Long-range first-passage percolation on the torus

TL;DR

We study long-range first-passage percolation on the $d$-dimensional torus, assigning edge-transmission times $T_e=\|e\|^{\\alpha}E_e$ with $E_e$ i.i.d. exponential and parameter $\\alpha\ge0$ in the instantaneous regime $\\alpha<d$. The authors develop a large-deviation exploration-process framework, yielding sharp 1-2-3-type scaling results for the typical distance, flooding time, and diameter, scaled by $R_n/\\log n$ where $R_n=\sum_{u\neq 0}\|u\|^{-\\alpha}$. They show that $X_{U,V}R_n/\\log n \xrightarrow{P}1$, $\max_v X_{u,v}R_n/\\log n \xrightarrow{P}2$, and $\max_{u,v} X_{u,v}R_n/\\log n \xrightarrow{P}3$, with $R_n$ growing like $n^{1-\\alpha/d}$ and a limiting constant $R=R(d,\\alpha)$ identified via integral representations. The work extends Janson’s complete-graph results to the spatial torus and provides quantitative connections to seed results in infinite lattices (Chatterjee-Dey). The findings advance understanding of how geometry and long-range penalties interact to control weighted distances in finite graphs, with potential applications to network design and percolation phenomena in spatial networks.

Abstract

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 $\mathcal K_n$ are embedded in the $d$-dimensional torus $\mathbb T_n^d$, and each edge $e$ is assigned an independent transmission time $T_e=\|e\|_{\mathbb T_n^d}^αE_e$, where $E_e$ is a rate-one exponential random variable associated with the edge $e$, $\|\cdot\|_{\mathbb T_n^d}$ denotes the torus-norm, and $α\geq0$ is a parameter. We are interested in the case $α\in[0,d)$, which corresponds to the instantaneous percolation regime for long-range first-passage percolation on $\mathbb Z^d$ studied by Chatterjee and Dey, and which extends first-passage percolation on the complete graph (the $α=0$ case) studied by Janson. 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 $\mathbb Z^d$.

Theorems & Definitions (6)

• Theorem 1.1: 1,2,3 times $\log{n}/R_n$ for long-range first-passage percolation on the torus
• Remark 1.2: Interpretation main result
• Remark 1.3: First-passage percolation on the complete graph
• Remark 1.4: More general distance functions
• Lemma 2.1: Distributional limit of $\tau_k$
• proof