Spectral signature of Breaking of ensemble equivalence

In this thesis we explore the concept of Breaking of Ensem- ble Equivalence (BEE) within the context of random graph models, focusing on spectral properties of adjacency matri- ces. Our research aims to identify spectral quantities that can distinguish between different random graph ensembles, thereby providing new insights into the structure and behav- ior of complex networks. We cover both theoretical aspects and practical implications, including simulations and sam- pling methods for random graph models. In Chapter 1 we introduce some basic notions of random graph theory, and discuss how maximum entropy graph models are fundamental in modeling real-world networks. We explain what BEE is, what is its characterization in the context of sta- tistical mechanics, and how it is intimately connected to dif- ferences that arise naturally between the canonical versus the microcanonical description of random graph ensembles. In order to do so, we delve into the spectral theory of random graphs and use it to investigate BEE. In Chapter 2 we formulate a conjecture on the equivalence of measure-BEE and the presence of a gap between the largest non-centered and non-scaled largest eigenvalues of the adja- cency matrix in the canonical and the microcanonical ensem- ble. We prove this conjecture in the setting of homogeneous graphs. In Chapter 3 we study the same question for Chung-Lu ran- dom graphs. In particular, we prove central limit theorems for the largest eigenvalue and its associated eigenvector.