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On Disagreement Percolation and Maximality of the Critical Value for iid Percolation
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Johan Jonasson, Chalmers University of Technology
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Abstract
Two different problems are studied:
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For an infinite locally finite
connected graph G, let pc(G)
be the critical value for
the existence of an infinite cluster in iid bond percolation on G
and let
Pc = sup{pc(G): G transitive,
pc(G)<1}.
Is Pc<1?
- Let G be transitive with pc(G)<1, take
p in [0,1]
and let X and Y
be iid bond percolations on G with retention parameters (1+p)/2
and
(1-p)/2 respectively.
Is there a q<1 such that p > q implies that for any monotone
coupling (X',Y') of X and Y the edges
for which
X' and Y' disagree form infinite connected component(s)
with positive probability?
Let pd(G) be the infimum of such q's (including
q=1) and let
Pd = sup{pd(G): G transitive,
pc(G)<1}.
Is the stronger statement Pd < 1 true?
On the other hand: Is it always true that pd(G)>pc
(G)?
It is shown that if one restricts attention to biregular planar graphs
then these two problems can be treated in a similar way and all the
above questions are positively answered.
We also give examples to show that if one drops the assumption of
transitivity, then the answer to the above two questions is no.
Furthermore it is shown that for any bounded-degree bipartite
graph G with pc(G)<1
one has pc(G) < pd(G).
Problem (2) arises naturally from [6] where an example is given
of a coupling of the distinct plus- and minus measures for
the Ising model on a quasi-transitive graph at super-critical
inverse temperature. We give an example of such a coupling on
the r-regular tree, Tr, for r > 1.
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Full text: PDF
Pages: 1-13
Published on: June 15, 2001
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Electronic Journal of Probability. ISSN: 1083-6489 |
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