JOURNAL OF MATHEMATICAL PHYSICS
VOLUME 41, NUMBER 3
MARCH 2000
Some remarks on AB-percolation in high dimensions
Harry Kestena)
Department of Mathematics, Malott Hall, Cornell University, Ithaca, New York 14853
Zhonggen Sub)
Department of Mathematics, Zhejiang University, Zhejiang 310028,
People’s Republic of China
共Received 18 November 1999; accepted for publication 7 December 1999兲
d
alt d
and Zd . Let p H
(Z ) be
In this paper we consider the AB-percolation model on Z⫹
alt d
d
(Z )
the critical probability for AB-percolation on Z . We show that p H
2
2
⬃1/(2d ). If the probability of a site to be in state A is ␥ /(2d ) for some fixed
␥ ⬎1, then the probability that AB-percolation occurs converges as d→⬁ to the
unique strictly positive solution y( ␥ ) of the equation y⫽1⫺exp(⫺␥y). We also
d
. In
find the limit for the analogous quantities for oriented AB-percolation on Z⫹
alt d
2
particular, p H (Z⫹ )⬃2/d . We further obtain a small extension to the two parameter problem in which even vertices of Zd have probability p A of being in state A
and odd vertices have probability p B of being in state B 共but without relation
between p A and p B 兲. The principal tools in the proofs are a method of Penrose
共1993兲 for asymptotics of percolation on graphs with vertices of high degree and
the second moment method. © 2000 American Institute of Physics.
关S0022-2488共00兲01303-7兴
I. INTRODUCTION AND MAIN RESULTS
Let G be an infinite connected graph with edge set E and vertex set V. To each vertex of G
assign one of two states, say A and B, with probability p and 1⫺p, respectively, independently of
all other vertices. The corresponding product probability on the configurations of sites is denoted
by P p ;E p denotes expectation with respect to P p . The state of v will be denoted by X( v ). We
A
denote by 兵 C↔ D 其 the event that there exists a self-avoiding path v 0 , v 1 ,..., v n with initial point
v 0 in C and endpoint v n in D and all of whose vertices are in state A. We call such a path an
A-path. If D⫽ 兵 ⬁ 其 we require the path to be infinite instead of the requirement v n 苸D. For a fixed
A
vertex v 0 , we say that A-percolation occurs from the vertex v 0 if v 0 ↔ ⬁. Write ␪ v 0 (p,G) for the
probability that A-percolation occurs from v 0 , that is
A
␪ v 0 共 p,G兲 ⫽ P p 兵 v 0 ↔ ⬁ on G其 .
It is well known that for reasonable graphs G there is a critical probability p c (G) strictly
between 0 and 1, such that
␪ v 0 共 p,G兲
再
⬎0
if p⬎p c 共 G兲
⫽0
if p⬍p c 共 G兲
.
For background, see Kesten 共1982兲1 or Grimmett 共1999兲.2 The model which we just described is
usually called 共Bernoulli兲 site-percolation. The AB-percolation model is the following variant of
this classical model. An AB-path is a self-avoiding path v 0 , v 1 ,..., v n , the state of whose vertices
a兲
Electronic mail: kesten@math.cornell.edu
Electronic mail: zgsu@mail.hz.zj.cn
b兲
0022-2488/2000/41(3)/1298/23/$17.00
1298
© 2000 American Institute of Physics
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J. Math. Phys., Vol. 41, No. 3, March 2000
Some remarks on AB-Percolation in high dimensions
1299
alternates between A and B, starting with an A, that is, v 2i is in state A and v 2i⫹1 in state B for
i⭓0. The definition of a BA-path is obtained by interchanging A and B in the preceding definition.
alt
兵 C↔ D 其 denotes the event that there exists an AB-path or a BA-path which starts in C and ends in
D 共‘alt’ here stands for alternating兲. Again we make the obvious modifications when D⫽ 兵 ⬁ 其 . For
alt
a given vertex v 0 , we say that AB-percolation occurs when v 0 ↔ ⬁. We denote the probability of
AB-percolation from v 0 by
alt
␪ alt
v 0 共 p,G 兲 ⫽ P p 兵 v 0 ↔ ⬁ on G其 ,
共1.1兲
and define the AB-critical probability by
alt
pH
共 G兲 ⫽inf兵 p: ␪ alt
v 共 p,G 兲 ⬎0 其 .
共1.2兲
0
alt
undefined. This
The set in the right-hand-side here could be empty. In such a case we leave p H
alt
will not arise in the high-dimensional situations which interest us here. It is easily seen that p H
does not depend on v 0 . We use here the notation p H rather than p c for the critical probability
because the equality of several differently defined critical probabilities has not yet been proven for
AB-percolation. Even on G⫽Zd we do not know that the probability for an AB-path of length n
alt d
(Z ).
from v 0 decays exponentially in n when p⬍ p H
One fundamental difference between the A-percolation or classical site-percolation model and
the AB-percolation model is the latter’s lack of monotonicity. Analysis of the classical model
depends heavily on certain correlation inequalities, including the FKG and BK inequalities 共see
Secs. 2.2 and 2.3 of Grimmet 共1999兲2 for these inequalities兲, which in turn depend on the fact that
A
A
the classical model is ‘‘increasing:’’ The occurrence of events such as 兵 v 0 ↔ ⬁ 其 or 兵 u↔ v 其 can
only be helped if the state of any collection of vertices is changed from B to A. This monotonicity
is, in general, absent in the AB-model and can lead to rather unexpected phenomena. For example,
Appel and Wierman 共1987兲3 proved that AB-percolation does not occur for any value of the
parameter p on a class of bipartite graphs, including G⫽Z2 . Also Łuczak and Wierman 共1988兲4
explicitly constructed a graph which exhibits multiple AB-percolation phase transitions. Thus the
definition 共1.2兲 is somewhat arbitrary. It is in general not true that AB-percolation occurs for all
alt
(G),1/2 兴 共note that the set of p values for which AB-percolation occurs is symmetric about
p苸(p H
1/2, by the symmetry between A and B兲.
The AB-percolation model evidently has some appeal as a model of physical phenomena. In
fact, this model was introduced independently by Mai and Halley 共1980兲5 in the context of
chemisorption and by Sevšek et al. 共1983兲6 as antipercolation in the study of the model on a Bethe
lattice in connection with antiferromagnetism. Also, Wilkinson 共1987兲7 looked at a more general
two parameter problem as a model for gelation processes; AB percolation is a special case of his
model. So far the study of AB-percolation model in the references focuses on the question whether
it is possible to have AB-percolation on specific graphs G 共such as Zd , the hexagonal, triangular
lattice etc.兲 for any parameter value, and if so, for what values of p AB-percolation does occur. In
addition to the result of Appel and Wierman 共1987兲3 which we mentioned above, Wierman and
Appel 共1987兲8 and Wierman 共1989兲9 proved that AB-percolation does occur on the triangular
lattice for p in some open interval around the value 1/2. If G is bipartite, let H1 and H2 be the
following graphs: The vertex set of Hi consists of all vertices v for which there exists a path
( v 0 , v 1 ,..., v n ⫽ v ) on G with n odd for i⫽1 and n even for i⫽2, from the distinguished v 0 to v ;
two vertices u and v , (u⫽ v ) are adjacent on Hi if and only if they are both adjacent on G to some
vertex w of G. Clearly the occurrence of AB-percolation on G implies the occurrence of
A-percolation on H1 or on H2 . Therefore
alt
pH
共 G兲 ⭓min共 p c 共 H1 兲 ,p c 共 H2 兲兲 .
