statistical mechanics - How and why can random matrices answer physical problems?

Random matrix theory pops up regularly in the context of dynamical systems.

I was, however, so far not able to grasp the basic idea of this formalism. Could someone please provide an instructive example or a basic introduction to the topic?

I would also appreciate hints on relevant literature.

Answer

The basic idea is that statistical properties of complex physical systems fall into a small number of universal classes. A very known example of this phenomenon is the universal law implied by the central limit theorem where the sum of a large number of random variables belonging to a large class of distrubutions converges to the normal distribution. Please see Percy Deift's article for a historical and motivational review of the subject. Of course, one of the most motivating examples (mentioned Percy Deift's article) in is the original Wigner's explanation of the heavy nuclei spectra. Wigner "conjectured" the universality and looked for a model which can explain the repulsion between the energy levels of the large nuclei (two close energy levels are unlikely) which led him to the Gaussian orthogonal ensemble having this property built in. Now, the heavy nuclei Hamiltonian matrix elements are not random, but since by universality , for large N, the distribution of the eigenvalues does not depend on the matrix elements distribution, then the random matrix eigenvalue distribution approximates that of the Hamiltonian.