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# Résultats de recherche

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## Emergence of extended states at zero in the spectrum of sparse random graphs (workshop ERC Nemo Processus ponctuels et graphes aléatoires unimodulaires)

We confirm the long-standing prediction that c=e≈2.718 is the threshold for the emergence of a non-vanishing absolutely continuous part (extended states) at zero in the limiting spectrum of the Erdős-Renyi random graph with average degree c. This is achieved by a detailed second-order analysis of the resolvent (A−z)−1 near the singular point z=0, where A is the adjacency operator of the Poisson-Galton-Watson tree with mean offspring c. More generally, our method applies to arbitrary unimodular Galton-Watson trees, yielding explicit criteria for the presence or ... Voir la vidéole (55m56s)

## A notion of entropy for limits of sparse marked graphs (workshop ERC Nemo Processus ponctuels et graphes aléatoires unimodulaires)

Bordenave and Caputo (2014) defined a notion of entropy for probability distributions on rooted graphs with finite expected degree at the root. When such a probability distribution \rho has finite BC entropy \Sigma(\rho), the growth in the number of vertices n of the number of graphs on n vertices whose associated rooted graph distribution is close to \rho is as d/2 n \log n + \Sigma(\rho) n + o(n), where d is expected degree of the root under \rho. We develop the parallel result for probability distributions on marked rooted graphs. Our graphs have vertex marks drawn from a finite ... Voir la vidéole (59m49s)

## Sampling cluster point processes: a review (workshop ERC Nemo Processus ponctuels et graphes aléatoires unimodulaires)

The theme of this talk is the sampling of cluster and iterated cluster point processes. It is partially a review, mainly of the Brix–Kendall exact sampling method for cluster point processes and its adaptation by Moller and Rasmussen to Hawkes branching point processes on the real line with light-tail fertility rate. A formal proof via Laplace transforms of the validity of the method in terms of general clusters that are not necessarily point processes fits this purpose and allows to include the exact sampling of Boolean ... Voir la vidéole (52m52s)

## Absence of percolation for Poisson outdegree-one graphs (workshop ERC Nemo Processus ponctuels et graphes aléatoires unimodulaires)

A Poisson outdegree-one graph is a directed graph based on a marked Poisson point process such that each vertex has only one outgoing edge. We state the absence of percolation for such graphs satisfying two assumptions. The Shield assumption roughly says that the graph is locally determined with possible random horizons. The Loop assumption ensures that any forward branch merges on a loop provided that the Poisson point process is augmented with a finite collection of well-chosen points. This result allows to solve a ... Voir la vidéole (53m51s)

## Subdiffusivity of random walks on random planar maps, via stationarity (workshop ERC Nemo Processus ponctuels et graphes aléatoires unimodulaires)

Random planar maps have been the subject of numerous studies over the last years. They are instance of stationary and reversible random planar maps exhibiting a non-conventional geometry at large scale. Because of their “fractal” geometry, the simple random walk on these random graphs is believed to be subdiffusive, i.e. it displaces slowler than in the regular grid case. We will propose an approach to such results strongly based on the stationary of these random graphs, i.e. the fact that their distributions is invariant under ... Voir la vidéole (46m11s)

## Central Limit theorem for quasi-local statistics of point processes with fast decay of correlations (workshop ERC Nemo Processus ponctuels et graphes aléatoires unimodulaires)

We shall consider Euclidean stationary point processes which have fast decay of correlations i.e., their correlation functions factorize upto an additive error decaying exponentially in the separation distance. By a quasi-local statistic of the point process, we refer to statistics that can be expressed as sum of contributions from the points and the contribution of every point being determined by a random ball around the point whose radius has an exponential tail. There are many well-known point processes and statistics that satisfy these conditions ... Voir la vidéole (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 ... Voir la vidéole (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 ... Voir la vidéole (43m39s)