共1.3兲
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1300
J. Math. Phys., Vol. 41, No. 3, March 2000
H. Kesten and Z. Su
Wierman 共1988兲10 shows that under very mild conditions this inequality is in fact a strict inequality.
Also, if G is bipartite and p c (G)⬍1/2 then for any p苸(p c (C),1⫺ p c (C)), AB-percolation
occurs 关see Wierman 共1989兲兴.9 Further references can be found in Wierman 共1989兲.
AB-percolation in high dimensions seems to have received little attention. In this paper we
shall mainly consider the graphs T d 共this is a rooted regular d-ary tree, i.e., a tree with a distinguished vertex v 0 , called the root, and such that each vertex other than the root has d⫹1
neighbors, while v 0 has d neighbors; all edges are oriented away from the root, so that there are
d
viewed as a directed graph 共each edge is oriented
d outgoing edges incident to each vertex兲, Z⫹
from a vertex v to v ⫹ ␰ i for some 1⭐i⭐d, where ␰ i is the positive ith coordinate vector兲, and Zd .
d
, and Zd we always take the root and origin for the distinguished vertex v 0 , respecFor T d , Z⫹
alt d
alt d
(Z⫹ ) and p H
(Z ). The result is not
tively. Our first result gives the asymptotic behaviors of p H
unexpected, since a well-known branching process argument tells us that on T d , AB-percolation
occurs if and only if p(1⫺p)⬎1/d 2 . Theorem 1 says that asymptotically for large d the ABd
and Zd differ only by a multiplicative constant
percolation critical probability for T d and for Z⫹
共but are of the same order in d兲. Throughout this paper, f (d)⬃g(d) will mean f (d)/g(d)→1 as
d→⬁.
Theorem 1:
2
d2
共1.4兲
1
.
2d 2
共1.5兲
alt d
pH
共 Z⫹ 兲 ⬃
and
alt d
pH
共 Z 兲⬃
d
and Zd when
The second theorem deals with the limit of the AB-percolation probability in Z⫹
2
2
we take p⫽2 ␥ /d and p⫽ ␥ /2d , respectively, for some fixed ␥ ⬎1, and let d go to ⬁. This will
follow from the fact that around a fixed site v the number of neighbors of v connected to ⬁ is
close to its expected value 共when d is large兲.
Theorem 2: Let y( ␥ ) be the unique strictly positive solution of the equation
y⫽1⫺e ⫺ ␥ y .
共1.6兲
Then, for ␥ ⬎1, in the AB-percolation model
␪ alt
0
冉
2␥ d
2␥y共␥兲
,Z ⬃
d2 ⫹
d
冊
共1.7兲
␪ alt
0
冉
␥
␥y共␥兲
d
.
2 ,Z ⬃
2d
d
冊
共1.8兲
and
Remarks: 共i兲 The proof which we give below actually proves the following, somewhat more
d
d
even(odd) if ⌺ i⫽1
general, result. Call a vertex v ⫽( v (1),..., v (d)) of Zd or Z⫹
v (i) is even
共respectively, odd兲. Assume that each even site can be in two possible states, A or C, while each
odd site can be in the states B or D. Let X( v ) again denote the state of v and let
P 兵 X 共 v 兲 ⫽A 其 ⫽p A ⫽1⫺ P 兵 X 共 v 兲 ⫽C 其 if v is even
共1.9兲
P 兵 X 共 v 兲 ⫽B 其 ⫽ p B ⫽1⫺ P 兵 X 共 v 兲 ⫽D 其 if v is odd.
共1.10兲
and
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J. Math. Phys., Vol. 41, No. 3, March 2000
Some remarks on AB-Percolation in high dimensions
1301
Assume the X( v ) are independent as before and let ␥ ⬎1 be fixed. Then, if
p A⭓
2␥
,
d2
p B →1 as d→⬁,
共1.11兲
d
for large d. Similarly, if
AB-percolation occurs on Z⫹
p A⭓
␥
,
2d 2
p B →1 as d→⬁,
共1.12兲
then AB-percolation occurs on Zd for large d.
共ii兲: As mentioned, Wierman 共1989兲9 showed that AB-percolation occurs on a bipartite graph
alt
d
(G)⭐p c (G). In the cases G⫽Z⫹
and G⫽Zd this gives
G if p c (G)⬍p⬍1⫺ p c (G), so that p H
alt d
alt d
asymptotically as d→⬁, p H (Z⫹ )⭐(1⫹o(1))/d and p H (Z )⭐(1⫹o(1))/(2d) 关see Bollobás
and Kohayakawa 共1994兲,11 Cox and Durrett 共1983兲,12 Gordon 共1991兲,13 Hara and Slade 共1990兲,14
and Kesten 共1990兲兴.15 It turns out that in these cases the lower bound in 共1.3兲 is closer to the truth.
It follows from Theorem 1 and Remark 共i兲 that asymptotically as d→⬁:
alt d
pH
共 Z⫹ 兲 ⬃ p c 共 H⫹
2 兲,
alt d
pH
共 Z 兲 ⬃ p c 共 H2 兲 ,
d
d
where H⫹
2 (H2 ) are the graphs described before 共1.3兲 for G⫽Z⫹ (G⫽Z ).
The next section will largely be devoted to the proof of Theorem 1. For AB-percolation on
d
, we shall closely follow the argument of Cox and Durrett 共1983兲12 which uses the second
Z⫹
moment method. On this graph one can explicitly find the expected number of AB-paths starting
at 0 and of length n, and the second moment of this number can be well estimated. On Zd we shall
use the method of Penrose 共1993兲.16 Before we give these proofs we shall look at AB-percolation
on T d in the hope that the calculation for this simple example will give us insight into more
general situations. Related calculations on Galton–Watson trees 共without asymptotics such as d
→⬁兲 have been carried out by Appel and Wierman 共1992兲.17 In fact, they even consider the two
parameter problem of Remark 共i兲 on Galton–Watson trees. In Sec. III we indicate how Theorem
2 can be proven along the lines of Kesten 共1991兲18 or Penrose 共1993兲.16
II. RESULTS ON TREES AND PROOF OF THEOREM 1
Let us begin with AB-percolation on T d for which exact calculations are possible.
