File Name: lecture notes on particle systems and percolation .zip
It is proved that the derivative at the endpoint of the critical curve for percolation exists and its absolute value coincides with the critical rate for the corresponding contact process. This is a preview of subscription content, access via your institution. Rent this article via DeepDyve.
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Journal Help. Article Tools Indexing metadata. How to cite item. Font Size. User Username Password Remember me. Current Issue. Abstract We consider a two-type contact process on the integers. Both types have equal finite range and supercritical infection rate. We show that a given type becomes extinct with probability 1 if and only if, in the initial configuration, it is confined to a finite interval and surrounded by infinitely many individuals of the other type.
Additionally, we show that if both types are present in finite number in the initial configuration, then there is a positive probability that they are both present for all times. Finally, it is shown that, starting from the configuration in which all sites to the left of the origin are occupied by type 1 particles and all sites to the right of the origin are occupied by type 2 particles, the process defined by the size of the interface area between the two types is stochastically tight.
References Andjel, Enrique D. Review Andjel, Enrique D. Bernoulli 16, Number 4 , Math. Review not available Belhaouari, S. Convergence results and sharp estimates for the voter model interfaces. Review Belhaouari, S. Tightness for the interfaces of one-dimensional voter models. Review Cox, J. Hybrid zones and voter model interfaces.
Bernoulli 1 , no. Review Durrett, Richard Lecture notes on particle systems and percolation. Review Durrett, Rick Probability: theory and examples. Fourth edition. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge, ISBN: , Math. Stochastic Process. Review Kuczek, Thomas The central limit theorem for the right edge of supercritical oriented percolation.
Review Lawler, Gregory F. Cambridge Studies in Advanced Mathematics, ISBN: Math. Review Levin, David A. Markov chains and mixing times. With a chapter by James G. Propp and David B.
Review Liggett, Thomas M. Interacting particle systems. Springer-Verlag, New York, Stochastic interacting systems: contact, voter and exclusion processes. Springer-Verlag, Berlin, Review Mountford, Thomas S. An extension of Kuczek's argument to nonnearest neighbor contact processes. Review Neuhauser, Claudia Ergodic theorems for the multitype contact process.
Theory Related Fields 91 , no. Review Spitzer, Frank Principles of random walks. Second edition. Graduate Texts in Mathematics, Vol. Springer-Verlag, New York-Heidelberg, Tightness of voter model interfaces. Review Remember me.
Percolation transitions and wetting transitions in stochastic models. Stochastic models with irreversible elementary processes are introduced, and their macroscopic behaviors in the infinite-time and infinite-volume limits are studied extensively, in order to discuss nonequilibrium stationary states and phase transitions. The Domany-Kinzel model is a typical example of such an irreversible particle system. We first review this model, and explain that in a certain parameter region, the nonequilibrium phase transitions it exhibits can be identified with directed percolation transitions on the spatio-temporal plane. We then introduce an interacting particle system with particle conservation called friendly walkers FW. We show that FW can be considered as a model of interfacial wetting transitions, and that the phase transitions and critical phenomena of FW can be studied using Fisher's theory of phase transitions in linear systems. The FW model may be the key to constructing a unified theory of directed percolation transitions and wetting transitions.
Two of the simplest interacting particle systems are the coalescing random walks and the voter model. Results of a recent paper by Sawyer are applied. This is a preview of subscription content, access via your institution. Rent this article via DeepDyve. Arratia, R.
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All processes are independent of one another and of the random period of time sites remains occupied. Therefore, the process survives whenever the number of particles is positive for all times, which corresponds to the case that there are always infected individuals in the second one. This means that each site waits an exponential time with the corresponding rate, and then flips so 0 becomes 1 and vice versa.
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