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A necessary and sufficient condition for global convergence of the complex zeros of random orthogonal polynomials
| Content Provider | Semantic Scholar |
|---|---|
| Author | Dauvergne, Duncan |
| Copyright Year | 2019 |
| Abstract | Consider random polynomials of the form $G_n = \sum_{i=0}^n \xi_i p_i$, where the $\xi_i$ are i.i.d. non-degenerate complex random variables, and $\{p_i\}$ is a sequence of orthonormal polynomials with respect to a regular measure $\tau$ supported on a compact set $K$. We show that the normalized counting measure of the zeros of $G_n$ converges weakly almost surely to the equilibrium measure of $K$ if and only if $\mathbb E \log(1 + |\xi_0|) e^n) = o(n^{-1})$. Our proofs rely on results from small ball probability and exploit the structure of general orthogonal polynomials. Our methods also work for sequences of asymptotically minimal polynomials in $L^p(\tau)$, where $p \in (0, \infty]$. In particular, sequences of $L^p$-minimal polynomials and (normalized) Faber and Fekete polynomials fall into this class. |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | https://arxiv.org/pdf/1901.07614v1.pdf |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
