Random matrices is an active field of mathematics and physics. Initiated in
the 1920s–1930s by statisticians and introduced in physics in the 1950s–1960s by
Wigner and Dyson, the field, after about two decades of the "normal science" de-
velopment restricted mainly to nuclear physics, has became very active since the
end of the 1970s under the flow of accelerating impulses from quantum field theory,
quantum mechanics (quantum chaos), statistical mechanics, and condensed mat-
ter theory in physics, probability theory, statistics, combinatorics, operator theory,
number theory, and theoretical computer science in mathematics, and also telecom-
munication theory, qualitative finances, structural mechanics, etc. In addition to
its mathematical richness random matrix theory was successful in describing vari-
ous phenomena of these fields, providing them with new concepts, techniques, and
Random matrices in statistics have arisen as sample covariance matrices and
have provided unbiased estimators for the population covariance matrices. About
twenty years later physicists began to use random matrices in order to model the
energy spectra of complex quantum systems and later the systems with complex
dynamics. These, probabilistic and spectral, aspects have been widely represented
and quite important in random matrix theory until the present flourishing state of
the theory and its applications to a wide variety of seemingly unrelated domains,
ranging from room acoustics and financial markets to zeros of the Riemann ζ-
One more aspect of the theory concerns integrals over matrix measures de-
fined on various sets of matrices of an arbitrary (mostly large) dimension. Matrix
integrals proved to be partition functions of models of quantum field theory and
statistical mechanics and generating functions of numerical characteristics of com-
binatorial and topological objects; they satisfy certain finite-difference and differ-
ential identities connected to many important integrable systems. However, the
matrix integrals themselves, their dependence on parameters, etc., can often be
interpreted in spectral terms related to random matrices whose probability law is
a matrix measure in the integral.
Thus, random matrix theory can be viewed as a branch of random spectral
theory, dealing with situations where operators involved are rather complex and
one has to resort to their probabilistic description. It is worth noting that ap-
proximately at the same time as Wigner and Dyson, i.e., in the 1950s, Anderson,
Dyson, and Lifshitz proposed to use finite-difference and differential operators with
random coefficients, i.e., again certain random matrices, to describe the dynamics
of elementary excitations in disordered media (crystals with impurities, amorphous
substances), thereby creating another branch of random spectral theory, known
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