Mixing time and isoperimetry in random geometric graphs

TL;DR

This work determines the precise order of the mixing and relaxation times for the simple random walk on the giant component of supercritical random geometric graphs in dimension $d\ge2$ built from a unit-rate Poisson process. The authors establish a sharp upper bound via a novel high-probability isoperimetric inequality that lower-bounds the edge boundary across large vertex-sets, combined with the Lovász–Kannan average-conductance method, and prove a matching lower bound using concentration of graph distances and a distance-to-origin argument. They show $\tau_{mix}$ and $\tau_{rel}$ are both $\Theta\big(n^{2/d}/r^{2}\big)$ for all $r\ge(1+\varepsilon)r_g$, across the full supercritical regime, and they additionally prove that no cutoff occurs. The analysis hinges on a three-regime decomposition of the radius and a renormalization/containers approach to control isoperimetric boundaries in the random geometric setting, with a complementary one-dimensional treatment. The results bridge gaps with percolation and spatial random graphs, and provide tools of independent interest, notably the isoperimetric inequality for large vertex sets in random geometric graphs.

Abstract

In this paper we study the mixing time of the simple random walk on the giant component of supercritical $d$-dimensional random geometric graphs generated by the unit intensity Poisson Point Process in a $d$-dimensional cube of volume $n$. With $r_g$ denoting the threshold for having a giant component, we show that for every $ε> 0$ and any $r \ge (1+ε)r_g$, the mixing time of the giant component is with high probability $Θ(n^{2/d}/r^{2})$, thereby closing a gap in the literature. The main tool is an isoperimetric inequality which holds, w.h.p., for any large enough vertex set, a result which we believe is of independent interest. Our analysis also implies that the relaxation time is of the same order.

Figures (4)

• Figure 1: Vertices of $\mathcal{G}_n$ are represented as circles. Small circles in gray represent vertices that do not belong to $\mathcal{L}_n$. (a) Elements of $A'$ are shown as blue circles while elements of $A\setminus A'$ are shown in red. (b) The tiles that are colored (cyan or magenta) are the tiles in $L_A$. (c) Tiles in $T_A$ are colored in magenta.
• Figure 2: Vertices of $\mathcal{G}_n$ are represented as circles. Small circles in gray represent vertices that do not belong to $\mathcal{L}_n$. Solid vertices shown in colors correspond to elements of $A'$. (a) Distinct elements of $\mathcal{A}'$ (components induced by $A'$ in $\mathcal{L}_n$) are shown in different colors. (b) For illustration purposes, we consider as elements of $\mathcal{A}'_{\hbox{$\succeq$}}$ components of $\mathcal{A}'$ of size at least $20$ which are illustrated as blue and orange colored vertices. (c) Vertices in colored tiles correspond to $A"$. (d) Again, for the sake of illustration, we consider $k=3$ and color the tiles belonging to each of the two $3$-lattice animals in $\mathfrak{L}(A)$ in green and cyan, respectively. (e) Uncolored tiles are *-connected and contain elements of $\mathcal{L}_n\setminus A$ (see Figure \ref{['fig:first']} for a depiction of the set $A$). Hence, in this example, $\mathfrak{K}(A)$ contains a unique element, consisting of the collection of uncolored tiles. (f) For the illustrated instance, for each $L\in\mathfrak{L}(A)$, it holds that $\mathfrak{M}(L)=\{L^c\}$.
• Figure 3: Vertices $v'$ and $v"$ are connected by an edge of length $r$ traversing the tile $\tau$ of side length $r_d$, which is contained in either $B_{v'}(r)$ or $B_{v"}(r)$
• Figure 4: Tile $\tau$ is useful, implying that both $\tau$ and $\tau'$ are good. The red lines represent the crossings of the smaller sub-tiles, and thus a part of the largest component inside $\tau'$. $x$ is a vertex of $\mathcal{L}_n$, the green line represents a path to the outside of the tile $\tau'$. As the part of this path in $\tau'\setminus \tau$ is of length at least one fifth of the side length of $\tau'$, it must belong to the largest component of $\tau'$ (by definition of good tile): the connection to the largest component of $\tau'$ is represented by the magenta line.

Theorems & Definitions (62)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• proof
• Theorem 2.1
• Lemma 2.2
• proof
• Theorem 2.3
• Lemma 2.4
• proof