholar 

  • J. T. Cox, R. Durrett and R. Schinazi, The critical contact process seen from the right edge, Probab. Th. Rel. Fields 87 (1991), 325–332.

    MathSciNet  MATH  Google Scholar 

  • J. T. Cox and A. Greven, On the long term behavior of some finite particle systems, Probab. Th. Rel. Fields 85 (1990), 195–237.

    MathSciNet  MATH  Google Scholar 

  • R. Dickman, Nonequilibrium lattice models: series analysis of steady states, J. Statist. Phys. 55 (1989), 997–1026.

    MathSciNet  Google Scholar 

  • R. Durrett, The contact process, 1974–1989, Proceedings of the 1989 AMS Seminar on Random Media (W. E. Kohler and B. S. White, ed.), vol. 27, AMS Lectures in Applied Mathematics, 1991, pp. 1–18.

    Google Scholar 

  • R. Durrett, Stochastic growth models — bounds on critical values, J. Appl. Prob. 29 (1992), 11–20.

    MathSciNet  MATH  Google Scholar 

  • R. Durrett and D. Griffeath, Contact processes in several dimensions, Z. Wahrsch. verw. Gebiete 59 (1982), 535–552.

    MathSciNet  MATH  Google Scholar 

  • R. Durrett and X. Liu, The contact process on a finite set, Ann. Probab. 16 (1988), 1158–1173.

    MathSciNet  MATH  Google Scholar 

  • R. Durrett and E. Perkins, Rescaled contact processes converge to super Brownian motion for d ≥ 2, Probab. Theory Rel. Fields (1999).

    Google Scholar 

  • R. Durrett and R. Schinazi, Intermediate phase for the contact process on a tree, Ann. Probab. 23 (1995), 668–673.

    MathSciNet  MATH  Google Scholar 

  • R. Durrett and R. Schonmann, Stochastic growth models, Percolation Theory and Ergodic Theory of Infinite Particle Systems (H. Kesten, ed.), Springer, 1987, pp. 85–119.

    Google Scholar 

  • R. Durrett and R. Schonmann, The contact process on a finite set II, Ann. Probab. 16 (1988a), 1570–1583.

    MathSciNet  MATH  Google Scholar 

  • R. Durrett and R. Schonmann, Large deviations for the contact process and two dimensional percolation, Probab. Th. Rel. Fields 77 (1988b), 583–603.

    MathSciNet  MATH  Google Scholar 

  • R. Durrett, R. Schonmann and N. Tanaka, The contact process on a finite set III The critical case, Ann. Probab. 17 (1989), 1303–1321.

    MathSciNet  MATH  Google Scholar 

  • S. N. Evans and E. A. Perkins, Measure-valued branching diffusions with singular interactions, Canad. J. Math. 46 (1994), 120–168.

    MathSciNet  MATH  Google Scholar 

  • M. Fiocco, Statistical estimation for the supercritical contact process, Thesis, Leiden, 1997.

    Google Scholar 

  • A. Galves, F. Martinelli and E. Olivieri, Large density fluctuations for the one dimensional supercritical contact process, J. Statist. Phys. 55 (1989), 639–648.

    MathSciNet  MATH  Google Scholar 

  • A. Galves and E. Presutti, Edge fluctuations for the one dimensional supercritical contact process, Ann. Probab. 15 (1987a), 1131–1145.

    MathSciNet  MATH  Google Scholar 

  • A. Galves and E. Presutti, Travelling wave structure of the one dimensional contact process, Stoch. Proc. Appl. 25 (1987b), 153–163.

    MathSciNet  MATH  Google Scholar 

  • A. Galves and R. Schinazi, Approximations finis de la mesure invariante du processus de contact sur-critique vu par la premiere particule, Probab. Th. Rel. Fields 83 (1989), 435–445.

    MathSciNet  MATH  Google Scholar 

  • L. Gray, Is the contact process dead?, Proceedings of the 1989 AMS Seminar on Random Media (W. E. Kohler and B. S. White, ed.), vol. 27, AMS Lectures in Applied Mathematics, 1991, pp. 19–29.

    Google Scholar 

  • D. Griffeath, Limit theorems for nonergodic set-values Markov processes, Ann. Probab. 6 (1978), 379–387.

    MathSciNet  MATH  Google Scholar 

  • C. Grillenberger and H. Ziezold, On the critical infection rate of the one dimensional basic contact process: numerical results, J. Appl. Probab. 25 (1988), 1–8.

    MathSciNet  MATH  Google Scholar 

  • G. Grippenberg, A lower bound for the order parameter in the one-dimensional contact process, Stoch. Proc. Appl. 63 (1996), 211–219.

