I have been lately studying the random matrices and their application that
widely defaults for the JT gravity. Though, random matrices need to be started
with Wigner's idea of the random matrix in nuclear theory. Right now, random
matrix theory can be considered an important subject, at least from my
learning view. In the following (short), I present my rough ideas of random
matrix theory, extracted from
here.
Random matrix theory (RMT) is a classic example of statistical group theory in
general physics. The most recent development for RMT is the equivalence of JT
gravity with RMT, see
here. From the
correspondence of AdS/CFT, one learns that a bulk theory with gravity lives on
the boundary of a quantum system. However, the equivalence of JT gravity is
not given to a boundary theory (or a bulk theory). In fact, RMT shares the
correspondence with JT gravity; hence JT gravity is dual to random matrix
integral of Hamiltonian $ H $, where $ H $ is a random matrix.
(RMT is mainly concerned about groups' statistics, at least for us, whose
applications are wide in physics, as indicated.)
Consider a matrix $ \sf M $, from linear algebra we know that $ \sf M$ holds
eigenvalues $ {\sf {m}}_{ij} $ (for $ 2 \times 2 $ matrix). Suppose the
elements of matrix $ \sf M $ are random variables, to which we say to obey
some specific outlined properties. In that case, the study of those
($\mathsf{m}_{ij} $) eigenvalues are called the "random matrix theory''
problem. Now one can ask what the practical application of RMT is. Actually,
there are many practical applications. Consider the well-studied example of
the nucleus using these random matrices, which Wigner (and Dyson) developed.
And the recent example of the success of RMT is JT gravity. I suggest the
reader to read the most interesting book on this subject by Madan Lal Mehta.
From a mathematical perspective, random matrices serve great as well. However, it tends that it is currently revolutionizing the physics. Nuclear physics had first exploitation of random matrices. And now, it is used as a tool in black hole information problems, JT gravity (see the Saad, Shenker, Stanford), and other statistics problems (include the pedastriation and all that researches). The deformations of these matrices were done by Witten, and he also did the volume problems for the subject.
Radom matrices can be classified into classes. See sec. 4 in
this and Witten's
paper on JT gravity. For a full study, consider the Mehta's book.