Random Matrix Theory is a branch of mathematics at the intersection of mathematical physics and probability theory that studies the properties of ensembles of matrices whose elements are randomly distributed. As a rule, the law of distribution of elements is specified. In this case, the statistics of eigenvalues of random matrices is studied, and sometimes also the statistics of their own vectors .
The theory of random matrices has many applications in physics, especially in applications of quantum mechanics to the study of disordered and classically chaotic dynamical systems . The fact is that the Hamiltonian of a chaotic system can often be thought of as a random Hermitian or symmetric real matrix , while the energy levels of this Hamiltonian will be the eigenvalues of a random matrix.
Wigner first applied the theory of random matrices in 1950 to describe the energy levels of an atomic nucleus . Subsequently, it turned out that the theory of random matrices describes many systems, including, for example, energy levels of quantum dots , energy levels of particles in complex potentials. As it turned out, the theory of random matrices is applicable to almost any quantum system whose classical analogue is not integrable . In this case, significant differences are observed in the distribution of energy levels: the distribution of energy levels in an integrable system is, as a rule, close to the Poisson distribution , while for a non-integrable system it has a different form characteristic of random matrices (see below).
The theory of random matrices turned out to be useful for seemingly extraneous branches of mathematics, in particular, the distribution of the zeros of the Riemann zeta-function on the critical line can be described using some ensemble of random matrices [1] .
Content
Basic ensembles of random matrices and their application in physics
There are three main types of ensembles of random matrices that are used in physics. These are a Gaussian orthogonal ensemble , a Gaussian unitary ensemble , a Gaussian symplectic ensemble .
A Gaussian unitary ensemble is the most general ensemble; it consists of arbitrary Hermitian matrices whose real and imaginary parts of the elements have a Gaussian distribution . The systems described by a Gaussian unitary ensemble are devoid of any symmetry - they are non-invariant with respect to time reversal (for example, systems in an external magnetic field have this property) and are non-invariant with respect to spin rotations.
A Gaussian orthogonal ensemble consists of symmetric real matrices. A Gaussian orthogonal ensemble describes systems that are symmetrical with respect to time reversal, which in practical cases means the absence of a magnetic field and magnetic impurities in such systems.
A Gaussian symplectic ensemble consists of Hermitian matrices whose elements are quaternions . A Gaussian symplectic ensemble describes a system containing magnetic impurities, but not located in an external magnetic field.
The most important characteristics of the spectrum of random matrices
Distribution of eigenvalues
Data obtained by diagonalizing a 1000 × 1000 Gaussian orthogonal matrix and a theoretical curve
The distribution of the eigenvalues of a sufficiently large Gaussian random matrix in the first approximation is a semicircle ( Wigner semicircle law ). The Wigner semicircle law is satisfied in the limit, to some extent corresponding to the semiclassical approximation in quantum mechanics , it is fulfilled the more precisely, the larger the size of the analyzed matrix. With a finite matrix size, the distribution of energy levels has Gaussian “tails”. Semicircles are obtained for all Gaussian ensembles; at this level, all three of the ensembles listed above give equivalent distributions. Qualitative differences between the three ensembles appear at the next level - at the level of pair correlation functions of eigenvalues.
Correlation function of eigenvalues
Notes
- ↑ Keating et al, 2000 .
Links
- Weisstein, Eric W. Random Matrix on Wolfram MathWorld .
Literature
- Mehta M. L. Random Matrices. - 3rd ed. - New York: Academic Press, 1991.
- Keating JP , Snaith NC Random matrix theory and (Eng.) // Commun. Math. Phys. : magazine. - 2000. - Vol. 214 . - P. 57–89 .