The diffusivity of supercritical Bernoulli percolation is infinitely differentiable

TL;DR

This paper proves that in supercritical Bernoulli percolation on integer lattices, the diffusivity and conductivity are infinitely differentiable functions, extending previous results and introducing new techniques for higher-order perturbations.

Contribution

It establishes the infinite differentiability of diffusivity and conductivity in supercritical Bernoulli percolation, using novel methods like cluster-growth decomposition and hole separation.

Findings

Diffusivity and conductivity are infinitely differentiable in supercritical regime.

Introduces new techniques for higher-order perturbation analysis in percolation.

Provides uniform estimates for finite-volume derivative approximations.

Abstract

We prove that, the diffusivity and conductivity on $\mathbb{Z}^d$-Bernoulli percolation ($d \geq 2$) are infinitely differentiable in supercritical regime. This extends a result by Kozlov [Uspekhi Mat. Nauk 44 (1989), no. 2(266), pp 79 - 120]. The key to the proof is a uniform estimate for the finite-volume approximation of derivatives, which relies on the perturbed corrector equations in homogenization theory. The renormalization of geometry is then implemented in a sequence of scales to gain sufficient degrees of regularity. To handle the higher-order perturbation on percolation, new techniques, including cluster-growth decomposition and hole separation, are developed.