I will discuss a model of interacting particles in continuous space which is reversible with respect to Poisson point measures with constant density. Similar discrete models are known to ”homogenize”, in the sense that the evolution of the particle density can be approximated by the solution to a partial differential equation over large scales. The goal of the talk is to present some results that make this approximation quantitative.
Based on joint works with Arianna Giunti, Chenlin Gu and Maximilian Nitzschner.[-]

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.[-]

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.[-]

The two-periodic Aztec diamond is a dimer or random tiling model with three phases, solid, liquid and gas. The dimers form a determinantal point process with a somewhat complicated but explicit correlation kernel. I will discuss in some detail how the Airy point process can be found at the liquid-gas boundary by looking at suitable averages of height function differences. The argument is a rather complicated analysis using the cumulant approach and subtle cancellations. Joint work with Vincent Beffara and Sunil Chhita.[-]