München 1997 – wissenschaftliches Programm
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MP: Theoretische und Mathematische Grundlagen der Physik
MP XVI: HV XVI
MP XVI.1: Hauptvortrag
Freitag, 21. März 1997, 14:45–15:30, HS 118
Finite-Size Scaling and Hyperscaling for Percolation — •C. Borgs1, J. Chayes2, J. Spencer3, and H. Kesten4 — 1Institut für Theoretische Physik, Universität Leipzig, Augustusplatz 10/11, 04109 Leipzig — 2Institute for Advanced Study, Princeton, NJ 08540 — 3Courant Institute, New York University, 251 Mercer Street, New York, NY 10012 — 4Department of Mathematics, Cornell University, Ithaca, NY 14853, USA
We address the question of finite-size scaling in percolation by studying bond percolation in a finite box of side length n. As a function of the size of the box, we determine the scaling window in which the system behaves critically. We characterize criticality in terms of the scaling of the sizes of the largest clusters in the box: incipient infinite clusters which give rise to the infinite cluster. Within the scaling window, we show that the size of the largest cluster behaves like ndf where df is the fractal dimension of the so-called incipient infinite cluster, and that there are typically many clusters of this scale. Below the window, we show that the size of the largest cluster scales like ξdf log(n/ξ), where ξ is the correlation length, and again, there are many clusters of this scale. Above the window, we show that the size of the largest cluster scales like ndP∞, where P∞ is the infinite cluster density, and that there is only one cluster of this scale. We establish these results axiomatically, and explicitly verify the axioms in d=2 dimensions. As a byproduct of our proofs, we obtain that uniform boundedness of critical crossing probabilities implies hyperscaling.