Is the critical percolation probability local?

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by  Itai Benjamini

IS THE CRITICAL PERCOLATION PROBABILITY LOCAL?
ITAI BENJAMINI,ASAFNACHMIASANDYUVAL PERES
9 Abstract. Weshowthatthecriticalprobabilityforpercolationonad-regularnon-
0 amenable graph of large girth is close to the critical probability for percolation on
0
an infinite d-regular tree. This is a special case of a conjecture due to O. Schramm
2
n on thelocality of pc. Wealso provea finiteanalogue of theconjecturefor expander
a graphs.
J

9
2

] 1. Introduction
R
P Denotebyp (G)thecritical probabilityforBernoullibondpercolation onaninfinite
c
.
h graph G, that is,
t
a
m p (G) = inf p ∈ [0,1] : P ∃ an infinite component > 0 .
c n p o
(cid:0) (cid:1)
[

Isthevalueof p determinedbythelocal geometry of thegraphor byglobal properties
1 c
v (such as volume growth and expansion)? In this note we show that the former is the
6
1 correct answer for non-amenable graphs with tree-like local geometry, and discuss a
6
conjecture of Schramm that p is locally determined in greater generality.
4 c
1. Recall that the girth g of a graph G is the minimum length of a cycle in G. Let P be
0
the transition matrix of the simple random walk (SRW) on G and let I be the identity
9
0 matrix. The bottom of the spectrum of I −P is defined to be the largest constant λ
1
:
v with the property that for all f ∈ ℓ2(G) we have
i
X
r hf,(I −P)fi≥ λ1hf,fi. (1.1)
a
Kesten ([9], [10]) proved that G is a non-amenable Cayley graph if and only if λ > 0.
1
This was extended by Dodziuk [7] to general infinite bounded degree graphs (for more
background on non-amenability see [11] and [14]).
Theorem 1.1. There exists an absolute constant C > 0 such that if G is a non-
amenable regular graph with degree d and girth g such that the bottom of spectrum of
I −P is λ > 0, then
1
Clog 1+ 1
p (G) ≤ 1 + (cid:0) λ21(cid:1) .
c
d−1 dg
1
2 ITAIBENJAMINI,ASAFNACHMIASANDYUVALPERES
Recall that p (T ) = 1 where T is an infinite d-regular tree and that for any d-
c d d−1 d
regular graph G we have p (G) ≥ 1 . Thus, Theorem 1.1 asserts that non-amenable
c d−1
graphs with large girth and degree d have p close to the lowest possible value, p (T ).
c c d
It is easy to construct non-amenable graphs with arbitrary girth, for example, take
the Cayley graph of ha,b,c | cn = 1i. Olshanskii and Sapir [13] constructed for any
k ≥ 2a group Γ with the following property. For any ℓ > 0 there is a set S consisting
k ℓ
of k generators for Γ , such that the Cayley graph G(Γ ,S ) has girth at least ℓ and
k k ℓ
inf λ (G(Γ ,S )) > 0(asremarkedin[13],fork ≥ 4suchgroupswerealsoconstructed
ℓ 1 k ℓ
by Akhmedov [2]).
Let G be a graph and v a vertex in G. Denote by B (v,R) the ball of radius R in
G
G centered at v, in the graph metric, with its induced graph structure. We say that a
sequence of transitive graphs G converges to G if for any integer R > 0 there exists
n
N such that B (v ,R) and B (v,R) are isomorphic as rooted graphs, for all n ≥ N
Gn n G
(note that the choices of v and v are irrelevant due to transitivity). Oded Schramm
n
(personal communication) suggested the following conjecture.
