Absence of WARM percolation in the very strong reinforcement regime

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 $\mathit{\alpha }\gg 1$ (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.