Existence and Convergence of Interacting Particle Systems on Graphs
Kuldeep Guha Mazumder
TL;DR
This work establishes existence and convergence of interacting particle systems on locally finite graphs under weakened assumptions, notably relaxing uniform degree bounds and uniformly bounded jump rates. The authors hinge on a graphical (Poisson) construction and a non-percolation criterion defined via double and simple jump-trail rates, ensuring that high-degree or fast-jump vertices remain sufficiently separated to prevent blow-up. They extend the framework to random graphs, providing verifiable conditions on SAW growth and jump-rate moments that guarantee almost-sure existence of dynamics and convergence from finite windows to the infinite graph. The paper also demonstrates concrete models (sandpiles, consensus, contact processes, urns, Bak-Sneppen evolution) and two key random-graph families (long-range percolation and geometric graphs on Delone sets) where the theory applies, illustrating broad applicability to complex networks with unbounded degrees or rates.
Abstract
We give a general existence and convergence result for interacting particle systems on locally finite graphs with possibly unbounded degrees or jump rates. We allow the local state space to be Polish, and the jumps at a site to affect the states of its neighbours. The two common assumptions on interacting particle systems are uniform bounds on degrees and jump rates. In this paper, we relax these assumptions and allow for vertices with high degrees or rapid jumps. We introduce new assumptions ensuring that such vertices are placed sufficiently apart from each other and hence the process does not blow up. Our assumptions involve finitude of certain weighted connective constants on the square graph of the underlying graph and our proofs proceed by showing that these assumptions imply non-percolation of the Poisson graphical construction. For some random graph models, we give practically verifiable sufficient conditions under which our assumptions hold almost surely. These conditions involve exponential growth of a fractional moment sum of probabilities of self-avoiding walks from each vertex and that of product moments of fixed powers of jump rates. Using these conditions, we show the existence of interacting particle systems with possibly unbounded jump rates like contact processes, consensus formation models, evolutionary models, etc., on random graphs which can lack uniform bounds on degrees almost surely, e.g., long-range percolations on quasi-transitive graphs, and geometric random graphs on Delone sets.