Conjecture1.2. LetG besequenceofvertextransitiveinfinitegraphswithsup p (G ) <
n n c n
1 such that G converges to a graph G. Then p (G ) → p (G).
n c n c
Thisconjecture is open for infinitegraphseven if weassume thatthey are uniformly
nonamenable. Wecanprovethefollowinganalogueoftheconjectureforfiniteexpander
graphs, by extending the analysis of [1], Proposition 3.1. For each n ≥ 1, let G be
n
a finite graph and let U be a uniformly chosen random vertex in G . We say that
n n
the sequence of finite graphs {G } converges weakly to an infinite rooted graph (G,ρ)
n
(where ρ is a fixed vertex of G) if for each R > 0 we have
P B (U ,R)6= B (ρ,R) → 0 as n → ∞,
(cid:16) Gn n G (cid:17)
where the event above means that the balls are not isomorphic as rooted graphs. This
is a special case of the graph limits defined in [5]. For two sets of vertices A and B,
write E(A,B) for the set of edges with one endpoint in A and the other in B. Recall
that the Cheeger constant h(G) of a finite graph G = (V,E) is defined by
|E(A,V \A)|
h(G) = min : 0< |A| ≤ |V|/2 .
n o
A⊂V |A|
Theorem 1.3. Let (G,ρ) be an infinite bounded degree rooted graph and let G be a
n
sequence of finite graphs with uniform Cheeger constant h > 0 and a uniform degree
bound d, such that G → G weakly. Let p ∈ [0,1] and write G (p) for the graph of open
n n
IS THE CRITICAL PERCOLATION PROBABILITY LOCAL? 3
edges obtained from G by performing bond percolation with parameter p. If p < p (G),
n c
then for any constant α > 0 we have
P G (p) contains a component of size at least α|G | → 0 as n→ ∞,
(cid:16) n n (cid:17)
and if p > p (G), then there exists some α > 0 such that
c
P G (p) contains a component of size at least α|G | → 1 as n→ ∞.
(cid:16) n n (cid:17)
For the reader’s convenience, we present the proof of Theorem 1.3 in Section 4.
1.1. Further discussion. Conjecture 1.2 suggests that the critical percolation prob-
ability is locally determined. This contrasts with critical exponents which are believed
to be universal and depend only on global properties of the graph. For instance, the
value of p on the the two dimensional square lattice is 1, but on the two dimensional
c 2
triangular lattice it is 2sin(π/18); however, the critical exponents are believed to be
the same.
It is worth noting another example of the locality of p where the limit graph is the
c
lattice Zd. For d> 1 and n> 1, write Zd for the d-dimensional torus with side n. The
n
following theorem is an immediate corollary of a theorem of Grimmett and Marstrand
[8] combined with the fact that the critical probability of a quotient graph is always
at least the critical probability of the original graph (see [4], [6] or [11]).
Theorem 1.4 (Grimmett, Marstrand [8]). For any d > 1 and k satisfying 2 ≤ k < d
we have
p (Zk ×Zd−k)→ p (Zd) as n → ∞.
c n c
Observe that this is theorem is a special case of Conjecture 1.2. For background
and further conjectures regarding percolation on infinite graphs see [4, 11].
2. Uniform escape probability
In this section we prove a useful lemma.
Lemma 2.1. Consider a reversible irreducible Markov chain {X } on a countable
t
state space V, with infinite stationary measure π and transition matrix P, such that
the bottom of the spectrum of I −P is λ > 0 (that is, (1.1) holds for any f ∈ ℓ2(π)).
1
Let A ⊂ V be a nonempty set of states with π(A) < ∞ and let π (·) = π(·)/π(A) be
A
the normalized restriction of π to A. Then
P X never returns to A ≥ λ .
πA(cid:16) t (cid:17) 1
4 ITAIBENJAMINI,ASAFNACHMIASANDYUVALPERES
Proof. Let B ⊂ V be disjoint from A such that V \(A∪B) is finite. Define
τ = min{t ≥ 0: X ∈ A∪B} and τ+ = min{t > 0: X ∈ A∪B}.
t t
The irreducibility assumption and the finiteness of the complement of A∪ B imply
that τ+ <∞ a.s. for any starting state. We will show that for all sets B as above,
P X ∈B ≥ λ . (2.1)
πA(cid:0) τ+ (cid:1) 1
The assertion of the lemma then follows by enumerating V \A as v ,v ,v ,..., taking
1 2 3
B = B = {v : j ≥ k} and intersecting the events in (2.1) over all these sets B = B
k j k
for k ≥ 1. Let
f(x)= P (X ∈ A).