    Google Scholar 

  • I. Heuter, Anisotropic contact process on homogeneous trees, (2000).

    Google Scholar 

  • R. Holley and T. M. Liggett, The survival of contact processes, Ann. Probab. 6 (1978), 198–206.

    MathSciNet  MATH  Google Scholar 

  • I. Jensen and R. Dickman, Time-dependent perturbation theory for nonequilibrium lattice models, J. Statist. Phys. 71 (1993), 89–127.

    MathSciNet  MATH  Google Scholar 

  • S. Jitomirskaya and A. Klein, hing model in a quasiperiodic transverse field, percolation, and contact processes in quasiperiodic environments, J. Statist. Phys. 73 (1993), 319–344.

    MathSciNet  MATH  Google Scholar 

  • M. Katori, Rigorous results for the diffusive contact process in d ≥ 3, J. Phys. A 27 (1994), 7327–7341.

    MathSciNet  MATH  Google Scholar 

  • M. Katori and N. Konno, Correlation inequalities and lower bounds for the critical value λ c of contact processes, J. Phys. Soc. Japan 59 (1990), 877–887.

    MathSciNet  Google Scholar 

  • M. Katori and N. Konno, Applications of the Harris-FKG inequality to upper bounds for order parameters in the contact process, J. Phys. Soc. Japan 60 (1991), 430–434.

    MathSciNet  Google Scholar 

  • M. Katori and N. Konno, Three point Markov extension and an improved upper bound for survival probability of the one dimensional contact process, J. Phys. Soc. Japan 60 (1991), 418–429.

    MathSciNet  Google Scholar 

  • M. Katori and N. Konno, Correlation identities for nearest-particle systems and their applications to one-dimensional contact process, Modern Phys. Lett. B 5 (1991), 151–159.

    MathSciNet  Google Scholar 

  • M. Katori and N. Konno, An upper bound for survival probability of infected region in the contact process, J. Phys. Japan 60 (1991), 95–99.

    MathSciNet  Google Scholar 

  • M. Katori and N. Konno, Upper bounds for the survival probability of the contact process, J. Statist. Phys. 63 (1991), 115–130.

    MathSciNet  Google Scholar 

  • M. Katori and N. Konno, Bounds on the critical line of the θ-contact processes with 1 ≤ θ ≤ 2, J. Phys. A 26 (1993), 6597–6614.

    MathSciNet  MATH  Google Scholar 

  • A. Klein, Extinction of contact and percolation processes in a random environment, Ann. Probab. 22 (1994), 1227–1251.

    MathSciNet  MATH  Google Scholar 

  • N. Konno, Asymptotic behavior of basic contact processes with rapid stirring, J. Th. Probab. 8 (1995), 833–876.

    MathSciNet  MATH  Google Scholar 

  • N. Konno, Lecture Notes on Harris Lemma and Particle Systems, Publicacoes do Instituto de Matematica e Estatistica da Universidade de Sao Paulo, 1996.

    Google Scholar 

  • N. Konno, Lecture Notes on Interacting Particle Systems, Rokko Lectures in Mathematics #3 Kobe University, 1997.

    Google Scholar 

  • N. Konno and K. Sato, Upper bounds on order parameters of diffusive contact processes, J. Phys. Soc. Japan 64 (1995), 2405–2412.

    MathSciNet  Google Scholar 

  • S. M. Krone, The two-stage contact process, Ann. Appl. Probab. 9 (1999).

    Google Scholar 

  • T. Kuczek, The central limit theorem for the right edge of supercritical oriented percolation, Ann. Probab. 17 (1989), 1322–1332.

    MathSciNet  MATH  Google Scholar 

  • S. P. Lalley, Growth profile and invariant measures for the weakly supercritical contact process on a homogeneous tree, Ann. Probab. 27 (1999), 206–225.

    MathSciNet  MATH  Google Scholar 

  • S. P. Lalley and T. Sellke, Limit set of a weakly supercritical contact process on a homogeneous tree, Ann. Probab. 26 (1998), 644–657.

    MathSciNet  MATH  Google Scholar 

  • J. L. Lebowitz and R. H. Schonmann, On the asymptotics of occurrence times of rare events for stochastic spin systems, J. Statist. Phys. 48 (1987), 727–751.

    MathSciNet  MATH  Google Scholar 

  • T. M. Liggett, Spatially inhomogeneous contact processes, Spatial Stochastic Processes. A Festschrift in honor of the Seventieth Birthday of Ted Harris, Birkhauser, 1991a, pp. 105–140.

