Abstract
Random plane wave is conjectured to be a universal model for high-energy eigenfunctions of the Laplace operator on generic compact Riemanian manifolds. This is known to be true on average. In the present paper we discuss one of important geometric observable: critical points. We first compute one-point function for the critical point process, in particular we compute the expected number of critical points inside any open set. After that we compute the short-range asymptotic behaviour of the two-point function. This gives an unexpected result that the second factorial moment of the number of critical points in a small disc scales as the fourth power of the radius.
Highlights
Introduction and Main Results1.1 Random Gaussian functionsStudying the Laplace eigenfunctions and their geometry is a classical subject going back to at least XIX century
For a generic chaotic domain it was conjectured by Berry [3] that the high energy functions behave like a random superposition of monochromatic plane waves propagating in different directions, usually referred to as the random plane wave, rigorously defined below
For a compact Riemannian manifold M we can consider an orthonormal basis of eigenfunctions φi satisfying φi + ti2φi = 0 with t0 ≤ t1 ≤ . . . , and define the band-limited functions fT =
Summary
Studying the Laplace eigenfunctions and their geometry is a classical subject going back to at least XIX century. It is easy to evaluate ψ explicitly as ψ(z, w) = J0(k|z − w|), where J0 is the Bessel J function of order 0 From this representation it follows that is stationary (i.e., its law is translation invariant), and isotropic, (i.e., its law is invariant under rotations); by the standard abuse of notation we write ψ(z, w) = ψ(z − w). The geometric properties considered below are related to the nodal lines (i.e., −1(0)), nodal domains (i.e., connected components of the complement of the nodal set), as well as the level curves ( −1(c)), and excursion sets (connected components of {z : (z) > c}) The geometry of these sets is closely related to that of the set of critical points of. The critical points and values and their applications appear a lot ([7, 8, 11,12,13,14] to mention a few) in the literature on nodal domains of random plane waves and, more generally, smooth Gaussian fields
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