Bibliographic Details
| Title: |
Expected number of nodal components for cut‐off fractional Gaussian fields. |
| Authors: |
Rivera, Alejandro |
| Superior Title: |
Journal of the London Mathematical Society; Jun2019, Vol. 99 Issue 3, p629-652, 24p |
| Subject Terms: |
BETTI numbers, EULER characteristic, RIEMANNIAN manifolds, RANDOM variables, EIGENVALUES, MESHFREE methods, EIGENFUNCTIONS |
| Abstract: |
Let (X,g) be a closed Riemannian manifold of dimension n>0. Let Δ be the Laplacian on X, and let (ek)k be an L2‐orthonormal and dense family of Laplace eigenfunctions with respective eigenvalues (λk)k. We assume that (λk)k is non‐decreasing and that the ek are real valued. Let (ξk)k be a sequence of independent and identically distributed N(0,1) random variables. For each L>0 and s∈R, possibly negative, set fLs=∑0<λj⩽Lλj−s2ξjej. Then, fLs is almost surely regular on its zero set. Let NL be the number of connected components of its zero set. If s0 such that NL∼νVolg(X)Ln/2 in L1 and almost surely. In particular, E[NL]≍Ln/2. On the other hand, we prove that if s=n2, then E[NL]≍Ln/2lnL1/2.In the latter case, we also obtain an upper bound for the expected Euler characteristic of the zero set of fLs and for its Betti numbers. In the case s>n/2, the pointwise variance of fLs converges so it is not expected to have universal behavior as L→+∞. [ABSTRACT FROM AUTHOR] |
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| Database: |
Complementary Index |
