The diffusivity of supercritical Bernoulli percolation is infinitely differentiable
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866908398931410944 |
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| author | Gu, Chenlin Zhao, Wenhao |
| author_facet | Gu, Chenlin Zhao, Wenhao |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_07158 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The diffusivity of supercritical Bernoulli percolation is infinitely differentiable Gu, Chenlin Zhao, Wenhao Probability Mathematical Physics Analysis of PDEs 35B27, 60K37, 60K35 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. |
| title | The diffusivity of supercritical Bernoulli percolation is infinitely differentiable |
| topic | Probability Mathematical Physics Analysis of PDEs 35B27, 60K37, 60K35 |
| url | https://arxiv.org/abs/2506.07158 |