    Google Scholar 

  • T. M. Liggett, The periodic threshold contact process, Random Walks, Brownian Motion and Interacting Particle Systems, A Festschrift in honor of Frank Spitzer (R. Durrett and H. Kesten, ed.), Birkhauser, 1991b, pp. 339–358.

    Google Scholar 

  • T. M. Liggett, The survival of one dimensional contact processes in random environments, Ann. Probab. 20 (1992), 696–723.

    MathSciNet  MATH  Google Scholar 

  • T. M. Liggett, Survival and coexistence in interacting particle systems, Probability and Phase Transition, Kluwer, 1994a, pp. 209–226.

    Google Scholar 

  • T. M. Liggett, Improved upper bounds for the contact process critical value, Ann. Probab. 23 (1995a), 697–723.

    MathSciNet  MATH  Google Scholar 

  • T. M. Liggett, Multiple transition points for the contact process on the binary tree, Ann. Probab. 24 (1996a), 1675–1710.

    MathSciNet  MATH  Google Scholar 

  • T. M. Liggett, Branching random walks and contact processes on homogeneous trees, Probab. Th. Rel. Fields 106 (1996b), 495–519.

    MathSciNet  MATH  Google Scholar 

  • T. M. Liggett, Branching random walks on finite trees, Perplexing Problems in Probability: Papers in Honor of Harry Kesten, Birkhauser, 1999.

    Google Scholar 

  • N. Madras and R. Schinazi, Branching random walks on trees, Stoch. Proc. Appl. 42 (1992), 255–267.

    MathSciNet  MATH  Google Scholar 

  • N. Madras, R. Schinazi and R. Schonmann, On the critical behavior of the contact process in deterministic inhomogeneous environment, Ann. Probab. 22 (1994), 1140–1159.

    MathSciNet  MATH  Google Scholar 

  • G. Morrow, R. Schinazi and Y. Zhang, The critical contact process on a homogeneous tree, J. Appl. Probab. 31 (1994), 250–255.

    MathSciNet  MATH  Google Scholar 

  • T. S. Mountford, A metastable result for the finite multidimensional contact process, Can. Math. Bull. 36 (1993), 216–226.

    MathSciNet  MATH  Google Scholar 

  • T. S. Mountford, Existence of constant for finite system extinction, J. Statist. Phys. (1999).

    Google Scholar 

  • T. S. Mountford and T. D. Sweet, An extension of Kuczek’s argument to nonnearest neighbor contact processes, (1999).

    Google Scholar 

  • C. Mueller and R. Tribe, A phase transition for a stochastic PDE related to the contact process, Probab. Th. Rel. Fields 100 (1994), 131–156.

    MathSciNet  MATH  Google Scholar 

  • C. Mueller and R. Tribe, Stochastic p.d.e. ’s arising from the long range contact and long range voter processes, Probab. Th. Rel. Fields 102 (1995), 519–545.

    MathSciNet  MATH  Google Scholar 

  • C. M. Newman and S. Volchan, Persistent survival of one-dimensional contact processes in random environments, Ann. Probab. 24 (1996), 411–421.

    MathSciNet  MATH  Google Scholar 

  • C. M. Newman and C. C. Wu, Percolation and contact processes with low dimensional inhomogeneity, Ann. Probab. 25 (1997), 1832–1845.

    MathSciNet  MATH  Google Scholar 

  • R. Pemantle, The contact process on trees, Ann. Probab. 20 (1992), 2089–2116.

    MathSciNet  MATH  Google Scholar 

  • R. Pemantle and A. M. Stacey, The branching random walk and contact process on non-homogeneous and Galton-Watson trees, (2000).

    Google Scholar 

  • M. D. Penrose, The threshold contact process: a continuum limit, Probab. Th. Rel. Fields 104 (1996), 77–95.

    MathSciNet  MATH  Google Scholar 

  • A. Puha, A reversible nearest particle system on the homogeneous tree, J. Th. Probab. 12 (1999), 217–254.

    MathSciNet  MATH  Google Scholar 

  • A. Puha, Critical exponents for a reversible nearest particle system on the binary tree, Ann. Probab. (2000).

    Google Scholar 

  • M. Salzano and R. H. Schonmann, The second lowest extremal invariant measure of the contact process, Ann. Probab. 25 (1997), 1846–1871.

    MathSciNet  MATH  Google Scholar 

  • M. Salzano and R. H. Schonmann, A new proof that for the contact process on homogeneous trees local survival implies complete convergence, Ann. Probab. 26 (1998), 1251–1258.

    MathSciNet  MATH  Google Scholar 

  • M. Salzano and R. H. Schonmann, The second lowest extremal invariant measure of the contact process II, Ann. Probab. 27 (1999).

