# Systematics of spectral shifts in random matrix ensembles

# Systematics of spectral shifts in random matrix ensembles

A joint Princeton - Rutgers seminar

**Zoom link:****https://princeton.zoom.us/j/91814557028**

**Password: 058258**

We chose a $N\times N$ Hermitian matrix randomly picked from one of the random Gaussian matrix ensembles $(\beta =1,2,4)$ - the reference matrix. Perturbing it with a sequence of rank $t$ matrices, with $t$ taking the values $1\le t \le N$, we study the expected difference between the spectra of the perturbed and the reference matrices as a function of $t$, and its dependence on the random matrix ensemble and the kind of rank $t$ perturbations. In particular, we consider a "mild" perturbation which either permutes or randomizes $t$ \emph{diagonal} elements. We derive universal expressions in the scaled parameter $\tau =t/N$ for the expectation of the variance of the spectral shift functions, and show that it is proportional to $\tau^{\frac{1}{2}}$ (subdifusive ) in the limit of large $N$ and $t$, and fixed ratio $\tau = t/N$ as long as $\tau < \tau_{max} (\beta)$.