Scaling limits of random walks and their related parameters on critical random trees and graphs

In this thesis we study random walks in random environments, a major area in Probability theory. Within this broad topic, we are mainly focused in studying scaling limits of random walks on random graphs at criticality, that is precisely when we witness the emergence of a giant component that has size proportional to the number of vertices of the graph. Critical random graphs of interest include critical Galton-Watson trees and maximal components that belong to the Erd}os- R_enyi universality class. The first part of the thesis expands upon using analytic and geometric properties of those random graphs to establish distributional convergence of certain graph parameters, such as the blanket time. Our contribution refines the previous existing results on the order of the mean blanket time. The study of this problem can be seen as a stepping stone to deal with the more delicate problem of establishing convergence in distribution of the rescaled cover times of the discrete-time walks in each of the applications of our main result. Relying on powerful resistance techniques developed in recent years, another part of the thesis investigates random walks in random enviroments on tree-like spaces and their scaling limits in a certain regime, that is when the potential of the random walk in random environment converges. Results include novel scaling continuum limits of a biased random walk on large critical branching random walk and a self-reinforced discrete process on size-conditioned critical Galton-Watson trees. In both cases the diffusions that are not on natural scale are identified as Brownian motions on a continuum random fractal tree with its natural metric replaced by a distorted resistance metric