Random matrix theory and symmetric spaces

Abstract

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.

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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• Complete random matrix classification of SYK models with $\mathcal{N}=0$, $1$ and $2$ supersymmetry
• Random matrices beyond the Cartan classification
• Anderson localization and the topology of classifying spaces
• Energy correlations for a random matrix model of disordered bosons
• Weingarten calculus for matrix ensembles associated with compact symmetric spaces
• Orbit measures, random matrix theory and interlaced determinantal processes
• Infinite systems of non-colliding generalized meanders and Riemann-Liouville differintegrals
• Random-matrix theory of thermal conduction in superconducting quantum dots
• Localization from sigma-model geodesics
• Noncolliding processes, matrix-valued processes and determinantal processes
• Chern-Simons matrix models, two-dimensional Yang-Mills theory and the Sutherland model
• Anderson Transitions: Criticality, Symmetries, and Topologies
• Moments of characteristic polynomials for compact symmetric spaces and Jack polynomials
• Localization for Yang-Mills Theory on the Fuzzy Sphere
• Beta-functions for a SU(2) Matrix Model in 2 + epsilon Dimensions
• Large deviations for random matrix ensembles in mesoscopic physics
• From the Anderson model on a strip to the DMPK equation and random matrix theory
• Phase space methods for Majorana fermions
• Matrix oscillator and Calogero-type models
• Matrix Quantum Mechanics on $S^1/{\mathbb{Z}}_2$
• Riemannian Gaussian distributions, random matrix ensembles and diffusion kernels
• Extended Studies of Separability Functions and Probabilities and the Relevance of Dyson Indices
• Generalized Two-Qubit Whole and Half Hilbert-Schmidt Separability Probabilities
• Finite Volume Gauge Theory Partition Functions in Three Dimensions
• The tenfold way redux: Fermionic systems with $N$-body interactions
• Resolution of unity for fermionic Gaussian operators
• A Generalization of Random Matrix Ensemble II: Concrete Examples and Integration Formulae
• Random ballistic growth and diffusion in symmetric spaces
• Gaussian ensembles distributions from mixing quantum systems
• Symmetric space description of carbon nanotubes
• Ratios of maximal concurrence-parameterized separability functions, and generalized Peres-Horodecki conditions
• On the Cartan Decomposition for Classical Random Matrix Ensembles
• On the realization of Riemannian symmetric spaces in Lie groups
• Deformations of anti-de Sitter black holes
• Infinite symmetry on the boundary of SL(3)/SO(3)
• Calogero model with Yukawa like interaction
• A Generalization of Random Matrix Ensemble I: General Theory
• The smoothness of orbital measures on noncompact symmetric spaces
• Spin-mixing-tunneling network model for Anderson transitions in two-dimensional disordered spinful electrons
• Higher-order mesoscopic fluctuations in quantum wires: Conductance and current cumulants
• The superspin approach to a disordered quantum wire in the chiral-unitary symmetry class with an arbitrary number of channels
• Jastrow-like ground states for quantum many-body potentials with near-neighbors interactions
• $T\bar{T}$ deformation of Calogero-Sutherland model via dimensional reduction
• Topological Tenfold Classification and the Entanglement of Two Qubits
• On Controllability of Driftless Control System on Symmetric Spaces
• A Criterion for the Covering Condition of Generalized Random Matrix Ensembles