Percolation on Soft Random Geometric Graphs

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The goal of this project is to investigate percolation results on Soft Random Geometric Graphs (SRGGs). The SRGG is a random connection model embedded in some metric space, usually we choose the standard Euclidean space.More precisely, we build a graph G in our space in two steps. We first choose the vertices by a homogenous Poisson point process. We then randomly connect points together based on their distance from each other. We can describe the probability of connection by a connection function. One such example would be a function which prescribes probability p to all pairs of points which have distance less than 1 to each other, and probability 0 else.Given such a graph we can now investigate its properties. This research project aims to study large-scale connectivity properties of such models, what mathematicians call percolation theory (and physicists call statistical physics). One classical result in this field is understanding what conditions are required for infinite components to exist - this property is called percolation. The percolation probability is then the probability that a specific node is connected to the infinite cluster (usually the node at zero). More specifically, critical parameters exist, which if increased always lead to infinite clusters, and if decreased always lead to their absence. This creates three explicit regimes to study the model under: supercritical, subcritical and critical. Other results include showing exponential decay of components for subcritical parameters, exponential decay of finite components for supercritical parameters, and more generally developing analogous tools to the FKG inequality, the BK inequality and Russo's formula, amongst others. I plan to show these results by combining existing methods from the study of Random Geometric Graphs and continuum percolation from the likes of Mathew Penrose, Geoffrey Grimmett, Ronald Meester and Rahul Roy newer methods developed by Vincent Tassion, Hugo Dumil-Copin, Gabor Pete, Ioan Manolescu and many others.Applications of this work might include networking, in which case the percolation property corresponds to the probability that a user can send a message to another user in the system with unreliable connections. In physics percolation theory has been used to understand quantum spin systems - thus it is possible that the continuum analogue has similar applications.