Random matrix theory and symmetric spaces

摘要: Abstract In this review we discuss the relationship between random matrix theories and symmetric spaces. We show that integration manifolds of theories, eigenvalue distribution, Dyson boundary indices characterizing ensembles are in strict correspondence with spaces intrinsic characteristics their restricted root lattices. Several important results can be obtained from identification. particular Cartan classification triplets positive, zero negative curvature gives rise to a new ensembles. The is organized into two main parts. Part I theory reviewed emphasis on ideas relevant for appreciating theories. II various applications disordered systems derived also how mapping integrable Calogero–Sutherland models used matrices, consequences quantum transport problems. conclude indicating some interesting directions research based these identifications.