Abstract:
This thesis consists of five chapters. The first chapter is the current one where we
informally discuss the problem we address and introduce the content of the future chapters.
The second chapter has, by now well established, preliminaries about the random
operator families and their spectral properties. In this chapter, we introduce the Canopy
tree as the correct object to consider for discussing the local statistics on the Bethe lattice
as done by Aizenman and Warzel in [5].
The following chapter has one of the original results proved by the author in a joint
work with Mallick [8]. Here we show that, in the Canopy graph context and for a choice
of the {P n }s, some part of the spectrum has multiplicity bigger than one [Theorem 3.0.1],
if the rank of Pn is larger than one. This is done by explicitly constructing more than one
mutually orthogonal eigenfunctions associated with eigenvalues of the random operator
that we consider.
Chapter 4 has the main theorem [4.1.2] on the local statistics and proofs of the different
components needed there. The main idea is to show that a limiting random variable
associated with a sequence given in equation (1.0.2) is a Compound Poisson random
variable.
In the fifth and concluding chapter, we discuss open problems for further enquiry that
came up during our study into the problems that this thesis discusses.
