scholarly journals Hurwitz and the origins of random matrix theory in mathematics

2017 ◽  
Vol 06 (01) ◽  
pp. 1730001 ◽  
Author(s):  
Persi Diaconis ◽  
Peter J. Forrester
Keyword(s):  
Invariant Measure ◽  
Explicit Form ◽  
Random Matrix Theory ◽  
Random Matrix ◽  
Matrix Theory ◽  
Euler Angles ◽  
Subsequent Work ◽  
Matrix Groups ◽  
The Matrix ◽  
Invariant Group

The purpose of this paper is to put forward the claim that Hurwitz’s paper [Über die Erzeugung der invarianten durch integration, Nachr. Ges. Wiss. Göttingen 1897 (1897) 71–90.] should be regarded as the origin of random matrix theory in mathematics. Here Hurwitz introduced and developed the notion of an invariant measure for the matrix groups [Formula: see text] and [Formula: see text]. He also specified a calculus from which the explicit form of these measures could be computed in terms of an appropriate parametrization — Hurwitz chose to use Euler angles. This enabled him to define and compute invariant group integrals over [Formula: see text] and [Formula: see text]. His main result can be interpreted probabilistically: the Euler angles of a uniformly distributed matrix are independent with beta distributions (and conversely). We use this interpretation to give some new probability results. How Hurwitz’s ideas and methods show themselves in the subsequent work of Weyl, Dyson and others on foundational studies in random matrix theory is detailed.

scholarly journals KAPPA-DEFORMED RANDOM-MATRIX THEORY BASED ON KANIADAKIS STATISTICS

2012 ◽  
Vol 26 (10) ◽  
pp. 1250059 ◽  
Author(s):  
A. Y. ABUL-MAGD ◽  
M. ABDEL-MAGEED
Keyword(s):  
Random Matrix Theory ◽  
Random Matrix ◽  
Matrix Theory ◽  
Distribution Functions ◽  
Spacing Distribution ◽  
Initial Stage ◽  
The Matrix ◽  
Gibbs Entropy ◽  
Non Gaussian ◽  
Mixed Systems

We present a possible extension of the random-matrix theory, which is widely used to describe spectral fluctuations of chaotic systems. By considering the Kaniadakis non-Gaussian statistics, characterized by the index κ (Boltzmann–Gibbs entropy is recovered in the limit κ → 0), we propose the non-Gaussian deformations (κ ≠ 0) of the conventional orthogonal and unitary ensembles of random matrices. The joint eigenvalue distributions for the κ-deformed ensembles are derived by applying the principle maximum entropy to Kaniadakis entropy. The resulting distribution functions are base invariant as they depend on the matrix elements in a trace form. Using these expressions, we introduce a new generalized form of the Wigner surmise valid for nearly-chaotic mixed systems, where a basis-independent description is still expected to hold. We motivate the necessity of such generalization by the need to describe the transition of the spacing distribution from chaos to order, at least in the initial stage. We show several examples about the use of the generalized Wigner surmise to the analysis of the results of a number of previous experiments and numerical experiments. Our results suggest the entropic index κ as a measure for deviation from the state of chaos. We also introduce a κ-deformed Porter–Thomas distribution of transition intensities, which fits the experimental data for mixed systems better than the commonly-used gamma-distribution.

Random matrix theory and ribonucleic acid (RNA) folding

2018 ◽  
Author(s):  
Antonia Tulino ◽  
Sergio Verdu
Keyword(s):  
Random Matrix Theory ◽  
Physical Interpretation ◽  
Random Matrix ◽  
Mechanical Model ◽  
Rna Folding ◽  
Matrix Theory ◽  
Three Dimensions ◽  
Large N ◽  
Rna Pseudoknots ◽  
The Matrix

This article discusses a series of recent applications of random matrix theory (RMT) to the problem of RNA folding. It first provides a schematic overview of the RNA folding problem, focusing on the concept of RNA pseudoknots, before considering a simplified framework for describing the folding of an RNA molecule; this framework is given by the statistic mechanical model of a polymer chain of L nucleotides in three dimensions with interacting monomers. The article proceeds by presenting a physical interpretation of the RNA matrix model and analysing the large-N expansion of the matrix integral, along with the pseudoknotted homopolymer chain. It extends previous results about the asymptotic distribution of pseudoknots of a phantom homopolymer chain in the limit of large chain length.

scholarly journals Octonions in random matrix theory

2017 ◽  
Vol 473 (2200) ◽  
pp. 20160800 ◽  
Author(s):  
Peter J. Forrester
Keyword(s):  
Random Matrices ◽  
Random Matrix Theory ◽  
Random Matrix ◽  
Quantum Physics ◽  
Matrix Theory ◽  
Analytic Theory ◽  
Division Algebras ◽  
Number Systems ◽  
The Matrix ◽  
Real Complex

The octonions are one of the four normed division algebras, together with the real, complex and quaternion number systems. The latter three hold a primary place in random matrix theory, where in applications to quantum physics they are determined as the entries of ensembles of Hermitian random matrices by symmetry considerations. Only for N =2 is there an existing analytic theory of Hermitian random matrices with octonion entries. We use a Jordan algebra viewpoint to provide an analytic theory for N =3. We then proceed to consider the matrix structure X † X , when X has random octonion entries. Analytic results are obtained from N =2, but are observed to break down in the 3×3 case.

SSA, Random Matrix Theory, and Noise-Reduced Correlations

2016 ◽  
Author(s):  
Jan W Dash ◽  
Xipei Yang ◽  
Mario Bondioli ◽  
Harvey J. Stein
Keyword(s):  
Random Matrix Theory ◽  
Random Matrix ◽  
Matrix Theory

History—an overview

2018 ◽  
Author(s):  
Oriol Bohigas ◽  
Hans A. Weidenmüller
Keyword(s):  
Random Matrix Theory ◽  
Random Matrix ◽  
Matrix Theory ◽  
Rapid Development ◽  
The Past ◽  
History Of ◽  
And Mathematics

An overview of the history of random matrix theory (RMT) is provided in this chapter. Starting from its inception, the authors sketch the history of RMT until about 1990, focusing their attention on the first four decades of RMT. Later developments are partially covered. In the past 20 years RMT has experienced rapid development and has expanded into a number of areas of physics and mathematics.

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