A system in Quantum Mechanics can be characterized by a self-adjoint linear operator in Hilbert space, its Hamiltonian , which we may think informally of as a matrix of infinitely many dimensions. This suggests that input to such theory should be very general properties of the underlying generic Hamiltonians, most importantly such global features as the Hermiticity, the time-inversion invariance as well as other symmetries Hamiltonians may obey from general principles. Wigner hoped that the output of the model will be universal, that is independent on the detail and common to majority of systems sharing the corresponding symmetries.

Along those lines Wigner succeeded in calculating the simplest nontrivial spectral characteristics of random real symmetric matrices with independent, identically distributed entries - the mean density of eigenvalues, and demonstrated that in the limit of large matrix size it is given quite generally by the so called Semicircular Law. Wigner also provided insights into statistics of separations between the neighbouring eigenvalues of such matrices Wigner surmise.

### Random matrices, theory and applications (opens 2018-12222)

Boosted by those works RMT problems attracted considerable attention and within the next few years many essential tools helping to analyse properties of random matrices were developed, most notably the method of orthogonal polynomials by Mehta and Gaudin [5]. Wigner's ideas were further substantiated by the seminal Dyson works [6] who gave important symmetry classification of Hamiltonians implying the existence of three major symmetry classes of random matrices - Orthogonal, Unitary and Symplectic, which cover the most relevant classical ensembles.

Dyson also introduced the abovementioned circular versions of random matrix ensembles, developed a detailed theory of their spectra, and suggested a model of Brownian motion in random matrices ensembles which proved to be conceptually important and established a link to exactly soluble systems, such as the Calogero-Sutherland-Moser model.

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Starting from the 's and further into 's the interest in RMT ideas, methods and results became widespread in the Theoretical Physics community. Motivated initially mainly by applications in Nuclear Physics see [7] for an early influential review RMT was further influenced by advances in the field of Quantum Chaos , such as the conjecture of Bohigas, Giannoni, and Schmit BGS [8] claiming statistical similarity between RMT spectra and highly excited energy levels of generic quantum systems whose classical counterparts show chaotic dynamical behaviour see [9] for recent works towards justification of the BGS conjecture.

From a different angle, somewhat similar ideas were substantiated and quantified by applications of powerful methods coming from the field of disordered systems and the Anderson localization, most importantly by supermatrix approach due to Efetov [10] and its further adaptation to RMT by Weidenmueller, Verbaarschot, and Zirnbauer [11] see [12] for a few recent reviews.

Also influential were a highly successful maximal-entropy based RMT approach to statistics of electronic transport in quantum-coherent mesoscopic samples introduced by Mello and collaborators [13], as well as the idea of parametric dependence of spectra of disordered and chaotic systems, and their RMT counterparts, developed by Simons and Altshuler [14] see also related works [15] on statistics of level curvatures, as well as early papers [16] on ensembles interpolating between GOE and GUE and [17] on an impressively accurate verification of many aspects of RMT in acoustic waves experiments.

Those and related developments resulted in a considerable broadening of the list of random matrix ensembles relevant to physical applications and amenable to analytical treatment beyond the classical Wigner-Dyson list.

### Random Matrices and Their Applications

In particular, ensembles of random matrices which are not invariant with respect to changes of the basis attracted considerable attention, e. Band Random Matrices [18], which allowed for the effects of the Anderson localization to be correctly incorporated at the level of RMT. Moreover, the Dyson three-fold symmetry classification of underlying Hamiltonians was discovered to be insufficient for describing spectra appearing in Quantum Chromodynamics [19] as well as in disordered superconducting structures and was replaced by a more comprehensive list [20].

Around the same time systematic investigations of non-Hermitian deformations of random matrix ensembles were undertaken due to their relevance for description of statistics of S-matrix poles resonances in quantum chaotic scattering [21] and in Quantum Chromodynamics with non-zero chemical potential [22], which also lead to a general boost of interest in their properties [23].

On the other hand, from a very different perspective deep connections to the Random Matrix Theory were discovered in the models of 2-dimensional Quantum Gravity [24]. That line of research could be traced back to the t'Hooft's idea that in field theories with a large gauge group as e. U N the diagrammatic expansion will be dominated in the large-N limit by planar Feynman diagrams.

Until the end of 's the research on properties of random matrices in Mathematics community was seemingly not as nearly as intensive as in Theoretical Physics. On the other hand, needs of numerical analysis stimulated interests in the condition numbers of random matrices [32]. The situation changed considerably in 's after a few independent developments in various branches of Mathematics among others, development of the "free probability" theory by D. Voiculescu and its relations to random matrices of infinite size [33] lead to an essential boost of interest in various aspects of the Random Matrix Theory.

First of all, the wealth of information obtained by heuristic methods of theoretical physics called for a proper understanding by mathematical standards.

## Dr Pierpaolo Vivo

To that end, considerable efforts were directed towards proving conjectured universality [34] of correlations between eigenvalues of random matrices beyond classical Gaussian or Circular Wigner-Dyson ensembles, that is independence of their properties of the choice of the distribution of matrix elements for general RMT ensembles in the limit of large dimension. First, such a universality was verified for a rather general class of invariant ensembles [35], the line of research resulting, in particular, in development of the powerful Riemann-Hilbert approach to the asymptotics of orthogonal polynomials [36].

More recently similar universality was extended also to a broad class of ensembles with independent, identically distributed entries [37].

Deserve mentioning also works [38] on the large deviation statistics of spectral measure and on distribution of the number of eigenvalues in a given interval of spectrum, as well as the paper [39] which nontrivially extends the Dyson work on Brownian motion of eigenvalues by relating it to non-intersecting random walks. Around the same time the statistics of the largest eigenvalue in Gaussian Ensembles attracted considerable attention [40]. For topics on particular articles, maintain the dialogue through the usual channels with your editor.

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## Random Matrices: Theory and Application

World Scientific Publishing Co. Pte Ltd. Random Matrix Theory RMT has a long and rich history and has, especially in recent years, shown to have important applications in many diverse areas of mathematics, science, and engineering. The scope of RMT and its applications include the areas of classical analysis, probability theory, statistical analysis of big data, as well as connections to graph theory, number theory, representation theory, and many areas of mathematical physics.

## Products of Random Matrices with Applications to Schrödinger Operators (Progress in Probability)

Applications of Random Matrix Theory continue to present themselves and new applications are welcome in this journal. Some examples are orthogonal polynomial theory, free probability, integrable systems, growth models, wireless communications, signal processing, numerical computing, complex networks, economics, statistical mechanics, and quantum theory. Special issues devoted to single topic of current interest will also be considered and published in this journal.

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