Variance of the volume of random real algebraic submanifolds II

Abstract

Let $\mathcal{X}$ be a complex projective manifold of dimension $n$ defined over the reals and let $M$ be its real locus. We study the vanishing locus $Z\_s\_d$ in $M$ of a random real holomorphic section $s\_d$ of $\mathcal{E}\otimes {\mathcal{L}}^d$, where $\mathcal{L}\to \mathcal{X}$ is an ample line bundle and $\mathcal{E}\to \mathcal{X}$ is a rank $r$ Hermitian bundle, $r\in \{1,\dots ,n\}$. We establish the asymptotic of the variance of the linear statistics associated with $Z\_s\_d$, as $d$ goes to infinity. This asymptotic is of order $d^{r-\frac{n}{2}}$. As a special case, we get the asymptotic variance of the volume of $Z\_s\_d$. The present paper extends the results of [20], by the first-named author, in essentially two ways. First, our main theorem covers the case of maximal codimension ($r=n$), which was left out in [20]. And second, we show that the leading constant in our asymptotic is positive. This last result is proved by studying the Wiener--It{ō} expansion of the linear statistics associated with the common zero set in ${\mathbb{RP}}^n$ of $r$ independent Kostlan--Shub--Smale polynomials.

Final version, published in Indiana Math. Univ. J

References in corpus (2)

• Phase Singularities in Complex Arithmetic Random Waves
• Expected number of real roots for random linear combinations of orthogonal polynomials associated with radial weights

Cited by in corpus (2)

• Zeros of smooth stationary Gaussian processes
• Universality of the nodal length of bivariate random trigonometric polynomials