Contemporary Mathematics

Volume 41, 1985

INEQUALITIES FOR CRITICAL PROBABILITIES IN PERXLATION

Massimo Campanino

ABSTRACT. We examine methods of comparison between critical

percolation probabilities in two cases : a) planar graphs

related by inclusion ; b) graphs related by identification

of sites.

1. INTRODUCTION. Percolation theory deals with the study of geo-

metrical properties of random systems and in particular with the

distribution of clusters. Since the paper ofT. E. Harris ([2])

appeared, there has been remarkable progress in the understanding

of two-dimensional systems, whereas most of the important problems

are still open in higher dimensions.

One of the interesting objects in percolation theory are criti-

cal probabilities of specific models, i.e. the

probabiliti~s

for

the occupation of single sites or bonds above which infinite

clusters start to arise. In some planar graphs with particular

symmetry properties the critical probabilities can be exactly

computed. This is in general not possible and a natural problem

is to find methods of comparing critical probabilities in diffe-

rent graphs.

It is quite intuitive that the critical probability of a given

model strictly decreases if we add, in a periodic way, new connec-

tions to a periodic graph. A first result in this direction was

obtained by Y. Higushi

([3])

who proved that the critical proba-

bility for site percolation on the square lattice with connection

is strictly less than that on the square lattice with nearest

neighbour connection. The conjecture has then been proved by

H. Kesten ([5]) under wide conditions

tor

planar graphs.

1980 Mathematics Subject Classification. Primary 60K35,

secondary 82A67.

1

© 1985 American Mathematical Society

0271-4132/85 $1.00

+

$.25 per page

http://dx.doi.org/10.1090/conm/041/814699

Volume 41, 1985

INEQUALITIES FOR CRITICAL PROBABILITIES IN PERXLATION

Massimo Campanino

ABSTRACT. We examine methods of comparison between critical

percolation probabilities in two cases : a) planar graphs

related by inclusion ; b) graphs related by identification

of sites.

1. INTRODUCTION. Percolation theory deals with the study of geo-

metrical properties of random systems and in particular with the

distribution of clusters. Since the paper ofT. E. Harris ([2])

appeared, there has been remarkable progress in the understanding

of two-dimensional systems, whereas most of the important problems

are still open in higher dimensions.

One of the interesting objects in percolation theory are criti-

cal probabilities of specific models, i.e. the

probabiliti~s

for

the occupation of single sites or bonds above which infinite

clusters start to arise. In some planar graphs with particular

symmetry properties the critical probabilities can be exactly

computed. This is in general not possible and a natural problem

is to find methods of comparing critical probabilities in diffe-

rent graphs.

It is quite intuitive that the critical probability of a given

model strictly decreases if we add, in a periodic way, new connec-

tions to a periodic graph. A first result in this direction was

obtained by Y. Higushi

([3])

who proved that the critical proba-

bility for site percolation on the square lattice with connection

is strictly less than that on the square lattice with nearest

neighbour connection. The conjecture has then been proved by

H. Kesten ([5]) under wide conditions

tor

planar graphs.

1980 Mathematics Subject Classification. Primary 60K35,

secondary 82A67.

1

© 1985 American Mathematical Society

0271-4132/85 $1.00

+

$.25 per page

http://dx.doi.org/10.1090/conm/041/814699