Deterministic and random Growth Models.

(Some remarks on Laplacian growth).

S.Rohde (University of Washington)

M.Zinsmeister (MAPMO,Université d’Orléans et PMC, Ecole Polytechnique)

Some physical phenomena are modelized by random growth processes: cluster at time n+1 is obtained by choosing at random a point on the boundary of the cluster at time n and adding at this point some object

Here are some examples:

.Electrodeposition

More examples with different voltages:

Voltage: a:2V, b:3V, c:4V, d:6V, e:10V, f:

12V, g:16V

Voltage: a:2V, b:3V, c:4V, d:6V, e:10V, f:

12V, g:16V

Formation of conducting regions inside isolating matter submitted to high electric potential.

Lightnings:

Bacteria colonies with various quantities of nutriments:

D) Croissance des mégapoles

These pictures indicate the need of a unique model with parameter

The model must consist of:

1) A probability law for the choice of the boundary point.

2) An object to attach.

Dielectric breakdown models

A) Eden ’s model.

•Model used in biology:

•Growth of bacteria colonies with abundance of nutriments

•Growth of tumors.

DLA Model (Diffusion-limited aggregation)

The study of the growth process consists in comparing the diameter Dn of the cluster at time n and its length Ln.

An important remark is that in the case of HL(0) Cn=Cn for some C>1.

The HL(0) process

DETERMINISTIC MODELS

We consider growth models for which the size of the added objects is infinitesimally small with appropriate time change.

Loewner processes

Conformal mapping

The fact that the process is increasing translates into

Which implies the existence of measures (µt ) such that

We get Loewner equation:

And every (reasonnable) family (µt ) of positive measures can be obtained in this way .

Re(A(t,z))=

C(t) is the capacity of Kt

Case alpha=2; Hele-Shaw flows, supposedly modelising introduction of a non-viscous fluid into a viscous one.

Picture= experience with coloured water into oil.

REGULARIZATION

Proof: