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Isoperimetric Inequalities and Supercritical Percolation on High-dimensional Graphs

Sahar Diskin, Joshua Erde, Mihyun Kang, Michael Krivelevich

Abstract

It is known that many different types of finite random subgraph models undergo quantitatively similar phase transitions around their percolation thresholds, and the proofs of these results rely on isoperimetric properties of the underlying host graph. Recently, the authors showed that such a phase transition occurs in a large class of regular high-dimensional product graphs, generalising a classic result for the hypercube. In this paper we give new isoperimetric inequalities for such regular high-dimensional product graphs, which generalise the well-known isoperimetric inequality of Harper for the hypercube, and are asymptotically sharp for a wide range of set sizes. We then use these isoperimetric properties to investigate the structure of the giant component $L_1$ in supercritical percolation on these product graphs, that is, when $p=\frac{1+ε}{d}$, where $d$ is the degree of the product graph and $ε>0$ is a small enough constant. We show that typically $L_1$ has edge-expansion $Ω\left(\frac{1}{d\ln d}\right)$. Furthermore, we show that $L_1$ likely contains a linear-sized subgraph with vertex-expansion $Ω\left(\frac{1}{d\ln d}\right)$. These results are best possible up to the logarithmic factor in $d$. Using these likely expansion properties, we determine, up to small polylogarithmic factors in $d$, the likely diameter of $L_1$ as well as the typical mixing time of a lazy random walk on $L_1$. Furthermore, we show the likely existence of a path of length $Ω\left(\frac{n}{d\ln d}\right)$. These results not only generalise, but also improve substantially upon the known bounds in the case of the hypercube, where in particular the likely diameter and typical mixing time of $L_1$ were previously only known to be polynomial in $d$.

Isoperimetric Inequalities and Supercritical Percolation on High-dimensional Graphs

Abstract

It is known that many different types of finite random subgraph models undergo quantitatively similar phase transitions around their percolation thresholds, and the proofs of these results rely on isoperimetric properties of the underlying host graph. Recently, the authors showed that such a phase transition occurs in a large class of regular high-dimensional product graphs, generalising a classic result for the hypercube. In this paper we give new isoperimetric inequalities for such regular high-dimensional product graphs, which generalise the well-known isoperimetric inequality of Harper for the hypercube, and are asymptotically sharp for a wide range of set sizes. We then use these isoperimetric properties to investigate the structure of the giant component in supercritical percolation on these product graphs, that is, when , where is the degree of the product graph and is a small enough constant. We show that typically has edge-expansion . Furthermore, we show that likely contains a linear-sized subgraph with vertex-expansion . These results are best possible up to the logarithmic factor in . Using these likely expansion properties, we determine, up to small polylogarithmic factors in , the likely diameter of as well as the typical mixing time of a lazy random walk on . Furthermore, we show the likely existence of a path of length . These results not only generalise, but also improve substantially upon the known bounds in the case of the hypercube, where in particular the likely diameter and typical mixing time of were previously only known to be polynomial in .
Paper Structure (17 sections, 32 theorems, 111 equations, 1 figure)

This paper contains 17 sections, 32 theorems, 111 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background and motivation
  3. Main results
  4. Outline of the proofs
  5. Preliminary lemmas
  6. Isoperimetric inequalities
  7. Expansion and Expanders
  8. Expansion of subsets of the early giant
  9. Structure of subsets in the residue
  10. Proof of Theorems \ref{['th: connected expansion']}\ref{['i:medium']} and \ref{['th: expansion']}\ref{['i:large']}
  11. Consequences of expansion in the giant component
  12. Proof of Theorem \ref{['th: consequences']}\ref{['i: diameter']}
  13. Proof of Theorem \ref{['th: consequences']}\ref{['i: mixing time']}
  14. Discussion and open questions
  15. Isoperimetry in product graphs
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $d \in \mathbb{N}$. For every $k\in\left[2^d\right]$ Furthermore, the only sets which achieve equality in the above estimate are subcubes.

Figures (1)

  • Figure 1: An illustration of the sets and matchings in Lemma \ref{['l: disjoint paths']}. The matchings $M_1$ through $M_5$, in purple, are ordered according to the order they are constructed in the proof. In dark blue, one can see the properties of vertices in $A_2, B_2, A_1$ and $B_1$, with respect to their set of neighbours in $A_1, B_1, A$ and $B$, respectively. Observe that if the first matching $M_1$ had many endpoints in $A'\setminus A_2$ (or $B'\setminus B_2$), we could continue in the same manner with fewer matchings required.

Theorems & Definitions (58)

  • Theorem 1.1: H64, see also L64B67H76
  • Theorem 1.2: Theorem 2 in DEKK22
  • Theorem 1.3: Theorem 2 of CT98
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Remark 1.4
  • Theorem 6
  • ...and 48 more