Random matrix

In probability theory and statistics (with main applications in theoretical physics ) a random matrix is ​​a matrix-valued random variable (English: Random matrix).

Many important properties of physical systems can be mathematically formulated with matrices and random matrices appear so on in problems of statistical mechanics. For example, the thermal conductivity of a crystalline solid can be calculated directly from the so-called dynamic matrix of the particle-particle interaction in the crystal lattice.

For motivation: Disordered Systems

In the case of a disordered physical system (eg, so-called amorphous material ) the relevant matrix elements are random variables. The physics of these systems can be detected mainly by the parameters of the respective matrices, eg by the mean and variation of the respective size. Of particular interest are the eigenvectors and eigenvalues ​​of the random matrix.

Spectral theory of random matrices

Mathematicians and physicists have developed many remarkable theoretical connections and empirical evidence to the theory of random matrices. One of the most important results is the so-called Wigner'sche law (see Eugene Wigner ): It states that the spectral measure of the eigenvalues ​​of a symmetric random matrix, in physics known as the so-called density of states, a characteristic semicircle distribution is sufficient. This involves N × N matrices with Gaussian distributed elements in the limit. The Wigner'sche law applies not only to symmetric matrices (so-called orthogonal ensemble) but with slight modifications for unitary or symplectic matrices (so-called unitary or symplektisches ensemble). For example, the latter case was observed in current simulations of lattice gauge theory of quantum chromodynamics.

Applications

• In physics, there are applications, inter alia, in magnetic systems, such as multi-layer thin-film systems, magnetic systems, the quantum Hall effect, the so-called quantum dots and superconductors.

• Applications in nuclear physics relate, inter alia, the above-mentioned Gaussian orthogonal, unitary and symplectic ensemble which: energy spectrum and cross section of a nucleus are indeed extremely complex, but that is why the theory of the so-called chaotic behavior accessible.
• Other applications relating to the signal processing and wireless networks
• And the so-called quantum chaos and mesoscopic physics.
• Furthermore, there are applications in the so-called quantum gravity in two-dimensional systems.
• Recent evidence obtained as the assumption that the theory of random matrices could lead to improvements in Web search engines.