Euclidean random matrix

Within mathematics, an N×N Euclidean random matrix  is defined with the help of an arbitrary deterministic function f(r, r′) and of N points {ri} randomly distributed in a region V of d-dimensional Euclidean space.

[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 Aii, motivated by the physical context in which they studied the matrix.

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

For N points distributed randomly in a cube of side L and volume V = L3, one can show[7] 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).

The probability distribution of Λ can be found analytically[8] if the density of point ρ = N/V obeys ρλ03 ≤ 1 and 9N/(8k0 R)2 < 1 (see figure).

Example 1
Example of the probability distribution of eigenvalues Λ of the Euclidean random matrix generated by the function f ( r , r ′) = sin( k 0 ǀ r - r ′ǀ)/( k 0 ǀ r - r ′ǀ), with k 0 = 2π/λ 0 . 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 ρλ 0 3 = 0.1.
Example 2
Example of the probability distribution of eigenvalues Λ of the Euclidean random matrix generated by the function f ( r , r ′) = exp( ik 0 ǀ r - r ′ǀ)/( k 0 ǀ r - r ′ǀ), with k 0 = 2π/λ 0 and f ( r = r ′) = 0.