Premium
Elliptic Regularity and Quantitative Homogenization on Percolation Clusters
Author(s) -
Armstrong Scott,
Dario Paul
Publication year - 2018
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.21726
Subject(s) - mathematics , homogenization (climate) , euclidean space , mathematical analysis , renormalization , exponential function , percolation threshold , elliptic operator , statistical physics , pure mathematics , mathematical physics , quantum mechanics , physics , electrical resistivity and conductivity , biodiversity , ecology , biology
We establish quantitative homogenization, large‐scale regularity, and Liouville results for the random conductance model on a supercritical (Bernoulli bond) percolation cluster. The results are also new in the case that the conductivity is constant on the cluster. The argument passes through a series of renormalization steps: first, we use standard percolation results to find a large scale above which the geometry of the percolation cluster behaves (in a sense, made precise) like that of euclidean space. Then, following the work of Barlow [8], we find a succession of larger scales on which certain functional and elliptic estimates hold. This gives us the analytic tools to adapt the quantitative homogenization program of Armstrong and Smart [7] to estimate the yet larger scale on which solutions on the cluster can be well‐approximated by harmonic functions on ℝ d . This is the first quantitative homogenization result in a porous medium, and the harmonic approximation allows us to estimate the scale on which a higher‐order regularity theory holds. The size of each of these random scales is shown to have at least a stretched exponential moment. As a consequence of this regularity theory, we obtain a Liouville‐type result that states that, for each k ∊ ℕ, the vector space of solutions growing at most like o (| x | k +1 ) as | x | → ∞ has the same dimension as the set of harmonic polynomials of degree at most k , generalizing a result of Benjamini, Duminil‐Copin, Kozma, and Yadin from k ≤ 1 to k ∊ ℕ. © 2018 Wiley Periodicals, Inc.