| Abstract: |
| Consider two generalizations of the famous Anderson model defined on a $d$-dimensional integer lattice of linear size $L$. The first generalization is the random band matrix model. In this model, the entries are independent centered complex Gaussian random variables, and the element $H_{xy}$ is nonzero only when the distance $|x-y|$ is less than the band width $W$. The second generalization is the block Anderson model. In this model, the i.i.d. diagonal potential in the Anderson model is replaced by an i.i.d. diagonal block potential with a coupling strength parameter $\lambda>0$, and the blocks have a linear size of $W$. Both models are non-mean-field models, where the parameter $W$ describes the length of local interactions. Furthermore, it is conjectured that these models exhibit Anderson transitions as $W$ or $\lambda$ varies.
In this talk, I will present some of our recent results on the Anderson delocalization of these two models when $d\ge 7$ and $W\ge L^\delta$, where $\delta>0$ is a small constant. Additionally, I will discuss the quantum diffusion conjecture related to the delocalization of these models. The research is based on joint works with Changji Xu, Horng-Tzer Yau, and Jun Yin. |
|