Theorem 3: 共i兲 For d⭓2
alt
d
d
p alt
c 共 T 兲共 1⫺ p c 共 T 兲兲 ⫽
1
.
d2
共2.1兲
共ii兲 Define
E Av ⫽ 兵 at least one neighbor of v 0 is connected to ⬁
0
by an AB-path which does not pass through v 0 其
and
E Bv ⫽ 兵 at least one neighbor of v 0 is connected to ⬁
0
by a BA-path which does not pass through v 0 其 .
Then for ␥ ⬎1, and y( ␥ ) as in 共1.6兲
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1302
J. Math. Phys., Vol. 41, No. 3, March 2000
␪ alt
v0
H. Kesten and Z. Su
冉 冊
␥ d
␥y共␥兲
,T ⬃ P ␥ /d 2 兵 E Av 其 ⬃
0
d2
d
共2.2兲
and
P ␥ /d 2 兵 E Bv 其 →y 共 ␥ 兲 ,
0
d→⬁.
共2.3兲
Moreover, if
d 2 p→⬁ but d p remains bounded,
共2.4兲
A
d
⫺dp
␪ alt
v 0 共 p,T 兲 ⬃ P p 兵 E v 0 其 ⬃1⫺e
共2.5兲
lim P p 共 E Bv 兲 ⫽1.
0
共2.6兲
then
and
d→⬁
Proof of Theorem 3: Let v 0 be the root. For any vertex v 苸T there exists a unique oriented 共and
hence selfavoiding兲 path ( v 0 , v 1 ,..., v h( v ) ⫽ v ) from v 0 to v in T. We denote this path by ␲ ( v ).
h( v ) is the graph distance of v from the root. In order to find the asymptotic behavior of P p 兵 E Bv 其
0
as d→⬁ and p⫽ ␥ /d 2 or when 共2.4兲 holds we first need a crude lower bound for P p 兵 E Bv 其 . We
0
derive this bound by the second moment method. Let R2n be the set of all oriented BA-paths from
some neighbor v 1 of v 0 and of length 2n, that is the set of all ␳ ⫽( v 1 ,..., v 2n ) with v 1 adjacent
to v 0 ,h( v i )⫽i and
X共 vi兲⫽
再
B
if i is odd
A
.
if i is even
关Recall that X(u) is the state of the vertex u.兴 Denote the cardinality of a set A by 兩A兩. Then it is
not hard to see that
E p 兩 R2n 兩 ⫽d 2n 关 p 共 1⫺ p 兲兴 n ,
共2.7兲
and for d 2 p(1⫺p)⬎1
E p 兩 R2n 兩 2 ⫽
兺
␳ ⫽ 共 v 1 ,..., v 2n 兲
P p 兵 ␳ 苸R2n and ␳ ⬘ 苸R2n 其
␳ ⬘ ⫽ 共 v ⬘1 ,..., v 2n
⬘ 兲
2n
⫽
兺
l⫽0
兺
P p 兵 ␳ 苸R2n and ␳ ⬘ 苸R2n 其
v 1 ,..., v 2n
⬘
v 1⬘ ,..., v 2n
⬘
v i ⫽ v ⬘i ,i⭐l but v l⫹1 ⫽ v l⫹1
n
⭐ 关 d 2n p n 共 1⫺ p 兲 n 兴 2
兺
m⫽0
关 d 2 p 共 1⫺ p 兲兴 ⫺m
n⫺1
⫹ 关 d 2n p n 共 1⫺ p 兲 n 兴 2
兺
m⫽0
关 d 2 p 共 1⫺ p 兲兴 ⫺m 关 d 共 1⫺ p 兲兴 ⫺1
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Some remarks on AB-Percolation in high dimensions
J. Math. Phys., Vol. 41, No. 3, March 2000
冉
⭐ 共 E p 兩 R2n 兩 兲 2 1⫹
冊
d 2 p 共 1⫺ p 兲
1
.
2
d 共 1⫺ p 兲 d p 共 1⫺ p 兲 ⫺1
1303
共2.8兲
In particular, by the Cauchy–Schwarz inequality
P p 兵 E Bv 其 ⫽ lim P p 兵 兩 R2n 兩 ⬎0 其 ⭓lim inf
0
n→⬁
n→⬁
d 2 p 共 1⫺ p 兲 ⫺1
1
共 E p 兩 R2n 兩 兲 2
, 共2.9兲
2 ⭓
E p 兵 兩 R2n 兩 其 1⫹ 关 1/d 共 1⫺ p 兲兴 d 2 p 共 1⫺ p 兲
and this is bounded away from 0, uniformly on 兵d 2 p⭓1⫹ ␦ , p⭐ ␦ /2其, for any fixed 0⬍ ␦ ⭐1/2.
A
as the number of vertices v with h( v )⫽2n such
Now set Z A0 ⫽Z B0 ⫽1. For n⭓1 define Z 2n
that for the corresponding path ␲ ( v )⫽( v 0 , v 1 ( v ),..., v 2n ( v )⫽ v ) one has
X 共 v i 共 v 兲兲 ⫽
再
A
if i is odd
B
if i is even,i⫽0
共2.10兲
.
B
A
by interchanging A and B in 共2.10兲. It is easy to check that Z 2n
,n⭓0, is a Galton–
Define Z 2n
Watson process whose offspring distribution has generating function
f A 共 s 兲 ⫽ f A 共 s,p 兲 ⫽ 关 1⫺ p⫹ p 共 p⫹ 共 1⫺ p 兲 s 兲 d 兴 d .
共2.11兲
B
,n⭓0, is a Galton–Watson process whose offspring distribution has the generating
Similarly, Z 2n
function
f B 共 s 兲 ⫽ f B 共 s,p 兲 ⫽ 关 p⫹ 共 1⫺ p 兲共 1⫺ p⫹ ps 兲 d 兴 d .
共2.12兲
The expected number of children per individual in each of these branching processes is d 2 p(1
⫺ p). AB-percolation occurs from v 0 if and only if at least one of these branching processes has
a strictly positive survival probability, that is, when
d 2 p 共 1⫺ p 兲 ⬎1,
关see Athreya and Ney 共1972兲,19 Theorem I.5.1兴. This proves 共2.1兲
To prove 共2.2兲 and 共2.3兲 we define q A and q B as the extinction probabilities of the processes
A
B
兵 Z 2n 其 and 兵 Z 2n
其 , respectively. It is well known 关see Athreya and Ney 共1972兲,19 Theorem I.5.1兴
that these are the smallest solutions in 关0,1兴 of the equations
q *⫽ f *共 q * 兲 ,
* ⫽A or B.
共2.13兲
Note that by our definitions
* ⬎0 for all n 其 ⫽ P p 兵 E *v 其 .
