Percolation Today is an online seminar aiming to promote the exchange of exciting new ideas on both Bernoulli and dependent percolation processes, at a level appropriate for specialists in the area. We hope to be a focal point to foster new connections and collaborations between the global community of percolation lovers.
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Distances in Scale-free Percolation
Date: Tuesday June 22, 16:15 CEST
Speakers: Nannan Hao & Markus Heydenreich (LMU Munich)
We study an inhomogeneous version of long-range percolation known as scale-free percolation. Every vertex in
is equipped with an i.i.d. weight, and two vertices are linked by an edge with probability depending on the Euclidean distance and the weight of the two vertices involved. This model, introduced by Deijfen, van der Hofstad and Hooghiemstra in 2013, exhibits a rich phase diagram w.r.t. graph distances and percolation properties.
In the first part of the talk, we introduce the model, describe some of the recent developments and discuss a far-reaching generalisation. The main focus of the talk is then on graph distances, in particular in a regime where the graph distance between two vertices is (poly-)logarithmic w.r.t. the Euclidean distance. We explain how a hierarchical argument developed by Biskup (2004) can be extended to the inhomogeneous setting.
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Adèle Ivain and Laurie Quétel