A Quantitative Central Limit Theorem for the Effective Conductance on the Discrete Torus

作者：Gloria Antoine^*; Nolen James

来源：Communications on Pure and Applied Mathematics, 2016, 69(12): 2304-2348.

DOI：10.1002/cpa.21614

We study a random conductance problem on a d-dimensional discrete torus of size L>0. The conductances are independent, identically distributed random variables uniformly bounded from above and below by positive constants. The effective conductance A(L) of the network is a random variable, depending on L, that converges almost surely to the homogenized conductance A(hom). Our main result is a quantitative central limit theorem for this quantity as L. In particular, we prove there exists some sigma>0 such that
d(K) (L-d/2 A(L) - A(hom)/sigma, G) less than or similar to L-d/2 log(d) L.
where d(K) is the Kolmogorov distance and G is a standard normal variable. The main achievement of this contribution is the precise asymptotic description of the variance of A(L).

• 出版日期2016-12
• 单位INRIA