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Deterministic and random Growth Models. (Some
remarks on Laplacian growth).
S.Rohde (University of Washington) M.Zinsmeister
(MAPMO,Université dOrléans et PMC, Ecole
Polytechnique)
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Some physical phenomena are modelized by random
growth processes cluster at time n1 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
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Electrodeposition
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More examples with different voltages
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Voltage a2V, b3V, c4V, d6V, e10V, f 12V,
g16V
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Lightnings
Formation of conducting regions inside isolating
matter submitted to high electric potential.
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Bacteria colonies with various quantities of
nutriments
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D) Croissance des mégapoles
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These pictures indicate the need of a unique
model with parameter

• The model must consist of
• A probability law for the choice of the boundary
  point.
• 2) An object to attach.

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Dielectric breakdown models
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A) Eden s model.

• Model used in biology
• Growth of bacteria colonies with abundance of
  nutriments
• Growth of tumors.

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DLA Model (Diffusion-limited aggregation)
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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)
CnCn for some Cgt1.
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The HL(0) process
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DETERMINISTIC MODELS
We consider growth models for which the size of
the added objects is infinitesimally small with
appropriate time change.
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Loewner processes
Conformal mapping
C(t) is the capacity of Kt
The fact that the process is increasing
translates into
We get Loewner equation
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 .
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Case alpha2 Hele-Shaw flows, supposedly
modelising introduction of a non-viscous fluid
into a viscous one. Picture experience with
coloured water into oil.
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REGULARIZATION
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Proof
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