Analysis of the Anderson operator
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866908343272996864 |
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| author | Bailleul, I. Dang, N. V. Mouzard, A. |
| author_facet | Bailleul, I. Dang, N. V. Mouzard, A. |
| contents | We consider the continuous Anderson operator $H=Δ+ξ$ on a two dimensional closed Riemannian manifold $\mathcal{S}$. We provide a short self-contained functional analysis construction of the operator as an unbounded operator on $L^2(\mathcal{S})$ and give almost sure spectral gap estimates under mild geometric assumptions on the Riemannian manifold. We prove a sharp Gaussian small time asymptotic for the heat kernel of $H$ that leads amongst others to strong norm estimates for quasimodes. We introduce a new random field, called Anderson Gaussian free field, and prove that the law of its random partition function characterizes the law of the spectrum of $H$. We also give a simple and short construction of the polymer measure on path space and relate the Wick square of the Anderson Gaussian free field to the occupation measure of a Poisson process of loops of polymer paths. We further prove large deviation results for the polymer measure and its bridges. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2201_04705 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Analysis of the Anderson operator Bailleul, I. Dang, N. V. Mouzard, A. Probability Mathematical Physics Analysis of PDEs 35P15 35R60 60H15 58J50 58C40 58J65 81T16 We consider the continuous Anderson operator $H=Δ+ξ$ on a two dimensional closed Riemannian manifold $\mathcal{S}$. We provide a short self-contained functional analysis construction of the operator as an unbounded operator on $L^2(\mathcal{S})$ and give almost sure spectral gap estimates under mild geometric assumptions on the Riemannian manifold. We prove a sharp Gaussian small time asymptotic for the heat kernel of $H$ that leads amongst others to strong norm estimates for quasimodes. We introduce a new random field, called Anderson Gaussian free field, and prove that the law of its random partition function characterizes the law of the spectrum of $H$. We also give a simple and short construction of the polymer measure on path space and relate the Wick square of the Anderson Gaussian free field to the occupation measure of a Poisson process of loops of polymer paths. We further prove large deviation results for the polymer measure and its bridges. |
| title | Analysis of the Anderson operator |
| topic | Probability Mathematical Physics Analysis of PDEs 35P15 35R60 60H15 58J50 58C40 58J65 81T16 |
| url | https://arxiv.org/abs/2201.04705 |