It is known, since the work of Dynkin building on pioneering ideas of Symanzik, that Markov processes and Markovian random fields are deeply related. In the simplest discrete setup of graphs, on which we shall focus in this talk, this relation links random walk paths to the discrete Gaussian free field.

At the formal level, this relation stems from the fact that a same operator, the discrete Laplacian, generates two types of processes: (1) by means of its semi-group (this is the random walk), (2) by means of its Dirichlet form (this is the discrete Gaussian free field).

At a refined probabilistic level, it is known that the local time (i.e. the time spent at each vertex) of random walk paths is related to the square of the discrete Gaussian free field. This is Dynkin's isomorphism, which admits further variations and extensions, including theorems of Eisenbaum, Le Jan, and Sznitman.

In this talk, I will consider more general functionals of a path than its local time. These encode more faithfully the actual geometry of the trajectory and I will explain how they are related to vectorial versions of the Gaussian free field. Time permitting, I will say a few words about the related topics of gauge theory. No prior knowledge will be assumed.

Joint work with Thierry Lévy (Univ. Paris 6).