On the chemical distance in critical percolation
On the chemical distance in critical percolation
复制标题
临界渗滤中的化学距离
DOI：
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发表时间：
2015
期刊：
影响因子：
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通讯作者：
Philippe Sosoe
中科院分区：
文献类型：
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作者：
M. Damron;Jack Hanson;Philippe Sosoe
我们考虑二维临界键渗流。条件的存在下，在一个环形的开路，我们表明，最短的开路的预期大小的最内层电路的预期大小的比率趋于零的环形的边长趋于无穷大，纵横比保持固定。同样的证明对于矩形的最低开交叉也得到类似的结果。在这最后一种情况下，我们回答了一个问题的凯斯滕和张显示，此外，该比率的长度最短的交叉长度最低的概率趋于零。这表明临界渗流中的化学距离由严格小于最低路径的指数给出。
We consider two-dimensional critical bond percolation. Conditioned on the existence of an open circuit in an annulus, we show that the ratio of the expected size of the shortest open circuit to the expected size of the innermost circuit tends to zero as the side length of the annulus tends to infinity, the aspect ratio remaining fixed. The same proof yields a similar result for the lowest open crossing of a rectangle. In this last case, we answer a question of Kesten and Zhang by showing in addition that the ratio of the length of the shortest crossing to the length of the lowest tends to zero in probability. This suggests that the chemical distance in critical percolation is given by an exponent strictly smaller than that of the lowest path.