Euclidean random matrix

A N×N Euclidean random matrix  is defined with the help of an arbitrary deterministic function f(r, r′) and of N points {r_i} randomly distributed in a region V of d-dimensional Euclidean space. The element A_ij of the matrix is equal to f(r_i, r_j): A_ij = f(r_i, r_j).

History

Euclidean random matrices were first introduced by M. Mézard, G. Parisi and A. Zee in 1999. They studied a special case of functions f that depend only on the distances between the pairs of points: f(r, r′) = f(r - r′) and imposed an additional condition on the diagonal elements A_ii,
A_ij = f(r_i - r_j) - u δ_ij∑_kf(r_i - r_k),
motivated by the physical context in which they studied the matrix. Euclidean distance matrix is a particular example of Euclidean matrix with either f(r_i - r_j) = |r_i - r_j|² (as on the Wikipedia page about Euclidean distance matrices) or f(r_i - r_j) = |r_i - r_j|.

Properties

Because the positions of the points {r_i} are random, the matrix elements A_ij are random too. Moreover, because the N×N elements are completely determined by only N points and, typically, one is interested in N≫d, strong correlations exist between different elements.

Hermitian Euclidean random matrices

Example of the probability distribution of eigenvalues Λ of Euclidean random matrix generated by the function f(r, r′) = sin(k₀ǀr-r′ǀ)/(k₀ǀr-r′ǀ), with k₀ = 2π/λ₀. The Marchenko-Pastur distribution (red) is compared with the result of numerical diagonalization of a set of randomly generated matrices of size N×N. The density of points is ρλ₀³ = 0.1.

Hermitian Euclidean random matrices appear in various physical contexts, including supercooled liquids, phonons in disordered systems, and waves in random media.

Example: Consider the matrix A generated by the function f(r, r′) = sin(k₀|r-r′|)/(k₀|r-r′|), with k₀ = 2π/λ₀. For N points distributed randomly in a cube of volume V = R³, one can show that the probability distribution of eigenvalues Λ of A is approximately given by the Marchenko-Pastur law, if the density of point ρ = N/V obeys ρλ₀³ << 1 and 2.8N/(k₀ L)² < 1 (see figure).

Non-Hermitian Euclidean random matrices

The theory for the eigenvalue density of large (N≫1) non-Hermitian Euclidean random matrices was developed by Goetschy and Skipetrov and has been applied to study the problem of random laser.