Extreme and smooth gradient percolation
Bernard Sapoval
Ecole polytechnique
Subtitle:
Fractal exponents (like 7/4) without fractals and
without SLE
and critical fluctuations without critical parameter
Agnès Desolneux (MAP 5, Université Paris 5)
Andrea Baldassarri (Università La Sapienza, Roma)
Diffuse distribution of particles on a lattice:
gradient percolation
Particles
source
Lg
Lg+1
lines
Particles
sink
L columns
Smooth gradient percolation:
At each occupied site a continuous gaussian distribution
with a variance s2 is attached.
F(x,y) =(1/2s2) occup. sites i exp-{[(x-xi)2 +(y-yi)2] / 2s2}
Then the contributions of the occupied sites are summed:
The contributions of the occupied sites are summed:
500 x 500
s=2
Find the constant level lines:
Whatever the level,
one observes similar
fluctuations.
Critical fluctuations
but around an
arbitrary level 
Physical problems at the origin of
gradient percolation:
• Diffusion fronts: geometry of diffuse
contacts and soldering.
• Structure of fuzzy images.
• Corrosion fronts in the etching of
random solids.
• Erosion fronts: sea-coasts geometry.
DIFFUSE STRUCTURE:
xf
Lg
2
The front is
characterized by
its position xf ,
its width f
and its length Nf.
A source of particles is kept
at a constant concentration =1
At time t > 0, the particle concentration at a distance x is
given by the complementary error function:
p(x) 1
where

lD  2(Dt)
1
2
2

1
2
 u2
 e du
x
lD
0
is the diffusion length at time t
If nf(x) is the mean number per unit
horizontal length of points of the front:
Diffusion fronts: geometry of
diffuse contacts and soldering.
What is gradient percolation?
Random
distribution of particles
with a gradient
of concentration.

The gradient percolation
front is the frontier
of the infinite cluster
Mathematical aspects:
P. Nolin
 ≈ Lg4/7
Numerical observations
The average front position is such that,
for large Lg, p(xf) is close to pc
pc = 0.59280 ± 10-5
Rosso, Gouyet, BS(1985)
pc = 0.592745 ± 2. 10-6
Ziff, BS (1986)
pc = 0.5927460 ± 5. 10-7
Ziff and Stell (1988)
First measurements of the front fractal dimension…
Df = 1.76 ± 0.02.
(1984)
1984
The front width f followed a power law
f  Lgwith ≈ 0.57…
The front length Nf followed a power law
Nf  Lg with  ≈ 0.42…
An island is a finite cluster because
it is situated at a position x where p(x) < pc
 f  K. (x f   f )
 (x)   0 p(x)  pc

  K p(x   )  p
f
0
f
f

c

dp
 K 0 . f
(x f )
dx


 f 
 K 0  
Lg 
 = 4/3 (den Nijs, 1983)

