In a closed manifold of positive dimension $n$, we estimate the expected volume and Euler characteristic for random submanifolds of codimension $r\in \{1,...,n\}$ in two different settings. On one hand, we consider a closed Riemannian manifold and some positive $\lambda$. Then we take $r$ independent random functions in the direct sum of the eigenspaces of the Laplace-Beltrami operator associated to eigenvalues less than $\lambda$ and consider the random submanifold defined as the common zero set of these $r$ functions. We compute asymptotics for the mean volume and Euler characteristic of this random submanifold as $\lambda$ goes to infinity. On the other hand, we consider a complex projective manifold defined over the reals, equipped with an ample line bundle $\mathcal{L}$ and a rank $r$ holomorphic vector bundle $\mathcal{E}$ that are also defined over the reals. Then we get asymptotics for the expected volume and Euler characteristic of the real vanishing locus of a random real holomorphic section of $\mathcal{E}\otimes\mathcal{L}^d$ as $d$ goes to infinity. The same techniques apply to both settings.Comment: Final version, accepted for publication in J. Funct. Anal., 50 pages.A change in notational convention impacts the statement of the main theorems and most formula
Let 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 Zs d in M of a random real holomorphic section s d of E ⊗ L d , where L → X is an ample line bundle and E → X is a rank r Hermitian bundle, r ∈ {1, . . . , n}. We establish the asymptotic of the variance of the linear statistics associated with Zs d , as d goes to infinity. This asymptotic is of order d r− n 2 . As a special case, we get the asymptotic variance of the volume of Zs 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-Itō expansion of the linear statistics associated with the common zero set in RP n of r independent Kostlan-Shub-Smale polynomials.Remark 1.1. If n = r then Z s is a finite subset of M for almost every s. In this case, |dV s | is the sum of the unit Dirac masses on the points of Z s . Let s d be a standard Gaussian vector in RH 0 (X , E ⊗ L d ). Then |dV s d | is a random positive Radon measure on M . We set Z d = Z s d and |dV d | = |dV s d | in order to simplify notations. We are interested in the asymptotic distribution of the linear statisticsAs usual, we denote by E[X] the mathematical expectation of the random vector X. The asymptotic expectation of |dV d | , φ was computed in [19, sect. 5.3].Theorem 1.2 ([19]). Let X be a complex projective manifold of positive dimension n defined over the reals, we assume that its real locus M is non-empty. Let E → X be a rank r ∈ {1, . . . , n} Hermitian vector bundle and let L → X be a positive Hermitian line bundle, both equipped with compatible real structures. For every d ∈ N, let s d be a standard Gaussian vector in RH 0 (X , E ⊗L d ). Then the following holds as d → +∞:The asymptotic variance of |dV d | , φ , as d goes to infinity, was proved to be a O d r− n 2 when the codimension of Z d is r < n (see [20, thm. 1.6]). Our first main theorem (Thm. 1.6 below) extends this result to the maximal codimension case.Statement of the main results. Before we state our main result, let us introduce some more notations. We denote by] the covariance of the real random variables X and Y . Let Var(X) = Cov(X, X) denote the variance of X. Finally, we call variance of |dV d | and we denote by Var(|dV d |) the symmetric bilinear form on C 0 (M ) defined by:Definition 1.3. Let φ ∈ C 0 (M ), we denote by φ its continuity modulus, which is defined by:where ρ g (·, ·) stands for the geodesic distance on (M, g).We denote by M rn (R) the space of matrices of size r × n with real coefficients.Definition 1.4. Let L : V → V be a linear map between two Euclidean spaces, we denote by det ⊥ (L) the Jacobian of L:where L * : V → V is the adjoint operator of L. Similarly, let A ∈ M rn (R), we define its Jacobian to be: det ⊥ (A) = det (AA t ).2 in (1.2) do not dep...
We study the number of real roots of a Kostlan (or elliptic) random polynomial of degree d in one variable. More generally, we are interested in the distribution of the counting measure of the set of real roots of such a polynomial. We compute the asymptotics of the central moments of any order of these random variables, in the large degree limit. As a consequence, we prove that these quantities satisfy a strong Law of Large Numbers and a Central Limit Theorem. In particular, the real roots of a Kostlan polynomial almost surely equidistribute as the degree diverges. Moreover, the fluctuations of the counting measure of this random set around its mean converge in distribution to the Gaussian White Noise. We also prove that the random variables we study concentrate in probability around their mean faster than any negative power of d. More generally, our results hold for the real zeros of a random real section of a line bundle of degree d over a real projective curve, in the complex Fubini-Study model.
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