Euclidean random matrix
 | This article, Euclidean random matrix, has recently been created via the Articles for creation process. Please check to see if the reviewer has accidentally left this template after accepting the draft and take appropriate action as necessary.
Reviewer tools: Inform author |
Definition
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]
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)
- ^ S.E. Skipetrov and A. Goetschy, Eigenvalue distributions of large Euclidean random matrices for waves in random media, J. Phys. A 44, 065102 (2011)
- ^ 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)