x τ
Observe that f ≡ 1 on A and f ≡ 0 on B. For all x ∈G,
(Pf)(x) = P X ∈A .
x τ+
(cid:0) (cid:1)
In particular, f is harmonic (satisfies Pf = f) on G\(A∪B). Thus ((I −P)f)(x) =
P X ∈ B for x∈ A and ((I −P)f)(x) = 0 for x∈ G\(A∪B). Therefore
x τ+
(cid:0) (cid:1)
hf,(I −P)fi = π(x)P X ∈ B = π(A)P X ∈B .
X x(cid:0) τ+ (cid:1) πA(cid:0) τ+ (cid:1)
x∈A
On the other hand, clearly,
hf,fi≥ π(x)f(x)2 = π(A).
X
x∈A
The claim (2.1) follows by inserting the last two formulas in (1.1). (cid:3)
3. Proof of Theorem 1.1
We return to the setting of Theorem 1.1. Let G be regular graph of degree d and
girth g and write g := ⌈g/2⌉−1. Given a set of vertices A in G and α ∈ (0,1), we say
that an edge (x,ue) is (α,A)-good if x ∈ A and at least an α fraction of the (d−1)eg
non-backtracking paths of length g emanating from u, for which the first step is not
x, avoid A (in particular, u 6∈ A.) eThe following lemma is a corollary of Lemma 2.1.
Corollary 3.1. Let G be a regular graph with degree d and girth g. If G is nona-
menable, i.e., it satisfies λ > 0, then for any finite set A ⊂ V(G), there exist at least
1
λ1d|A| edges (x,u) which are (λ /2,A)-good.
2 1
IS THE CRITICAL PERCOLATION PROBABILITY LOCAL? 5
Proof. For an edge (x,u) with x ∈ A, let β = P (X = u and ∀t > 0 X 6∈ A),
(x,u) x 1 t
where {X } is a SRW in G, started at x. Let τ := min{t : dist(u,X ) = g}. Since
t t
the ball B (u,g) is a spherically symmetric tree, the loop erasure of (X )τ e yields a
G t t=0
uniform randome non-backtracking path of length g from u. Thus if β ≥ α, then
(x,u)
the edge (x,u) is (α,A)-good. By Lemma 2.1, e
1
β ≥ λ ,
d|A| X (x,u) 1
(x,u):x∈A
and we conclude that at least λ1d|A| edges (x,u) with x ∈ A must satisfy β ≥
2 (x,u)
λ /2. (cid:3)
1
Proof of Theorem 1.1. Let ǫ > 0 be a small number and set p = 1 +ǫ. For each
d−1
edge e we draw two independent Bernoulli random variables X (p) and Y (ǫ) with
e e
means p and ǫ respectively. We say that an edge is open if one of these variables takes
the value 1 and closed otherwise. We also say that the edge e is p-open if X (p) = 1
e
and ǫ-open if Y (ǫ) = 1. For a vertex v we write C(v) for the open cluster of v.
e
The probability that an edge is closed is (1−p)(1−ǫ), hence |C(v)| is dominated by
the cluster size in (p + ǫ)-bond percolation. Our goal is to show that with positive
probability |C(v)| = ∞.
We perform the following exploration process, which will produce an increasing
sequence {A } of connected vertex sets in which A ⊂ C(v) for all t. At each step,
t t
some of the edges touching A will beǫ-closed and some will beǫ-unchecked. We begin
t
by setting A to bethe p-cluster of v (that is, all the vertices connected to v by p-open
0
paths) and all the edges touching A are ǫ-unchecked. We assume that A is finite
0 0
(otherwise we are finished). At step t > 1 let E be the set of ǫ-unchecked edges
t−1
(x,u) such that (x,u) is (λ /2,A )-good. If E is empty, the process ends. If not,
1 t−1 t−1
we choose (x,u) ∈ E according to some prescribed ordering of the edges and check
t−1
whether the edge is ǫ-open. If it is ǫ-closed we put A = A and continue to the next
t t−1
step of the process. Otherwise, we let
A = A ∪V,
t t−1
where V is the set of vertices v of distance at most g from u such that the unique path
of length at most g between u and v avoids A aend is p-open.
