"Quantum ergodicity" usually deals with the study of eigenfunctions of the Laplacian on Riemannian manifolds, in the high-frequency asymptotics. The rough idea is that, under certain geometric assumptions (like negative curvature), the eigenfunctions should become spatially uniformly distributed, in the high-frequency limit. I will review the many conjectures in the subject, some of which have been turned into theorems recently. Physicists like Uzy Smilansky or Jon Keating have suggested looking for similar questions and results on large (finite) discrete graphs. Take a large graph  and an eigenfunction ψ� of the discrete Laplacian -- normalized in . What can we say about the probability measure (x\in V)? Is it close to uniform, or can it, on the contrary, be concentrated in small sets? I will talk about ongoing work with Etienne Le Masson, in the case of large regular graphs.

Motivated by a celebrated work of Erdös and Rényi, we consider Bernoulli bond percolation on a tree with size , where the parameter of percolation depends on the size of the tree.
Our purpose is to investigate the asymptotic behavior of the sizes of the largest clusters for appropriate regimes.
We shall first provide a simple characterization of tree families and percolation regimes which yield giant clusters, answering a question raised by David Croydon.
In the second part, we will review briefly recent results concerning two natural families of random trees with logarithmic heights, namely recursive trees and scale-free trees.
We shall see that the next largest clusters are almost giant, in the
sense that their sizes are of order , and obtain precise limit theorems in terms of certain Poisson random measures. A common feature in the analysis of percolation for these models is that, even though one addresses a static problem, it is useful to consider dynamical versions in which edges are removed, respectively vertices are inserted, one after the other in certain order as time passes.

Heuristically, one can give arguments why the fluctuations of classical models of statistical mechanics near criticality are typically expected to be described by nonlinear stochastic PDEs. Unfortunately, in most examples of interest, these equations seem to make no sense whatsoever due to the appearance of infinities or of terms that are simply ill-posed. I will give an overview of a new theory of "regularity structures" that allows to treat such equations in a unified way, which in turn leads to a number of natural conjectures. One interesting byproduct of the theory is a new (and rigorous) interpretation of "renormalisation group techniques" in this context. At the technical level, the main novel idea involves a complete rethinking of the notion of "Taylor expansion" at a point for a function or even a distribution. The resulting structure is useful for encoding "recipes" allowing to multiply distributions that could not normally be multiplied. This provides a robust analytical framework to encode renormalisation procedures.