Universality in phase transitions in random graphs

ongoing

GraPhTra aims to advance the rigorous theoretical understanding of phase transitions in random graphs and their universal properties, driven by questions in neuromorphic computing, physics, and epidemiology. We will work on three work packages. -We introduce a new model for random graphs, that are sampled with local energy considerations, inspired by statistical physics and motivated by material science for neuromorphic computing. We use a Gibbs measure to sample a spanning tree of a graph, and by varying the temperature we thus interpolate between the uniform spanning tree and the minimal spanning tree. We demonstrate that the global structure of Gibbsian trees (in the Gromov-Hausdorff-Prokhorov topology) exhibits a phase transition that is universal across various underlying graphs. -We find conditions under which the position of the percolation phase transition on random unimodular graphs can be reliably estimated using local information. On the way, we resolve a key conjecture on the limiting threshold for Poisson–Voronoi percolation in hyperbolic space as the density of points goes to 0. -We prove that the spatial random graph models ‘long-range percolation’ and ‘scale-free percolation’ possess a phase where neighbourhoods grow exponentially, ensuring a well-defined reproductive number for spatial SIR (Susceptible → Infected → Recovered) epidemics, answering a long-standing open question.