4
 f  Lg with  

1  7

= 0.5714…
Front length Nf:
The front width  being a
correlation length:
xf
Lg
2
Nf is of order (L / ).Df
Nf ≈ L. (Df-1) ≈ L. Lg 
 = (Df -1)/(1 + )
 ≈ 0.426
The fact that  is the horizontal correlation length
has been shown recently by Pierre Nolin (arXiv:math/0610682)
but one had
+≈1
/(1 + ) + (Df -1)/(1 + ) = 1
Df  / (1+ ) ≈ 1
Percolation cluster hull
Df = (1+ )/ 
Df = 7/4 ???
conjecture
(1984)
But now we know (Duplantier, 1987, Smirnov, 2001)
that for sure Df = 7/4:
The number of particles in a box of lateral width  is :
(Nf /L).  ≈  Df ≈ Lg Df . / (1+ ) ≈ Lg
• But (Nf /L).  is the number of particles in a box
of size  where 
is the statistical width of the frontier.
This width is defined independently of
the fractal character of the frontier.
Fractality does not appear in this statement
For diffusion, it means that the
correlated surface is of the
order of the diffusion length at
time t:
lD(t) =
1/2
2(Dt)
What if the gradient is so large that the
frontier is no more fractal?
Small Lg values?
The same power laws are observed
Is there a conservation law?
?
??
Other situation with gradient percolation: fuzzy image
a: original object
b: fuzzy photograph
c, d: gradient percolation
Df = 7/4
Df = 4/3
e, f: smoothing and filtering
B. Sapoval And M. Rosso, Fractals, 3, 23-31 (1995)
Statistique des fractures: empirique ou théorique?
Attaque corrosif d’une couche mince d'aluminium plongé dans une solution
Expériences: L.Balazs (1996) PMC Ecole Polytechnique.
CCD Camera
Aluminium
Solution
Verre
Lumière
Other situation with gradient percolation:
pit corrosion of an aluminum film.
L. Balasz (Ecole polytechnique,1997)
Time evolution of the corrosion picture:
• The first circular pit
grows with time.
• It roughens
progressively and
slows down.
• It finally stops on a
fractal frontier with
dimension 4/3.
QuickTime™ et un
décompresseur TIFF (non compressé)
sont requis pour visionner cette image.
The corrosion model:
•Andrea Baldassari
•Andrea Gabrielli
Width of the front as a function of the gradient
but where is the gradient?
• EROSION OF ROCKY COASTS:
Rocky coast-line erosion has marine and
atmospheric causes which act on random
‘lithologic’units:
random
rocks
Random means that the ‘mechano-chemical’
properties of the rocks (due to structure and
composition) are unknown and exhibit some
dispersion.
Sea-coasts could be fractal because their geometry damps the
sea-waves (and currents …) in such manner that, for a given
‘sea power ’, the erosion is minimized.
In that sense it is not only the coast which is eroded but the
effective erosion force of the sea which is diminished by the
geometry of the coast.
And this is why one observes fractal sea-coasts ????
Phys. Rev. Lett. 2004.
http://www.nature.com/nsu/031124/031124-4.html
The sea power and erosion force is a decreasing
function of the coast perimeter, for example:
1
f (t) 
g Lp (t)
1
Lp (t  0)
Empirical breakwater construction recipe:
The ‘resisting’ earth is represented
by a square lattice where each site presents a
random lithology or resistance to the sea.
x11 x12 x13 x14 x15
x6 x7 x8 x9 x10
x1 x2 x3 x4 x5
Model representation of the erosion process
x11 x12 x13 x14 x15
x6 x7 x8 x9 x10
x1 x2 x3 x4 x5
5 , f(t=0)
Lp = L
57p ,=f(t=1)
=
f(t=2)
< f(t=0)
The weaker sites are
eroded and, at the
same time:
1- new sites (strong
or weak) are
uncovered.
2- the coastal length
is modified.
Time evolution: fractal morphology is a
statistical geometrical attractor
North Sardinia real and numerical
Df, geo. = 1.33
Df, num. = 4/3
Box-counting determination
of the fractal dimension
Df = 4/3
The value 4/3 is the
dimension of the
percolation cluster
accessible
perimeter.
(Grossman, Aharony,
1985)
(Duplantier et al. 1999)
Scaling behavior of the coast width
4/7 is related
to gradient
percolation
Final coast morphologies
The erosion model in itself is more general.
It could apply to rough but non-fractal coastline:
What if the gradient is so large that the
frontier is no more fractal?
Small Lg values?
The same power laws are observed
Extreme gradient percolation or fractal
exponents without fractals
A. DESOLNEUX, B. SAPOVAL, and A. BALDASSARRI,
Self-Organised Percolation Power Laws with and without Fractal Geometry in
the Etching of Random Solids,
in “Fractal Geometry and Applications”
Proceedings of Symposia
in Pure Mathematics (PSPUM).

American Mathematical Society
In print. See cond-mat/0302072.


