Long-Range First-Passage Percolation on the Torus.

We study a geometric version of first-passage percolation on the complete graph, known as long-range first-passage percolation. Here, the vertices of the complete graph K n are embedded in the d-dimensional torus T n d , and each edge e is assigned an independent transmission time T e = ‖ e ‖ T n d α E e , where E e is a rate-one exponential random variable associated with the edge e, ‖ · ‖ T n d denotes the torus-norm, and α ≥ 0 is a parameter. We are interested in the case α ∈ [ 0 , d) , which corresponds to the instantaneous percolation regime for long-range first-passage percolation on Z d studied by Chatterjee and Dey [14], and which extends first-passage percolation on the complete graph (the α = 0 case) studied by Janson [24]. We consider the typical distance, flooding time, and diameter of the model. Our results show a 1, 2, 3-type result, akin to first-passage percolation on the complete graph as shown by Janson. The results also provide a quantitative perspective to the qualitative results observed by Chatterjee and Dey on Z d . [ABSTRACT FROM AUTHOR]