TY - JOUR

T1 - Superposition of random plane waves in high spatial dimensions

T2 - Random matrix approach to landscape complexity

AU - Lacroix-A-Chez-Toine, Bertrand

AU - Fyodorov, Yan V.

AU - Fedeli, Sirio Belga

N1 - Funding Information:
We would like to thank Pierre Le Doussal for his interest in the work and, in particular, for pointing out a few relevant references. The research by Bertrand Lacroix-A-Chez-Toine and Yan V. Fyodorov was supported by the EPSRC under Grant No. EP/V002473/1 (Random Hessians and Jacobians: theory and applications). Sirio Belga Fedeli acknowledges the support from the EPSRC Centre for Doctoral Training in Cross-Disciplinary Approaches to Non-Equilibrium Systems through Grant No. EP/L015854/1.
Publisher Copyright:
© 2022 Author(s).

PY - 2022/9/1

Y1 - 2022/9/1

N2 - Motivated by current interest in understanding statistical properties of random landscapes in high-dimensional spaces, we consider a model of the landscape in RN 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 = KTKT, 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.

AB - Motivated by current interest in understanding statistical properties of random landscapes in high-dimensional spaces, we consider a model of the landscape in RN 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 = KTKT, 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.

UR - http://www.scopus.com/inward/record.url?scp=85137546101&partnerID=8YFLogxK

U2 - 10.1063/5.0086919

DO - 10.1063/5.0086919

M3 - Article

AN - SCOPUS:85137546101

SN - 0022-2488

VL - 63

JO - JOURNAL OF MATHEMATICAL PHYSICS

JF - JOURNAL OF MATHEMATICAL PHYSICS

IS - 9

M1 - 093301

ER -