论文标题
在$ \ mathbb {z}^d $上进行关键渗透的两臂指数上的上限
An upper bound on the two-arms exponent for critical percolation on $\mathbb{Z}^d$
论文作者
论文摘要
考虑在$ \ mathbb {z}^d $上使用$ d \ geq 2 $上的关键网站渗透。 Cerf（2015）指出，从Aizenman，Kesten和Newman（1987）和Gandolfi，Grimmett和Russo（1988）的古典作品中，人们可以获得两臂指数至少为$ 1/2 $。 CERF的纸略微改善了下限。 除了$ d = 2 $，对于高$ d $，到目前为止，在文献中似乎没有任何上限（甚至没有意义）。我们表明，距离 - $ n $两臂概率至少为$ c n^{ - （d^2 + 4 d -2）}} $（$ c> 0 $ a candant取决于$ d $），从而为上述指数提供了上限$ d^2 + 4 d -2 $。
Consider critical site percolation on $\mathbb{Z}^d$ with $d \geq 2$. Cerf (2015) pointed out that from classical work by Aizenman, Kesten and Newman (1987) and Gandolfi, Grimmett and Russo (1988) one can obtain that the two-arms exponent is at least $1/2$. The paper by Cerf slightly improves that lower bound. Except for $d=2$ and for high $d$, no upper bound for this exponent seems to be known in the literature so far (not even implicity). We show that the distance-$n$ two-arms probability is at least $c n^{-(d^2 + 4 d -2)}$ (with $c >0$ a constant which depends on $d$), thus giving an upper bound $d^2 + 4 d -2$ for the above mentioned exponent.
