Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Andjel, E.D. T.M. Liggett, and T. Mountford (1992) Clustering in one dimensional threshold voter models. Stoch. Processes Appl. 42, 73–90
Aronson, D.G. and H.F. Weinberger (1978) Multidimensional diffusion equations arising in population genetics. Advances in Math. 30, 33–76
Asmussen, S. and N. Kaplan (1976) Branching random walks, I, Stoch. Processes Appl. 4, 1–13
Bezuidenhout, C. and L. Gray (1993) Critical attractive spin systems. Ann. Probab., to appear
Bezuidenhout, C. and G. Grimmett (1990) The critical contact process dies out. Ann. Probab. 18, 1462–1482
Bezuidenhout, C. and G. Grimmett (1991) Exponential decay for subcritical contact and percolation processes. Ann. Probab. 19, 984–1009
Boerlijst, M.C. and P. Hogeweg (1991) Spiral wave structure in pre-biotic evolution: hypercycles stable against parasites. Physica D 48, 17–28
Bramson, M. (1983) Convergence of solutions of the Kolmogorov equation to travelling waves. Memoirs of the AMS, 285
Bramson, M. and R. Durrett (1988) A simple proof of the stability theorem of Gray and Griffeath. Probab. Th. Rel. Fields 80, 293–298
Bramson, M., R. Durrett, and G. Swindle (1989) Statistical mechanics of Crabgrass. Ann. Prob. 17, 444–481
Bramson, M. and L. Gray (1992) A useful renormalization argument. Pages??? in Random Walks, Brownian Motion, and Interacting Particle Systems, edited by R. Durrett and H. Kesten, Birkhauser, Boston
Bramson, M. and D. Griffeath (1987) Survival of cyclic particle systems. Pages 21–30 in Percolation Theory and Ergodic Theory of Infinite Particle Systems edited by H. Kesten, IMA Vol. 8, Springer
Bramson, M. and D. Griffeath (1989) Flux and fixation in cyclic particle systems Ann. Probab. 17, 26–45
Bramson, M. and C. Neuhauser (1993) Survival of one dimensional cellular automata. Preprint
Chen, H.N. (1992) On the stability of a population growth model with sexual reproduction in Z2. Ann. Probab. 20, 232–285
Cox, J.T. (1988) Coalescing random walks and voter model consensus times on the torus in Z d. Ann. Probab.
Cox, J.T. and R. Durrett (1988) Limit theorems for the spread of epidemics and forest fires. Stoch. Processes Appl. 30, 171–191
Cox, J.T. and R. Durrett (1992) Nonlinear voter models. Pages 189–202 in Random Walks, Brownian Motion, and Interacting Particle Systems, edited by R. Durrett and H. Kesten, Birkhauser, Boston
Cox, J.T. and D. Griffeath (1986) Diffusive clustering in the two dimensional voter model. Ann. Probab. 14, 347–370
DeMasi, A., P. Ferrari, and J. Lebowitz (1986) Reaction diffusion equations for interacting particle systems. J. Stat. Phys. 44, 589–644 *** DIRECT SUPPORT *** A00I6B63 00003
DeMasi, A. and E. Presutti (1991) Mathematical Methods for Hydrodynamic Limits. Lecture Notes in Math 1501, Springer, New York.
Durrett, R. (1980) On the growth of one dimensional contact processes. Ann. Probab. 8, 890–907.
Durrett, R. (1984) Oriented percolation in two dimensions. Ann. Probab. 12, 999–1040.
Durrett, R. (1988) Lecture Notes On Particle Systems And Percolation, Wadsworth, Belmont, CA
Durrett, R. (1991a) Stochastic models of growth and competition. Pages 1049–1056 in Proceedings of the International Congress of Mathematicians, Kyoto, Springer, New York.
Durrett, R. (1991b) The contact process, 1974–1989. Pages 1–18 in Proceedings of the AMS Summer seminar on Random Media. Lectures in Applied Math 27, AMS, Providence, RI
Durrett, R. (1991c) Some new games for your computer. Nonlinear Science Today Vol. 1, No. 4, 1–7.
Durrett, R. (1992a) Multicolor particle systems with large threshold and range. J. Theoretical Prob., 5 (1992), 127–152.
Durrett, R. (1992b) A new method for proving the existence of phase transitions. Pages 141–170 in Spatial Stochastic Processes, edited by K.S. Alexander and J. C. Watkins, Birkhauser, Boston
Durrett, R. (1992c) Stochastic growth models: bounds on critical values. J. Appl. Prob. 29
Durrett, R. (1992d) Probability: Theory and Examples. Wadsworth, Belmont, CA
Durrett, R. (1993) Predator-prey systems. Pages 37–58 in Asymptotic Problems in Probability Theory: Stochastic Models and Diffusions on Fractals, edited by K.D. Elworthy and N. Ikeda, Pitman Research Notes in Math 283, Longman, Essex, England.
Durrett, R. and D. Griffeath (1993) Asymptotic behavior of excitable cellular automata. Preprint.
Durrett, R. and S. Levin (1993) Stochastic spatial models: A user's guide to ecological applications. Phil. Trans. Roy. Soc. B, to appear
Durrett, R. and A.M. Moller (1991) Complete convergence theorem for a competition model. Probab. Th. Rel. Fields 88, 121–136.
