Transcript rsws2 6842

Percolation Density Contours
Robert M. Ziff
University of Michigan
Collaborators: Peter Kleban, Jacob Simmons*,
University of Maine (*Oxford)
Kevin Dahlberg, Univ. of Michigan
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.
• 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.)
A system slightly below the percolation point
Simulation results - contours of the density of clusters
touching the point on the bottom.
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
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
D1
M(r

dr)

M(r)
r
d D
5 / 96
 (r) 


r

r
cr d 1dr
cr d 1
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–
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.
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
Conformal transformation to a strip with
periodic b.c. (cylinder)
z=x+iy
i
w = e2z
w=u+iv
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).
• 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).
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 h = 5/96 and h = 1/3. This gives:
ContourPlot[y^(11/48) (x^2 + y^2)^(-1/3), {x, -1, 1}, {y, 0, 1}]
Comparison with simulation:
Transformation to a square:
Comparison (rotated)
Two anchoring points on the
boundary
• Simulations showed that the density satisfies
a kind of square-root superposition relation:
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
bb
c
d
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 ≈ 1.030 ± 0.001, and
furthermore it was found that C is universal!
Conformal transformation to a square theoretical prediction for clusters
anchored at two points
Prediction of constant C (Thesis of Jacob
Simmons, defended April 18, 2007)
= 1.0299267867...
It is related to the coefficient in Cardy’s formula for
the crossing of a rectangle with open boundaries
Two internal points and a boundary anchor. The
square-root superposition seems to hold only far from
anchor points (simulations)
Anchoring points on opposite sides
of a square.
(z,0.5)
(z,0.5  i)
(z,0.5,0.5  i)
Superposition of densities for finding density of
clusters simultaneously touching two opposite

anchors.

(z,0.5,0.5  i)  (z,0.5)(z,0.5  i)
•
Comparison of simulations and theory for density of clusters touching
two opposite points on the square.
Let upper point go to infinity: get infinite clusters touching a point in the halfinfinite 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:
(z)  1 (z)  (z) 

y11/ 48 11/ 48 y11/ 48
y
 1/ 3
2/3
r
r
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.)
Density of clusters touching two
boundaries (rather than two points)
simultaneously
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.
500
400
5/8
3/4
7/8
15/16
31/32
63/64
127/128
1
300
200
100
0
0
100
200
300
400
500
(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.
• 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.)
•
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):
• ,
Numerical test
•
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).
•
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.
Density at the ends of half-infinite strips
Density of clusters vertically crossing the strip:
(Density at the ends of half-infinite strips)
Density of clusters vertically crossing the strip:
Compare with the density far from the end: