96-436 Michael Aizenman
On the Number of Incipient Spanning Clusters (182K, LaTeX(2e)) 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 $L^{d-6}$, 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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