##### Department of Mathematics,

University of California San Diego

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### Math 288 - Probability Seminar

## Yuriy Nemish

#### UCSD

## Spectral properties of polynomials and rational functions of Wigner matrices

##### Abstract:

We consider general self-adjoint polynomials and rational expressions in several independent random matrices whose entries are centered and have constant variance. Under some numerically checkable conditions, we establish for these models the optimal local law, i.e., we show that the empirical spectral distribution on scales just above the eigenvalue spacing follows the global density of states which is determined by free probability theory. We show that the above results can be applied to prove the optimal bulk local law for two concrete families of polynomials: general quadratic forms in Wigner matrices and symmetrized products of independent matrices with i.i.d. entries. Moreover, in the framework of the developed theory for rational expressions in random matrices, we study the density of transmission eigenvalues in the random matrix model for transport in quantum dots coupled to a chaotic environment. This is a joint work with Laszlo Erd$\ddot{\text{o}}$s and Torben Kr$\ddot{\text{u}}$ger.

### October 10, 2019

### 11:00 AM

### AP&M 6402

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