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We introduce and analyze a simple probabilistic cellular automaton which emulates the flow of cars along a highway. Our Traffic CA captures the essential features of several more complicated algorithms, studied numerically by K. Nagel and others over the past decade as prototypes for the emergence of traffic jams. By simplifying the dynamics, we are able to identify and precisely formulate the self-organized critical evolution of our system. We focus here on the Cruise Control case, in which well-spaced cars move deterministically at maximal speed, and we obtain rigorous results for several special cases. Then we introduce a symmetry assumption that leads to a two-parameter model, described in terms of acceleration () and braking () probabilities. Based on the results of simulations, we map out the (, ) phase diagram, identifying three qualitatively distinct varieties of traffic which arise, and we derive rigorous bounds to establish the existence of a phase transition from free flow to jams. Many other results and conjectures are presented. From a mathematical perspective, Traffic CA provides local, particle-conserving, one-dimensional dynamics which cluster, and converge to a mixture of two distinct equilibria. 相似文献
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Summary We prove large deviation theorems for occupation time functionals of independent random walks started from a Poisson field on Z
d. In dimensions 1 and 2 the large deviation tails are larger than exponential. Exact asymptotics are derived.Partially supported by the National Science Foundation under Grant MCS 81-02131 and MCS 81-00256Alfred P. Sloan Research Fellow 相似文献
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Maury Bramson J. Theodore Cox David Griffeath 《Probability Theory and Related Fields》1988,77(3):401-413
This paper is a sequel to [5] and [6]. We continue our study of occupation time large deviation probabilities for some simple infinite particle systems by analysing the so-called voter model t (see e.g., [11] or [8]). In keeping with our previous results, we show that the large deviations are classical in high dimensions (d5 for t) but fat in low dimensions (d4). Interaction distinguishes the voter model from the independent particle systems of [5] and [6], and consequently exact computations no longer seem feasible. Instead, we derive upper and lower bounds which capture the asymptotic decay rate of the large deviation tails.Dedicated to Frank Spitzer on his 60th birthdayPartially supported by the National Science Foundation under Grant DMS-831080Partially supported by the National Science Foundation under Grant DMS-841317Partially supported by the National Science Foundation under Grant DMS-830549 相似文献
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Maury Bramson J. Theodore Cox David Griffeath 《Probability Theory and Related Fields》1986,73(4):613-625
Summary This paper is a sequel of a paper of Cox and Griffeath “diffusive clustering in the two dimensional voter model”. We continue
our study of the voter model and coalescing random walks on the two dimensional integer lattice. Some exact asymptotics concerning
the rate of clustering in the former process and the coalescence rate of the latter are derived. We use these results to prove
a limit law, announced in that earlier paper, concerning the size of the largest square centered at the origin which is of
solid color at a large time t.
Partially supported by the National Science Foundation under Grant DMS-831080
Partially supported by the National Science Foundation under Grant DMS-841317
Partially supported by the National Science Foundation under Grant DMS-830549 相似文献
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Upper bounds on [ ) are derived for those p-functions such thatp() = m is minimum, and p' (+) = q(l-m). 相似文献
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