1⫺q * ⫽ P p 兵 Z 2n
0
共2.14兲
It is not difficult to see from 共2.13兲 and the explicit expression 共2.12兲 for f B , that if we set p
⫽ ␥ /d 2 for some fixed ␥ ⬎1 and let d→⬁, then
冋
q B ⫽ f B 共 q 兲 ⫽ 1⫺
册
d
␥
共 1⫺q B 兲 ⫹O 共 d ⫺2 兲 .
d
Therefore, if d→⬁ along some subsequence for which q B has a limit, Q say, then Q must satisfy
Q⫽e ⫺ ␥ 共 1⫺Q 兲 .
共This is the equation for the extinction probability of a branching process with a mean ␥ Poisson
distribution.兲 Moreover, 1⫺Q⫽lim P ␥ /d 2 兵 E Bv 其 does not vanish, by virtue of 共2.9兲. Therefore, 1
0
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1304
J. Math. Phys., Vol. 41, No. 3, March 2000
H. Kesten and Z. Su
⫺Q⫽y(␥), as defined in 共1.6兲. 共This is the only place in the proof of 共2.3兲 for which we need
共2.9兲.兲 This is independent of the subsequence through which d→⬁. Therefore, 共2.3兲 holds for
␥ ⬎1.
It is obvious that for any fixed ␥ and d→⬁
P ␥ /d 2 兵 E Av 其 ⭐ P ␥ /d 2 兵 X 共 v 兲 ⫽A for some neighbor v of v 0 其 ⭐d
0
␥
→0.
d2
In the other direction
P ␥ /d 2 兵 E Av 其 ⭓ P ␥ /d 2 兵 X 共 v 兲 ⫽A for some neighbor v of v 0 其 P ␥ /d 2 兵 E Bv 其 ⭓
0
0
C1
,
d
共2.15兲
for some constant C 1 ⬎0 共independent of d兲. Then P ␥ /d 2 兵 E Av 其 ⫽1⫺q A must satisfy
0
d 共 1⫺q A 兲 ⫽d 共 1⫺ f A 共 q A 兲兲
再 冋
再 冋
冉 冉 冊 冊册冎
冉 冉 冊 冊 册冎
⫽d 1⫺ 1⫺
␥
␥
␥
A
2 ⫹ 2 1⫺ 1⫺ 2 共 1⫺q 兲
d
d
d
⫽d 1⫺ 1⫺
␥ ␥
␥
1⫺ 1⫺ 2 共 1⫺q A 兲
⫹
d d
d
d d
d
⫹O 共 d•d 2 共 ␥ d ⫺2 兲 2 兲
⫽ ␥ 共 1⫺exp关 ⫺ 共 1⫺ ␥ /d 2 兲 d 共 1⫺q A 兲兴 兲 ⫹o 共 1 兲 .
This shows that if d(1⫺q A ) converges along some subsequence to z, say, then z⫽ ␥ (1⫺e ⫺z ),
that is, z⫽0 or z⫽ ␥ y( ␥ ). By virtue of 共2.15兲 only the second value is possible, so that
lim d P ␥ /d 2 兵 E Av 其 ⫽ ␥ y 共 ␥ 兲 .
0
共2.16兲
d→⬁
Finally, for p⫽ ␥ /d 2
␪ alt
v0
冉 冊
冉 冊
␥ d
␥
␥
␥y共␥兲
,
⫽ 2 P ␥ /d 2 兵 E Bv 其 ⫹ 1⫺ 2 P ␥ /d 2 兵 E Av 其 ⬃
2 ,T
0
0
d
d
d
d
共2.17兲
which proves 共2.2兲.
We also want to give a slightly different derivation of 共2.3兲 which uses a consistency relation
for the limit of P p 兵 E Bv 其 . Even though this basically is the same argument as used in the standard
0
derivation of 共2.13兲, it may be useful to help the reader understand the derivation of analogous
d
and Zd in Theorem 2. Let I 关v兴 be the indicator function of the event 兵at least one
results for Z⫹
neighbor of v is connected to ⬁ by an oriented BA-path ( v 1 , v 2 ,...) with h( v i )⭓3 for all i⭓1其.
If h( v )⫽2, then
P p 兵 I 关v兴 ⫽1 其 ⫽ P p 兵 E Bv 其 ,
0
共2.18兲
because the tree of descendants of v is isomorphic to T d Let
M⫽
兺
w:h 共 w 兲 ⫽2
I关w兴,
and let w 1 ,...,w M be all the vertices with h(w)⫽2 and I 关 w 兴 ⫽1. Denote by w ⬘ the parent of w in
T d , that is the vertex which just precedes w on ␲ (w). Then
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Some remarks on AB-Percolation in high dimensions
J. Math. Phys., Vol. 41, No. 3, March 2000
1305
P ␥ /d 2 兵 E Bv 其 ⫽ P ␥ /d 2 兵 for some i⭐M ,X 共 w i 兲 ⫽A and X 共 w ⬘i 兲 ⫽B 其
0
⫽ P ␥ /d 2 兵 for some i⭐M ,X 共 w i 兲 ⫽A 其
⫺O 共 P ␥ /d 2 兵 X 共 u 兲 ⫽A for some u with h 共 u 兲 ⫽1 其 兲 .
Clearly the last probability is at most d( ␥ /d 2 )→0 as d→⬁. Now all the I 关 w 兴 and the X(w) with
h(w)⫽2 are independent. Therefore
P ␥ /d 2 兵 for some i⭐M ,X 共 w i 兲 ⫽A 其
⫽E 兵 P ␥ /d 2 兵 for some i⭐M ,X 共 w i 兲 ⫽A 其 兩 w 1 ,...,w M 其
⫽1⫺E
再冉 冊 冎
1⫺
␥
d2
M
.
Also by the independence of the I 关 w 兴 with h(w)⫽2 and by 共2.18兲 and the fact that P ␥ /d 2 兵 E Bv 其 is
0
bounded away from 0 关see 共2.9兲兴, we have
P
M
B
⫺
P
2
E
→
0.
兵
其
␥ /d
v0
d2
共2.19兲
Consequently, if d→⬁ along any subsequence for which the limit of P ␥ /d 2 兵 E Bv 其 exists, then this
0
subsequential limit must be a solution of 共1.6兲. Because of 共2.9兲 this solution must be y( ␥ ) and not
the zero solution. This leads to 共2.3兲 as before.
Next we turn to the case of 共2.4兲. The relation 共2.6兲 is now immediate from 共2.9兲. Moreover
d 共 1⫺q A 兲 ⫽d P p 兵 E Av 其 →⬁,
0
by the argument for 共2.15兲. We then find from the relation
q A ⫽1⫺ P p 兵 E Av 其 ⫽ f A 共 q A 兲 ⫽e ⫺dp 共 1⫹o 共 1 兲兲 ,
0
that
P p 兵 E Av 其 ⬃1⫺e ⫺dp .
