arXiv subjects
Publications and source records attributed to Grégory Marc Miermont.
We give a new proof of a theorem by Le Gall & Paulin, showing that scaling limits of random planar quadrangulations are homeomorphic to the 2-sphere. The main geometric tool is a reinforcement of the notion of Gromov-Hausdorff convergence, called 1-regular convergence, that preserves topological properties of metric surfaces.
We prove that critical multitype Galton-Watson trees converge after rescaling to the Brownian continuum random tree, under the hypothesis that the offspring distribution has finite covariance matrices. Our study relies on an ancestral decomposition for marked multitype trees. We then couple the genealogical structure with a spatial motion, whose step distribution may depend on the structure of the tree in a local way, and show that the resulting discrete spatial trees converge once suitably rescaled to the Brownian snake, under some suitable moment assumptions.
We consider a generalized version of Knuth's parking problem, in which caravans consisting of a number of cars arrive at random on the unit circle. Then each car turns clockwise until it finds a free space to park. Extending a recent work by Chassaing and Louchard, we relate the asymptotics for the sizes of blocks formed by occupied spots with the dynamics of the additive coalescent. According to the behavior of the caravan's size tail distribution, several qualitatively different versions of eternal additive coalescent are involved.