Griffeath, D. (1979). Additive and Cancellative Interacting Particle Systems. Lecture Notes in Mathematics no. 724, Springer-Verlag, New York.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... Z d and t 0 : The rate parameters (fi fi fi; ffi ffi ffi) of such a system are called additive. Systems with additive rates are closely connected with percolation models, so it is not surprising that many results from percolation theory more easily generalize to such systems. See the book by Griffeath (1979) or the more recent and somewhat less formal book by Durrett (1988) for conditions on the rates that lead to additive systems. Critical processes There are several notions of criticality for systems f A t : t 0; A Z d g with attractive rates (fi fi fi; ffi ffi ffi) For A Z d , we ....

Griffeath, D. (1979). Additive and Cancellative Interacting Particle Systems. Lecture Notes in Mathematics no. 724, Springer-Verlag, New York.

....from [CDP98] serves as an important tool for these results. 1. Introduction. The voter model was introduced independently by Clifford and Sudbury in [CS73] where it was called the invasion process) and by Holley and Liggett in [HL75] It is one of the simplest interacting particle systems (see [Gr79] and [Li85] but one which exhibits a wide range of interesting phenomena. The process is easily described. One supposes that at each site x of the d dimensional integer lattice Z d there is a voter who randomly changes opinion. In the two type model, each voter holds one of two opinions, say ....

Griffeath, D. (1979) Additive and cancellative interacting particle systems. Lec. Notes in Math. 724, Springer, New York.

....from [CDP98] serves as an important tool for these results. 1. Introduction. The voter model was introduced independently by Clifford and Sudbury in [CS73] where it was called the invasion process) and by Holley and Liggett in [HL75] It is one of the simplest interacting particle systems (see [Gr79] and [Li85] but one which exhibits a wide range of interesting phenomena. The process is easily described. One supposes that at each site x of the d dimensional integer lattice Z d there is a voter who randomly changes opinion. In the two type model, each voter holds one of two opinions, say ....

Griffeath, D. (1979) Additive and cancellative interacting particle systems. Lec. Notes in Math. 724, Springer, New York.

....e.g. 12] We note, however, that our proof of Theorem 1.3 depends on Theorem 1.1, which in turn depends on Sawyer s limit law. In the multitype voter model with mutation, particles mutate at a positive rate . The multitype voter model with mutation has a unique stationary distribution (see e.g. [7]) A theorem also due to Sawyer [13] states that the rescaled size of the type at the origin in equilibrium approaches an exponential distribution as the mutation rate goes to zero. This result can now be obtained by probabilistic methods from the limit law (1.9) as is explained in [6] The ....

....P(#(u R ) # 2 for some # u # 1) 0. The claim of Corollary 4.4 follows as # # 0. # Before presenting our final proposition that shows (in high dimensions) that any limiting point process of T t # t satisfies a strong independence property, we introduce a construction of Gri#eath ([7], proof of Th. II.2.6) This construction makes precise the notion that (# x,r u ) u#0 and (# y,s u ) u#0 evolve independently up until the time the distance between the two clusters reaches one. For B, C # Z d let d(B, C) min z 1 #B,z 2 #C z 1 z 2 . Let # be an independent ....

Griffeath, D. (1979) Additive and cancellative interacting particle systems, Lecture Notes in Math. 724, Springer, New York.

....representation. That is to say, we are given appropriate families of Poisson processes which may be used to couple together the different epidemic processes corresponding to different initial conditions. Such constructions are standard, and may be found in Bezuidenhout and Grimmett (1990) Griffeath (1979), Harris (1974) Liggett (1985) and elsewhere. We shall continue to use the notation P ff;fi to denote the relevant probability measure; this notation is not entirely appropriate, since the initial configuration (3.1) is not germane to the following discussion. Let A be a finite subset of Z d ....

Griffeath, D. (1979). Additive and Cancellative Interacting Particle Systems. Lecture Notes in Mathematics no. 724, Springer-Verlag, Berlin.

.... Aldous Department of Statistics University of California Berkeley, CA 94720 James Allen Fill Department of Mathematical Sciences The Johns Hopkins University Baltimore, MD 21218 2692 March 10, 1994 There is a well established topic interacting particle systems , treated in the books by Griffeath [14], Liggett [17] and Durrett [11] which studies different models for particles on the infinite lattice Z d . All these models make sense, but mostly have not been systematically studied, in the context of finite graphs. Some of these models the voter model, the antivoter model, and the ....

....v (t) v 2 G) is a continuous time Markov chain on state space f Gamma1; 1g G . So, provided this chain is irreducible, there is a unique stationary distribution (j v ; v 2 G) for the antivoter model. This model on infinite lattices was studied in the interacting particle systems literature [14, 17], and again the key idea is duality. In this model the dual process consists of annihilating random walks. We will not go into details about the duality relation, beyond the following definition we need later. For vertices v; w, consider independent continuous time random walks started at v and at ....

D. Griffeath. Additive and Cancellative Interacting Particle Systems, volume 724 of Lecture Notes in Math. Springer-Verlag, 1979. 37

....the alarm clock next goes off, for each state i in the stack, Map[i] gets set to be the state at that time. Perfectly Random Sampling and Tree Generation 18 much attention in the continuous time case, usually on a grid where p i;j is nonzero only if i and j are neighbors on the grid. See [36] [32], and [43] for background on the voter model. These two models are dual to one another, in the sense that each can be obtained from the other by simply reversing the direction of time (see [3] Specifically, suppose that for each t between t 1 and t 2 , for each state we choose in advance a ....

David Griffeath. Additive and Cancellative Interacting Particle Systems. SpringerVerlag, 1979. Lecture Notes in Mathematics, #724.

....neighbor will vote for, and at the next time step changes his choice to be that candidate. These updates are done in parallel. The voter model has been studied quite a lot in continuous time case, and usually on a grid where p i;j is nonzero only if i and j are neighbors on the grid. See [45] [39], and [61] for background on the voter model. These two models are dual to one another, in the sense that each can be obtained from the other by simply reversing the direction of time (see [5] Specifically, suppose that for each t between t 1 and t 2 , for each state we choose in advance a ....

David Griffeath. Additive and Cancellative Interacting Particle Systems. SpringerVerlag, 1979. Lecture Notes in Mathematics, #724.

....(z i ) by the usual graphical representation: Give each site 2 Z an independent rate one Poisson process of possible jump times, and declare that server z k jumps to Gamma 1 at time t if z k (t Gamma) t is a possible jump time at site , and z k Gamma1 (t Gamma) Gamma 1. See [Gr] for details of this construction. Through (1.8) we also obtain a construction of the customer process for any initial queues. Fix a site , and consider starting the server process from the configuration x i = for i 0, x i = Gamma1 for i 0. Think of an infinitely long queue of customers ....

D. Griffeath, Additive and Cancellative Interacting Particle Systems, Springer-Verlag, Lecture Notes in Mathematics 724, 1979.

....algorithm; see [60] Time reversal also shows up in the theory of coalescent duality for interacting particle systems. A standard example of this duality is the relationship between the coalescing random walk model and the voter model generalized to weighted directed graphs (see [32] [27], 41] Given a continuoustime Markov chain on n states, in which transitions from state i to state j occur at rate p i;j , one defines a coalescing random walk by placing a particle on each state and decreeing that particles must move according to the Markov chain statistics; particles must ....

David Griffeath. Additive and Cancellative Interacting Particle Systems. Springer-Verlag, 1979. Lecture Notes in Mathematics, #724.

....random walk, therefore, the density in the annihilating random walk should be one half the density of the coalescing random walk. This one half thinning is contained in the following proposition. This has actually been made rigorous for a class of coalescing annihilating random walks by Griffeath (1979) (see, Proposition 5.4, Chapter III) and Arratia (1981) It turns out that the proof of our result needs a different argument, primarily since in our process there is a source of particles at the origin. Proposition 2. lim x 1 P (x 2 j 1 ) P (x 2 1 ) 1 2 : Combining the two ....

Griffeath, D. (1979). Additive and Cancellative Interacting Particle Systems, Lecture Notes in Mathematics 724, Springer, New York.

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Griffeath, D.: Additive and Cancellative Interacting Particle Systems, Springer Lect. Notes in Math., 724, 1979.

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D. Griffeath, Additive and Cancellative Interacting Particle Systems (Springer Lecture Notes in Mathematics, Vol. 724, 1979).

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D. Griffeath. Additive and Cancellative Interacting Particle Systems, volume 724 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1979.

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D. Griffeath. Additive and Cancellative Interacting Particle Systems, volume 724 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1979.

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Griffeath, D. (1979). Additive and Cancellative Interacting Particle Systems. Springer Verlag.

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