 96436 Michael Aizenman
 On the Number of Incipient Spanning Clusters
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Sep 24, 96

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Abstract. In critical percolation models, in a large cube there will
typically be more than one cluster of comparable diameter.
In 2D, the probability of $k>>1$ spanning clusters is of the
order $ e^{\alpha \ k^{2}}$. In dimensions $d>6$,
when $\eta = 0$ the spanning clusters proliferate:
for $L\to \infty$ the spanning probability tends to one,
and the number of robust spanning clusters grows as
$L^{d6}$, with $\C_{max} \approx L^4$.
The rigorous results confirm a generally accepted
picture for $d>6$, but also correct some
misconceptions concerning the uniqueness of
the dominant cluster. We distinguish between two
related concepts: the Incipient Infinite Cluster,
which by its construction is unique,
and the Incipient Spanning Clusters, which are not.
The scaling limits of the ISC exhibit interesting differences
between low ($d=2$) and high dimensions. Notably,
in the latter case ($d>6 ?$) the double limit: infinite
volume and zero lattice spacing, will exhibit
both percolation at the critical state,
and infinitely many infinite clusters.
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