Random sections of line bundles over real Riemann surfaces

Roots of Kostlan polynomials: moments, strong Law of Large Numbers and Central Limit Theorem

“…These estimates still hold if we replace L d by E ⊗ L d , where E is any fixed real Hermitian line bundle (see [21,Theorem 4.2.1]). Thus, all the results in [2] are still valid for random real sections of E ⊗ L d → X .…”

“…[16]) that for all d ∈ N, the average number of roots of this random polynomial is E[Card(Z d )] = d 1 2 , where Card(Z d ) is the cardinality of Z d . It was later proved by Dalmao (see [10]) that Var(Card(Z d )) ∼ σ 2 d 1 2 as d → +∞, where σ is some explicit positive constant. Dalmao also proved that Card(Z d ) satisfies a Central Limit Theorem as d → +∞.…”

“…We denote by Z d = Z s d and by ν d = ν s d for simplicity. In this setting, the linear statistics of ν d where studied in [2,13,18,20], among others. In particular, the exact asymptotics of their expected values and their variances are known.…”

“…) in Theorem 1.3 can be improved (see [20,Theorem 1.6]). In [2], Ancona derived a two terms asymptotic expansion of the non-central moments of the linear statistics of ν d . As a consequence, he proved the following (cf.…”

“…In Ancona's paper the line bundle E is trivial. The results of [2] rely on precise estimates for the Bergman kernel of L d . These estimates still hold if we replace L d by E ⊗ L d , where E is any fixed real Hermitian line bundle (see [21,Theorem 4.2.1]).…”