Random sections of line bundles over real Riemann surfaces

Let $\mathcal{L}$ be a positive line bundle over a Riemann surface $\mathrm{\Sigma }$ defined over $\mathbb{R}$. We prove that sections $s$ of ${\mathcal{L}}^d$, $d\gg 0$,whosenumberofrealzeros$#Z_s$ 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}$[#Z^k_s]=a_k\sqrt{d}^{k}+b_k\sqrt{d}^{k-1}+O(\sqrt{d}^{k-1-\alpha})}$forallthe(central)momentsofthenumberofrealzeros\mathrm{.}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.