Euclidean random matrix
A 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. The element Aij of the matrix is equal to f(ri, rj): Aij = f(ri, rj).
History
Euclidean random matrices were first introduced by M. Mézard, G. Parisi and A. Zee 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 δij∑kf(ri - rk),
motivated by the physical context in which they studied the matrix.
Euclidean distance matrix is a particular example of Euclidean matrix with either f(ri - rj) = |ri - rj|2 [2] or f(ri - rj) = |ri - rj|.[3]
Properties
Because the positions of the points {ri} 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,[4] phonons in disordered systems,[5] and waves in random media.[6]
Example: Consider the matrix A generated by the function f(r, r′) = sin(k0|r-r′|)/(k0|r-r′|), with k0 = 2π/λ0. For N points distributed randomly in a cube of volume V = R3, one can show[6] that the probability distribution of eigenvalues Λ of A is approximately given by the Marchenko-Pastur law, if the density of point ρ = N/V obeys ρλ03 << 1 and 2.8N/(k0 L)2 < 1 (see figure).
Non-Hermitian Euclidean random matrices
The theory for the eigenvalue density of large (N≫1) non-Hermitian Euclidean random matrices was developed by Goetschy and Skipetrov[7] and has been applied to study the problem of random laser.[8]
References
- ^ M. Mézard, G. Parisi, and A. Zee, Spectra of euclidean random matrices, Nucl. Phys. B 559, 689 (1999)
- ^ http://en.wikipedia.org/wiki/Euclidean_distance_matrix
- ^ E. Bogomolny, O. Bohigas, and C. Schmit, Spectral properties of distance matrices, J. Phys. A: Math. Gen. 36, 3595 (2003)
- ^ T.S. Grigera, V. Martin-Mayor, G. Parisi, and P. Verrocchio, Phonon interpretation of the 'boson peak' in supercooled liquids, Nature (London) 422, 289 (2003)
- ^ A. Amir, Y. Oreg, and Y. Imry, Localization, anomalous diffusion, and slow relaxations: A random distance matrix approach, Phys. Rev. Lett. 105, 070601 (2010)
- ^ a b Attention: This template ({{cite doi}}) is deprecated. To cite the publication identified by doi:10.1088/1751-8113/44/6/065102, please use {{cite journal}} (if it was published in a bona fide academic journal, otherwise {{cite report}} with
|doi=10.1088/1751-8113/44/6/065102instead. - ^ A. Goetschy and S.E. Skipetrov, Non-Hermitian Euclidean random matrix theory, Phys. Rev. E 84, 011150 (2011)
- ^ A. Goetschy and S.E. Skipetrov, Euclidean matrix theory of random lasing in a cloud of cold atoms, EPL 96, 34005 (2011)