Random matrices beyond the Cartan classification

Details Random matrices beyond the Cartan classification Random matrices beyond the Cartan classification

2007-07-03

arxiv.org/abs/0707.0418v2

Physics

Mathematical Physics

41 pages, no figures. References and comments added; the representation of ensemble 15 changed to quaternion real. Version acc

Scientific paper

10.1088/1751-8113/41/4/045203

It is known that hermitean random matrix ensembles can be identified with symmetric coset spaces of Lie groups, or else with tangent spaces of the same. This results in a classification of random matrix ensembles as well as applications in practical calculations of physical observables. In this paper we show that a large number of non-hermitean random matrix ensembles defined by physically motivated symmetries - chiral symmetry, time reversal invariance, space rotation invariance, particle-hole symmetry, or different reality conditions - can likewise be identified with symmetric spaces. We give explicit representations of the random matrix ensembles identified with lateral algebra subspaces, and of the corresponding symmetric subalgebras spanning the group of invariance. Among the ensembles listed we identify as special cases all the hermitean ensembles identified with Cartan classes of symmetric spaces and the three Ginibre ensembles with complex eigenvalues.

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