Elliptic regularity and quantitative homogenization on percolation clusters

Abstract

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, 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 to estimate the yet larger scale on which solutions on the cluster can be well-approximated by harmonic functions on ${\mathbb{R}}^d$. 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 $k\in \mathbb{N}$, the vector space of solutions growing at most like $o(|x{|}^{k+1})$ as $|x|\to \infty$ 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 $k\le 1$ to $k\in \mathbb{N}$.

102 pages, revised version, to appear in Comm. Pure Appl. Math

References in corpus (8)

• Quenched invariance principle for simple random walk on percolation clusters
• Random walks on supercritical percolation clusters
• An optimal error estimate in stochastic homogenization of discrete elliptic equations
• Recent progress on the Random Conductance Model
• Quenched invariance principles for random walks with random conductances
• The additive structure of elliptic homogenization
• Lipschitz regularity for elliptic equations with random coefficients
• Disorder, entropy and harmonic functions

Cited by in corpus (16)

• Quantitative homogenization of the parabolic and elliptic Green's functions on percolation clusters
• An informal introduction to quantitative stochastic homogenization
• Disconnection and entropic repulsion for the harmonic crystal with random conductances
• Optimal corrector estimates on percolation clusters
• Homogenization, linearization and large-scale regularity for nonlinear elliptic equations
• Quantitative nonlinear homogenization: control of oscillations
• Ratio convergence rates for Euclidean first-passage percolation: Applications to the graph infinity Laplacian
• Asymptotic behavior of the Dirichlet energy on Poisson point clouds
• Computing homogenized coefficients via multiscale representation and hierarchical hybrid grids
• Homogenization of Boundary Value Problems in Perforated Lipschitz Domains
• Tiny fluctuations of the averaging process around its degenerate steady state
• Large-Scale Lipschitz Estimates for Elliptic Systems with Periodic High-Contrast Coefficients
• Decay of semigroup for an infinite interacting particle system on continuum configuration spaces
• Homogenization theory of random walks among deterministic conductances
• Scaling limit of the directional conductivity of random resistor networks on simple point processes
• Coupling between Brownian motion and random walks on the infinite percolation cluster