Random matrices

Collections of large matrices, chosen at random from some ensemble. Random-matrix theory is a branch of mathematics which emerged from the study of complex physical problems, for which a statistical analysis is often more enlightening than a hopeless attempt to control every degree of freedom, or every detail of the dynamics. Although the connections to various parts of mathematics are very rich, the relevance of this approach to physics is also significant.

Random matrices were introduced by Eugene Wigner in nuclear physics in 1950. In quantum mechanics the discrete energy levels of a system of particles, bound together, are given by the eigenvalues of a hamiltonian operator, which embodies the interactions between the constituents. This leads to the Schrödinger equation which, in most cases of interest in the physics of nuclei, cannot be solved exactly, even with the most advanced computers. For a complex nucleus, instead of finding the location of the nuclear energy levels through untrustworthy approximate solutions, Wigner proposed to study the statistics of eigenvalues of large matrices, drawn at random from some ensemble. The only constraint is to choose an ensemble which respects the symmetries that are present in the forces between the nucleons in the original problem, and to select a sequence of levels corresponding to the quantum numbers that are conserved as a consequence of these symmetries, such as angular momentum and parity. The statistical theory does not attempt to predict the detailed sequence of energy levels of a given nucleus, but only the general properties of those sequences and, for instance, the presence of hidden symmetries. In many cases this is more important than knowing the exact location of a particular energy level. This program became the starting point of a new field, which is now widely used in mathematics and physics for the understanding of quantum chaos, disordered systems, fluctuations in mesoscopic systems, random surfaces, zeros of analytic functions, and so forth. See Conservation laws (physics), Eigenvalue (quantum mechanics), Quantum mechanics

The mathematical theory underlying the properties of random matrices overlaps with several active fields of contemporary mathematics, such as the asymptotic behavior of orthogonal polynomials at large-order, integrable hierachies, tau functions, semiclassical expansions, combinatorics, and group theory; and it is the subject of active research and collaboration between physics and mathematics.

random matrices

[‚ran·dəm ′mā·tri‚sēz]

(mathematics)

Collections of large matrices, chosen at random from some ensemble.

References in periodicals archive

Given any uniformly random matrix A [member of] [Z.sup.nxm.sub.q], a zero vector 0 [member of] [Z.sup.n.sub.q], and any syndrome u [member of] [Z.sup.n.sub.q], a q -ary lattice [[and].sup.[perpendicular to].sub.q] (A) and a coset [[and].sup.u.sub.q] (A) of [[and].sup.[perpendicular to].sub.q] (A) are defined as follows:

Transitive Signature Schemes for Undirected Graphs from Lattices

The symposium was held during July 2016 in Charlotte, North Carolina and the 16 papers provide a snapshot of rapid developments in the emerging research field of topological recursion, which developed independently in random matrix theory/matrix models and in geometry.

Topological Recursion and Its Influence in Analysis, Geometry, and Topology

where F is a smooth convex function for every realization of [xi] on [S.sup.n.sub.+], [xi] is a random matrix whose probability distribution P is supported on set [OMEGA] [member of] [R.sup.rxr], A : [S.sup.n] [right arrow] [R.sup.m] is a linear map, b [member of] [R.sup.m] and X [greater than or equal to] 0 mean that X [member of] [S.sup.+.sub.n], and [S.sup.n] is the space of n x n real symmetric endowed with the standard trace inner product <x, x> and Frobenius norm [mathematical expression not reproducible] is the set of positive semidefinite matrices in [S.sup.n].

Inexact SA Method for Constrained Stochastic Convex SDP and Application in Chinese Stock Market

Recently, there are some measurement matrices that satisfy the restricted isometry principle, including the Gauss random matrix, Bernoulli random matrix, Hadamard matrix, Toeplitz matrix, and random Fourier matrix [16-19].

Composite Plate Phased Array Structural Health Monitoring Signal Reconstruction Based on Orthogonal Matching Pursuit Algorithm

(iii) The norm of random matrix is defined in the form of

Numerical Study of Random Periodic Lipschitz Shadowing of Stochastic Differential Equations

In these classic reference sharing mechanism-based self-embedding schemes, the binary random matrix is used as the encoding matrix.

Reference Sharing Mechanism-Based Self-Embedding Watermarking Scheme with Deterministic Content Reconstruction

Here [m.sub.1] (i, j) is the [M.sub.1] random matrix element, and [m.sub.2] (i, j) is the [M.sub.2] random matrix element.

Search of Fuzzy Periods in the Works of Poetry of Different Authors

Random matrix theory (see the classical text [1]) first appeared in physics in Wigner's work on the level spacing in large nuclei.

A Renormalisation Group Approach to the Universality of Wigner's Semicircle Law for Random Matrices with Dependent Entries

Since the late 1990s, there have been several candidates for such measures: the induced measures [1, 2], the Bures measure [3], or random matrix product states [4], just to name a few.

On the Separability of Unitarily Invariant Random Quantum States: The Unbalanced Regime

In recent years, the method of random matrix theory has been applied to the spectrum sensing [5, 6], which uses the eigenvalue of the signal covariance matrix as the statistic and then derives the corresponding threshold to judge.

A Spectrum Sensing Method Based on Signal Feature and Clustering Algorithm in Cognitive Wireless Multimedia Sensor Networks