Strict inequalities for the time constant in first passage percolation
Strict inequalities for the time constant in first passage percolation
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DOI：
10.1214/aoap/1031863179
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发表时间：
2002-08
影响因子：
1.8
通讯作者：
R. Marchand
中科院分区：
文献类型：
--
作者：
R. Marchand
在这项工作中，我们感兴趣的是根据通过时间分布在Z2上的第一通道渗透的渐近形状的变化。我们的主要定理扩展了van den Berg和Kesten证明的结果，即当通过时间的分布以某种方式修改时（根据凸阶扩展随机比较），时间常数严格减小。Van den Berg和Kesten的结果要求，当通过时间分布的最小支撑r为严格正时，给予r的质量小于嵌入式定向渗流模型的临界阈值。在二维情况下，我们去掉了这个假设，为了实现这个目标，我们完全确定了当质量r大于临界阈值时出现的平边，作为超临界嵌入定向渗流过程渐近速度的函数，并给出了时间常数的相关上界。
In this work we are interested in the variations of the asymptotic shape in first passage percolation on Z2 according to the passage time distribution. Our main theorem extends a result proved by van den Berg and Kesten, which says that the time constant strictly decreases when the distribution of the passage time is modified in a certain manner (according to a convex order extending stochastic comparison). Van den Berg and Kesten’s result requires, when the minimum r of the support of the passage time distribution is strictly positive, that the mass given to r is less than the critical threshold of an embedded oriented percolation model. We get rid of this assumption in the two-dimensional case, and to achieve this goal, we entirely determine the flat edge occurring when the mass given to r is greater than the critical threshold, as a functional of the asymptotic speed of the supercritical embedded oriented percolation process, and we give a related upper bound for the time constant.