On discrete surfaces: Enumerative geometry, matrix models and universality classes via topological recursion

The main objects under consideration in this thesis are called maps, a certain class of graphs embedded on surfaces. Our problems have a powerful relatively recent tool in common, the so-called topological recursion (TR) introduced by Chekhov, Eynard and Orantin. We call a map fully simple if it has non self-intersecting disjoint boundaries, and ordinary if such a restriction is not imposed. We study the combinatorial relation between fully simple and ordinary maps with the topology of a disk or a cylinder, which reproduces relations between moments and free cumulants established in the context of free probability. We propose a combinatorial interpretation of the exchange symplectic transformation of TR. We provide a matrix model interpretation for fully simple maps via the formal hermitian matrix model with external field and deduce a universal relation between generating series of fully simple and ordinary maps, which involves double monotone Hurwitz numbers. In particular, we obtain an ELSV-like formula for double $2$-orbifold strictly monotone Hurwitz numbers. We consider ordinary maps endowed with an $O(mathsf{n})$ loop model, which is a classical model in statistical physics, and determine which shapes are more likely to occur by looking at the nesting properties of the loops decorating the maps. We want to study the limiting objects when the number of vertices becomes arbitrarily large, which can be done by studying the generating series at dominant singularities. We analyze the nesting statistics in the $O(mathsf{n})$ loop model on random maps of arbitrary topologies in the presence of large and small boundaries, relying on previous results for disks and cylinders and TR for this model. We study the generating series of maps which realize a fixed nesting graph and characterize their critical behavior in the dense and dilute phases.