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.[-]

Consider two ancestral lineages sampled from a system of two-dimensional branching random walks with logistic regulation in the stationary regime. We study the asymptotics of their coalescence time for large initial separation and find that it agrees with well known results for a suitably scaled two-dimensional stepping stone model and also with Malécot's continuous-space approximation for the probability of identity by descent as a function of sampling distance.
This can be viewed as a justification for the replacement of locally fluctuating population sizes by fixed effective sizes. Our main tool is a joint regeneration construction for the spatial embeddings of the two ancestral lineages.[-]

We study an undirected polymer chain living on the one-dimensional integer lattice and carrying i.i.d. random charges. Each self-intersection of the polymer chain contributes an energy to the interaction Hamiltonian that is equal to the product of the charges of the two monomers that meet. The joint probability distribution for the polymer chain and the charges is given by the Gibbs distribution associated with the interaction Hamiltonian. We analyze the annealed free energy per monomer in the limit as the length of the polymer chain tends to infinity. We derive a spectral representation for the free energy and use this to show that there is a critical curve in the (charge bias, inverse temperature)-plane separating a ballistic phase from a subballistic phase. We show that the phase transition is first order, identify the scaling behaviour of the critical curve for small and for large charge bias, and also identify the scaling behaviour of the free energy for small charge bias and small inverse temperature. In addition, we prove a large deviation principle for the joint law of the empirical speed and the empirical charge, and derive a spectral representation for the associated rate function. This in turn leads to a law of large numbers and a central limit theorem. What happens for the quenched free energy per monomer remains open. We state two modest results and raise a few questions. Joint work with F. Caravenna, N. Petrelis and J. Poisat[-]

We analyse patterns of genetic variability of populations in the presence of a large seed bank with the help of a new coalescent structure called seed bank coalescent. This ancestral process appears naturally as scaling limit of the genealogy of large populations that sustain seed banks, if the seed bank size and individual dormancy times are of the same order as the active population. Mutations appear as Poisson process on the active lineages, and potentially at reduced rate also on the dormant lineages. The presence of ‘dormant' lineages leads to qualitatively altered times to the most recent common ancestor and non-classical patterns of genetic diversity. To illustrate this we provide a Wright-Fisher model with seed bank component and mutation, motivated from recent models of microbial dormancy, whose genealogy can be described by the seed bank coalescent. Based on our coalescent model, we derive recursions for the expectation and variance of the time to most recent common ancestor, number of segregating sites, pairwise differences, and singletons. Commonly employed distance statistics, in the presence and absence of a seed bank, are compared. The effect of a seed bank on the expected site-frequency spectrum is also investigated. Our results indicate that the presence of a large seed bank considerably alters the distribution of some distance statistics, as well as the site-frequency spectrum. Thus, one should be able to detect the presence of a large seed bank in genetic data. Joint work with Bjarki Eldon, Adrián González Casanova, Noemi Kurt, Maite Wilke-Berenguer[-]

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.[-]