t−1
This finishes theedescription of the exploration process. To analyze this process we
introduce the following random variable
Z = e: e is an ǫ-closed and ǫ-checked edge touching A .
t t
(cid:12)(cid:8) (cid:9)(cid:12)
(cid:12) (cid:12)
6 ITAIBENJAMINI,ASAFNACHMIASANDYUVALPERES
Let τ be the stopping time
2t
τ = min t :|A |< .
n t o
λ d
1
At each step we check the ǫ-status precisely one edge, hence Z ≤ t for all t. Thus, by
t
Corollary 3.1, if |A |> 2t there must exist at least one ǫ-unchecked edge (x,u) which
t λ1d
is (λ /2,A )-good. Write F for the σ-algebra generated by the ǫ and p status of the
1 t t
edges we examined in the exploration process up to time t and let ξ = |A |−|A |.
t t+1 t
By the discussion above we have that
ǫλ eg λ [(1+ǫ(d−1))eg −1]
E[ξ |F ,τ > t] ≥ 1 (1+ǫ(d−1))j ≥ 1 . (3.1)
t t−1 2 X 2(d−1)
j=1
To see the first inequality in (3.1), recall that (x,u) is ǫ-open with probability ǫ. Also,
for any j ≤ g the expected number of vertices of distance j from u such that the path
between theem and u avoids A and is p-open is at least λ1(p(d−1))j = λ1(1+ǫ(d−1))j.
t 2 2
We now assume that
log 1+ 8
g ≥ (cid:0) λ21(cid:1) , (3.2)
log(1+ǫ(d−1))
e
sothatE[ξ |F ,τ > t]≥ 4d−1λ−1 by(3.1). Since|ξ | ≤ (d−1)eg, Azuma-Hoeffding’s
t t−1 1 t
inequality (see Chapter 7 of [3]) gives that for any t > 1
t
2t
P τ = t+1 | A ≤ P ξ ≤ ≤ e−ct, (3.3)
(cid:0) 0(cid:1) (cid:16)X i dλ (cid:17)
1
i=1
where c = 2λ−2d−2(d−1)−2eg > 0. Since |A | is a non-decreasing sequence we have
1 t
that τ > λ1d|A0|. For any K > 0 there is some positive probability (depending on K)
2
of having |A |≥ K and we infer from (3.3) that
0
P(τ = ∞) ≥ P(|A |≥ K)P τ =∞ | |A | ≥ K
0 0
(cid:0) (cid:1)
≥ P(|A |≥ K) 1− e−ct > 0,
0 h X i
t≥λ1dK
2
as long as we choose K = K(g,λ ,d) to be large enough. The event τ = ∞ implies
1
that |C(v)| = ∞, and hence, by (3.2) when ǫ ≥ C(dg)−1log 1+ 1 there is positive
(cid:0) λ21(cid:1)
probability of an infinite component in 1 +2ǫ -bond percolation (for ǫ ≤ (d−1)−1
(cid:0)d−1 (cid:1)
one can take C = 128 using the inequalities log(1+8x)≤ 8log(1+x) and log(1+x)≥
x/2 valid for x∈ (0,1)). This concludes the proof of the theorem. (cid:3)
IS THE CRITICAL PERCOLATION PROBABILITY LOCAL? 7
4. Proof of Theorem 1.3
Without loss of generality assume that |G | = n. We first take p < p (G) and fix
n c
α > 0. Since p < p (G), for any ǫ > 0 there exists R = R(ǫ) large enough such that in
c
G
P ρ↔ ∂B(ρ,R) < ǫ.
p
(cid:0) (cid:1)
Thus for large enough n we have in G
n
L×P U ↔ ∂B(U ,R) ≤ ǫ,
p(cid:16) n n (cid:17)
where L is the law of U . Since G has bounded degree, we deduce that for any ǫ > 0
n
there exists n large enough such that
L×P |C(U )| ≥ dR+1 ≤ ǫ.
