Other growth models

Date: 2015-10-07; view: 440.

SLE is in fact just one very special, solvable, example of an approach to growth processes in two dimensions using conformal mappings which has been around for a number of years. For a recent review see [43]. The prototypical problem of this type is diffusion-limited aggregation (DLA). In this model of cluster formation, particles of finite radius diffuse in, one by one, from infinity until they hit the existing cluster, where they stick. The probability of sticking at a given point is proportional to the local electric field, if we imagine the cluster as being charged. The resultant highly branched structures are very similar to those observed in smoke particles, and in viscous fingering experiments where one fluid is forced into another in which it is immiscible. Hastings and Levitov [44] proposed an approach to this problem using conformal mappings. At each time t, the boundary of the cluster is described by the conformal mapping f_t (z) which takes it to the unit disc. The cluster is grown by adding a small semicircular piece to the boundary. The way this changes f_t is well known according to a theorem of Hadamard. The difficulty is that the probability of adding this piece at a given point depends on the local electric field which itself depends on . The equation of motion for f_t is therefore more complicated than in SLE. Moreover, it may be shown that almost all initially smooth boundary curves evolve towards a finite-time singularity: this is thought to be responsible for branching, but just at this point the equations must be regularised to reflect the finite size of the particles (or, in viscous fingering, the effects of finite surface tension.)

It is also possible to generate branching structures by making the driving term a_t in Loewner's equation discontinuous, for example taking it to be a Levy process. Unfortunately this does not appear to describe a physically interesting model.

Finally, Hastings [45] has proposed two related growth models which each lead, in the continuum limit, to SLE. These are very similar to DLA, except that growth is only allowed at the tip. The first, called the arbitrary Laplacian random walk, takes place on the lattice. The tip moves to one of the neighbouring unoccupied sites r with relative probability E (r)^η, where E (r) is the lattice electric field, that is the potential difference between the tip and r, and η is a parameter. The second growth model takes place in the continuum via iterated conformal mappings, in which pieces of length ℓ₁ are added to the tip, but shifted to the left or right relative to the previous growth direction by a random amount ±ℓ₂. This model depends on the ratio ℓ₂/ℓ₁, and leads, in the continuum limit, to SLE_κ with κ = ℓ₂/4ℓ₁. For the lattice model there is no universal relation between κ and η, except for η = 1, which is the same as the loop-erased random walk (Section 2.2) and converges to SLE₂.