Random walk on the incipient infinite cluster for oriented percolation in high dimensions
Random walk on the incipient infinite cluster for oriented percolation in high dimensions
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DOI：
10.1007/s00220-007-0410-4
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发表时间：
2008-03-01
影响因子：
2.4
通讯作者：
Slade, Gordon
中科院分区：
文献类型：
--
作者：
Barlow, Martin T.;Jarai, Antal A.;Slade, Gordon
本文考虑Z（d）x Z（+）上定向渗流扩散模型的初始无限团簇上的简单随机游动。在d > 6维的情况下，我们得到了退出时间、转移概率和随机游走范围的界，这确定了初始无限簇的谱维数是4/3，从而证明了亚历山大-奥巴赫猜想在这种情况下的一个版本。证明分为两部分。第一部分给出了任意无限随机图上简单随机游动的一般估计，给出了随机图的体积和有效电阻的适当界。第二部分，然后提供这些边界上的体积和有效阻力的初始无限簇在尺寸d > 6，通过扩展的结果，通过花边扩展先前获得的临界定向渗流。
We consider simple random walk on the incipient infinite cluster for the spread-out model of oriented percolation on Z(d) x Z(+). In dimensions d > 6, we obtain bounds on exit times, transition probabilities, and the range of the random walk, which establish that the spectral dimension of the incipient infinite cluster is 4/3, and thereby prove a version of the Alexander-Orbach conjecture in this setting. The proof divides into two parts. One part establishes general estimates for simple random walk on an arbitrary infinite random graph, given suitable bounds on volume and effective resistance for the random graph. A second part then provides these bounds on volume and effective resistance for the incipient infinite cluster in dimensions d > 6, by extending results about critical oriented percolation obtained previously via the lace expansion.