random matrices

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]
Collections of large matrices, chosen at random from some ensemble.
References in periodicals archive ?
Koch, "Bayesian approach to extended object and cluster tracking using random matrices," IEEE Transactions on Aerospace and Electronic Systems, vol.
Their topics include elementary convex analysis, quantum mechanics, for mathematicians, metric entropy and concentration of measure in classical spaces, Gaussian processes and random matrices, the geometry of the set of mixed states, and Bell inequalities and the Grothendieck-Tsirelson inequality.
Suppose that the joint density of the random matrices [W.
We also derive the density function of the matrix quotient of two independent random matrices having confluent hypergeometric function kind 1 and gamma distributions.
Furthermore, these random matrices are often difficult or expensive in hardware implementation.
The proposed research, which lies at the interface of probability, integrable systems, random matrices, statistical physics, automorphic forms, algebraic combinatorics and representation theory, will make novel contributions in all of these areas.
In the context of sparse matrices, a fast algorithm is required to apply the perturbation term PQ*; the random matrices can be constructed and applied using, for example, the fast Johnson-Lindenstrauss transform (FJLT) [1] or the subsampled randomized Fourier transform (SRFT) [29].
Outlining a connection from random matrices to the six-vertex model of statistical physics, Bleher and Liechty focus on the Riemann-Hilbert method for both continuous and discrete orthogonal polynomials, and applications of this approach to matrix models as well as to the six-vertex model.
The Tracy-Widom distributions were found by Tracy and Widom as the limiting law of the largest eigenvalue of certain random matrices [22].
Conjecturally, the features of this model should be similar to those of lozenge tilings: we expect the formation of a limit shape and various connections with random matrices.
When the measurement matrix is constructed by Gaussian random matrices or random partial Fourier basis, it performs well for RIP [40].