The study of fine geometric properties of sparse random graphs and of random maps has been a very active field for decades. More recently, a whole branch of the theory has been devoted to the study of random processes living on these objects. In particular random walks, percolation models, Ising and Potts model, or interacting particle systems like the contact process have been considered with motivations ranging from theoretical physics to biological sciences. The project ProGraM gathers young mathematicians as well as a graduate student and a postgraduate student that are experts in probability theory, combinatorics and theoretical physics. It aims at understanding the behavior of processes coming from statistical mechanics on local limits of random graphs and random maps, and how such processes can shed some light about the geometry of the underlying objects through four axes. 1- Random maps coupled with matter. We will study the geometric properties of random maps coupled with an Ising model. Their asymptotic properties are conjectured to be identical to those of the Brownian map when the Ising model is not critical. On the other hand, taking a critical Ising model should bring these models out of this universality class. 2- Exploration algorithms. The breadth and depth first search algorithms on a connected graph can be seen as random walks whose trace form a spanning tree of the graph. On a large number of classical models of random graphs, we want to show that the profile of this spanning tree converges towards a deterministic limit. The study of these limits and variants of these algorithms will then allow us to obtain information on long paths in the graph. 3- Adjacency spectrum. The study of the spectral properties of large random graphs is in full swing. We want to contribute via the information on long paths that we will get with the previous point. We also intend to study the influence of vertices of abnormally large degrees on the presence of outliers in the spectrum of a graph. 4- Interacting particles. The behavior of interacting particle systems modeling the propagation of an infection or a rumor on a graph is highly dependent on the geometric properties of the graph. In particular, the contact process is very sensitive to the distribution of vertices of high degrees in the graph if these degrees are unbounded. We will explore this link further by using an original percolation process introduced by two members of the project.