Quantitative homogenization of the parabolic and elliptic Green's functions on percolation clusters

Abstract

We study the heat kernel and the Green's function on the infinite supercritical percolation cluster in dimension d ≥ 2 and prove a quantitative homogenization theorem for these functions with an almost optimal rate of convergence. These results are a quantitative version of the local central limit theorem proved by Barlow and Hambly in [23]. The proof relies on a structure of renormalization for the infinite percolation cluster introduced in [12], Gaussian bounds on the heat kernel established by Barlow in [21] and tools of the theory of quantitative stochastic homogenization. An important step in the proof is to establish a C 0,1 -large-scale regularity theory for caloric functions on the infinite cluster and is of independent interest.

Quantitative homogenization of interacting particle systems

“…Our strategy is inspired by recent developments in the quantitative analysis of elliptic equations with random coefficients, and in particular on the renormalization approach developed in [14,13,9,10,11,7,8]; see also [58] for a gentle introduction, and [59,56,38,39,36,40,37] for another approach based on concentration inequalities. This renormalization approach has shown its versatility in a number of other settings, covering now the homogenization of parabolic equations [5], finite-difference equations on percolation clusters [6,24,26], differential forms [25], the "∇φ" interface model [23,12], and the Villain model [27].…”

Decay of semigroup for an infinite interacting particle system on continuum configuration spaces

“…We refer to the excellent monographs [32,33,34,42] for a panorama of the field. In recent years, many works in probability and stochastic processes illustrate the diffusion universality in various models: a well-understood model is the random conductance model, see [14] for a survey, and especially the heat kernel bound and invariance principle is established for the percolation clusters in [13,37,41,36,11,12,40]; from the view point of stochastic homogenization, the quantitative results are also proved in a series of work [9,6,10,7,26,27,23,24,25], and the monograph [8], and these techiques also apply on the percolation clusters setting, as shown in [5,19,28,20]; for the system of hard-spheres, Bodineau, Gallagher and Saint-Raymond prove that Brownian motion is the Boltzmann-Grad limit of a tagged particle in [16,15,17]. All these works make us believe that the model in this work should also have diffusive behavior in large scale or long time.…”

Non-uniformly parabolic equations and applications to the random conductance model

“…The main achievement of Lemma 2 is estimate (24), where at the expense of increasing the domain of integration we control u α in terms of u να with ν < 1. The factor on the right-hand side involving norm of u α (and not u να ) to a small power will be dealt with later.…”

“…Estimate (26) follows directly from ( 27) and (32). To show (24), we combine ( 27) and ( 29) to obtain…”

“…A quantitative quenched invariance principle (under quantified ergodicity assumptions) with degenerate conductances can be found in [6]. Very recently, building on [8], an almost optimal quantitative local limit theorem in the percolation setting was proven by Dario and Gu [24]. Further results in the stationary & ergodic setting under moment conditions include large-scale regularity [14], homogenization in the sense of Γ-convergence [34], or spectral homogenization [25].…”