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- ItemPhase transitions for chase-escape models on Poisson–Gilbert graphs([Madralin] : EMIS ELibEMS, 2020) Hinsen, Alexander; Jahnel, Benedikt; Cali, Elie; Wary, Jean-PhilippeWe present results on phase transitions of local and global survival in a two-species model on Poisson–Gilbert graphs. Initially, there is an infection at the origin that propagates on the graph according to a continuous-time nearest-neighbor interacting particle system. The graph consists of susceptible nodes and nodes of a second type, which we call white knights. The infection can spread on susceptible nodes without restriction. If the infection reaches a white knight, this white knight starts to spread on the set of infected nodes according to the same mechanism, with a potentially different rate, giving rise to a competition of chase and escape. We show well-definedness of the model, isolate regimes of global survival and extinction of the infection and present estimates on local survival. The proofs rest on comparisons to the process on trees, percolation arguments and finite-degree approximations of the underlying random graphs.
- ItemConnection intervals in multi-scale dynamic networks(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Hirsch, Christian; Jahnel, Benedikt; Cali, ElieWe consider a hybrid spatial communication system in which mobile nodes can connect to static sinks in a bounded number of intermediate relaying hops. We describe the distribution of the connection intervals of a typical mobile node, i.e., the intervals of uninterrupted connection to the family of sinks. This is achieved in the limit of many hops, sparse sinks and growing time horizons. We identify three regimes illustrating that the limiting distribution depends sensitively on the scaling of the time horizon.