    Google Scholar 

  • R. Schinazi, On multiple phase transitions for branching Markov chains, J. Statist. Phys. 71 (1993), 521–525.

    MathSciNet  Google Scholar 

  • R. Schinazi, The asymmetric contact process on a finite set, J. Statist. Phys. 74 (1994), 1005–1016.

    MathSciNet  MATH  Google Scholar 

  • R. Schinazi, A contact process with a single inhomogeneous site, J. Statist. Phys. 83 (1996), 767–777.

    MathSciNet  MATH  Google Scholar 

  • R. H. Schonmann, Metastability for the contact process, J. Statist. Phys. 41 (1985), 445–464.

    MathSciNet  MATH  Google Scholar 

  • R. H. Schonmann, Central limit theorem for the contact process, Ann. Probab. 14 (1986a), 1291–1295.

    MathSciNet  MATH  Google Scholar 

  • R. H. Schonmann, The asymmetric contact process, J. Statist. Phys. 44 (1986b), 505–534.

    MathSciNet  MATH  Google Scholar 

  • R. H. Schonmann, A new look at contact processes in several dimensons, Percolation Theory and Ergodic Theory of Infinite Particle Systems (H. Kesten, ed.), vol. 8, IMA Series in Mathematics and its Applications, 1987a, pp. 245–250.

    Google Scholar 

  • R. H. Schonmann, A new proof of the complete convergence theorem for contact processes in several dimensions with large infection parameter, Ann. Probab. 15 (1987b), 382–387.

    MathSciNet  MATH  Google Scholar 

  • R. H. Schonmann, The triangle condition for contact processes on homogeneous trees, J. Statist. Phys. 90 (1998), 1429–1440.

    MathSciNet  MATH  Google Scholar 

  • R. H. Schonmann and M. E. Vares, The survival of the large dimensional basic contact process, Probab. Th. Rel. Fields 72 (1986), 387–393.

    MathSciNet  MATH  Google Scholar 

  • A. Simonis, Metastability for the d-dimensional contact process, J. Statist. Phys. 83 (1996), 1225–1239.

    MathSciNet  MATH  Google Scholar 

  • A. Simonis, Filling in the hypercube in the supercritical contact process in equilibrium, Markov Proc. Rel. Fields 4 (1998), 113–130.

    MathSciNet  MATH  Google Scholar 

  • A. M. Stacey, Bounds on the critical probabilities in oriented percolation models, Cambridge University thesis (1994).

    Google Scholar 

  • A. M. Stacey, The existence of an intermediate phase for the contact process on trees, Ann. Probab. 24 (1996), 1711–1726.

    MathSciNet  MATH  Google Scholar 

  • A. M. Stacey, The contact process on a finite tree, (2000).

    Google Scholar 

  • T. Sweet, The asymmetric contact process at its second critical value, J. Statist. Phys. 86 (1997), 749–764.

    MathSciNet  MATH  Google Scholar 

  • G. Swindle, A mean field limit of the contact process with large range, Probab. Th. Rel. Fields 85 (1990), 261–282.

    MathSciNet  MATH  Google Scholar 

  • A. Y. Tretyakov, V. Belitsky, N. Konno and T. Yamaguchi, Numerical estimation on correlation inequalities for Holley-Liggett bounds, Mem. Muroran Inst. Tech. 48 (1998), 101–105.

    MathSciNet  Google Scholar 

  • A. Y. Tretyakov, N. Inui and N. Konno, Phase transition for the one-sided contact process, J. Phys. Soc. Jap. 66 (1997), 3764–3769.

    Google Scholar 

  • A. Y. Tretyakov and N. Konno, Phase transition of the contact process on the binary tree, J. Phys. Soc. Japan 64 (1995), 4069–4072.

    MathSciNet  MATH  Google Scholar 

  • C. C. Wu, The contact process on a tree: behavior near the first phase transition, Stoch. Proc. Appl. 57(1995), 99–112.

    MATH  Google Scholar 

  • C. C. Wu, Inhomogeneous contact processes on trees, J. Statist. Phys. 88 (1997), 1399–1408.

    MathSciNet  MATH  Google Scholar 

  • Y. Zhang, The complete convergence theorem of the contact process on trees, Ann. Probab. 24 (1996), 1408–1443.

    MathSciNet  MATH  Google Scholar 

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  1. Mathematics Department, University of California, Los Angeles, CA, 90095-1555, USA

    Thomas M. Liggett

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Liggett, T.M. (1999). Contact Processes. In: Stochastic Interacting Systems: Contact, Voter and Exclusion Processes. Grundlehren der mathematischen Wissenschaften, vol 324. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-03990-8_2

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  • DOI: https://doi.org/10.1007/978-3-662-03990-8_2

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