论文标题
安德森操作员的分析
Analysis of the Anderson operator
论文作者
论文摘要
我们考虑连续的安德森操作员$ h =δ+ξ$在二维封闭的Riemannian歧管$ \ Mathcal {S} $上。我们在$ l^2（\ Mathcal {s}）上提供了一个简短的独立功能分析构建，作为无限运算符，并在Riemannian歧管上的轻度几何假设下给出了几乎确定的光谱差距估计值。我们证明，对于$ h $的加热核是一个急剧的高斯小时渐近，这导致了对准莫丁的高规范估计。我们引入了一个新的随机字段，称为Anderson Gaussian Free Field，并证明其随机分区功能的定律是$ h $的频谱定律的特征。我们还对路径空间上的聚合物度量进行了简单而简短的结构，并将安德森高斯自由场的灯芯正方形与聚合物路径环的泊松过程的占用度量相关联。我们进一步证明了聚合物度量及其桥梁的较大偏差结果。
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.
