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arXiv 0907.2843math.PR
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二维接触过程中的渗流相变

Sharpness of the percolation transition in the two-dimensional contact process

J. van den Berg

备注：
Published in at http://dx.doi.org/10.1214/10-AAP702 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)
发表信息：
Annals of Applied Probability 2011, Vol. 21, No. 1, 374-395
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摘要：

对于一大类晶格上的普通（独立）渗流，众所周知，在临界渗流参数$p_c$以下，团簇尺寸分布具有指数衰减，并且这种分布的幂律行为只能发生在$p_c$处。这种行为通常被称为“渗透转变的锐度”。由于理论上的原因，以及应用研究的推动，人们对具有（弱）依赖性的渗流模型越来越感兴趣。例如，生物学家和农业研究人员使用某些二维接触式过程（的静态分布）来模拟干旱景观中的植被模式（见[20]）。在这种情况下，被占用的集群被解释为植被斑块。对于这些模型中的一些，在[20]中报告了计算机模拟表明模型参数的正长度的某个区间内的幂律行为。这意味着在这些模型中，渗流转变并不尖锐。这促使我们研究具有参数$λ$的普通（“基本”）$2D$接触过程的类似问题。我们表明，使用技术从Bollobás和Riordan [8，11]，对于上不变测度${\barν}_λ$的这个过程的渗滤过渡是尖锐的。如果$λ$使得（${\barν}_λ$ -a.s.）不存在无限簇，则对于所有低于$λ$的参数值，簇大小分布具有指数衰减。

英文摘要：

For ordinary (independent) percolation on a large class of lattices it is well known that below the critical percolation parameter $p_c$ the cluster size distribution has exponential decay and that power-law behavior of this distribution can only occur at $p_c$. This behavior is often called ``sharpness of the percolation transition.'' For theoretical reasons, as well as motivated by applied research, there is an increasing interest in percolation models with (weak) dependencies. For instance, biologists and agricultural researchers have used (stationary distributions of) certain two-dimensional contact-like processes to model vegetation patterns in an arid landscape (see [20]). In that context occupied clusters are interpreted as patches of vegetation. For some of these models it is reported in [20] that computer simulations indicate power-law behavior in some interval of positive length of a model parameter. This would mean that in these models the percolation transition is not sharp. This motivated us to investigate similar questions for the ordinary (``basic'') $2D$ contact process with parameter $λ$. We show, using techniques from Bollobás and Riordan [8, 11], that for the upper invariant measure ${\barν}_λ$ of this process the percolation transition is sharp. If $λ$ is such that (${\barν}_λ$-a.s.) there are no infinite clusters, then for all parameter values below $λ$ the cluster-size distribution has exponential decay.

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