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 randomly distributed in a region V of d-dimensional Euclidean space. The element Aij of the matrix is equal to f(ri, rj): Aij = f(ri, rj).
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 Aii,
Aij = f(ri - rj) - u δijΣkf(ri - rk), 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) = |ri - rj|2 or f(ri - rj) = |ri - rj|.[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] [4]
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 appear in various physical contexts, including supercooled liquids,[5] phonons in disordered systems,[6] and waves in random media.[7]
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[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).
A theory for the eigenvalue density of large (N≫1) non-Hermitian Euclidean random matrices has been developed[8] and has been applied to study the problem of random laser.[9]
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[8] if the density of point ρ = N/V obeys ρλ03 ≤ 1 and 9N/(8k0 R)2 < 1 (see figure).