Abstract I:
We investigate natural generalizations of the notion of Benjamini-Schramm local weak limit, to the settings of dynamical graphs, which for us means graphs that evolve over time but keep a fixed vertex set. We obtain a convergence result for the natural family of dynamical inhomogeneous random graphs, determined by arbitrary connection and updating kernels. The limit is a dynamical version of the (multitype) Poisson Galton-Watson tree.
Abstract II:
The Brownian sphere is a random metric space, homeomorphic to the two-dimensional sphere, which arises as the universal scaling limit of many types of random planar maps. The direct construction of the Brownian sphere is via a continuous analogue of the Cori-Vauquelin-Schaeffer (CVS) bijection. The CVS bijection maps labeled trees to planar maps, and the continuous version maps Aldous' continuum random tree with Brownian labels (the Brownian snake) to the Brownian sphere. In this work, we describe the inverse of the continuous CVS bijection, by constructing the Brownian snake as a measurable function of the Brownian sphere. Special care is needed to work with the orientation of the Brownian sphere.
Biography:
Emmanuel Jacob is a Visiting Associate Professor at NYU Shanghai. He has been an Associate Professor at École Normale Supérieure de Lyon in France, after he received his PhD in 2010 from Université Pierre et Marie Curie in Paris, France. Jacob's research interests span several areas of probability theory, particularly in connection with complex systems, graphs, and geometry. His recent work includes continuum percolation models featuring long-range and inhomogeneity effects, interacting particle systems such as the contact process on static or dynamic scale-free networks, and the study of random maps and the Brownian sphere.
Seminar by the NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai
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