Zeros of random polynomials and random partition functions

We will be interested in the distribution of complex zeros of the random polynomials of the form $$P_n(z)=\sum _{k=0}^n{f}_{k,n}{\xi }_k{z}^k,$$ where $f_{k,n}$ are deterministic complex coefficients and ${\xi }_1,{\xi }_2,\dots$ are independent identically distributed random variables such that $\mathbb{E}\log |{\xi }_k|<\mathrm{\infty }$. Under suitable conditions on the coefficients $f_{k,n}$ we will derive a formula for the limiting distribution of complex zeros of $P_n$ as $n\to \mathrm{\infty }$. For example, for the Weyl polynomials of the form $$P_n(z)=\sum _{k=0}^n{\xi }_k\frac{z^k}{\sqrt{k!}},$$ the limiting distribution of zeros is the uniform distribution on the unit disc. This is an analogue of the celebrated circular law for random matrices.

In the last part of the talk we will briefly consider the distribution of the partition function zeros (Lee--Yang--Fisher zeros) for two models of spin glass: the Random Energy Model and the Generalized Random Energy Model.