On the jump of the cover time in random geometric graphs

TL;DR

The paper analyzes the cover time of the simple random walk on the giant component of sparse, supercritical $d$-D random geometric graphs on a Poisson vertex set, revealing a sharp jump at the connectivity radius $r_c$ from $\Theta(n \log^2 n)$ to $\Theta(n \log n)$. It develops an electrical-network-based approach with a renormalization scheme to create a backbone of crossings, enabling precise bounds on the effective resistance and, through standard connections between resistance and cover time, the claimed scaling across radius regimes. The results unify and extend prior findings by proving the $\Theta(n \log n)$ upper bound right above $r_c$ in dimension $d=2$ and establishing $\Theta(n \log^2 n)$ behavior in the giant-component regime for $r$ below connectivity, with a detailed analysis of the transition vicinity. The techniques, especially the backbone construction and flow-based resistance bounds, provide a framework for analyzing cover times in geometric and percolation-like graphs with varying density.

Abstract

In this paper we study the cover time of the simple random walk on the giant component of supercritical $d$-dimensional random geometric graphs on $\mathrm{Poi}(n)$ vertices. We show that the cover time undergoes a jump at the connectivity threshold radius $r_c$: with $r_g$ denoting the threshold for having a giant component, we show that if the radius $r$ satisfies $(1+\varepsilon)r_g \le r \le (1-\varepsilon)r_c$ for $\varepsilon > 0$ arbitrarily small, the cover time of the giant component is asymptotically almost surely $Θ(n \log^2 n$). On the other hand, we show that for $r \ge (1+\varepsilon)r_c$, the cover time of the graph is asymptotically almost surely $Θ(n \log n)$ (which was known for $d=2$ only for a radius larger by a constant factor). Our proofs also shed some light onto the behavior around $r_c$.

Figures (12)

• Figure 1: The central cube $C_i$ and its surrounding paths. The colored lines represent vertex paths of $G \in \mathcal{G}(n,r,d)$.
• Figure 2: A $2$-dimensional slice $\mathcal{K}_n^2$, with a red $\alpha$-logarithmic strip.
• Figure 3: A pending path in 2 dimensions with root at $x$ ($x$ is thus $m$-special for some $m$); all cubes surrounding the path are empty; $y$ is the vertex associated to $x$ that belongs to $\mathcal{R}$. Observe that a vertex in the path could be connected by an edge to more than 2 vertices (the maximum being 5 vertices).
• Figure 4: Path $\mathcal{L}$ of parallelepipeds $\mathrm{T}$. The path changes direction when visiting the parallelepipeds $\mathrm{T}_{a_i,b_i}$
• Figure 5: Position of $y_X$ relative to the parallelepipeds

Theorems & Definitions (35)

• Theorem 1.1
• Theorem 1.2
• Definition 2.1
• Remark 2.2
• Theorem 2.3: Rayleigh, Theorem 9.12 of levinperes
• Lemma 2.4: Corollary 10.8 of levinperes
• Lemma 2.5: Proposition 9.15 of levinperes
• Theorem 2.6: Theorem 2.4 of chandra1989electrical
• Theorem 2.7: Theorem 1.3 of kahn2000cover
• Theorem 2.8: Theorem 2.2 of Chatterjee