Spectrum of large euclidean random matrices generated from l_p ellipsoids

Abstract

In this paper, we study the probability distribution of eigenvalues of the $n\times n$ Euclidean random matrix whose $(i,j{)}^{th}$ entry is of the form $f(\parallel {\mathbf{x}}_i-{\mathbf{x}}_j{\parallel }^2)-u_n{\delta }_{ij}\sum _{k}f(\parallel {\mathbf{x}}_i-{\mathbf{x}}_k{\parallel }^2)$ for some real function f. Random points $x_i$'s are independently distributed in the $l_p$ ellipsoid or on its surface in ${\mathbb{R}}^N$ including the unit sphere $S^{N-1}$, the simplex, the ordinary ellipsoid and the hyper-cube. Here $u_n$ is allowed to be any real number which includes the two most interesting cases $u_n=0$ and $u_n=1$. The limits of the empirical distribution of its eigenvalues are derived in two high dimensional settings: $n/N\to \gamma \in (0,\mathrm{\infty })$ and $n/N\to 0$ as both n and N go to infinity. By taking $f\left(x\right)$ to be suitable functions, we also give the explicit limiting spectral distributions for some distance matrices whose entries are based on the Euclidean distance and the geodesic distance.

Keywords:

• Euclidean random matrices
• empirical distribution of eigenvalues
• Laplacian
• $l_p$ ellipsoids
• geodesic distance

AMS CLASSIFICATIONS:

• 15B52
• 15A18
• 60B10
• 60F15

Acknowledgments

The author would like to thank the referees for careful reviewing of the manuscript and giving many helpful comments on the paper which improved the quality of this article.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by the Scientific Research Fund of Hunan Provincial Education Department under [grant number 18B019], Natural Science Foundation of Hunan Province under [grant number 2017JJ3216], National Natural Science Foundation of China under [grant number 11401203] and the Construct Program of the Key Discipline in Hunan Province.