Durrett, R. and C. Neuhauser (1991) Epidemics with recovery in d=2. Ann. Applied Probab. 1, 189–206.
Durrett, R. and C. Neuhauser (1993) Particle systems and reaction diffusion equations. Ann. Probab., to appear
Durrett, R. and R. Schinazi (1993) Asymptotic critical value for a competition model. Ann. Applied Probab., to apear
Durrett, R. and J. Steif (1993) Fixation results for threshold voter models. Ann. Probab., to appear
Durrett, R. and G. Swindle (1991) Are there bushes in a forest? Stoch. Proc. Appl. 37, 19–31.
Durrett, R. and G. Swindle (1993) Coexistence results for catalysts. Preprint
Eigen, M. and P. Schuster (1979) The Hypercycle: A Principle of Natural Self-Organization, Springer, New York.
Fife, P.C. and J.B. McLeod (1977) The approach of solutions of nonlinear diffusion equations to travelling front solutions. Arch. Rat. Mech. Anal. 65, 335–361
Fisch, R. (1990a) The one dimensional cyclic cellular automaton: a system with deterministic dynamics which emulates a particle system with stochastic dynamics. J. Theor. Prob. 3, 311–338.
Fisch, R. (1990b) Cyclic cellular automata and related processes. Physica D 45, 19–25.
Fisch, R. (1992). Clustering in the one dimensional 3-color cyclic cellular automaton, Ann. Probab. 20, 1528–1548.
Fisch, R., J. Gravner and D. Griffeath (1991) Threshold range scaling of excitable cellular automata. Statistics and Computing 1, 23–39.
Fisch, R., J. Gravner and D. Griffeath (1992) Cyclic cellular automata in two dimensions. In Spatial Stochastic Processes edited by K. Alexander and J. Watkins, Birkhauser, Boston
Fisch, R., J. Gravner, and D. Griffeath (1993) Metastability in the Greenberg Hastings model. Ann. Applied. Probab., to appear
Grannan, E. and G. Swindle (1991) Rigorous results on mathematical models of catalyst surfaces. J. Stat. Phys. 61, 1085–1103
Gravner, J. and D. Griffeath (1993) Threshold growth dynamics. Transactions A.M.S., to appear
Gray, L. and D. Griffeath (1982) A stability criterion for attractive nearest neighbor spin systems on Z. Ann. Probab. 10, 67–85
Gray, L. (1987) Behavior of processes with statistical mechanical properties. Pages 131–168 in Percolation Theory and Ergodic Theory of Infinite Particle Systems edited by H. Kesten, IMA Vol. 8, Springer
Griffeath, D. (1979) Additive and Cancellative Interacting Particle Systems. Lecture Notes in Math 724, Springer
Harris, T.E. (1972) Nearest neighbor Markov interaction processes on multidimensional lattices. Adv. in Math. 9, 66–89.
Harris, T.E. (1976) On a class of set valued Markov processes. Ann. Probab. 4, 175–194.
Hassell, M.P., H.N. Comins and R.M. May (1991) Spatial structure and chaos in insect population dynamics. Nature 353, 255–258.
Hirsch, M.W. and S. Smale (1974) Differential Equations, Dynamical Systems, and Linear Algebra, Academic Press, New York.
Holley, R.A. (1972) Markovian interaction processes with finite range interactions. Ann. Math. Stat. 43, 1961–1967.
Holley, R.A. (1974) Remarks on the FKG inequalities. Commun. Math. Phys. 36, 227–231.
Holley, R.A. and T.M. Liggett (1975) Ergodic theorems for weakly interacting systems and the voter model. Ann. Probab. 3, 643–663.
Holley, R.A. and T.M. Liggett (1978) The survival of contact processes. Ann. Probab. 6, 198–206.
Kinzel, W. and J. Yeomans (1981) Directed percolation: a finite size renormalization approach. J. Phys. A 14, L163–L168.
Liggett, T.M. (1985) Interacting Particle Systems. Springer, New York.
Liggett, T.M. (1993) Coexistence in threshold voter models. Ann. Probab., to appear
Neuhauser, C. (1992) Ergodic theorems for the multitype contact process. Probab. Theory Rel. Fields. 91, 467–506
Redheffer, R., R. Redlinger and W. Walter (1988) A theorem of La Salle-Lyapunov type for parabolic systems. SIAM J. Math. Anal. 19, 121–132
Schonmann, R.H. and M.E. Vares (1986) The survival of the large dimensional basic contact process. Probab. Th. Rel. Fields 72, 387–393.
Spohn, H. (1991) Large Scale Dynamics of Interacting Particle Systems, Springer, New York
Zhang, Yu (1992) A shape theorem for epidemics and forest fires with finite range interactions. Preprint.
Editor information
Rights and permissions
Copyright information
© 1995 Springer-Verlag
About this chapter
Cite this chapter
Durrett, R. (1995). Ten lectures on particle systems. In: Bernard, P. (eds) Lectures on Probability Theory. Lecture Notes in Mathematics, vol 1608. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0095747
Download citation
DOI: https://doi.org/10.1007/BFb0095747
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-60015-2
Online ISBN: 978-3-540-49402-7
eBook Packages: Springer Book Archive