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# 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).