Random sections of line bundles over real Riemann surfaces
Year: 2018
Type: preprint
Abstract: Let $\mathcal{L}$ be a positive line bundle over a Riemann surface $Σ$ defined over $\mathbb{R}$. We prove that sections $s$ of $\mathcal{L}^d$, $d\gg 0$, whose number of real zeros $\#Z_s$ deviates from the expected one are rare. We also provide asymptotics of the form $\mathbb{E}[(\#Z_s-\mathbb{E}[\# Z_s])^k]=O(\sqrt{d}^{k-1-α})$ and ${\mathbb{E}[\#Z^k_s]=a_k\sqrt{d}^{k}+b_k\sqrt{d}^{k-1}+O(\sqrt{d}^{k-1-α})}$ for all the (central) moments of the number of real zeros. Here, $α$ is any number in $(0,1)$, and $a_k$ and $b_k$ are some explicit and positive constants.Finally, we obtain similar asymptotics for the distribution of complex zeros of random sections. Our proof involves Bergman kernel estimates as well as Olver multispaces.
Source: arXiv (Cornell University)
Author: Ancona, Michele
Institution: Institut Camille Jordan
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4289016715