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Mixing time of the random walk on the giant component of the random geometric graph

Magnus H. Haaland, Anđela Šarković

TL;DR

This work analyzes the simple random walk on the giant component of a random geometric graph in dimensions d ≥ 2 within the supercritical radius regime. By combining evolving-sets techniques with sharp isoperimetric bounds obtained through a renormalized block construction (and, in 2D, a streamlined percolation-based argument), the authors establish that both the mixing time and the relaxation time on the giant component scale as Θ(n^{2/d}) whp, implying no cutoff. The results hinge on showing robust lower bounds for edge boundaries of large connected subsets, enabling precise control over the isoperimetric profile Φ_*(u). The paper provides a general d ≥ 2 treatment and a specialized, simpler proof for d = 2, together with polylogarithmic isoperimetric bounds for moderately large subsets, contributing to the broader understanding of diffusion on geometric random graphs.

Abstract

We consider a random geometric graph obtained by placing a Poisson point process of intensity 1 in the d-dimensional torus of side length n^(1/d) and connecting two points by an edge if their distance is at most r. We consider the case of d>=2 and r in [r_min, r_max], where r_min<r_max are any constants with r_min>r_g and r_g is a constant above which this graph has a giant component with high probability. We show that, with high probability, the mixing time and the relaxation time of the simple random walk on the giant component in this case are both of order n^(2/d) and that therefore there is no cutoff. We also obtain bounds for the isoperimetric profile of subsets of the giant component of at least polylogarithmic size.

Mixing time of the random walk on the giant component of the random geometric graph

TL;DR

This work analyzes the simple random walk on the giant component of a random geometric graph in dimensions d ≥ 2 within the supercritical radius regime. By combining evolving-sets techniques with sharp isoperimetric bounds obtained through a renormalized block construction (and, in 2D, a streamlined percolation-based argument), the authors establish that both the mixing time and the relaxation time on the giant component scale as Θ(n^{2/d}) whp, implying no cutoff. The results hinge on showing robust lower bounds for edge boundaries of large connected subsets, enabling precise control over the isoperimetric profile Φ_*(u). The paper provides a general d ≥ 2 treatment and a specialized, simpler proof for d = 2, together with polylogarithmic isoperimetric bounds for moderately large subsets, contributing to the broader understanding of diffusion on geometric random graphs.

Abstract

We consider a random geometric graph obtained by placing a Poisson point process of intensity 1 in the d-dimensional torus of side length n^(1/d) and connecting two points by an edge if their distance is at most r. We consider the case of d>=2 and r in [r_min, r_max], where r_min<r_max are any constants with r_min>r_g and r_g is a constant above which this graph has a giant component with high probability. We show that, with high probability, the mixing time and the relaxation time of the simple random walk on the giant component in this case are both of order n^(2/d) and that therefore there is no cutoff. We also obtain bounds for the isoperimetric profile of subsets of the giant component of at least polylogarithmic size.
Paper Structure (7 sections, 9 theorems, 38 equations, 4 figures)

This paper contains 7 sections, 9 theorems, 38 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Proof overview and preliminary results
  3. Isoperimetric bound for d ge 2
  4. Bounds on the mixing and relaxation time
  5. Isoperimetric bound for d=2
  6. Proof of Lemma \ref{['lemma:sumsofdeginboxes']}
  7. Proof of Claim \ref{['claim:numberofsizeksetswithnottoomanystarcomps']}

Key Result

Theorem 1.3

Let $d\geq 2$ and let $r_g<r_{\min}\le r_{\max}$. Then there exist positive constants $c$ and $C$ so that with high probability as $n\to\infty$ the mixing time of the simple random walk and of the lazy simple random walk on the giant component $L_1(G)$ satisfies

Figures (4)

  • Figure 1: Illustration of $s$-boxes and blocks on $\Lambda_n^{(d)}$. Here $d=2$, $L=10$ and $\frac{\sqrt[d]n}{Ls}=4$.
  • Figure 2: Illustration of boxes, triangles and openness. Here the red edge is open, the other grid edges are not. The condition $s\in\left[\frac{2}{\sqrt{13}}r, \sqrt{\frac{2}{5}}\,r\right]$ ensures $r$ is bounded by the lengths of the oblique gray segments.
  • Figure 3: Elementary geometric observations used in the proof of \ref{['prop:largeconnsetshavelargeboundary']}.
  • Figure 4: The three cases where we use the additional edges. Suppose two red boxes in different *-connected components are connected by an $L_1(G)$-edge $e$ with endpoints in $A$. Then $e$ must intersect a perpendicular bisector of a white cell $c$ (in the third case, this follows by $r\le \frac{\sqrt{13}}{2}s$). So $c\in W^+$ and the edges of $c$, shown in blue, are in $\mathcal{E}^+$.

Theorems & Definitions (28)

  • definition 1.1: Random Geometric Graph (RGG) on the torus
  • definition 1.2
  • Theorem 1.3
  • Proposition 1.4
  • definition 2.1
  • Proposition 2.2
  • Lemma 2.3
  • Lemma 2.4
  • definition 3.1: Boxes and blocks
  • definition 3.2: Good blocks
  • ...and 18 more