Absence of WARM percolation in the very strong reinforcement regime

Christian Hirsch; Mark Holmes; Victor Kleptsyn

doi:10.1214/20-AAP1587

Absence of WARM percolation in the very strong reinforcement regime

Christian Hirsch^*, Mark Holmes, Victor Kleptsyn

^*Corresponding author for this work

Research output: Contribution to journal/Conference contribution in journal/Contribution to newspaper › Journal article › Research › peer-review

5 Citations (Scopus)

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 sufficiently large, all connected components are in fact trees. If the Poisson firing rates are constant over the vertices, then these trees are of diameter at most 3. The proof of nonpercolation relies on coupling with a percolation-type model that may be of interest in its own right.

Original language	English
Journal	Annals of Applied Probability
Volume	31
Issue	1
Pages (from-to)	199-217
Number of pages	19
ISSN	1050-5164
DOIs	https://doi.org/10.1214/20-AAP1587
Publication status	Published - Feb 2021
Externally published	Yes

Keywords

Reinforcement
Polya's urn
percolation
coupling
Poisson point process

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10.1214/20-AAP1587

Cite this

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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 sufficiently large, all connected components are in fact trees. If the Poisson firing rates are constant over the vertices, then these trees are of diameter at most 3. The proof of nonpercolation relies on coupling with a percolation-type model that may be of interest in its own right.",

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T1 - Absence of WARM percolation in the very strong reinforcement regime

AU - Hirsch, Christian

AU - Holmes, Mark

AU - Kleptsyn, Victor

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N2 - 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 sufficiently large, all connected components are in fact trees. If the Poisson firing rates are constant over the vertices, then these trees are of diameter at most 3. The proof of nonpercolation relies on coupling with a percolation-type model that may be of interest in its own right.

AB - 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 sufficiently large, all connected components are in fact trees. If the Poisson firing rates are constant over the vertices, then these trees are of diameter at most 3. The proof of nonpercolation relies on coupling with a percolation-type model that may be of interest in its own right.

KW - Reinforcement

KW - Polya's urn

KW - percolation

KW - coupling

KW - Poisson point process

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DO - 10.1214/20-AAP1587

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SP - 199

EP - 217

JO - Annals of Applied Probability

JF - Annals of Applied Probability

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ER -

Absence of WARM percolation in the very strong reinforcement regime

Abstract

Keywords

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