This thesis studies spectral properties of self-adjoint operators associated with complex (quantum) systems.

In the first chapter, we introduce a model for a randomly perturbed graph: independently at every vertex x of a finite graph G, we draw a random family of rooted graphs and connect x to the root of each graph in the family. We study the density of states of the resulting "sprinkled" graph in the limit of large G.

In the second chapter, we develop a framework that allows us to study the bottom of the spectrum of a self-adjoint operator H using renewal theory. We apply this framework in the subsequent chapters to two models of matter-field interaction: the polaron model and the spin-boson model. For the polaron, we study the energy-momentum relation and the effective mass. For the spin-boson model, we study the existence and absence of ground states.