Quantization of the Hall conductance and delocalization in ergodic Landau Hamiltonians

Non-Ergodic Delocalization in the Rosenzweig-Porter Model

We consider the Rosenzweig-Porter model $H = V + \sqrt{T}\, Φ$, where $V$ is a $N \times N$ diagonal matrix, $Φ$ is drawn from the $N \times N$ Gaussian Orthogonal Ensemble, and $N^{-1} \ll T \ll 1$. We prove that the eigenfunctions of $H$ are typically supported in a set of approximately $NT$ sites, thereby confirming the existence of a previously conjectured non-ergodic delocalized phase. Our proof is based on martingale estimates along the characteristic curves of the stochastic advection equation satisfied by the local resolvent of the Brownian motion representation of $H$.

具有无界随机变量的随机Landau Hamilton算子的离域化

Delocalization for Random Landau Hamiltonians with Unbounded Random Variables

本文证明了在振幅无界的电势随机扰动下Landau哈密顿量的定域/离域跃迁的存在性。特别是，当随机势开启时，朗道能隙的存在概率为1，能隙闭合，局部态填满。在离域区有最小的输运速率。要做到这一点，我们利用霍尔电导的先验量子化和扩展最近的Wegner估计的情况下，无界的随机变量。

$\u210d^d$上Dirichlet截断算子的量子遍历性

Quantum ergodicity for Dirichlet-truncated operators on $\mathbb{Z}^d$

本文证明了$\u210d^d$上邻接矩阵Dirichlet截断的量子遍历性，并推广到有限程观测和周期至多为二的周期Schrödinger算子，部分回答了McKenzie和Sabri提出的问题。

Quantum ergodicity for the Anderson model on regular graphs

We prove a result of delocalization for the Anderson model on the regular tree (Bethe lattice). When the disorder is weak, it is known that large parts of the spectrum are a.s. purely absolutely continuous, and that the dynamical transport is ballistic. In this work, we prove that in such AC regime, the eigenfunctions are also delocalized in space, in the sense that if we consider a sequence of regular graphs converging to the regular tree, then the eigenfunctions become asymptotically uniformly distributed. The precise result is a quantum ergodicity theorem.

特征函数与量子输运及其在Γ-修剪薛定谔算子中的应用

Eigenfunctions and Quantum Transport with Applications to Trimmed Schrodinger Operators

本文利用预解恒等式和Combes-Thomas估计，在解满足增长条件时证明离散薛定谔算子的动力学离域，并将其应用于Γ-修剪薛定谔算子及Anderson模型，得到同时具有局域化与离域现象的随机遍历模型。

Local Limit Theorems for the Random Conductance Model and Applications to the Ginzburg-Landau $\nablaϕ$ Interface Model

We study a continuous-time random walk on $\mathbb{Z}^d$ in an environment of random conductances taking values in $(0,\infty)$. For a static environment, we extend the quenched local limit theorem to the case of a general speed measure, given suitable ergodicity and moment conditions on the conductances and on the speed measure. Under stronger moment conditions, an annealed local limit theorem is also derived. Furthermore, an annealed local limit theorem is exhibited in the case of time-dependent conductances, under analogous moment and ergodicity assumptions. This dynamic local limit theorem…