Speed of Convergence and Moderate Deviations of FPP on Random Geometric Graphs

TL;DR

This work studies first-passage percolation on random geometric graphs in the supercritical regime, focusing on geodesic behavior, moderate deviations, and shape convergence. The authors develop an approximation scheme using an augmented graph and truncated times, enabling precise control of deviations and enabling a quantitative shape theorem with rates $\sim \frac{\log t}{\sqrt{t}}$. They establish exponential moderate-deviation tails, derive tight bounds on $\mathbb{E}[T(x)]$ and $\mathrm{Var}\,T(x)$, and show that the asymptotic shape is approached at a computable rate. They further analyze the fluctuations and directions of semi-infinite geodesics and prove $f$-straightness of spanning trees, demonstrating robust omnidirectional behavior in this geometric FPP setting.

Abstract

This study delves into first-passage percolation on random geometric graphs in the supercritical regime, where the graphs exhibit a unique infinite connected component. We investigate properties such as geodesic paths, moderate deviations, and fluctuations, aiming to establish a quantitative shape theorem. Furthermore, we examine fluctuations in geodesic paths and characterize the properties of spanning trees and their semi-infinite paths.

Figures (5)

• Figure 1: A geodesic path with its fluctuations in a cylindrical region of space given by \ref{['thm:fluctuations.of.geodesics']}.
• Figure 2: Section of a spanning tree $\mathlcal{T}_x$ (left) and the subtree given by $\mathlcal{T}_x^{\mathsf{out}}(v)$ (right).
• Figure 3: Geodesic paths crossing $v \in \mathop{\mathrm{\mathcal{H}}}\nolimits \setminus \mathscr{F}_x$ within the conic region described in \ref{['thm:f.straight.spanning.trees']}.
• Figure 4: The image depicts the same region of a standard RGG, denoted as $\mathcal{G}$ (left), alongside an RGG with extra vertices and extra edges $\mathcal{G}^t$ (right).
• Figure 5: For $v$ ranging over the gray region $\mathsf A'$, the value of $f(v):=\|x-v\|+\|y-v\|$ is minimized by $v=v_2$. To see this, first note that, along the red curve $\mathcal{L}_1$, $f(v)$ is minimized by $v_1$ (since $\|x-v\|=R$ along this curve, and $\|y-v\|$ clearly increases as we move $v$ towards the left). Next, the problem of minimizing $f(v)$ for $v$ along the blue line $\mathcal{L}_2$ is the well-known Calculus exercise where we seek the triangle of smallest perimeter, when the base and the height are kept fixed; the minimizer is the isosceles triangle.

Theorems & Definitions (53)

• Theorem 1.1: Quantitative shape theorem for FPP on RGGs
• Theorem 1.2: Moderate deviations of first-passage times
• Theorem 1.3: Asymptotic expectation and variance of first-passage times
• Theorem 1.4: Fluctuations of geodesics
• Theorem 1.5: Asymptotic behaviour of geodesics
• Corollary 1.6
• Proposition 2.1
• Proposition 2.2
• Lemma 2.3
• Lemma 2.4