Random Matrices: a historical point of view

The early history (1955-1968)

During the fifties, as a consequence of the making of the atomic bomb, nuclear physicists were involved into measuring the fission resonances in various nuclei. It became clear very soon that the resonance spectra were complex. The high number of resonances was preventing to compute them from diagonalizing the nucleus Hamiltonian. In the mid fifties, under the influence of Wigner, Porter & Thomas, statistical methods were applied to analyze them. Porter and Thomas showed, from the experimental datas that the resonance width was the square of a gaussian random variable. This is the so called Porter-Thomas distribution. In 1955, Wigner realized that the statistical distribution of the resonances energies was given by the eigenvalues of a random matrix sharing the same properties as the Hamiltonian of the nucleus.  He investigated first a model in which the matrix elements are independent identically distributed random variables, taking on values +1 or -1 with equal probabilities. By a combinatorial argument, he proved that the moments of the density of states were given by the Catalan numbers, namely that the DOS was a semicircle distribution. It became also clear almost immediately that the energy level , namely the eigenvalues of such matrices, were repelling each other. In 1956, based upon the result of 2x2 gaussian random matrices, Wigner surmized that the level spacing distribution was approximately given by the formula

The comparison with the experimental datas proved convincing almost immediately. Wigner advocated the work of Wishart which was giving the joint distribution of the eigenvalues to confirm his finding.

In the early sixties, Mehta produced the first analytic result for the Wigner ensemble: he could compute the DOS and the correlations functions.  Simultaneously, Dyson,  realized  that the Wishart distribution could be interpreted in term of the Gibbs state of a one-dimensional gas of point particles interacting through  2D Coulomb forces. Moreover, he showed that all random matrix ensembles fall into three universality classes classified by their symmetries, called the orthogonal ensemble (OE), unitary ensemble (UE) or symplectic ensemble (SE).  For gaussian matrices, this gives the GOE, GUE or GSE. At last he also introduced the circular ensembles COE, CUE, CSE, making the analytic calculations easier. In the end of the sixties, Mehta and Gaudin could compute analytically the level spacing distribution and showed that the Wigner surmise was accurate to few persent if compared with the exact result.  Over the years, the statistics available for a comparison between the predictions of the Random Matrix Theory and the nuclear resonance datas, shown  an excellent agreement.  But no new progress came after the sixties, so that the subject didnot progress any more.

Quantum Chaos and RMT(1983-1990)

In 1983, Bohigas, Giannoni and Schmit  published a result showing that the Random Matrix Theory used in Nuclear Physics, could be used to describe the eigenvalue spectrum of the quantized version of the motion of a ball moving in a Sinai billiard. From then on, a large number of numerical calculations confirmed that the RMT was accounting for the level spacing distribution and the correlation for classically chaotic quantum systems. This observations are not yet understood yet starting from first principle. However, during the eighties, the semiclassical approach proposed by M. Gutzwiller in 1968 and later by Balian and Bloch in 1972, was producing formulae for the calculations of the eigenvalue spectrum. The sequence of eigenvalues, produced in this way were compared successfully to the prediction of the RMT by several physicists like M. Berry.  The accuracy was such that it became possible to reach the regime for which the RMT was not valid anymore.  The analogy with disordered quantum systems, as described by Thouless in the seventies through the finite size scaling theory, lead to the notion of Thouless energy as the borderline between the RMT regime and the semiclassical one. It was later shown that the non RMT regime is well described by the Gutswiller formula and the contribution of the short classical periodic orbits of the system.

Mesoscopic Physics (1986-)
In 1986, Atshuler and Shklovskii proposed to used the RMT as a substitute to graph expansion to treat the coherent transport in mesoscopic disordered systems. ....