Understanding 2D critical percolation
from Harris to Smirnov and beyond
Oded Schramm
http://research.microsoft.com/∼schramm
Site percolation
Let G = (V, E) be an infinite connected graph and let p ∈ [0, 1].
In (Bernoulli) p site percolation, we select every vertex of G with probability
p, independently, and consider the induced subgraph on these vertices.
1
Bond percolation
In (Bermoulli) p bond percolation, we select every edge with probability p,
independently, and consider the subgraph on V with these edges.
2
Other models
For example:
3
Critical Percolation
There is some number pc ∈ [0, 1] such that there is an infinite component
with probability 1 if p > pc and with probability 0 if p < pc.
The large-scale behaviour changes drastically when p increases past pc. This
is perhaps the simplest model for a phase transition.
4
The big picture; sample questions
Questions about general graphs: when is pc(G) = 1?
Questions away from p = pc generally look at behaviour as p → pc.
In Zd: is there an infinite cluster at p = pc?
Harris 1960 / Kesten 1980: no for d = 2.
Hara & Slade 1990: no if d > 19.
5
Specialize to critical percolation in planar lattices
Z2, bond percolation:
6
Triangular lattice, site percolation (TG):
7
Triangular lattice, site percolation (TG):
8
Triangular lattice, site percolation (TG):
9
The Harris-Kesten Theorem
Theorem (Harris 1960). At p = 1/2 there are no inifinite clusters a.s.
Therefore, pc > 1/2.
Theorem (Kesten 1980). pc = 1/2.
10
Crossing probabilities & duality
In Z2, bond, p = 1/2 the probability to left-right cross an (n + 1) × n
rectangle is exactly 1/2.
11
Similar for a rhombus in TG:
12
The Russo-Seymour-Welsh Theorem 1978
∀ρ > 0
n
o
lim inf P left-right cross [0, ρ L] × [0, L] > 0 .
L→∞
13
The lowest crossing
14
Smirnov’s proof of RSW
Enough to prove:
a
2a
a
P
b > s =⇒
b P
a
b
aa
2a
b
2
>
s
/16
b
a
b
15
because,
16
a
b
a a
a
b
P>s
=⇒
P > s/2
17
b
a
b
a
aa
a
a
P > s/2
b
P > s/4
or
P > s/4 > s2/16
18
b
b
b
P > s/4
P > s/4
=⇒
P > s2/16
19
Harris’ theorem follows
20
A large critical cluster
At pc, there are no infinite clusters. If we condition on the event that the
cluster of the origin has more than 1000 vertices, then here’s what it looks
like.
21
Predictions from physics
Physicists have predicted some exponents describing asymptotics of critical
percolation in 2D.
For example, Nienhuis conjectured that the probability that the origin is in
a cluster of diameter > R is
R−5/48+o(1),
R→∞
and Cardy conjectured that the probability that the origin is connected to
distance R within the upper half plane is
R−1/3+o(1),
R → ∞.
22
Cardy’s formula
What is the probability of a white left-right crossing of a rectangle for
critical percolation?
Cardy predicted that in the limit as the mesh goes to zero the answer is
¡2¢
¡1 2 4 ¢
Γ 3
1
3
¡ 4 ¢ ¡ 1 ¢ η 2F1 , ; ; η .
3 3 3
Γ 3 Γ 3
23
¡2¢
¡1 2 4 ¢
Γ 3
1
3
¡
¢
¡
¢
Cardy’s formula = 4
1 η 2 F1 3 , 3 ; 3 ; η .
Γ 3 Γ 3
Here, 2F1 is the hypergeometric function and η is an explicit function of
the aspect ratio of the rectangle.
More precisely, η is the cross ratio of the images of the corners under any
conformal map mapping the rectangle to the upper half plane H.
24
Carleson’s version of Cardy’s formula
∀x ∈ [0, 1],
x
P
x
mesh
0
1
25
Smirnov’s Theorem 2001
For critical site percolation on the triangular lattice, Cardy’s formula and
Carleson’s version of it hold and crossing probabilities are asymptotically
conformally invariant, as the mesh tends to zero. In an appropriate sense,
this percolation process as a whole is asymptotically conformally invariant.
26
Sketch of Smirnov’s proof
A0




















f0(z) := P



















z
∂2




















∂1
(simple path)
∂0
A1
A2



















27
A0
A2
A1
fj ( ) − fj ( ) = P[ sep,
not] − P[ sep,
not].
fj (x, y) := P[x sep, y not].
Main Lemma: fj ( , ) = fj+1( , ).
28
Proof of main lemma
Produce a bijection!
29
Since fj (x, y) − fj (y, x) = fj (x) − fj (y) is a discrete derivative, the main
lemma gives a ternary Cauchy-Riemann like system. It follows that in
the limit fj + eπi/3 fj+1 is analytic, and the fj are harmonic. Boundary
behaviour then characterizes them.
30
Critical percolation interface
31
What shape?
32
Hit?
x0
x
A2
A1
A0
P[hit] ≈ Cardy(A0, A1, A2, x0) − Cardy(A0, A1, A2, x).
Gives conformal invariance.
33
Conformal Markov property
Apply a conformal map in the slitted half-plane to map back to the half-plane
gt
then in the limit the image of the continuation of the curve has the same
conditioned distribution as the original curve translated.
34
Loewner’s Theorem
A non self-crossing path γ from 0 to ∞ in the upper half plane is encoded
(up to time change) by a one dimensional path W : [0, ∞) → R, as follows.
gt : H \ γ[0, t] → H ,
2
dgt(z)
=
,
dt
gt(z) − Wt
Wt = gt(γ(t)).
g0(z) = z .
35
Stochastic Loewner Evolution
SLE(κ) is the path γ(t) where Wt = B(κ t), and B is one dimensional BM.
Schramm 1999: A path satisfying the conformal Markov property must be
SLE(κ) for some constant κ > 0. SLE(6) is the only SLE satisfying Cardy’s
formula.
36
Consequences
Lawler-Schramm-Werner 2001:
P[cluster of the origin has diam > R] = R−5/48+o(1).
Smirnov-Werner 2001: Many other percolation exponents.
Lawler-Schramm-Werner 2000, 2002: The outer boundary of the scaling
limit of percolation clusters is (essentially) the same as the outer boundary
of planar BM. Both have Hausdorff dimension 4/3.
37
Various random models
Loop erased random walk
Self avoiding walk
Critical Ising
Gaussian free field interface
TG percolation interface
UST Peano path
LSW 2001
conjectured
conjectured
SS 2006
Smirnov 2001
LSW 2001
SLE(2)
SLE(8/3)
SLE(3)
SLE(4)
SLE(6)
SLE(8)
38
LERW / SLE(2)
39
SAW / SLE(8/3)?
Half plane SAW
(by Tom Kennedy)
40
Ising / SLE(3)?
(Thanks David B. Wilson)
41
HE / SLE(4)
42
Percolation interface / SLE(6)
43
UST Peano path / SLE(8)
44