Motivated by current interest in understanding statistical properties of random landscapes in high-dimensional spaces, we consider a model of the landscape in [Formula: see text] obtained by superimposing M > N plane waves of random wavevectors and amplitudes and further restricted by a uniform parabolic confinement in all directions. For this landscape, we show how to compute the “annealed complexity,” controlling the asymptotic growth rate of the mean number of stationary points as N → ∞ at fixed ratio α = M/ N > 1. The framework of this computation requires us to study spectral properties of N × N matrices W = KTK T, where T is a diagonal matrix with M mean zero independent and identically distributed (i.i.d.) real normally distributed entries, and all MN entries of K are also i.i.d. real normal random variables. We suggest to call the latter Gaussian Marchenko–Pastur ensemble as such matrices appeared in the seminal 1967 paper by those authors. We compute the associated mean spectral density and evaluate some moments and correlation functions involving products of characteristic polynomials for such matrices.
Using the random matrix theory approach we derive explicit distributions of the real and imaginary parts for off-diagonal entries of the Wigner reaction matrix K for wave chaotic scattering in systems with and without time-reversal invariance, in the presence of an arbitrary uniform absorption. Whereas for time-reversal invariant system (β = 1) the scattering channels are assumed to be random and orthogonal on average, for broken time-reversal (β = 2) we consider the case of nontrivially correlated channel vectors.
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