0
Finally, 共2.5兲 now follows as in 共2.17兲.
䊏
d
: Probably this case can be proven together with the case of
Proof of Theorem 1 for Z⫹
AB-percolation on Zd by the method of Penrose 共1993兲16 as outlined in the next proof. Perhaps
some of the other methods for studying the asymptotic behavior of the critical probability of
high-dimensional percolation 关see Bollobás and Kohayakawa 共1994兲,11 Gordon 共1991兲,13 Kesten
共1990兲,15 Hara and Slade 共1990兲14兴 also apply, but we have not checked this. However, the method
of the second moment of Cox and Durrett 共1983兲12 is simpler in the oriented case, so we shall
illustrate this method here. For simplicity we restrict ourselves to the case of AB-percolation with
P p 兵 X( v )⫽B 其 ⫽1⫺p and leave the more general case of 共1.11兲 to the reader 关the proof on Zd
d
below does cover the case of 共1.12兲, though兴. First observe that Z⫹
is bipartite. Indeed, if one can
⫹
d
⫹
d
take V 1 ⫽ 兵 v 苸Z⫹ : 兩 v 兩 odd其 ,V 2 ⫽ 兵 v 苸Z⫹ : 兩 v 兩 even其 共with 兩 v 兩 denoting the l 1 -norm of v 兲, then
⫹
there are only edges between vertices of V ⫹
1 and of V 2 . Then, as in 共1.3兲
alt d
pH
共 Z⫹ 兲 ⭓ p c 共 H⫹
2 兲,
共2.20兲
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1306
J. Math. Phys., Vol. 41, No. 3, March 2000
H. Kesten and Z. Su
d
⫹
where H⫹
i is defined as in 共1.3兲 for G⫽Z⫹ and Hi is oriented in the obvious way. Note that
⫹
⫹
p c (H1 )⫽p c (H2 ) in this case. In turn, a standard Peierls argument shows that p c (H⫹
2 )
.
Therefore
it
is
enough
⭓2/关 d(d⫹1) 兴 , because each vertex has d(d⫹1)/2 outgoing edges in H⫹
2
alt d
(Z⫹ ).
to give an upper bound for p H
As we already stated we follow Cox and Durrett 共1983兲. We restrict ourselves to merely
sketching the necessary changes. Let R2n be the set of all oriented paths ␳ ⫽(0,v 1 ,..., v 2n ) of
length 2n with X( v 2i )⫽A and X( v 2i⫺1 )⫽B,i⫽1,2,...,n. Note that we do not specify X„0… here.
Then we have E p 兩 R2n 兩 ⫽d 2n p n (1⫺p) n and, as in 共2.9兲
alt
P p 兵 0↔ ⬁ 其 ⭓p P p 兵 兩 R2n 兩 ⬎0 for all n⭓1 其 ⫽ p lim P p 兵 兩 R2n 兩 ⬎0 其 ⭓ p lim inf
n→⬁
n→⬁
关 E p 兵 兩 R2n 兩 其 兴 2
.
E p 兩 R2n 兩 2
共2.21兲
It, therefore, suffices to bound E 兩 R2n 兩 2 . Observe that
E 兩 R2n 兩 2 ⫽
兺
␳ ⫽ 共 v 1 ,..., v 2n 兲
P p 兵 ␳ 苸R2n and ␳ ⬘ 苸R2n 其
␳ ⬘ ⫽ 共 v 1⬘ ,..., v 2n
⬘
⫽ 关 E 兵 兩 R2n 兩 其 兴 2 共 d 2n 兲 ⫺2
兺
␳,␳⬘
共 1⫺ p 兲 ⫺l 1 共 ␳ , ␳ ⬘ 兲 p ⫺l 2 共 ␳ , ␳ ⬘ 兲 ,
共2.22兲
where l i ( ␳ , ␳ ⬘ ) is the number of vertices common to ␳ and ␳ ⬘ in V ⫹
i . Now let S⫽(S 1 ,S 2 ,...)
d
, as in Cox and
and S ⬘ ⫽(S 1⬘ ,S 2⬘ ,...) be two independent oriented simple random walks on Z⫹
12
Durrett 共1983兲. Denote the probability measure which governs 兵 S t ,S ⬘t 其 by P, and let E be
expectation with respect to P. Let Fn be the ␴-field generated by S t ,S ⬘t ,t⭐n. Further, let t 1
⬍t 2 ⬍¯ denote the successive random indices for which S t i ⫽S t⬘ . Note that this sequence only
i
has finitely many members except on a P-null set when d⭓4. Define
␶ i⫽
再
0
if t i is even
1
if t i is odd
.
⫹
Since S t i ⫽S t⬘ 苸V ⫹
1 if and only if ␶ i ⫽1 and S t i ⫽S t⬘ 苸V 2 if and only if ␶ i ⫽0, we have
i
i
共 d 2n 兲 ⫺2
兺
␳,␳
⬘
共 1⫺ p 兲 ⫺l 1 共 ␳ , ␳ ⬘ 兲 p ⫺l 2 共 ␳ , ␳ ⬘ 兲
⬁
⭐
兺 E兵 共 1⫺ p 兲 ⫺⌺
m⫽0
⬁
⫽
兺
兺
m
i⫽1 ␶ i
m⫽0 ␩ 1 ,..., ␩ m 苸 兵 0,1其
m
p ⫺⌺ i⫽1 共 1⫺ ␶ i 兲 I 关 t m ⬍⬁ 兴 其
再兿
m⫺1
E
i⫽1
共共 1⫺ p 兲 ⫺ ␩ i p ⫺ 共 1⫺ ␩ i 兲 I 关 t m⫺1 ⬍⬁, ␶ i ⫽ ␩ i ,i⭐m⫺1 兴 兲
冎
⫻E兵 共 1⫺ p 兲 ⫺ ␩ m p ⫺ 共 1⫺ ␩ m 兲 I 关 t m ⬍⬁ 兴 , ␶ m ⫽ ␩ m 兩 Ft m⫺1 其 .
Now let ⌫( ␩ ,␭) be an upper bound for
E兵 共 1⫺ p 兲 ⫺␭ p ⫺ 共 1⫺␭ 兲 I 关 t i ⬍⬁, ␶ i ⫽␭ 兴 兩 Ft i⫺1 其 ,
on the event 兵 t i⫺1 ⬍⬁, ␶ i⫺1 ⫽ ␩ 其 . Then we can continue the preceding inequality to get
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Some remarks on AB-Percolation in high dimensions
J. Math. Phys., Vol. 41, No. 3, March 2000
共 d 2n 兲 ⫺2
兺
␳,␳⬘
⬁
⭐
兺
1307
共 1⫺ p 兲 ⫺l 1 共 ␳ , ␳ ⬘ 兲 p ⫺l 2 共 ␳ , ␳ ⬘ 兲
兺
再冉 兿
m⫺1
m⫽0 ␩ 1 ,..., ␩ m 苸 兵 0,1其
E
i⫽1
共 1⫺ p 兲 ⫺ ␩ i p ⫺ 共 1⫺ ␩ i 兲 I 关 t m⫺1 ⬍⬁, ␶ i ⫽ ␩ i ,i⭐m⫺1 兴
冊冎
⬁
⫻⌫ 共 ␩ m⫺1 , ␩ m 兲 ⭐¯⭐
兺 兺
␭苸 兵 0,1其 m⫽0
⌫ m 共 0,␭ 兲 .
