Fluctuations in the weakly coupled 4D Anderson Hamiltonian
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866908853352792064 |
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| author | Gabriel, Simon Rosati, Tommaso |
| author_facet | Gabriel, Simon Rosati, Tommaso |
| contents | We study the weak coupling limit of the Anderson Hamiltonian in the critical dimension $d=4$. In a perturbative sense, we prove Gaussian fluctuations about the Green's function of the Laplacian. The fluctuations are described by an explicit effective variance, up to a critical value of the coupling constant at which we expect a phase transition in the structure of the fluctuations. The proof is based on a combinatorial analysis of Feynman diagrams, and on a detailed study of the BPHZ renormalisation of the model. We characterise the limiting distribution in terms of primitive blow-ups, and prove that no Laplacian renormalisation is present. Our approach seems applicable to a broad class of equations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_22509 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Fluctuations in the weakly coupled 4D Anderson Hamiltonian Gabriel, Simon Rosati, Tommaso Probability Analysis of PDEs 60H15, 35R60 We study the weak coupling limit of the Anderson Hamiltonian in the critical dimension $d=4$. In a perturbative sense, we prove Gaussian fluctuations about the Green's function of the Laplacian. The fluctuations are described by an explicit effective variance, up to a critical value of the coupling constant at which we expect a phase transition in the structure of the fluctuations. The proof is based on a combinatorial analysis of Feynman diagrams, and on a detailed study of the BPHZ renormalisation of the model. We characterise the limiting distribution in terms of primitive blow-ups, and prove that no Laplacian renormalisation is present. Our approach seems applicable to a broad class of equations. |
| title | Fluctuations in the weakly coupled 4D Anderson Hamiltonian |
| topic | Probability Analysis of PDEs 60H15, 35R60 |
| url | https://arxiv.org/abs/2602.22509 |