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le (56m5s)

Eternal family trees and dynamics on unimodular random graphs (workshop ERC Nemo Processus ponctuels et graphes aléatoires unimodulaires)

This talk is centered on covariant dynamics on unimodular random graphs and random networks (marked graphs), namely maps from the set of vertices to itself which are preserved by graph or network isomorphisms. Such dynamics are referred to as vertex-shifts here. These dynamics have point-shifts on point processes as a subclass. First we give a classification of vertex-shifts on unimodular random networks. Each such vertex-shift partitions the vertices into a collection of connected components and foils. The latter are discrete analogues the stable manifold ...
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Conférences

le (51m43s)

On the notion of dimension of unimodular discrete spaces (workshop ERC Nemo Processus ponctuels et graphes aléatoires unimodulaires)

In this talk we will define notions of dimension for unimodular random graphs and point-stationary point processes. These notions are in spirit similar to the Minkowski dimension and the Hausdorff dimension. The key point in the definitions is the use of the mass transport principle which is used indispensably and distinguishes this view point from the previous notions which are defined in the literature. The connections of these definitions to volume growth and other notions of dimension are also discussed, which provide a toolset ...
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le (43m39s)

A stable marriage between order and disorder (workshop ERC Nemo Processus ponctuels et graphes aléatoires unimodulaires)

Stable matchings were introduced in a seminal paper by Gale and Shapley (1962) and play an important role in economics. Following closely Holroyd, Pemantle, Peres and Schramm (2009), we shall first discuss a few basic properties of stable matchings between two discrete point sets (resp. point processes) in Euclidean space, where the points prefer to be close to each other. For comparison we also discuss stable transports from Lebesgue measure to point processes. In the second part of the talk we consider a stable ...
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