Within mathematics, an ''N''×''N'' Euclidean random matrix Â 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 Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean sp ...

. 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. 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_ijΣ_k''f''(r_i - r_k), motivated by the physical context in which they studied the matrix. A

Euclidean distance matrix In mathematics, a Euclidean distance matrix is an matrix representing the spacing of a set of points in Euclidean space. For points x_1,x_2,\ldots,x_n in -dimensional space , the elements of their Euclidean distance matrix are given by squares ...

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, . 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 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. Fig sinc

Hermitian Euclidean random matrices

Hermitian {{Short description, none Numerous things are named after the French mathematician Charles Hermite (1822–1901): Hermite * Cubic Hermite spline, a type of third-degree spline * Gauss–Hermite quadrature, an extension of Gaussian quadrature m ...

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 Â generated by the function ''f''(r, r′) = sin(''k''₀, r-r′, )/(''k''₀, r-r′, ), with ''k''₀ = 2π/λ₀. This matrix is

and its

eigenvalue In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denot ...

s Λ are

real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (201 ...

. For ''N'' points distributed randomly in a cube of side ''L'' and volume ''V'' = ''L''³, 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 2.8''N''/(''k''₀ ''L'')² < 1 (see figure). Fig expc

Non-Hermitian Euclidean random matrices

A theory for the

density of large (''N''≫1) non-Hermitian Euclidean random matrices has been developed and has been applied to study the problem of

random laser A random laser (RL) is a laser in which optical feedback is provided by scattering particles. As in conventional lasers, a gain medium is required for optical amplification. However, in contrast to Fabry–Pérot cavities and distributed feedbac ...

. ''Example 2:'' Consider the matrix Â generated by the function ''f''(r, r′) = exp(''ik''₀, r-r′, )/(''k''₀, r-r′, ), with ''k''₀ = 2π/λ₀ and ''f''(r= r′) = 0. This matrix is not Hermitian and its eigenvalues Λ are

complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...

. The probability distribution of Λ can be found analytically if the density of point ρ = ''N''/''V'' obeys ρλ₀³ ≤ 1 and 9''N''/(8''k''₀ ''R'')² < 1 (see figure).

References

{{reflist Random matrices Mathematical physics