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Motivated by current interest in understanding statistical properties ofrandom landscapes in high-dimensional spaces, we consider a model of thelandscape in $\mathbb{R}^N$ obtained by superimposing $M>N$ plane waves ofrandom wavevectors and amplitudes. For this landscape we show how to computethe "annealed complexity" controlling the asymptotic growth rate of the meannumber of stationary points as $N\to \infty$ at fixed ratio $\alpha=M/N>1$. Theframework of this computation requires us to study spectral properties of$N\times N$ matrices $W=KTK^T$, where $T$ is diagonal with $M$ mean zero 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 GaussianMarchenko-Pastur Ensemble, as such matrices appeared in the seminal 1967 paperby those authors. We compute the associated mean spectral density and evaluatesome moments and correlation functions involving products of characteristicpolynomials for such and related matrices.

Superposition of Random Plane Waves in High Spatial Dimensions: Random Matrix Approach to Landscape Complexity