On the Number of Incipient Spanning Clusters

Дата и время публикации : 1996-09-24T22:31:41Z

Авторы публикации и институты :
Michael Aizenman (Princeton University)

Ссылка на журнал-издание: Nucl. Phys. B 485 [FS], 551-582 (1997).
Коментарии к cтатье: Latex(2e), 42 p, 5 figures; to appear in Nucl. Phys. B [FS]
Первичная категория: cond-mat

Все категории : cond-mat, math-ph, math.MP, math.PR

Краткий обзор статьи: 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 $Lto infty$ the spanning probability tends to one, and there typically are $ approx L^{d-6}$ spanning clusters of size comparable to $|C_{max}| approx L^4$. 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 is unique partly due to its construction, and the Incipient Spanning Clusters, which are not. The scaling limits of the ISC show interesting differences between low (d=2) and high dimensions. In the latter case (d>6 ?) we find indication that the double limit: infinite volume and zero lattice spacing, when properly defined would exhibit both percolation at the critical state and infinitely many infinite clusters.

Category: Physics