相关论文: Symmetry classes in random matrix theory

Physical systems exhibiting stochastic or chaotic behavior are often amenable to treatment by random matrix models. In deciding on a good choice of model, random matrix physics is constrained and guided by symmetry considerations. The…

In his seminal 1962 paper on the ``threefold way'', Freeman Dyson classified the spaces of matrices that support the random matrix ensembles deemed relevant from the point of view of classical quantum mechanics. Recently, Heinzner,…

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…

Building upon Dyson's fundamental 1962 article known in random-matrix theory as 'the threefold way', we classify disordered fermion systems with quadratic Hamiltonians by their unitary and antiunitary symmetries. Important examples are…

In 1962 Dyson used a physically based, macroscopic argument to deduce the first two terms of the large spacing asymptotic expansion of the gap probability for the bulk state of random matrix ensembles with symmetry parameter \beta. In the…

In these lectures we discuss some elementary concepts in connection with the theory of symmetric spaces applied to ensembles of random matrices. We review how the relationship between random matrix theory and symmetric spaces can be used in…

It is described how one comes to the Wigner-Dyson random matrix theory (RMT) starting from a model of a disordered metal. The lectures start with a historical introduction where basic ideas of the RMT and theory of disordered metals are…

In the last few years several new Random Matrix Models have been proposed and studied. They have found application in various different contexts, ranging from the physics of mesoscopic systems to the chiral transition in lattice gauge…

Recently, the theory of symmetric spaces has come to play an increased role in the physics of integrable systems and in quantum transport problems. In addition, it provides a classification of random matrix theories. In this paper we give a…

The first work of Dyson relating to random matrix theory, "The dynamics of a disordered linear chain", is reviewed. Contained in this work is an exact solution of a so-called Type I chain in the case of the disorder variables being given by…

Permutation matrices play an important role in understand the structure of magic squares. In this work, we use a class of symmetric permutation matrices than can be used to categorize magic squares. Many magic squares with a high degree of…

We propose a one-dimensional nonintegrable spin model with local interactions that covers Dyson's three symmetry classes (classes A, AI, and AII) depending on the values of parameters. We show that the nearest-neighbor spacing distribution…

We present a detailed classification of random Dirac hamiltonians in two spatial dimensions based on the implementation of discrete symmetries. Our classification is slightly finer than that of random matrices, and contains thirteen…

Modern deep learning models are highly overparameterized, resulting in large sets of parameter configurations that yield the same outputs. A significant portion of this redundancy is explained by symmetries in the parameter…

We use the order complex corresponding to a symmetric matrix (defined by Giusti et al in 2015). In this note, we use it to define a class of models of random graphs, and show some surprising experimental results, showing sharp phase…

Random matrices now play a role in many parts of computational mathematics. To advance these applications, it is desirable to have tools that are flexible, easy to use, and powerful. Over the last 25 years, researchers have developed a…

Quantum mechanical systems with some degree of complexity due to multiple scattering behave as if their Hamiltonians were random matrices. Such behavior, while originally surmised for the interacting many-body system of highly excited…

This manuscript presents a novel framework that integrates higher-order symmetries and category theory into machine learning. We introduce new mathematical constructs, including hyper-symmetry categories and functorial representations, to…

We introduce random matrix ensembles that correspond to the infinite families of irreducible Riemannian symmetric spaces of type I. In particular, we recover the Circular Orthogonal and Symplectic Ensembles of Dyson, and find other families…

Since their appearance in the 1950s, computational models capable of performing probabilistic choices have received wide attention and are nowadays pervasive in almost every areas of computer science. Their development was also inextricably…