Equilibrium distribution of zeros of random polynomials
| dc.creator | Shiffman, Bernard | |
| dc.creator | Zelditch, Steve | |
| dc.date | 2002-06-17 | |
| dc.date.accessioned | 2026-07-07T06:24:32Z | |
| dc.date.available | 2026-07-07T06:24:32Z | |
| dc.description | We consider ensembles of random polynomials of the form $p(z)=\sum_{j = 1}^N a_j P_j$ where $\{a_j\}$ are independent complex normal random variables and where $\{P_j\}$ are the orthonormal polynomials on the boundary of a bounded simply connected analytic plane domain $Ω\subset C$ relative to an analytic weight $ρ(z) |dz|$. In the simplest case where $Ω$ is the unit disk and $ρ=1$, so that $P_j(z) = z^j$, it is known that the average distribution of zeros is the uniform measure on $S^1$. We show that for any analytic $(Ω, ρ)$, the zeros of random polynomials almost surely become equidistributed relative to the equilibrium measure on $\partialΩ$ as $N\to\infty$. We further show that on the length scale of 1/N, the correlations have a universal scaling limit independent of $(Ω, ρ)$. | |
| dc.description | 19 pages, 3 figures | |
| dc.identifier | https://arxiv.org/abs/math/0206162 | |
| dc.identifier | http://arxiv.org/abs/math/0206162 | |
| dc.identifier | Int. Math. Res. Not. 2003, 25-49. | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/96576 | |
| dc.subject | Complex Variables | |
| dc.subject | Probability | |
| dc.title | Equilibrium distribution of zeros of random polynomials | |
| dc.type | text |