To study the limiting spectra of random block matrices, many authors have used themethod of Cauchy transform, e.g., [Girko, 1995], [Girko, 2000], [Wegner, 1979],
[Brezin and Zee, 1995], [Brezin et al., 1996], [Molinari, 1997], [Aktas et al., 2006], [Cottatellucci and Muller, 2007], [Far et al., 2006] and [Hastings et al., 1992]. See also [Bolla, 2004a] and [Bolla, 2004b]. Some of them are motivated by the applications of random block matrices in physics, graph theory, wireless communications and biology while others are motivated by the nobility of the mathematical problem.
In this thesis, we are going to study the existence of the limiting spectral measures of Hermitian random block matrices with Wigner blocks via the method of moments using tools that originate in free probability theory.
In addition, we are going to identify the limiting spectral measures of some pat-
terns of random block matrices. We will consider the symmetric circulant block structure, special cases of the symmetric Toeplitz block structure and a block structure that was first studied in [Girko, 2000].
Since most of the tools are from free probability theory, we have also studied some of its tools and results. In particular, we extend the asymptotic freeness for
independent Wigner matrices from the mean sense to the almost sure sense and
weaken the conditions on the parent distribution. We also compute the free additive
convolution of the semicircle and Marchenko-Pastur laws.
Finally, we will study products of correlated rectangular random matrices with
real-valued entries. Our results extend some results about symmetric matrices studied in [Mazza and Piau, 2002]. Products of large dimensional random matrices are also studied. Such products have applications in physics [Gudowska-Nowak et al., 2003] and [Isopi and Newman, 1992] and in wireless communications [Muller, 2002].