Random Sections of Line Bundles Over Real Riemann Surfaces

Abstract

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.