Superposition of random plane waves in high spatial dimensions: Random matrix approach to landscape complexity

Motivated by current interest in understanding statistical properties of random landscapes high-dimensional spaces, we consider a model the landscape $\mathbb{R}^N$ obtained superimposing $M>N$ plane waves wavevectors and amplitudes. For this show how to compute "annealed complexity" controlling asymptotic growth rate mean number stationary points as $N\to \infty$ at fixed ratio $\alpha=M/N>1$. The framework computation requires us study spectral $N\times N$ matrices $W=KTK^T$, where $T$ is diagonal with $M$ zero i.i.d. real normally distributed entries, all $MN$ entries $K$ are also normal variables. We suggest call latter Gaussian Marchenko-Pastur Ensemble, such appeared seminal 1967 paper those authors. associated density evaluate some moments correlation functions involving products characteristic polynomials for related matrices.