Euclidean random matrix

An 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 in 1999.^[1] 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. A Euclidean distance matrix is a particular example of Euclidean random matrix with either f(r_i - r_j) = |r_i - r_j|² or f(r_i - r_j) = |r_i - r_j|.^[2]

For example, in many biological networks, the strength of interaction between two nodes depends on the physical proximity of those nodes. Spatial interactions between nodes can be modelled as a Euclidean random matrix, if nodes are placed randomly in space.^[3]

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.

Example of the probability distribution of eigenvalues Λ of the 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

Hermitian Euclidean random matrices appear in various physical contexts, including supercooled liquids,^[4] phonons in disordered systems,^[5] and waves in random media.^[6]

Example 1: Consider the matrix  generated by the function f(r, r′) = sin(k₀|r-r′|)/(k₀|r-r′|), with k₀ = 2π/λ₀. This matrix is Hermitian and its eigenvalues Λ are real. For N points distributed randomly in a cube of side L and volume V = L³, one can show^[6] that the probability distribution of Λ is approximately given by the Marchenko-Pastur law, if the density of points ρ = N/V obeys ρλ₀³ ≤ 1 and 2.8N/(k₀ L)² < 1 (see figure).

Example of the probability distribution of eigenvalues Λ of the Euclidean random matrix generated by the function f(r, r′) = exp(ik₀ǀr-r′ǀ)/(k₀ǀr-r′ǀ), with k₀ = 2π/λ₀ and f(r= r′) = 0.

Non-Hermitian Euclidean random matrices

A theory for the eigenvalue density of large (N≫1) non-Hermitian Euclidean random matrices has been developed^[7] and has been applied to study the problem of random laser.^[8]

Example 2: Consider the matrix  generated by the function f(r, r′) = exp(ik₀|r-r′|)/(k₀|r-r′|), with k₀ = 2π/λ₀ and f(r= r′) = 0. This matrix is not Hermitian and its eigenvalues Λ are complex. The probability distribution of Λ can be found analytically^[7] if the density of point ρ = N/V obeys ρλ₀³ ≤ 1 and 9N/(8k₀ R)² < 1 (see figure).

References