I. This minicourse will start with a survey talk about the three main conjectures in quantum chaos. If we consider a chaotic classical hamiltonian system, and consider it from the point of view of quantum mechanics, we have : - the Bohigas-Gianonni-Schmit conjecture, according to which the spacing of eigenvalues should obey the Wigner statistics; - the quantum unique ergodicity conjecture, according to which the stationary wave functions should occupy the phase space uniformly; - the Berry conjecture, according to which the stationary wave functions should resemble, on a certain scale, a random gaussian process.
II. A detailed proof of the quantum ergodicity theorem (or Shnirelman theorem) will be given : when an ergodic classical system is quantized, then the eigenfunctions of the corresponding Schrödinger operator occupy the phase space uniformly in the semiclassical (small wavelength) limit -- except possibly for a scarce family.
III. We will then describe recent work of Anantharaman & Sabri, extending the scope of quantum ergodicity to graphs. Here we deal with the discrete laplacian (or more general Schrödinger operators) on a finite graph, and the semiclassical limit is to be understood as the size of the graph going to infinity. Under certain geometric and spectral assumptions, we prove that the eigenfunctions -- except possibly for a scarce family -- occupy the vertices uniformly.
IV. Finally, we discuss recent work of Backhausz & Szegedy, proving that eigenfunctions of the laplacian on random regular graphs are gaussian -- this proves a variant of the Berry conjecture on random regular graphs.