随机电网络有效电阻的次均值方差界
Submean variance bound for effective resistance of random electric networks
摘要：
本文研究了一类具有伯努利电阻的随机电网络模型。在格Z ^2的情况下，我们证明了0和顶点v之间的点到点有效电阻的方差至多为（log |v|）^（2/3）阶，而当v趋于无穷大时，其期望值为log |v|阶。当Z^d的维数不等于2时，期望和方差是同阶的。在p-电阻的情况下也得到了类似的结果。证明依赖于一个修改的庞加莱不等式由于Falik和Samorodnitsky。
英文摘要：
We study a model of random electric networks with Bernoulli resistances. In the case of the lattice Z^2, we show that the point-to-point effective resistance between 0 and a vertex v has a variance of order at most (log |v|)^(2/3) whereas its expected value is of order log |v|, when v goes to infinity. When the dimension of Z^d is different than 2, expectation and variance are of the same order. Similar results are obtained in the context of p-resistance. The proofs rely on a modified Poincare inequality due to Falik and Samorodnitsky.
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