Numerical Estimates of the Hausdorff Dimension of the Largest Cluster and Its Backbone in the Percolation Problem in Two Dimensions

Abstract

We present numerical estimates of the Hausdorff dimension D of the largest cluster and its " backbone" in the percolation problem on a square lattice as a function of the concentration £ . We fine that D is an approximately linear function of p in the region near P ~Pc (-0.59) with a dimension about equal to that of a self-avoiding walk whenp = 0.455. The dimension of the backbone, or biconnected part, of the largest cluster equals that of the self-avoiding walk when/? ^p c . Atp =p c the dimension of the largest cluster equals the anomalous dimension introduced by Stanley et al. In analysis of experiments on magnetic systemsBirgeneau et ah 1 and Stanley et al. 2 recently suggested a self-avoiding random walk (SAW) as a model for the largest cluster in a percolating net. Qualitative arguments were presented suggesting that the geometrical properties of the clusters were similar to those of the SAW. Here we study this interesting suggestion by estimating the dimension of each of the two structures numerically.The dimension which we estimate is the Hausdorff-Besicovitch 3 dimension D H which is defined so that a particular structure is covered by a minimum of N(r]) disks of radius rj and lim^0N(r])ri Du is finite. We have checked that direct application of this definition to numerically estimate D H for a self-avoiding walk (SAW) in two dimensions gave results consistent with the value 4 D S AW = 1»33 obtained by finding the mean end-to-end distance (r 2 ) as a function of the number of steps n. (Writing (r 2 ) =n 2Vs it is easy to show that D SAW -l/v SQ ) We have also checked the value D S AW = L33 by ap-plication of the second method.In the percolation system we estimate D H from the relation of the average size n c of the largest cluster to the total size n 2 of the percolating net. We find empirically that n c varies with n as n c -Kn 2y where K and y are constants (see Fig. 1). (This relation is expected when na is less than the coherence length |.) We note that n c can also be regarded as an upper bound on the number of disks of size r\-a required to cover the largest cluster in a net of size n 2 . For fixed a and for each n we introduce a change of length scale x' -x/na. In terms of this length scale the covering of the largest cluster in the net of size n 2 is by disks of radius 77' -r\/na =l/n. Thus each value of n corresponds to a different covering. Using n c =Kn 2y we haveiV(7? / ) = n c =Kn 2y =K(r] f y 2y or D = 2y. By the argument just given, these D's are an upper bound on the Hausdorff dimension. We do not have a quantitative estimate of the error involved in treating D as an estimate for D H . We have two indications that the error in using D as 740

Fractal Aggregation Near the Threshold

“…Otherwise, nearest-neighboring sites [i,j-l) and (i-l,j) (see Fig. 1) already compared with Y are needed to investigate to see whether they are occupied or not; if (at least) one of them is occupied the site (i,j) belongs to the cluster or earlier formed aggregate composed of neighboring sites and 1 is assigned to the site (i,j); if none of them is occupied, the site (i,j) is still assigned to be zero, even for p k > Y, and no deposition occurs or a colloid (only one site (i,j) is not appended to the already formed aggregate, then we say that it is dissolved (which is somewhat different from the backbone' 19 1 formation processes in standard percolation), and so forth. The computer simulations proceed on row by row from the bottom of a square lattice.…”