共2.23兲
Let ␥ 1 (d)⫽P兵 S j ⫽S ⬘j for some j⭓1 其 , ␥ 2 (d)⫽P兵 S 2 j ⫽S ⬘2 j but S 2 j⫺1 ⫽S ⬘2 j⫺1 for some j⭓1 其 .
Then according to the argument of Cox and Durrett 共1983兲12
1
␥ 1共 d 兲 ⬃ ,
d
␥ 2共 d 兲 ⬃
1
.
d2
Now we take for ⌫ the following matrix:
⌫⫽
冉
␥ 2共 d 兲
p
␥ 1共 d 兲
共 1⫺ p 兲
␥ 1共 d 兲
p
␥ 2共 d 兲
共 1⫺ p 兲
冊
共2.24兲
共2.25兲
.
A routine computation, together with 共2.24兲 shows that as p↓0
ªthe largest eigenvalue of ⌫, is asymptotically equivalent to
冋 冉
1
4
1 1
2⫹
2 2⫹
2 pd
pd 2
共 pd 兲
冊册
and d→⬁,
␭⌫
1/2
.
This is less than 1 whenever p⬎(2⫹ ⑀ )/d 2 for some fixed ⑀ ⬎0 and d sufficiently large. By the
Perron–Frobenius theorem 关see Gantmacher 共1960兲,20 Section 13.2兴 the strictly positive matrix ⌫
has an eigenvector v corresponding to the eigenvalue ␭ ⌫ , with all components strictly positive. In
fact, in our simple case we can explicitly calculate
v共 0 兲 ⫽
␥ 1共 d 兲
,
共 1⫺p 兲关 ␭ ⌫ ⫺ ␥ 2 共 d 兲 /p 兴
v共 1 兲 ⫽1.
Since ⌫ m v⫽ 关 ␭ ⌫ 兴 m v, it follows that:
兩 ⌫ m 共 0,␭ 兲 兩 ⭐
v共 0 兲
关 ␭ 兴 m.
v共 ␭ 兲 ⌫
Thus, the right-hand-side of 共2.23兲 is finite for p⬎(2⫹ ⑀ )/d 2 and d large. Together with 共2.21兲,
alt d
(Z⫹ )⭐(2⫹ ⑀ )/d 2 for all large d. Since we
共2.22兲 and 共2.23兲 this shows that for each fixed ⑀ ⬎0,p H
alt d
already have the lower bound p H (Z⫹ )⭓2/关 d(d⫹1) 兴 , 共1.4兲 follows.
Proof of Theorem 1 for Zd : The presence of circuits in Zd as well as the fact that we have to
deal with the two parameters p and 1⫺ p for AB-percolation makes exact calculations impossible.
Actually, we shall carry out this proof for the two parameter generalization described by 共1.9兲,
共1.10兲. In this proof P p will denote the measure under which the distribution of the vertices is
given by 共1.9兲, 共1.10兲.
For a lower bound on p H we note that Zd is also bipartite. If we take V 1 ⫽ 兵 v
d
苸Z : 兩 v 兩 odd其 ,V 2 ⫽ 兵 v 苸Zd : 兩 v 兩 even其 then there are only edges between vertices of V 1 and V 2 . If
H1 and H2 are defined as in 共1.3兲 for G⫽Zd , then H1 and H2 are isomorphic and there can be no
AB percolation unless
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1308
J. Math. Phys., Vol. 41, No. 3, March 2000
H. Kesten and Z. Su
p A ⭓ p c 共 H2 兲 ,
irrespective of the value of p B . This time each vertex has 2d 2 neighbors in H2 so that a Peierls
argument shows
p c 共 H2 兲 ⭓1/共 2d 2 兲 .
alt d
Thus, no percolation can occur when p A ⬍1/(2d 2 ). In particular, p H
(Z )⭓1/(2d 2 ).
In the other direction, we shall show that AB-percolation occurs under the assumption 共1.12兲.
We shall further assume that
p A⭐
␥
.
d2
共2.26兲
This is no restriction since the percolation probability is increasing in p A . To simplify notation we
derive the upper bound only for even d. This actually suffices, because percolation in dimension
d⫺1 implies percolation in d dimensions. We now more or less follow Penrose 共1993兲.16 Because
of our more complicated situation we have to redo a number of Penrose’s steps. It will still be
useful to the reader to consult Penrose 共1993兲16 for the details of some estimates. L will be the
subgraph of Z2 of all 共i,j兲 with i⭓0,兩 j 兩 ⭐i,i⫹ j even. We shall consider L as a directed graph, with
the edges between 共i,j兲 and (i⫹1,j⫾1) directed from the former to the latter. On this directed
graph we shall consider a dependent mixed bond-site percolation. Edges and sites will be open or
closed. We shall use a recursive procedure to decide the states of all sites and edges, starting with
the site 共0, 0兲. A site (i, j)苸L will be open if the edge from (i⫺1,j⫺1) to 共i,j兲 or the edge from
(i⫺1,j⫹1) to 共i,j兲 is open. In turn, the edge from (i⫺1,j⫺1) 关or from (i⫺1,j⫹1)兴 to 共i,j兲 can be
open only if (i⫺1,j⫺1) 关respectively, (i⫺1,j⫹1)兴 is open. From this we see that, once we know
the state of 共0, 0兲, the main step will be to decide when an edge is open. The states of the edges
will be determined one by one in the following order. An edge starting at (i 1 , j 1 ) precedes an edge
from (i 2 , j 2 ) if i 1 ⬍i 2 or if i 1 ⫽i 2 but j 1 ⬍ j 2 . Finally the edge from (i 1 , j 1 ) to (i 1 ⫹1,j 1 ⫺1)
precedes the edge from (i 1 , j 1 ) to (i 1 ⫹1,j 1 ⫹1). The state of an edge will be a function of some
of the states X(u) introduced before. For an edge e苸L we denote by F(e) the ␴-field generated
by those X( v ) which have been examined to determine the states of the edges preceding e in the
ordering of the edges of L introduced above. One of the difficulties in this proof is to keep track
of which vertices have been examined at any stage.
