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A Quantitative Central Limit Theorem for the Effective Conductance on the Discrete Torus
Author(s) -
Gloria Antoine,
Nolen James
Publication year - 2016
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.21614
Subject(s) - conductance , central limit theorem , torus , mathematics , bounded function , independent and identically distributed random variables , limit (mathematics) , random variable , variable (mathematics) , combinatorics , discrete mathematics , mathematical analysis , statistics , geometry
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 σ > 0 such that [ d K ( L d / 2A L − A homσ , G ) ≲ L − d / 2log 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 .© 2015 Wiley Periodicals, Inc.