Journal article

ABSENCE OF WARM PERCOLATION IN THE VERY STRONG REINFORCEMENT REGIME

Christian Hirsch, Mark Holmes, Victor Kleptsyn

ANNALS OF APPLIED PROBABILITY | INST MATHEMATICAL STATISTICS-IMS | Published : 2021

Abstract

We study a class of reinforcement models involving a Poisson process on the vertices of certain infinite graphs G. When a vertex fires, one of the edges incident to that vertex is selected. The edge selection is biased towards edges that have been selected many times previously, and a parameter α governs the strength of this bias. We show that for various graphs (including all graphs of bounded degree), if α 1 (the very strong reinforcement regime) then the random subgraph consisting of edges that are ever selected by this process does not percolate (all connected components are finite). Combined with results appearing in a companion paper, this proves that on these graphs, with α sufficient..

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University of Melbourne Researchers

Grants

Awarded by Danish Council for Independent Research-Natural Sciences


Awarded by Centre for Stochastic Geometry and Advanced Bioimaging - Villum Foundation


Awarded by Australian Research Council's Discovery Programme


Awarded by project ANR Gromeov


Funding Acknowledgements

The first author was supported by The Danish Council for Independent Research-Natural Sciences, grant DFF-7014-00074 Statistics for point processes in space and beyond, and by the Centre for Stochastic Geometry and Advanced Bioimaging, funded by grant 8721 from the Villum Foundation. The second author was supported under the Australian Research Council's Discovery Programme (Future Fellowship project number FT160100166). The third author was partially supported by the project ANR Gromeov (ANR-19-CE40-0007)