Random matrix theory and symmetric spaces

Physics Reports (Impact Factor: 20.03). 04/2003; 394(2). DOI: 10.1016/j.physrep.2003.12.004
Source: arXiv


In this review we discuss the relationship between random matrix theories and symmetric spaces. We show that the integration manifolds of random matrix theories, the eigenvalue distribution, and the Dyson and boundary indices characterizing the ensembles are in strict correspondence with symmetric spaces and the intrinsic characteristics of their restricted root lattices. Several important results can be obtained from this identification. In particular the Cartan classification of triplets of symmetric spaces with positive, zero and negative curvature gives rise to a new classification of random matrix ensembles. The review is organized into two main parts. In Part I the theory of symmetric spaces is reviewed with particular emphasis on the ideas relevant for appreciating the correspondence with random matrix theories. In Part II we discuss various applications of symmetric spaces to random matrix theories and in particular the new classification of disordered systems derived from the classification of symmetric spaces. We also review how the mapping from integrable Calogero--Sutherland models to symmetric spaces can be used in the theory of random matrices, with particular consequences for quantum transport problems. We conclude indicating some interesting new directions of research based on these identifications. Comment: 161 pages, LaTeX, no figures. Revised version with major additions in the second part of the review. Version accepted for publication on Physics Reports

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Available from: Michele Caselle, Sep 03, 2013
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    • "Finite dimensional symmetric spaces [20] [25] are rather well understood mathematical objects which have recently gained much importance in both mathematics and physics due to their intimate connections with random matrix theories, Reimannian geometries and their applications to many integrable systems, quantum transport phenomena (disordered system etc) [4] [5] [9] [22]. A compact irreducible symmetric space is either a compact simple Lie group G or a quotient G/K of a compact simple Lie group by the fixed point set of an involution ρ (or an open subgroup of it) and g = t ⊕ p is the decomposition of the Lie algebra g of group G into +1 and -1 eigenvalue spaces of ρ then K acts on g by adjoint representation leaving the decomposition invariant. "
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    • "These symmetries have been extended using the Cartan classification of symmetric spaces [7] (see also [8] [9] [10]). For example, the possible Jacobians of the transformations to radial coordinates are given by [8] "
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    • "Asymptotics in T = N → ∞ are studied in Section 5. Appendices are given to show proofs of formulae and lemmas used in the text. At the end of this introduction, we would like to refer to the papers [18] [8], which reported the further extensions of RM theory in physics and the representation theory. We hope that the present paper will demonstrate the fruitfulness of developing the probability theory of interacting infinite particle systems in connection with the extensive study of (multi-)matrix models in the RM theory. "
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