Transcript C n =C n - Université d`Orléans

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
Lightnings:
Formation of conducting regions inside isolating
matter submitted to high electric potential.
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
C(t) is the capacity of Kt
We get Loewner equation:
The fact that the process is increasing
translates into
Re(A(t,z))=
Which implies the existence of
measures (µt ) such that
And every (reasonnable)
family (µt ) of positive
measures can be obtained
in this way .
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: