Random Sections of Line Bundles Over Real Riemann Surfaces

Abstract

Let L be a positive line bundle over a Riemann surface Σ defined over R. We prove that sections s of L d , d ≫ 0, whose number of real zeros #Zs deviates from the expected one are rare. We also provide asymptotics of the form

Zeros of smooth stationary Gaussian processes

“…To conclude this section, let us mention that the setting of the present paper is related with that of [3,4,21,29]. Indeed, the Bargmann-Fock process introduced previously is the universal local scaling limit, as d → +∞, of a random section of degree d in the complex Fubini-Study ensemble.…”

“…However, note that Theorems 1.14 and 1.13 are more precise than their counterparts in [3]. For example, [3,Theorem 5.7] says that the k-point function ρ k vanishes along the diagonal ∆ k , while in Theorem 1.13 we also compute the vanishing order of ρ k along the diagonal ∆ k , also giving conditions on the process f for which this vanishing order is sharp. As explained in the last paragraph of Section 1.5, one of the fundamental parts of studying the k-point function is finding good expression for ρ k (x), depending on how the components of x are clustered.…”

“…Its proof relies on results of Ancona, who proved the counterpart of Theorem 1.14 in [3, Theorems 4.1 and 5.7 and Proposition 4.2]. However, note that Theorems 1.14 and 1.13 are more precise than their counterparts in [3]. For example, [3,Theorem 5.7] says that the k-point function ρ k vanishes along the diagonal ∆ k , while in Theorem 1.13 we also compute the vanishing order of ρ k along the diagonal ∆ k , also giving conditions on the process f for which this vanishing order is sharp.…”

“…As explained in the last paragraph of Section 1.5, one of the fundamental parts of studying the k-point function is finding good expression for ρ k (x), depending on how the components of x are clustered. The expressions used in the present article are different from those used in [3] (one should compare the expression appearing in Definition 6.14 with the one in [3,Proposition 5.21]). The new expressions used in the present paper turn out to be more precise for estimating ρ k along the diagonal.…”

“…The new expressions used in the present paper turn out to be more precise for estimating ρ k along the diagonal. The results of [3,4] apply to the number of real roots of a Kostlan polynomial of degree d, see [24]. In this case, the variance asymptotics and the Central Limit Theorem were proved by Dalmao [17].…”