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Résumé |
Many classes of random planar maps, i.e. planar graphs embedded in the
sphere modulo homeomorphisms, possess scaling limits (in a
Gromov-Hausdorff sense) described by a universal random continuum
metric space known as the Brownian map. One way to escape this
universality class is to consider certain Boltzmann planar maps with
carefully tuned weights on the faces. Le Gall and Miermont have shown
that the scaling limits of these fall into a larger class of continuum
random metric spaces, often referred to as the stable maps. In this
talk I will consider the geometry of the dual (in the planar map
sense) of these Boltzmann planar maps, which shows quite different
scaling behavior. In particular, I will discuss the growth of the
perimeter and volume of certain geodesic balls of increasing radius
based on the examination of a peeling process. For a particular range
of parameters, in the so-called the dilute phase, precise scaling
limits may be obtained for these processes, which suggests possibility
of a non-trivial scaling limit in the Gromov-Hausdorff sense. Finally I
will discuss a more detailed scaling limit in terms of growth-fragmentation
processes.
Based on work with Jean Bertoin, Nicolas Curien, and Igor Kortchemski. |
