Expected number and height distribution of critical points of smooth isotropic Gaussian random fields

Abstract

We obtain formulae for the expected number and height distribution of critical points of smooth isotropic Gaussian random fields parameterized on Euclidean space or spheres of arbitrary dimension. The results hold in general in the sense that there are no restrictions on the covariance function of the field except for smoothness and isotropy. The results are based on a characterization of the distribution of the Hessian of the Gaussian field by means of the family of Gaussian orthogonally invariant (GOI) matrices, of which the Gaussian orthogonal ensemble (GOE) is a special case. The obtained formulae depend on the covariance function only through a single parameter (Euclidean space) or two parameters (spheres), and include the special boundary case of random Laplacian eigenfunctions.

Topology trivialization transition in random non-gradient autonomous ODEs on a sphere

“…Those works demonstrated that calculating the mean number of stationary points in such a setting can be mapped onto a random matrix problem related to the standard GOE ensemble, and in this respect remains quite similar to the case of stationary fields in the Euclidean space, where such reduction was discovered originally, see [26,27,28,14]. Note also a recent work [30] giving a unified treatment of both cases which went beyond certain restrictions in the original papers.…”

Manifolds Pinned by a High-Dimensional Random Landscape: Hessian at the Global Energy Minimum

“…where we used (60). Hence we see that the averaged Green function decays exponentially ∼ e −|x−y|/Lc , with the characteristic length given by the Larkin length L c .…”

“…This relates to the broad effort of understanding the statistical structure of stationary points (minima, maxima and saddles) of random landscapes which is of steady interest in theoretical physics [30][31][32][33][34][35][36][37][38][39], with recent applications to statistical physics [10,[34][35][36][38][39][40][41], neural networks and complex dynamics [42][43][44][45][46], string theory [47,48] and cosmology [49,50]. It is also of active current interest in pure and applied mathematics [51][52][53][54][55][56][57][58][59][60], For the model (1)- (2) in the simplest case d = 0 (x is a single point), the mean number of stationary points and of minima of the energy function was investigated in the limit of large N 1 in [35,38,39], see also [37,50,52]. It was found that a sharp transition occurs from a 'simple' landscape for µ > µ c (the same µ...…”

“…Substituting this value of µ eff in (60) we thus obtain the final formula for the lower spectral edge λ (−) e of the Hessian (which we associate with the spectral gap) as a function of µ in the RS phase…”

On critical points of Gaussian random fields under diffeomorphic transformations

“…Let A be a nondegenerate N × N matrix, that is det(A) = 0. An anisotropic Gaussian random field {X(t), t ∈ R N } is defined as Using the isotropy property (3.2), it has been shown in Cheng and Schwartzman [5,6] that the exact formulae of the expected number and height distribution of critical points of the isotropic Gaussian field Z can be obtained by using GOI random matrices. We show below that the study of critical points of anisotropic Gaussian fields can be transferred to isotropic Gaussian fields, so that their expected number and height distribution can be obtained exactly.…”