Quantitative homogenization and large-scale regularity of Poisson point clouds
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866914032717398016 |
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| author | Armstrong, Scott Venkatraman, Raghavendra |
| author_facet | Armstrong, Scott Venkatraman, Raghavendra |
| contents | We prove quantitative homogenization results for harmonic functions on supercritical continuum percolation clusters--that is, Poisson point clouds with edges connecting points which are closer than some fixed distance. We show that, on large scales, harmonic functions resemble harmonic functions in Euclidean space with sharp quantitative bounds on their difference. In particular, for every point cloud which is supercritical (meaning that the intensity of the Poisson process is larger than the critical parameter which guarantees the existence of an infinite connected component), we obtain optimal corrector bounds, homogenization error estimates and large-scale regularity results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_12900 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Quantitative homogenization and large-scale regularity of Poisson point clouds Armstrong, Scott Venkatraman, Raghavendra Probability Analysis of PDEs We prove quantitative homogenization results for harmonic functions on supercritical continuum percolation clusters--that is, Poisson point clouds with edges connecting points which are closer than some fixed distance. We show that, on large scales, harmonic functions resemble harmonic functions in Euclidean space with sharp quantitative bounds on their difference. In particular, for every point cloud which is supercritical (meaning that the intensity of the Poisson process is larger than the critical parameter which guarantees the existence of an infinite connected component), we obtain optimal corrector bounds, homogenization error estimates and large-scale regularity results. |
| title | Quantitative homogenization and large-scale regularity of Poisson point clouds |
| topic | Probability Analysis of PDEs |
| url | https://arxiv.org/abs/2309.12900 |