| Autor: |
Bailleul, I., Dang, N. V., Mouzard, A. |
| Rok vydání: |
2022 |
| Předmět: |
|
| Druh dokumentu: |
Working Paper |
| Popis: |
We consider the continuous Anderson operator $H=\Delta+\xi$ 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. |
| Databáze: |
arXiv |
| Externí odkaz: |
|
