1. 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)

2. 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:

3. . Electrodeposition More examples with different voltages:

4. Voltage: a:2V, b:3V, c:4V, d:6V, e:10V, f: 12V, g:16V

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

6. Bacteria colonies with various quantities of nutriments:

7. D) Croissance des mégapoles

8. 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.

9. Dielectric breakdown models

10. A) Eden ’s model. Model used in biology: Growth of bacteria colonies with abundance of nutriments Growth of tumors.

11. DLA Model (Diffusion-limited aggregation)

15. The study of the growth process consists in comparing the diameter D n of the cluster at time n and its length L n. An important remark is that in the case of HL(0) C n =C n for some C>1.

16. The HL(0) process

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

21. Loewner processes Conformal mapping The fact that the process is increasing translates into Which implies the existence of measures (µ t ) such that We get Loewner equation: And every (reasonnable) family (µ t ) of positive measures can be obtained in this way. Re(A(t,z))= C(t) is the capacity of K t

23. 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.

25. REGULARIZATION

27. Proof: