Quantitative homogenization for the critical long-range random conductance model
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866917430421356544 |
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| author | Bou-Rabee, Ahmed Dario, Paul |
| author_facet | Bou-Rabee, Ahmed Dario, Paul |
| contents | We consider the long-range random conductance model on $\mathbb{Z}^d$ at the critical exponent: the jump rate between sites $x$ and $y$ decays as $\mathbf{a}(x,y) |x-y|^{-(d+2)}$, where $\mathbf{a}(x,y)$ are i.i.d. uniformly elliptic conductances. Below the critical exponent $(d+2)$ the walk converges to a stable process; above it, to Brownian motion with diffusive $\sqrt{t}$ scaling. At criticality the second moment of the jump kernel diverges logarithmically. We establish quantitative homogenization of the associated elliptic equation to the Laplacian at the rate $1/\sqrt{|\ln\varepsilon|}$. As a consequence, we deduce quenched convergence of the random walk to Brownian motion under the anomalous $\sqrt{t \log t}$ scaling. Unlike in standard homogenization, the effective diffusivity is determined by the mean conductance alone, with no corrector contribution at leading order. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_21143 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Quantitative homogenization for the critical long-range random conductance model Bou-Rabee, Ahmed Dario, Paul Probability Analysis of PDEs We consider the long-range random conductance model on $\mathbb{Z}^d$ at the critical exponent: the jump rate between sites $x$ and $y$ decays as $\mathbf{a}(x,y) |x-y|^{-(d+2)}$, where $\mathbf{a}(x,y)$ are i.i.d. uniformly elliptic conductances. Below the critical exponent $(d+2)$ the walk converges to a stable process; above it, to Brownian motion with diffusive $\sqrt{t}$ scaling. At criticality the second moment of the jump kernel diverges logarithmically. We establish quantitative homogenization of the associated elliptic equation to the Laplacian at the rate $1/\sqrt{|\ln\varepsilon|}$. As a consequence, we deduce quenched convergence of the random walk to Brownian motion under the anomalous $\sqrt{t \log t}$ scaling. Unlike in standard homogenization, the effective diffusivity is determined by the mean conductance alone, with no corrector contribution at leading order. |
| title | Quantitative homogenization for the critical long-range random conductance model |
| topic | Probability Analysis of PDEs |
| url | https://arxiv.org/abs/2604.21143 |