On an operator approach for hyperbolic random connection models

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This talk will be broadcast at 16:00 BST / 17:00 CEST / 18:00 EEST, May 28th, 2025 on MS Teams only.

Meeting ID: 393 527 089 878
Passcode: dsm7py

Please mind the change of the usual time due to the huge time difference from Vancouver to Europe.

Abstract

Random connection models form a family of random graph models. In our case the vertex set is distributed as a Poisson point process (with some intensity $\lambda>0$) on hyperbolic $d$-space, and an edges exists between vertices $x$ and $y$ independently with a given probability that is dependent only on $\mathrm{dist}(x,y)$. Under very mild conditions, these models exhibit a critical intensity $\lambda_c$ such that all connected components of the graph are almost surely finite for $\lambda<\lambda_c$, and there almost surely exists an infinite connected component for $\lambda>\lambda_c$. This talk will aim to show how formulating connection probabilities as an operator on different $L^p$ spaces can show: the existence of a phase with infinitely many infinite components, mean-field critical behaviour, and an asymptotic expansion for $\lambda_c$ in a long-range perturbative regime.

About Matthew

Matthew is currently a postdoc at the University of British Columbia. His main fields of interest cover various random marked point processes - including spatial random graph models and marked random connection, as well as bosonic loop soup models and random interlacements. Most interestingly, these models exhibit phase transitions in their percolation and condensation behaviour.