The birth of the infinite cluster: Finite-size scaling in percolation

We address the question of finite-size scaling in percolation by studying b ond percolation in a finite box of side length n, both in two and in higher dimensions. In dimension d = 2, we obtain a complete characterization of f inite-size scaling. In dimensions d > 2, we establish the same results unde r a set of hypotheses related to so-called scaling and hyperscaling postula tes which are widely believed to hold up to d = 6. As a function of the size of the box, we determine the scaling window in wh ich the system behaves critically. We characterize criticality in terms of the scaling of the sizes of the largest clusters in the box: incipient infi nite clusters which give rise to the infinite cluster. Within the scaling w indow, we show that the size of the largest cluster behaves like n(d)pi (n) , where pi (n) is the probability at criticality that the origin is connect ed to the boundary of a box of radius n. We also show that, inside the wind ow, there are typically many clusters of scale n(d)pi (n) and hence that "t he" incipient infinite cluster is not unique. Below the window, we show tha t the size of the largest cluster scales like xi (d)pi (xi) log(n/xi), wher e is the correlation length, and again, there are many clusters of this sca le. Above the window, we show that the size of the largest cluster scales l ike n(d) P-infinity, where P-infinity is the infinite cluster density, and that there is only one cluster of this scale. Our results are finite-dimens ional analogues of results on the dominant component of the Erdos-Renyi mea n-field random graph model.