The moles' labyrinth: a growth model

We present a simple model for the simultaneous growth of many clusters. A phenomenological theory and simulations in two and three dimensions for this model are presented which describe the critical point where, for the first time, an infinitely large cluster is formed by the coalescence of individual clusters. This critical point is not in the universality class of standard percolation. Geometrical growth models are of practical importance in many areas but only recently has one begun to study them in the context of critical phenomena (Stanley et a1 1983). For the growth of a single cluster several models that show the generic problems of the time dependence have been studied (Amit et a1 1983, Witten and Sander 1982, 1983, Savada et a1 1982, Chandler et a1 1982). For these models the critical phenomenon occurs in the limit of infinite time and is described by the fractal dimension of the cluster. The other case of practical interest is the simultaneous growth of many clusters (polymerisation, coagulation, antigen-antibody reaction etc). In this case the critical point is the time t , at which for the first time an infinitely large cluster is formed. The former belief that this critical point is always of percolation type has been ruled out by two different types of studies (Herrmann et a1 1982, 1983, Ziff et a1 1982a, b). But unfortunately the models studied there and subsequently investigated similar models for additive polymerisation (Bansil et a1 1983, Rushton et a1 1983, Pandey et a1 1983)-although they have explained many experimental details-are not very helpful for an understanding of the growth as a critical phenomenon because they are too complicated. So we present in this letter a simple model which already contains typical features encountered in previously studied growth models of many clusters. In a D-dimensional lattice of size LD we choose (regularly or at random) C,LD sites from each of which we start a random walk. The paths formed by the walks will be considered (figure 1) and the question is asked when the system formed by the paths percolates, i.e. when there is a cluster of paths formed which spans over the whole lattice. The case L + 00 and C, << 1 is of particular interest. This model can be interpreted in two dimensions as a description ofthe labyrinth formed by moles. One places moles with an (average) distance of 1/JC, in the earth at a given depth. Simultaneously they then dig runs in a random way and without changing the depth. The moles have a vital interest in the critical time t , at which for the first time their runs have intersected each other enough so that there exists one network of paths through which they could escape as far as they want. (At t,, of course, not yet all the moles will have direct connection to this network). Keeping the language of moles on the lattice we define one growth step of the labyrinth as follows: choose a mole randomly; choose one direction randomly and @ 1983 The Institute of Physics L611 L612 Letter to the Editor Figure 1. Different stages of the growth of a labyrinth on a 100 X 100 lattice with periodic boundary conditions. The moles are marked by a full circle. The initial positions of the moles (crosses) are regularly distributed with a concentration C,=O.O025. (a) , r = 100, m=0.121, Pm=0.073; ( b ) , r=280, m=0.273, P,=0.246; (c), r=440-fc, m=0.380, Pm=0.880; ( d ) , r=720, m=0.526, Pm=l.O. displace the mole in this direction by one lattice spacing. After CILD such growth steps on average every mole has moved once. So we define the ‘time’ by