# Jun Yin: Random band matrices and Anderson’s conjecture in high dimensions

**Time: **
Tue 2022-11-01 15.15 - 16.15

**Location: **
Zoom

**Video link: **
Meeting ID: 698 3346 0369

**Participating: **
Jun Yin (UCLA)

### Abstract

One famous conjecture in quantum chaos and random matrix theory is the so-called phase transition conjecture of random band matrices. It predicts that the eigenvectors’ localization-delocalization transition occurs at some critical bandwidth \(\mathrm{W}_{\mathrm{c}}(\mathrm{d})\), which depends on the dimension \(\mathrm{d}\). The well-known Anderson model and Anderson conjecture have a similar phenomenon. It is widely believed that $W_c(d)$ matches \(1 / \lambda_c(d)\) in the Anderson conjecture, where $\lambda_c(d)$ is the critical coupling constant. Furthermore, this random matrix eigenvector phase transition coincides with the local eigenvalue statistics phase transition, which matches the Bohigas-Giannoni-Schmit conjecture in quantum chaos theory. We proved the eigenvector's delocalization property for most of the general $d>=7$ random band matrix as long as the size of this random matrix does not grow faster than its bandwidth polynomially. In other words, delocalization occurs as long as bandwidth \(\mathrm{W}\) is larger than \(L \varepsilon\), for matrix size L, and some \(\varepsilon>0\).

It is joint work with H.T. Yau (Harvard), C.J. Xu (Harvard), and F. Yang (Upenn).