LA Probability Forum

Series: Caltech/UCLA Joint Probability Seminar Series

An Expansion for the Critical Intensity of Random Connection Models

Matthew Dickson, Department of Mathematics, University of British Columbia,

UCLA, Math Sciences Bldg., Rm 6627

Random connection models (RCMs) are random graph models where the vertices are given by a Poisson point process with a given intensity, $\lambda>0$, and the edges exist independently with a probability that depends upon the relative positions of the two vertices in question. These models include ``Poisson blob models", such as the Gilbert disc model. As we vary $\lambda$, we observe a percolation phase transition at a critical intensity $\lambda_c>0$. Finding $\lambda_c$ is only possible in very exceptional cases, so here we use the lace expansion for the RCM (as found by Heydenreich, van der Hofstad, Last, and Matzke 2022) to find a high-dimension asymptotic expansion for the critical intensity. This is based on arXiv:2309.08830 with Markus Heydenreich (Universität Augsburg).

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