From Statistical Mechanics to Large Deviations of Uniformly Random D-Regular Graphs.
Abstract
A Large system is difficult to study globally. Its study requires some random procedure that can
expose some of its properties to enable the identification of its typical behaviour. To this end,
uniformly random regular graphs were generated due to their fascinating properties and
applications. The Potts model was used to assign spin values to the vertices of the graph from a
finite spin space. Errors do occur in large system rarely but when they do, their impact may
cause some destruction. The randomness in rare events allows the use of probability theory to
assess it. Large deviations principle measures the probabilities of rare event and the rate at which
they occur. The aim of this thesis was to derive the Large Deviations Principle of the joint
empirical spin and bond measures of the uniformly random -regular graphs. Subsequently, the
rate function of the large deviation probabilities was derived. To achieve this, the method of
types of the joint empirical measure was used. The rate function was expressed in terms of sum
of the relative entropies of the joint empirical measures. The upper and lower bounds of the large
deviation probabilities were formulated using the types and type class of the measures. It is
recommended that to examine a large system, it is more parsimonious to find the large deviations
of the interaction of its components which would describe vividly the typical behaviour of the
system.
Description
Keywords
Statistical Mechanics, D-Regular Graphs., Potts model