Note that if percolation had been
defined in the first place through
gradient percolation :
…. pc = 1/2
Extreme gradients: is f 7/4 proportionnal to Lg ?
Lg = 2
Remark: for small Lg
One expects a law of the form 7/4 = a(Lg + b)
And Nf7/3 = c(Lg + d)
Is b = -1?
Numerical test of the exponent:
Is 7/4 the best exponent?
1- Choose arbitrary 1.6 < <1.9.
2- Find the best fit values of the law
a(Lg + b).
3- Then for each ,compute the distance:
2
1 50

d( ) 
  f (Lg )  a (Lg  b )
47 l  4
2
g
Result:
d()2

For very extreme gradients, one can
determine exact values:
DISCUSSION:
Comparison between the exact values
and the values extrapolated from the numerics
for Lg between 4 and 50.
Discussion:
Comparison between the exact values
and the values extrapolated from the numerics
for Lg = 4 and Lg = 5 for which one “knows” standard deviations
Extrapolation
from small to
large
Concluding remarks
• Exact values for Lg = 2 and 3 enters the confidence
interval deduced from Lg = 4 and 5.
• There exists a unique power law relating the width and
length of the gradient percolation front to the
gradient length for Lg between 2 and infinity. It does
not apply for Lg= 1.
• The same type of results are obtained for the
triangular lattice.
• There exists an implicit relation between width
and length of the type 7/4  Nf7/3. You may
measure the interface properties and find if it
was created, in the past, by a gradient percolation
mechanism.
• Here, 7/4 is not related to conformal invariance or
SLE nor to fractality …
• Its origin is combinatory …
• There is a conservation law in diffusion on a
lattice: The flux per column is proportional to the
correlated surface…
• However, nature does not live on a lattice and the
question remains of how to apply those results to
rough coatslines for example…
Smooth gradient percolation in 2D:
Other broadening functions creates
the same type
of fluctuations:
Gaussian
Exponential
Yukawa potential
…
Smooth gradient percolation in 2D:
Gaussian broadening
s=2
s=3
threshold lines = 0.1; 0.3; 0.5; 0.7; 0.9
Ordinary percolation on a lattice is defined
by the occupation or not
of sites or bonds with probability p.
In continuous percolation a Poisson
distribution of points is occupied by circles.
The macroscopic manifestations
of these elementary events appear
only in a narrow region near pc.
In smooth percolation,
critical fluctuations are found
around an arbitrary threshold.
Invariance of the geometry with
respect to the arbitrary threshold
apparent
fractal
dimension
QuickTime™ et un
décompresseur TIFF (non compressé)
sont requis pour visionner cette image.
threshold
Gaussian
width
Exponential broadening = exp(-r/)
(1s wave functions)
=3
=2
levels lines = 0.1; 0.3; 0.5; 0.7; 0.9
Yukawa potential or screened Coulomb
potential = [(1/(r+1)] exp(-r/)
 is the
Debye-Huckel
length
 =5
Yukawa potential or screened Coulomb
potential = [(1/(r+1)] exp(-r/)
 =10
Examples of Yukawa potentials
1- Distribution of radio frequencies antennas:
lines of constant electric field should
critical geometrical fluctuations.
2- Equipotential lines due to a charge
concentration distribution of ions in an
electrolyte. Electrochemical reaction paths?
1D Gradient percolation is simple
but not trivial
1D Percolation
dc= distance for the first empty site
dc  
k
k 0
pc 1
p
kp (1 p) 
1 p


k
 1
Gradient percolation in 1D
Starting at the origin,
where do we find the first empty site?
The size of the fluctuation zone is
Lg - 2E(T) of order Lg
Example:
Lg =100; s=3
Characteristic function
C(x)= 1 if
F(x) ≥  with =0.6
Average of Fs,(x)
Smooth gradient percolation in 1D
At each occupied site
a continuous gaussian distribution
with a variance s2 is attached.
Then the contributions of the occupied sites are summed:
1
(xk)2 /2s2
F(x) 
e

s 2 occupied k
Example:
Lg =1000; s=3
Characteristic function
C(x)= 1 if
F(x) ≥  with =0.6