论文标题
固定宽度的随机带矩阵的状态和局部特征值统计的密度
The density of states and local eigenvalue statistics for random band matrices of fixed width
论文作者
论文摘要
我们证明，$ d = 1 $随机带宽的局部特征值统计量，例如，高斯条目由泊松点过程给出，我们确定了该过程的强度。证明依赖于Schenker \ cite {Schenker}以及Wegner和Minami估算的定位范围的扩展。这两个估计是使用对角疾病的平均值证明的。新组件是均匀收敛性和状态功能密度平滑度的证明。限制函数已知为具有带宽度依赖性错误的半圆定律\ cite {bmp，dps，dl，mpk}，被识别为限制泊松点过程的强度。这些结果的证明依赖于新的结果，该结果简化并扩展了Dolai，Krishna和Mallick \ cite {dkm}所使用的一些想法。这些作者证明了在本地化制度中随机Schrödinger运算符（晶格和连续体）的状态密度的规律性特性。此处提供的证明适用于由\ cite {dkm}处理的一类无限图上的随机schrödinger算子，并将\ cite {dkm}的结果扩展到具有无界支持的概率度量。该方法还适用于$ d = 2,3 $的固定带宽rbm，提供某些定位范围。
We prove that the local eigenvalue statistics for $d=1$ random band matrices with fixed bandwidth and, for example, Gaussian entries, is given by a Poisson point process and we identify the intensity of the process. The proof relies on an extension of the localization bounds of Schenker \cite{schenker} and the Wegner and Minami estimates. These two estimates are proved using averaging over the diagonal disorder. The new component is a proof of the uniform convergence and the smoothness of the density of states function. The limit function, known to be the semicircle law with a band-width dependent error \cite{bmp,dps,dl,mpk}, is identified as the intensity of the limiting Poisson point process. The proof of these results for the density of states relies on a new result that simplifies and extends some of the ideas used by Dolai, Krishna, and Mallick \cite{dkm}. These authors proved regularity properties of the density of states for random Schrödinger operators (lattice and continuum) in the localization regime. The proof presented here applies to the random Schrödinger operators on a class of infinite graphs treated by in \cite{dkm} and extends the results of \cite{dkm} to probability measures with unbounded support. The method also applies to fixed bandwidth RBM for $d=2,3$ provided certain localization bounds are known.
