Euclidean random matrix
Within mathematics, an N×''N Euclidean random matrix  is defined with the help of an arbitrary deterministic function f'' and of N points randomly distributed in a region V of d-dimensional Euclidean space. The element Aij of the matrix is equal to f: Aij = f.
History
Euclidean random matrices were first introduced in 1999. They studied a special case of functions f that depend only on the distances between the pairs of points: f = f and imposed an additional condition on the diagonal elements Aii,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 = |ri - rj|2 or f = |ri - rj|.
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.
Properties
Because the positions of the points are random, the matrix elements Aij 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
Hermitian Euclidean random matrices appear in various physical contexts, including supercooled liquids, phonons in disordered systems, and waves in random media.Example 1: Consider the matrix  generated by the function f = sin/, 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 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''/2 < 1.
Non-Hermitian Euclidean random matrices
A theory for the eigenvalue density of large non-Hermitian Euclidean random matrices has been developed and has been applied to study the problem of random laser.Example 2: Consider the matrix  generated by the function f = exp/, with k0 = 2π/λ0 and f = 0. This matrix is not Hermitian and its eigenvalues Λ are complex. The probability distribution of Λ can be found analytically if the density of point ρ = N/''V obeys ρλ03 ≤ 1 and 9N''/2 < 1.