### Abstract

*n*random points in a

*d*-dimensional domain with Lipschitz boundary and with distance parameter decaying more slowly (as a function of

*n*) than the connectivity threshold, the Cheeger constant (under several possible definitions of surface and volume), also known as conductance, suitably rescaled, converges for large

*n*to an analogous Cheeger-type constant of the domain. Previously, Garca Trillos

*et al*had shown this for

*d*≥ 3 but had required an extra condition on the distance parameter when

*d*= 2.

Original language | English |
---|---|

Number of pages | 33 |

Journal | Annals of Applied Probability |

Early online date | 14 Sep 2019 |

Publication status | E-pub ahead of print - 14 Sep 2019 |

### Cite this

*Annals of Applied Probability*.

**Optimal Cheeger cuts and bisections of random geometric graphs.** / Penrose, Mathew; Müller, Tobias.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Optimal Cheeger cuts and bisections of random geometric graphs

AU - Penrose, Mathew

AU - Müller, Tobias

PY - 2019/9/14

Y1 - 2019/9/14

N2 - Let d ≥ 2. The Cheeger constant of a graph is the minimum surface-to-volume ratio of all subsets of the vertex set with relative volume at most 1/2. There are several ways to define surface and volume here: the simplest method is to count boundary edges (for the surface) and vertices (for the volume). We show that for a geometric (possibly weighted) graph on n random points in a d-dimensional domain with Lipschitz boundary and with distance parameter decaying more slowly (as a function of n) than the connectivity threshold, the Cheeger constant (under several possible definitions of surface and volume), also known as conductance, suitably rescaled, converges for large n to an analogous Cheeger-type constant of the domain. Previously, Garca Trillos et al had shown this for d ≥ 3 but had required an extra condition on the distance parameter when d = 2.

AB - Let d ≥ 2. The Cheeger constant of a graph is the minimum surface-to-volume ratio of all subsets of the vertex set with relative volume at most 1/2. There are several ways to define surface and volume here: the simplest method is to count boundary edges (for the surface) and vertices (for the volume). We show that for a geometric (possibly weighted) graph on n random points in a d-dimensional domain with Lipschitz boundary and with distance parameter decaying more slowly (as a function of n) than the connectivity threshold, the Cheeger constant (under several possible definitions of surface and volume), also known as conductance, suitably rescaled, converges for large n to an analogous Cheeger-type constant of the domain. Previously, Garca Trillos et al had shown this for d ≥ 3 but had required an extra condition on the distance parameter when d = 2.

UR - https://arxiv.org/abs/1805.08669

M3 - Article

JO - Annals of Applied Probability

JF - Annals of Applied Probability

SN - 1050-5164

ER -