LA Probability Forum

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).