Contemporary Mathematics
Volume 41, 1985
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
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
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
planar graphs.
1980 Mathematics Subject Classification. Primary 60K35,
secondary 82A67.
© 1985 American Mathematical Society
0271-4132/85 $1.00
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