Fragmentations and the convex hull of the Brownian motion in a disk

• Bénédicte Haas

Room: C516 (lunch in Espace 2/D206) Motivated by the study of the convex hull of the trajectory of a Brownian motion in the unit disk reflected orthogonally at its boundary, we study inhomogeneous fragmentation processes in which particles of mass $m \in (0,1)$ split at a rate proportional to $|\log m|^{-1}$. These processes do not belong to the well-studied family of self-similar fragmentation processes. Our main results characterize the Laplace transform of the typical fragment of such a process, at any time, and its large time behavior. We will then connect this asymptotic behavior to the prediction obtained by physicists for the growth of the perimeter of the convex hull of a Brownian motion in the disc reflected at its boundary, and also describe the large time asymptotic behavior of the whole fragmentation process. Based on a joint work with Bastien Mallein.

Hausdorff dimension measurements of 3D random geometries

• Alicia Castro

Room: C516 (lunch in Espace 2/D206) In this talk, I present results and techniques to measure the Hausdorff dimension of two types of random geometries. The first is a generalization of the mating of trees approach. The original mating of trees encodes Liouville Quantum Gravity on the 2-sphere in terms of a correlated Brownian motion describing a pair of random trees. We extended this approach to higher-dimensional correlated Brownian motions, leading to a family of non-planar random graphs that belong to new universality classes of scale-invariant random geometries. The second example of random geometry is the D-random feuilletage, which for D=2 agrees with a family of planar maps, while for D>2, these represent new universality classes of random geometries. We developed numerical methods to efficiently simulate these random graphs and explore their scaling limits through distance measurements, allowing us to estimate Hausdorff dimensions.

Scaling limit of first passage percolation geodesics on planar maps

• Emmanuel Kammerer

Room: C516 (lunch in Espace 2/D206) Consider the first passage percolation distance on random planar maps, which is obtained by putting i.i.d. exponential random lengths on each (dual) edge. The study of this distance is often simpler than the study of the (dual) graph distance. I will describe a time-reversal of the uniform peeling exploration, which enables to obtain the scaling limit of the number of faces along the geodesics to the root, to compare the metric balls for the first passage percolation and the dual graph distance and to upperbound the diameter of large random maps. Then, I will obtain the scaling limit of the tree of first passage percolation geodesics to the root via a stochastic coalescing flow of pure jump diffusions. This stochastic flow also enables to construct some random metric spaces which I conjecture to be the scaling limit of random planar maps with high degrees.

Slit-slide-sew bijections for oriented planar maps

• Jérémie Bettinelli

Room: C516 (lunch in Espace 2/D206) We construct growth bijections for bipolar oriented planar maps and for Schnyder woods. These give direct combinatorial proofs of several counting identities for these objects. Our method mainly uses two ingredients. First, a slit-slide-sew operation, which consists in slightly sliding a map along a well-chosen path. Second, the study of the orbits of natural rerooting operations on the considered classes of oriented maps.