Probability and Mathematical Physics

Date: October 7, 2022

Time: 09:00AM - 10:00AM

Location: Zoom

Speaker: Mostafa Sabri, Cairo University

Title: Quantum ergodicity for periodic graphs

Abstract: Quantum ergodicity for graphs is a delocalization result which says the following. Suppose that a sequence of finite graphs $Gamma_N$ converges to some infinite graph $\Gamma$. Then most eigenfunctions $\psi_j^{(N)}$ of the adjacency matrix $\mathcal{A}_N$ on $\Gamma_N$ become equidistributed on $\Gamma_N$ when $N$ gets large. More precisely, for most $j$, the probability measure $\sum_{v\in \Gamma_N} |\psi_j^{(N)}(v)|^2 \delta_v$ approaches the uniform measure $\frac{1}{|\Gamma_N|}\sum_{v\in \Gamma_N} \delta_v$, in a weak sense. Potentials $Q$ can sometimes be added, so that one now considers the eigenfunctions of $H_N = \mathcal{A}_N+Q_N$. Usually, the proof partially relies on certain nice properties of the infinite graph $\Gamma$. In particular, quantum ergodicity theorems have previously been established when $\Gamma$ is a tree. In this talk, I will present recent results of quantum ergodicity when $\Gamma$ is invariant under translations of some basis of $\mathbb{Z}^d$, and the ``fundamental block'' is endowed a potential Q which is copied across the blocks, so that $H = \mathcal{A}_\Gamma+Q$ is a periodic Schr\"odinger operator. This framework includes $\Gamma=\mathbb{Z}^d$, the honeycomb lattice, strips, cylinders, etc. I will first discuss the Bloch theorem and give some examples of its limitations, presenting along the way some very homogeneous graphs which violate quantum ergodicity. I will then discuss our main results, contrasting them with the tree case, give various examples of applications, and sketch the proof. An open problem concerning Schr\"odinger operators with a periodic potential on $\mathbb{Z}^d$, $d>1$, will also be presented. This talk is based on a joint work with Theo McKenzie (Harvard).