Random sections of line bundles over real Riemann surfaces

TL;DR

This paper studies the distribution of real and complex zeros of random sections of positive line bundles over real Riemann surfaces, showing that deviations from expected zeros are rare and providing asymptotic formulas for moments.

Contribution

It establishes new asymptotic formulas for the moments and distribution of zeros of random sections over real Riemann surfaces, using Bergman kernel estimates and Olver multispaces.

Findings

Deviations in the number of real zeros are rare for large degree d.

Asymptotic formulas for moments of the number of real zeros are derived.

Similar asymptotics are obtained for the distribution of complex zeros.

Abstract

Let $\mathcal{L}$ be a positive line bundle over a Riemann surface $\Sigma$ 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 $\mathbb{E}[(\#Z_s-\mathbb{E}[\# Z_s])^k]=O(\sqrt{d}^{k-1-\alpha})$ and ${\mathbb{E}[\#Z^k_s]=a_k\sqrt{d}^{k}+b_k\sqrt{d}^{k-1}+O(\sqrt{d}^{k-1-\alpha})}$ for all the (central) moments of the number of real zeros. Here, $\alpha$ 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.