p(cid:16) n (cid:17)
Write C (n) for the largest component of G (p) and note that as long as dR+1 ≤ αn
1 n
we have that
L×P |C(U )| ≥ dR+1 ≥ αP |C (n)| ≥ αn ,
p(cid:16) n (cid:17) p 1
(cid:0) (cid:1)
and we get that
P |C (n)| ≥ αn ≤ǫα−1,
p 1
(cid:0) (cid:1)
which proves the first assertion of the theorem.
Toprove thesecond assertion of thetheorem weuseasprinklingargument, as in [1].
Assumep > p (G)andforsomeǫ > 0letp = p (G)+ǫsuchthat1−p = (1−p )(1−ǫ).
c 1 c 1
We first consider G (p ). Since p > p (G), there exists some δ > 0 such that for all
n 1 1 c
R > 0 we have
P ρ↔ ∂B(ρ,R) ≥ δ.
p1(cid:0) (cid:1)
For v ∈ G , write B (v,R) for the set of vertices in G (p ) which are connected to v
n p1 n 1
in a p -open path of length at most R. We get that for any R > 0 there exists n such
1 0
that for n ≥ n we have in G
0
L×P |B (U ,R)| ≥ R ≥ δ/2.
(cid:16) p1 n (cid:17)
Let X denote the random variable
R
X = v ∈ G : |B (v,R)| ≥ R ,
R (cid:12)n n p1 o(cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
8 ITAIBENJAMINI,ASAFNACHMIASANDYUVALPERES
so that EX ≥ δn/2. On the other hand, note that changing the status of a single
R
edge can change X by at most dR, where d is the degree bound of G . The method
R n
of bounded differences (see Theorem 3.1 of [12] or Chapter 7 of [3]) gives that
δn δ2n
P X ≤ ≤ exp − . (4.1)
(cid:16) R 4 (cid:17) (cid:0) 8d2R(cid:1)
Assume now that X ≥ δn/4 and consider the connected components of G (p ) of
R n 1
size at least R. Their number m is at most n/R. We now consider the union of G (p )
n 1
with G (ǫ) and claim that many of these components join together by edges of G (ǫ)
n n
and create a component of linear size. Indeed, consider a partitition of the m large
components of G (p )into two sets, AandB, each spanningat least δn/12 vertices. If
n 1
for any such partition there is an open path in G (ǫ) connecting A and B, then there
n
exists a component of size at least δn/12 in G (p )∪ G (ǫ). Since G has Cheeger
n 1 n n
constant at least h> 0 we get by Menger’s Theorem that for any such A and B there
are at least hδn/12 edge disjoint paths connecting A to B. Since there are most dn/2
edges in G we have that at least a half of these paths must be of length at most 12d.
n hδ
The probability that all these paths are closed in G (ǫ) is at most
n
hδn
12d 24
h1−ǫhδ i .
There are at most 2m different possibilities for choosing A and B. Hence, the proba-
bility that there exists such A and B is at most
hδn
2mh1−ǫ1h2δdi 24 ≤ exp(cid:16)n/R−ǫ1h2δdhδn/24(cid:17),
which goes to 0 as long as R is chosen such that R−1 < ǫ1h2δdhδ. This together with
(4.1) shows that
δn
P G (p) contains a component of size at least → 1 as n→ ∞.
(cid:16) n (cid:17)
12
(cid:3)
Acknowledgements: We are indebted to Mark Sapir and Oded Schramm for useful
discussions.
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Itai Benjamini: itai.benjamini(at)weizmann.ac.il
The Weizmann Institute of Science,
Rehovot POB 76100, Israel.
Asaf Nachmias: asafn(at)microsoft.com
Microsoft Research, One Microsoft way,
Redmond, WA 98052-6399, USA.
Yuval Peres: peres(at)microsoft.com
Microsoft Research, One Microsoft way,
Redmond, WA 98052-6399, USA.