Scatter points randomly in ℝ^d from a stationary point process. Connect nearby points by the Delaunay triangulation — the dual of the Voronoi tessellation. Assign random weights (conductances) to the edges. The resulting random graph is a resistor network with both geometric and probabilistic structure.
The paper on moment bounds and exclusion processes on random Delaunay triangulations (arXiv: 2603.22562) establishes integrability conditions for weighted degrees under the Palm distribution, enabling three applications: random walks, resistor networks, and symmetric exclusion processes.
The Palm distribution conditions on a point being at the origin — it's the “typical point's view” of the point process. The weighted degree at a typical point (the sum of conductances on incident edges) must have finite moments for the random walk to be well-defined. The paper provides sufficient conditions in terms of the point process and conductance distribution.
For asymmetric jump rates, constructing the exclusion process (where particles occupy vertices and jump according to the conductances, with at most one particle per site) requires percolation criteria — the graph must percolate for the dynamics to be well-defined globally.
The through-claim: random geometry requires moment conditions to support dynamics. The Delaunay triangulation on a random point process has unbounded vertex degrees and random edge weights. For a random walk or exclusion process to make sense on this graph, the weighted degrees must be integrable — the geometry must be “tame enough” in a probabilistic sense. The moment condition is the bridge between random geometry and random dynamics.
2603.22562. Probability / random geometry / Delaunay triangulations / exclusion processes / percolation.