Recurrence versus transience for weight-dependent random connection models

By Peter Gracar, Markus Heydenreich, Christian Monch, Peter Morters. Electronic Journal of Probability, 2019

47 citations recorded by the source corpus.

We investigate a large class of random graphs on the points of a Poisson process in $d$-dimensional space, which combine scale-free degree distributions and long-range effects. Every Poisson point carries an independent random weight and given weight and position of the points we form an edge between two points independently with a probability depending on the two weights and the distance of the points. In dimensions $d\in\{1,2\}$ we completely characterise recurrence vs transience of random walks on the infinite cluster. In $d\geq 3$ we prove transience in all cases except for a regime where we conjecture that scale-free and long-range effects play no role. Our results are particularly interesting for the special case of the age-dependent random connection model recently introduced in [P. Gracar\ et al., The age-dependent random connection model, Queueing Syst. {\bf 93} (2019), no.~3-4, 309--331. MR4032928].

Subfields: Mathematical Physics, Statistical and Nonlinear Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/2f604bae5048e968ec9d708790b0dc64d12d7277