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

Abstract

Motivated by current interest in understanding statistical properties of random landscapes in high-dimensional spaces, we consider a model of the landscape in ${\mathbb{R}}^N$ obtained by superimposing $M>N$ plane waves of random wavevectors and amplitudes. For this landscape we show how to compute the "annealed complexity" controlling the asymptotic growth rate of the mean number of stationary points as $N\to \infty$ at fixed ratio $\alpha =M/N>1$. The framework of this computation requires us to study spectral properties of $N\times N$ matrices $W=KT{K}^T$, 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 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 and related matrices.

40 pages, 6 figures

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• Critical Behaviour of the Number of Minima of a Random Landscape at the Glass Transition Point and the Tracy-Widom distribution
• Superbosonization of invariant random matrix ensembles
• Triviality of the geometry of mixed $p$-spin spherical Hamiltonians with external field
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