The diffusivity of supercritical Bernoulli percolation is infinitely differentiable

2025/06/08 by Chenlin Gu, Gu, Chenlin, Wenhao Zhao +1
Computer Science · Mathematics · Physics and Astronomy · #35B27 #60K35 #60K37 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics

paper · pdf · doi:10.48550/arxiv.2506.07158

openalex publication_date 2025/06/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that, the diffusivity and conductivity on ℤ^d-Bernoulli percolation (d ≥ 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.

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