Random matrix theory and renormalization group: spectral theory for network ensembles

In this thesis we tackle the issue of spectral properties of networks introduced by Complex Systems Physics. Graph theory has developed methods to study the spectral properties of adjacency or stochastic matrices associated to networks based on algebraic techniques, whereas the applications to Complex Systems Theory have been essentially based on the methods of Statistical Mechanics. We use an approach to the problem using the results of the RMT in connection with some statistical mechanics techniques. RMT is of use in clarifying the physical meaning of Wigner law for large, random networks, and to compute its corrections. Then, we try to build a bridge between the sound results of RMT and Renormalisation Group methods in order to investigate the spectral properties of the Scale Free and Small World class of networks. In particular, we propose a mechanism who could help understanding the behaviour of self similar network structures with low diameter.