Samenvatting
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, García Trillos et al. had shown this for
d
≥
3
but had required an extra condition on the distance parameter when
d
=
2
.
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, García Trillos et al. had shown this for
d
≥
3
but had required an extra condition on the distance parameter when
d
=
2
.
Originele taal-2 | English |
---|---|
Pagina's (van-tot) | 1458–1483 |
Tijdschrift | The Annals of Applied Probability |
Volume | 30 |
Nummer van het tijdschrift | 3 |
DOI's | |
Status | Published - 1-jun.-2020 |