Systematics of spectral shifts in random matrix ensembles

Systematics of spectral shifts in random matrix ensembles

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Uzy Smilansky, Weizmann Institute

A joint Princeton - Rutgers seminar

Zoom link:

https://princeton.zoom.us/j/91814557028

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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 $\tau < \tau_{max} (\beta)$.