We will set things up so that for large d, for each edge e苸L
P p 兵 e is open兩 F共 e 兲 其 ⭓1⫺5 ⑀ ,
共2.27兲
on the event that the initial point of e is open. Here ⑀ ⬎0 is a fixed small number such that in
standard Bernoulli site percolation on L with each site open with probability 1⫺5 ⑀ , percolation
occurs. Under 共2.27兲 we can then couple our mixed bond-site percolation process with such a
Bernoulli site percolation to conclude that our process on L also percolates 关compare Lemma 1 in
Russo 共1982兲兴.21
To relate the mixed bond-site percolation process to the AB-percolation process we need some
more notation. For any vertex v ⫽( v (1),..., v (d))苸V 2 we define L( v )苸Z2 by
冉兺
d/2
L共 v 兲⫽
l⫽1
d
v共 l 兲 ,
兺
l⫽d/2⫹1
冊
v共 l 兲 .
We further define for (i, j)苸L
B 共 i, j 兲 ⫽L ⫺1 共 i, j 兲 ⫽ 兵 v 苸V 2 :L 共 v 兲 ⫽ 共 i, j 兲 其 .
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J. Math. Phys., Vol. 41, No. 3, March 2000
Some remarks on AB-Percolation in high dimensions
1309
We remind the reader that vertices in V 2 (V 1 ) are called even 共odd兲 vertices. Positive integers
k,m,k 2 will be chosen below such that 共2.49兲 and 共2.50兲 hold. 关We use k 2 instead of k 1 to
distinguish this constant from Penrose’s k 1 ; m and k are essentially the same as in Penrose
共1993兲.兴16 These integers will be independent of the dimension d. We shall also fix some arbitrary
ordering of the vertices of Zd . On a number of occasions it will be necessary to order some subsets
of Zd . This will always be done according to this fixed ordering. We begin by choosing 2m even
vertices u(0,0,1),...,u(0,0,2m) in B(0,0) satisfying
兩 u 共 0,0,l 兲 ⫺u 共 0,0,l ⬘ 兲 兩 ⭓4,
l⫽l ⬘ .
共2.28兲
These vertices are otherwise arbitrary. We declare (0,0)苸L open if X(u(0,0,l))⫽A for all 1⭐l
⭐2m. In the sequel, whenever we declare a site (i, j)苸L to be open we shall single out 2m
vertices u(i, j,1)...,u(i, j,2m) in B(i, j), which we call special vertices. All these special vertices
will be even and have X(u(i, j,l))⫽A and for each singled out vertex u(i, j,l) corresponding to an
open site 共i,j兲 there will be an AB-path on Zd from some u(0,0,l ⬘ ) to u(i, j,l). Thus if our mixed
bond-site model on L percolates, then AB-percolation occurs.
We now discuss how to determine the state of an edge. For the sake of definiteness, we
assume that the state of 共i,p兲 has been determined for all 兩 p 兩 ⭐i with (i,p)苸L, as well as of all
edges preceding the edge e from 共i,j兲 to (i⫹1,j⫺1). We now consider the state of this edge e. If
the site (i⫹1,j⫺1) is already open because the edge from (i, j⫺2) to (i, j⫺1) was declared
open, then the state of the edge from 共i,j兲 to (i⫹1,j⫺1) has no influence on the further evolution
of the states, so in this case, we may as well declare the edge e open. If the site 共i,j兲 is closed, then,
as we stated before, e is also declared closed. We, therefore, only need to consider the case when
共i,j兲 is open 关and the edge from (i, j⫺2) to (i⫹1,j⫺1) is closed兴. If 共i,j兲 is open, then we will
have chosen the special vertices u(i, j,l), 1⭐l⭐2m, in B(i, j). Roughly speaking, e will be open
if there exist 2m distinct vertices in B(i⫹1,j⫺1) which are connected by an AB-path to one of
the m special vertices u(i, j,l)苸B(i, j), 1⭐l⭐m. In order to prove 共2.27兲 recursively we need to
restrict these paths further. Our construction is such that when we come to decide on the state of
e, we will also have chosen a collection C(e) of AB-paths on Zd with the following properties
共2.29兲–共2.34兲:
Each path in C共 e 兲 starts at one of the vertices u 共 0,0,l 兲 ,1⭐l⭐2m.
共2.29兲
Let D be the collection of edges 共of Zd 兲 which appear in some path in C(e).
Then D is a forest.
共2.30兲
If ␲ ⫽(u 0 ⫽u(0,0,l),u 1 ,...,u p )苸C(e), then p⭐2k(i⫹1).
u r can be a special vertex only if r is a multiple of 2k.
If 2kq⬍p 共or equivalently, u 2kq is not the endpoint of ␲兲,
then u 2kq is a special vertex, say u 2kq ⫽u(q, j q ,l q ) for some (q, j q )苸L.
This also holds if 2kq⫽p and u p 苸B(s,t) with 共s,t兲 open.
Finally, it holds that j 0 ⫽0,兩 j q ⫺ j q⫺1 兩 ⫽1,1⭐l q ⭐2m.
共2.31兲
For any special vertex u(q, j q ,l q ), let ␯ (q, j q ,l q ) be the number of even vertices
w for which there exists a path (u 0 ,u 1 ,...,u r )苸C(e) which passes first
through u(q, j q ,l q ) and then reaches w in at most 2k more steps.
Then ␯ 共 q, j q ,l q 兲 ⭐k 2 .
共2.32兲
For any path ␲ ⫽(u 0 ⫽u(0,0,l),u 1 ,...,u p )苸C(e), whose endpoint u p
is not a special vertex, set ⳵␲ ⫽ 兵 v : v has l 1 -distance ⭐2 to some even
vertex of ␲ 其 . If the endpoint u p equals a special vertex, then set
⳵␲ ⫽ 兵 v : v has l 1 -distance ⭐2 to some even vertex of ␲ other than u p 其 .
Then X 共 v 兲 has been examined only for v 苸 ␲ or v 苸 ⳵␲ for some ␲ 苸C共 e 兲 .
共2.33兲
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1310
J. Math. Phys., Vol. 41, No. 3, March 2000
H. Kesten and Z. Su
Each path in C(e) is an AB-path. Each u(s,t,l), 1⭐l⭐2m, with
(s,t)苸L such that s⭐i or s⫽i⫹1,t⭐ j⫺1 and such that
共s,t兲 is already known to be open, is the endpoint of some path in C(e).
Moreover, 兩 u(s,t,l)⫺u(s,t,l ⬘ ) 兩 ⭓4 for all such 共s,t兲 and for l⫽l ⬘ .
Finally, for 1⭐l⭐m,u(i, j,l) does not belong to any path of C(e)
of which it is not the endpoint.
