The propagation of chaos phenomenon states roughly that a large system of weakly interacting particles will remain approximately independent for all times if initialized as such. This can be quantified in terms of the distance between low-dimensional marginal distributions and suitably chosen product measures. This talk will discuss some recent sharp quantitative results of this nature, both for classical mean field diffusions and for more recently studied non-exchangeable models. These results are driven by a new "local" relative entropy method, in which low-dimensional marginals are estimated iteratively by adding one coordinate at a time, leading to surprising improvements on prior results obtained by "global" arguments such as subadditivity inequalities. In the non-exchangeable setting, we exploit a surprising connection with first-passage percolation.[-]

The infinite bin model (IBM) is a family of ranked-biased branching random walks on the integers, parameterized by a probability distribution on positive integers. Alternatively it may be seen as a family of interacting particle systems, depicted as balls inside bins. The speed of the front of the IBM depends on the probability distribution which parameterizes it. I will first review some special cases that have been known for some time: the IBM parameterized by the uniform distribution on some finite interval of integers [1,N], which is nothing but a branching random walk with selection, and the IBM parameterized by a geometric distribution, which can be coupled with last passage percolation on the complete graph. Then I will discuss long memory properties of the IBM, in particular whether a site may reproduce infinitely often or not. Finally I will discuss a hydrodynamic limit of the IBM where one can explicitly compute the speed of the front. In that case, a wall-crossing phenomenon appears and Dyck paths come into play. The talk is based on joint works with Bastien Mallein (Université Toulouse III Paul Sabatier), Arvind Singh (CNRS and Université Paris-Saclay) and Benjamin Terlat (Université Paris-Saclay).[-]

The $O(n)$ model can be formulated in terms of loops living on the lattice, with n the fugacity per loop. In two dimensions, it is known to possess a rich critical behavior, involving critical exponents varying continuously with n. In this talk, we will consider the case where the model is ”coupled to 2D quantum gravity”, namely it is defined on a random map.
It has been known since the 90's that the partition function of the model can be expressed as a matrix integral, which can be evaluated exactly in the planar limit. A few years ago, together with G. Borot and E. Guitter, we revisited the problem by a combinatorial approach, which allows to relate it to the so-called Boltzmann random maps, which have no loops but faces of arbitrary (and controlled) face degrees. In particular we established that the critical points of the $O(n)$ model are closely related to the ”stable maps” introduced by Le Gall and Miermont.
After reviewing these results, I will move on to a more recent work done in collaboration with G. Borot and B. Duplantier, where we study the nesting statistics of loops. More precisely we consider loop configurations with two marked points and study the distribution of the number of loops separating them. The associated generating function can be computed exactly and, by taking asymptotics, we show that the number of separating loops grows logarithmically with the size of the maps at a (non generic) critical point, with an explicit large deviation function. Using a continuous generalization of the KPZ relation, our results are in full agreement with those of Miller, Watson and Wilson concerning nestings in Conformal Loop Ensembles.[-]

We introduce a new strategy for the solution of Mean Field Games in the presence of major and minor players. This approach is based on a formulation of the fixed point step in spaces of controls. We use it to highlight the differences between open and closed loop problems. We illustrate the implementation of this approach for linear quadratic and finite state space games, and we provide numerical results motivated by applications in biology and cyber-security.[-]