Starting from the works of Erdos, Yau, Schlein with coauthors, the significant
progress in understanding the universal behavior of many random graph and random matrix
models were achieved. However for the random matrices with a spacial structure our
understanding is still very limited. In this talk I am going to overview applications of another
approach to the study of the local eigenvalues statistics in random matrix theory based on
so-called supersymmetry techniques (SUSY) . SUSY approach is based on the representation
of the determinant as an integral over the Grassmann (anticommuting) variables.
Combining this representation with the representation of an inverse determinant as an integral
over the Gaussian complex field, SUSY allows to obtain an integral representation for the main
spectral characteristics of random matrices such as limiting density, correlation functions,
the resolvent's elements, etc. This method is widely (and successfully) used in the physics
literature and is potentially very powerful but the rigorous control of the integral representations,
which can be obtained by this method, is quite
difficult, and it requires powerful analytic and statistical mechanics tools.
In this talk we will discuss some recent progress in application of SUSY to the analysis
of local spectral characteristics of the prominent ensemble of random band matrices,
i.e. random matrices whose entries become negligible if their distance from the main diagonal
exceeds a certain parameter called the band width.