Long-Range First-Passage Percolation on the Torus

Abstract

AbstractWe 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 $$\mathcal {K}_n$$ K n are embedded in the d-dimensional torus $$\mathbb T_n^d$$ T n d , and each edge e is assigned an independent transmission time $$T_e=\Vert e\Vert _{\mathbb T_n^d}^\alpha E_e$$ T e = ‖ e ‖ T n d α E e , where $$E_e$$ E e is a rate-one exponential random variable associated with the edge e, $$\Vert \cdot \Vert _{\mathbb T_n^d}$$ ‖ · ‖ T n d denotes the torus-norm, and $$\alpha \ge 0$$ α ≥ 0 is a parameter. We are interested in the case $$\alpha \in [0,d)$$ α ∈ [ 0 , d ) , which corresponds to the instantaneous percolation regime for long-range first-passage percolation on $$\mathbb {Z}^d$$ Z d studied by Chatterjee and Dey [14], and which extends first-passage percolation on the complete graph (the $$\alpha =0$$ 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 $$\mathbb {Z}^d$$ Z d .