1. Percolation Density Contours
Robert M. Ziff
University of Michigan
Collaborators: Peter Kleban, Jacob Simmons*, University of Maine (*Oxford)
Kevin Dahlberg, Univ. of Michigan

2. Percolation density profiles
We are concerned with finding the density profiles of percolation clusters. Consider a large number N of samples of a system, and consider clusters satisfying a certain criterion (such as touching a point on the boundary). Then
(r) = the number of times the site at r is occupied (wetted) by the clusters, divided by N.

3. In the limit where the mesh size (lattice spacing) goes to zero, the density goes to zero, so we consider it renormalized so that it remains finite. (In general, we will ignore constant coefficients.)

4. A system slightly below the percolation point

5. Simulation results - contours of the density of clusters touching the point on the bottom.

6. Density around a single point in an infinite system
First, consider the* average density of occupied sites around
a) a single infinite critical cluster connected to a given site
b) all clusters (finite and infinite) touching a given site
*well-known

7. a) A single cluster in infinite space
For a single (infinite) cluster, the total mass within a radius R scales as
M(r) ~ rD
Where D = fractal dimension = 91/48 for d = spatial dimension = 2.
It follows that the density of this one cluster scales as

8. b) All clusters connected to a point in infinite space
The probability that a point belongs to a cluster of size s (meaning, containing s sites)
Ps = s ns ~ s1–
where ns ~ s–equals the number of clusters of size s, per lattice site. Therefore, the probability the point belongs to a cluster of size greater than s is given by
P≥s =∫s∞ s1– ds ~ s2–
and the probability that the point belongs to a cluster whose maximum radius is greater than r is given by
P≥r ~ rD(2–) = rD-d
Where we have used s ~ rD , and the hyperscaling relation
 – 1 = d/D.
Then,
(r) = P≥r ∞(r) ~ rD-d rD-d = r 2(D-d) = r–

9. Relation to crossing probability
Consider an annulus, and consider the probability
annulus(r) that a cluster crosses from the inner circle of radius 1 to an outer circle of radius r.
This is the same as the probability that in an infinite system, the maximum radius of a single cluster is greater than r. Thus:
annulus(r) = P≥r ~ rD-d r

10. Conformal transformation to a strip with periodic b.c. (cylinder)
z = x + i y
w = u + i v
w = e2z i 1
e2x
periodic b.c. x
Under a conformal transformation,
Crossing probabilities are invariant:
h(x) = annulus(e2x) = e–2(d-D)x = e–(5/24)x
(this is the horizontal crossing probability on the cylinder, for large x).

11. Densities transform as
where h = hΨ= 5/96. For annulus to rectangle, we find
h(x) = e–2(d–D)x = e–(5/24)x
h∞(x) = 1 (density is constant!)
Note:
h(x) = h(x) = Prob(xmax > x).
Interpretation: clusters are constant density up to their maximum extent. (Similar relation not true in annular geometry).

12. Single point at the edge of a half-infinite plane:
From boundary operators of conformal field theory, we find for clusters touching an “anchor point” at the origin,
Where hΨ = 5/96 and hφ = 1/3. This gives:

13. ContourPlot[y^(11/48) (x^2 + y^2)^(-1/3), {x, -1, 1}, {y, 0, 1}] Comparison with simulation:

14. Transformation to a square:

15. Comparison (rotated)

16. Two anchoring points on the boundary
Simulations showed that the density satisfies a kind of square-root superposition relation:

17. Simulations of density with (a) 1 anchor at y = 3/8, (b) 1 anchor at y = 5/8, (c) two anchors, and (d) the square root of the product of the one-anchor simulations, multiplied by a constant. a b b c d

18. Theoretical prediction
Define P(z1,z2) = the probability that the two points z1 and z2 are connected together, and P(z1,z2,z3) = the probability all three points are connected together. Then, it follows from boundary operator theory that if z1 and z2 are on the boundary,
Where C is a constant, valid everywhere except a few lattice spaces from the anchors (where C is effectively 1).
This was verified numerically with C ≈ ± 0.001, and furthermore it was found that C is universal!

19. Conformal transformation to a square - theoretical prediction for clusters anchored at two points

20. Prediction of constant C (Thesis of Jacob Simmons, defended April 18, 2007)=
It is related to the coefficient in Cardy’s formula for
the crossing of a rectangle with open boundaries

21. Two internal points and a boundary anchor
Two internal points and a boundary anchor. The square-root superposition seems to hold only far from anchor points (simulations)

22. Anchoring points on opposite sides of a square.

23. Superposition of densities for finding density of clusters simultaneously touching two opposite anchors.

24. Comparison of simulations and theory for density of clusters touching two opposite points on the square.

25. Let upper point go to infinity: get infinite clusters touching a point in the half-infinite plane.  = y^(11/48) (x^2 + y^2)^(-1/6) compared with  = y^(11/48) (x^2 + y^2)^(-1/3) for cluster just touching the bottom (rignt). Follows from sqrt superposition rule:

26. Density of clusters touching one boundary or an interval of a boundary
Half infinite system, clusters touching real axis:
(y) ~ y–
(Goes to infinity as y goes to zero.)

27. Density of clusters touching two boundaries (rather than two points) simultaneously

28. Density contours of clusters touching top and bottom boundaries, open b.c. on the sides (work with Kevin Dahlberg). Note density goes to zero at both open and fixed boundaries.

29. (contours) Here there are open boundaries on the sides.
Note that the density is highest in the center, because more paths cross there.
No theory yet, except for density at the boundaries and in the limits of strips.

30. Density of clusters simultaneously touching top and bottom:
Limit of upper boundary goes to infinity: density of infinite clusters touching upper half plane:
(y) ~ y
Here the density goes to zero at the boundary, rather than infinity as in the case of just touching one boundary! (Because of different fractal character near the boundary.)

31. Method of derivation: 0 = density for clusters touching lower boundary, 1 = density for clusters touching upper boundary, e = density for clusters touching either boundary (Burkhardt, Res and Straley), and 0 = density for clusters touching both boundaries): ,

32. Numerical test

33. Comparison of density contours on the same (arbitrary) logarithmic scale (equal density = same color), for crossing from lower point to opposite boundary (left), and crossing between two opposite points (right).

34. Clusters touching single point on lower boundary, and anywhere on upper boundary. Left: 566 crossing clusters out of 4,000 trials. Right: 29,151/205,000 trials. 256x256 lattice. Note that density goes to zero about the same way on left and right boundaries as top boundary. This is because if a cluster hits the top, it most likely will hit the sides too.

35. Density at the ends of half-infinite strips
Density of clusters vertically crossing the strip:

36. (Density at the ends of half-infinite strips)
Density of clusters vertically crossing the strip:
Compare with the density far from the end: