Title: Delocalization of two-dimensional random band matrices
Abstract: The delocalization-localization transition for random operators has been a central question in probability theory and mathematical physics since the introduction of the Anderson model, a discrete random Schr\"{o}dinger operator. A popular model conjectured to exhibit similar behavior to the random Schr\"{o}dinger operator is the d-dimensional random band matrix. In this talk, I will discuss recent results on the delocalization side of the transition in dimension two. We consider a random band matrix $H=(H_{xy})_{x,y}$ of dimension $N\times N$ with mean-zero complex Gaussian entries, where $x,y$ belong to the discrete torus $(\Z/\sqrt{N}\Z)^{2}$. The variance profile $\E|H_{xy}|^{2}=S_{xy}$ vanishes when $|x-y|\geq W$ for some band-width parameter $W$ depending on $N$. We show that if the band-width satisfies $W\geq N^\varepsilon$ for some $\varepsilon>0$, then in the large-$N$ limit the bulk eigenvectors of $H$ are delocalized.
Joint work with Kevin Yang, Horng-Tzer Yau, Jun Yin.