Evidence for the first-order phase transition in the multifractal spectrum for diffusion-limited aggregation

A new method of calculating the growth probabilities of the DLA clusters based on the Spitzer theorem is presented. It allows very accurate determination of the probabilities of hitting the random walkers by the perimeter of the cluster, even deep in the ‘fjords’ where the probabilities are small. The evidence for the first-order phase transition in the plots off(q) are found. The large fluctuations of the minimal growth probabilities between different clusters are also discussed. Recently there has been a growing interest in understanding the mechanism leading to formation of the ‘fjords’ in the diffusion-limited aggregation ( DLA) clusters (see Ball and Blunt 1989, 1990) and the related phase transition in the multifractal spectrum f ( q ) (Blumenfeld and Aharony 1989, Lee and Stanley 1988, Lee et a1 1989) as well as the breakdown of the scaling behaviour of the moments (Cohen and Harris 1990, Trunfio and Alstrom 1990). In this letter I present the results of very accurate computer calculations of the hitting probabilities of DLA clusters based on the Spitzer theorem (Spitzer 1964) which allow the firm detection of the first-order phase transition at q = 0. Diffusion-limited aggregation is a simple stochastic process leading to the formation of fractal patterns. It was proposed by Witten and Sander (1981) and now a large amount of literature devoted to this topic exists (see e.g. Stanley and Ostrovsky 1985, Vicsek 1988). In this model a single particle walks randomly on the square lattice until it reaches another particle (‘seed’), usually located in the centre of the lattice. Then, a new particle initiates its random walk. If the particle contacts the cluster (now consisting of two particles) it is incorporated into the cluster and the cluster grows. This process is repeated many times (-103-106) and leads to the formation of the ramified fractal structure for which the relation between the number, N ( R ) , of particles inside the circle of radius R is of the form N ( R ) R D ( 1 ) where D = 1.7 is the fractal dimension. A satisfactory theory of DLA is still missing. In particular, nobody has yet proved equation ( 1 ) although a few attempts were made (see Muthukumar 1984, Tokuyama and Kawasaki 1984 and Kolb 1987). The breakthrough occurred with the recognition of the role played by the set of the growth probabilities {p.;}csr., where p 5 is the probability that the perimeter site s is the next to grow and r is the set of the nodes on the perimeter of a given DLA cluster (Turkewich and Scher 1985). The customary 030S-4470/90/241287 + 05%03.SO @ 1990 IOP Publishing Ltd L1287 L1288 Letter to the Editor way of studying the properties of the set of probabilities { p s } is by means of the moments (Halsey et a1 1985, Amitrano et a1 1986) Z q ( R ) = .E p % (2) T where R is the linear size (radius of gyration) of the aggregate and q E R. In the early works, the power-like dependence of the moments on R was found: Z q ( R ) R-"". (3) The fact that the function T( q ) is not linear is called multifractality (Halsey et a1 1986) and the function f( q) f ( q ) = qa(q) 4 4 ) ( 4 b ) is called the multijructal spectrum. Sometimes the relation ( 4 a ) is inverted and substituted into (46) giving the Legendre transform f ( a ) of the function T(q). Blumenfeld and Aharony (1989) gave the theoretical arguments that the function f( q ) should display the first-order phase transition at qc = 0. The detection of the phase transition is a problem of a numerical nature-it is linked to the sites with very small hitting probabilities, p S , and to get reliable results the accuracy of the calculation of p,s should be many orders smaller thanpmi,. I have quite recently performed a numerical calculation using the Spitzer theorem and these results are reported here. This method is very accurate; for the completely screened sites (for which p c = 0 and which do not contribute to the moments) I have sometimes obtained ps of the order lo-*' instead of p S . There is also another check of the accuracy; for pairs (or triplets) of the sites, which by symmetry arguments should possess the same p r , I have obtained probabilities whose first 18 digits coincide. The Spitzer formula gives the hitting probabilities of the arbitrary finite set for the arbitrary aperiodic recurrent random walk in two dimensions. Because in the usual DLA the particles perform the symmetric random walk on a two-dimensional lattice Z2, I will describe here the Spitzer recipe for calculating the hitting probabilities of a simple random walk by points belonging to a finite set B. For the simple random walk the transition probability P(x, y ) is of the form i f x and y are nearest-neighbour sites in other cases. I (5) Let Pn(x, y ) denote the probability that a particle executing a random walk and starting at the point x will reach the point y after n steps: P(X, Y 1 = {; Pn(x, Y ) = C P ( x , x l ) P ( x , 9 ~ 2 ) . . P(xn-1, y ) . (6) x,EZ2,:= I , .... n I Let G n ( x , y ) denote the expected number of visits of the random walk starting at x to the point y within n steps: The crucial quantity in the Spitzer formula is the potential kernel defined as An(x , Y ) = Gn(0,O)Gn(x, Y ) . (8) Letter to the Editor L1289 Let A(x, y) denote the limit A(x, y ) = lim A,(x, y ) . (9) n i D It can be proved that the operator A(x, y ) is symmetric and, if restricted to any finite subset B of Z2, invertible. Let K B ( x , y) denote this inverse matrix Let H B ( x , y) denote the probability of first hitting the set B at the point y when starting point x iZ B. If the set B E Z2 consists of at least two points then the following formula holds In the diffusion-limited aggregation it is assumed that the particle starts from infinity; 1x1 + 00. For such a case it can be shown that (12) reduces to a simpler expression (see Spitzer 1964, theorem 14.1) Due to the translational symmetry of the simple random walk we have A(x, Y ) = a(x Y ) where the function a ( x ) is given by the following integral: Here the notation x = (m, n ) was introduced. The integral (14) can be calculated exactly only for points lying on the ‘diagonal’ x = ( n , n ) , but it suffices by proper use of the symmetry properties of the double integral (14) to obtain values of a( n, m) for arbitrary points on the plane (see Spitzer 1964). In this approach the natural parameter describing the size of the clusters is the number of perimeter sites, which I will denote as P. 1 have checked that there is a power-like dependence between P and N (or R ) : P N Y with y = 0.92 and it justifies the use of P instead of R or N. I have generated 400 clusters consisting of up to P=79, which corresponds to about 60 particles. At five stages of the growth process, P = 60. . .61, 64. . .65,. . . , 78 . . .79 (it should be remarked that the perimeter can also change by two sites), the actual p,s were calculated by means of the Spitzer theorem and recorded. The moments were averaged over clusters: