Related papers: Symmetry classes in random matrix theory
It is well-known that unitary irreducible representations of groups can be usefully classified in a 3-fold classification scheme: Real, Complex, Quaternionic. In 1962 Freeman Dyson pointed out that there is an analogous 10-fold…
We perform a systematic symmetry classification of the Markov generators of classical stochastic processes. Our classification scheme is based on the action of involutive symmetry transformations of a real Markov generator, extending the…
Random matrix ensembles are introduced that respect the local tensor structure of Hamiltonians describing a chain of $n$ distinguishable spin-half particles with nearest-neighbour interactions. We prove a central limit theorem for the…
In 1944 Dyson defined the rank of a partition as the largest part minus the number of parts, and conjectured that the residue of the rank mod 5 divides the partitions of 5n+4 into five equal classes. This gave a combinatorial explanation of…
We explore the role of group symmetries in binary classification tasks, presenting a novel framework that leverages the principles of Neyman-Pearson optimality. Contrary to the common intuition that larger symmetry groups lead to improved…
Since it was first applied to the study of nuclear interactions by Wigner and Dyson, almost 60 years ago, Random Matrix Theory (RMT) has developed into a field of its own within applied mathematics, and is now essential to many parts of…
We introduce a remarkable new family of norms on the space of $n \times n$ complex matrices. These norms arise from the combinatorial properties of symmetric functions, and their construction and validation involve probability theory,…
We consider real symmetric or complex hermitian random matrices with correlated entries. We prove local laws for the resolvent and universality of the local eigenvalue statistics in the bulk of the spectrum. The correlations have fast decay…
First we survey generating function methods for obtaining useful probability estimates about random matrices in the finite classical groups. Then we describe a probabilistic picture of conjugacy classes which is coherent and beautiful.…
Classical mathematics are founded within set theory, but sets don't have \emph{symmetries}. We conjecture that if we allow sets with symmetries, then many problems such as \emph{Mirror symmetry} or \emph{Homological mirror symmetry} can be…
The objective of the paper is to study accuracy of multi-class classification in high-dimensional setting, where the number of classes is also large ("large $L$, large $p$, small $n$" model). While this problem arises in many practical…
Statistical models that possess symmetry arise in diverse settings such as random fields associated to geophysical phenomena, exchangeable processes in Bayesian statistics, and cyclostationary processes in engineering. We formalize the…
Williamson's theorem is well known for symmetric matrices. In this paper, we state and re-derive some of the cases of Williamson's theorem for symmetric positive-semi definite matrices and symmetric matrices having negative index 1, due to…
Random matrix theory is a well-developed area of probability theory that has numerous connections with other areas of mathematics and its applications. Much of the literature in this area is concerned with matrices that possess many exact…
The notion of a topological Ramsey space was introduced by Carlson some 30 years ago. Studying the topological Ramsey space of variable words, Carlson was able to derive many classical combinatorial results in a unifying manner. For the…
We develop a comprehensive theory of the stable representation categories of several sequences of groups, including the classical and symmetric groups, and their relation to the unstable categories. An important component of this theory is…
A "mysterious" relation between the number variance and the variance of the $L$-th ordered eigenvalue, first suggested by French et al. [Ann. Phys. 113, 277 (1978)], is revisited and proven to be asymptotically exact for the $\beta=2$ Dyson…
We study the symmetry classes for the random Dirac fermions in 2 dimensions. We consider $N_f$ species of fermions, coupled by different types of disorder. We analyse the renormalisation group flow at the order of one loop. At $N_f$ large,…
Symmetry is a key feature observed in nature (from flowers and leaves, to butterflies and birds) and in human-made objects (from paintings and sculptures, to manufactured objects and architectural design). Rotational, translational, and…
Multivariate statistical analysis is concerned with observations on several variables which are thought to possess some degree of inter-dependence. Driven by problems in genetics and the social sciences, it first flowered in the earlier…