共2.34兲
A few explanatory comments to these conditions may be helpful. Equation 共2.30兲 means that
there are no circuits in D. More explicitly, there cannot be two vertices v and w with two disjoint
paths made up from edges in D from v to w. Roughly speaking, condition 共2.31兲 says that exactly
every 2k steps a path ␲ 苸C(e) passes through a special point. The last part of 共2.31兲 says that
(q, j q ), q⭓0, runs through an oriented path on L. Finally, 共2.32兲 gives an upper bound on the
number of descendants in generations 2kq⫹1,...,2k(q⫹1) of u(q, j q ,l q ) in the tree made up of
edges of D which contains u(q, j q ,l q ). We remind the reader that all these properties are assumed
only when 共i,j兲 is already known to be open.
The recursive step must be such that at the end we have a new collection of paths C(e ⬘ ) which
can be used for the examination of the next edge e ⬘ 关which runs from 共i,j兲 to (i⫹1,j⫹1)兴. This
C(e ⬘ ) must have the properties 共2.29兲–共2.33兲 with e replaced by e ⬘ . It should also satisfy 共2.34兲
with the final condition modified to ‘‘for m⫹1⭐l⭐2m,u(i, j,l) does not belong to any path of
C(e ⬘ ) of which it is not the endpoint.’’
As motivation for the steps to follow, note that we cannot choose the same X( v ) twice, so that
we must avoid visiting a site v for which X( v ) has already been examined in a previous step. This
will be achieved by only moving in ‘‘new coordinate directions,’’ as we explain now. First, let
␲ ⫽(u 0 ,...,u 2ki ) be a path from some u(0,0,l)⫽u 0 to some special vertex u(i, j,p)⫽u 2ki
苸B(i, j). Let ␲
˜ ⫽(u 0 ,...,u 2ki ,u 2ki⫹1 ,...u 2ki⫹r ) be an extension by r⭐2k steps of ␲. Denote the
piece (u 2ki⫹1 ,...,u 2ki⫹r ) which was added on by û. We want to know to which sites in ␲ ⬘ 艛 ⳵␲ ⬘
with ␲ ⬘ 苸C(e), u 2ki⫹r can be equal. If u 2ki⫹r 苸 ␲ ⬘ 艛 ⳵␲ ⬘ , then there exists an even w苸 ␲ ⬘ such
that 兩 u 2ki⫹r ⫺w 兩 ⭐2 共by definition of ⳵␲ ⬘ 兲. Let ␲ ⬘ ⫽(w 0 ,...,w q ) and w⫽w s with 2kn⬍s
⭐2k(n⫹1) 共when s⫽0 take n⫽0兲. Then, by 共2.31兲, w 2kn ⫽u(n, j ⬘ ,p ⬘ ) is a special vertex and we
must have
兩 u 共 i, j, p 兲 ⫺u 共 n, j ⬘ , p ⬘ 兲 兩 ⭐ 兩 u 2ki⫹r ⫺u 共 i, j,p 兲 兩 ⫹ 兩 u 2ki⫹r ⫺w 兩 ⫹ 兩 w⫺u 共 n, j ⬘ ,p ⬘ 兲 兩 ⭐4k⫹2.
Therefore,
兩 i⫺n 兩 ⫹ 兩 j⫺ j ⬘ 兩 ⭐4k⫹2.
Thus, there are at most (8k⫹3) 2 2m choices for u(n, j ⬘ ,p ⬘ ). Moreover, w is a descendant in
generation s of u(n, j ⬘ , p ⬘ ) in the tree of edges from D which contains u(n, j, p ⬘ ). Hence, by
virtue of 共2.32兲, there are at most k 2 possibilities for w once u(n, j ⬘ ,p ⬘ ) has been fixed. In total
there are at most K⫽(8k⫹3) 2 2mk 2 possibilities for w. Each such w has l 1 -distance to u(i, j,p)
of at most 兩 u 2ki⫹r ⫺u 2ki 兩 ⫹2⭐2k⫹2 from u 2ki . Therefore, there exists a set ⌳ 0
⫽⌳ 0 (i, j, p)傺 兵 1,2,...,d 其 of cardinality at most (2k⫹2)K such that each possible w satisfies
w 共 l 兲 ⫽u 2ki 共 l 兲 for all l苸⌳ 0 ,
共w(l) is the lth coordinate of w兲. Now take
2m
⌳⫽⌳ 共 i, j 兲 ⫽ 艛 ⌳ 0 共 i, j,p 兲 .
p⫽1
Then 兩 ⌳ 兩 ⭐2m(2k⫹2)K. Moreover, any even w which is within distance 2 of an extension of
length⭐2k of the path in C(e) to a u(i, j,p), 1⭐ p⭐2m, must satisfy
w 共 l 兲 ⫽u 共 i, j,p 兲共 l 兲 for l苸⌳.
共2.35兲
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J. Math. Phys., Vol. 41, No. 3, March 2000
Some remarks on AB-Percolation in high dimensions
1311
Note that such a set ⌳ can be determined by just knowing C(e), that is, by information from the
past. It will turn out that we can obtain 共2.27兲 by restricting ourselves to extensions û which satisfy
for some 1⭐p⭐2m
u 2ki⫹t 共 l 兲 ⫺u 共 i, j,p 兲共 l 兲 ⫽0 for l苸⌳,1⭐t⭐r,
共2.36兲
that is to extensions which move only in coordinate directions outside ⌳.
We claim that if the extension û of a path ␲ to u(i, j,p) satisfies 共2.36兲 and u 2ki⫹r
⫽u(i, j, p), then
u 2ki⫹r 苸 ␲ ⬘ 艛 ⳵␲ ⬘ for all ␲ ⬘ 苸C共 e 兲 .
共2.37兲
To see this, assume that u 2ki⫹r ⫽w or u 2ki⫹r ⫽w⫾ ␰ a or u 2ki⫹r ⫽w⫾ ␰ a ⫾ ␰ b with w苸 ␲ ⬘ , w
⫽w s an even vertex, as in the preceding paragraph, and ␰ a the ath coordinate vector 共⫾ ␰ a and
⫾ ␰ a ⫾ ␰ b represent the generic vectors of l 1 -norm 1 and 2兲. Now u 2ki⫹r (l)⫺u(i, j,p)(l)⫽0 for
l苸⌳. On the other hand, by our choice of ⌳, w(l)⫺u 2ki (l)⫽w(l)⫺u(i, j,p)(l)⫽0 for l苸⌳. By
projecting on the coordinates outside ⌳ we obtain that
u 2ki⫹r ⫺u 共 i, j, p 兲 ⫽projection of 0
or of⫾ ␰ ␣ or of ⫾ ␰ a ⫾ ␰ b .
共2.38兲
This is impossible if 兩 u 2ki⫹r ⫺u(i, j, p) 兩 ⭓3. If 兩 u 2ki⫹r ⫺u(i, j,p) 兩 ⫽2, then this is possible only if
⫾ ␰ a ⫾ ␰ b ⫽u 2ki⫹r ⫺u(i, j, p) and then w⫽u 2ki⫹r ⫺(⫾ ␰ a ⫹⫾ ␰ b )⫽u(i, j,p). However, this case is
excluded, because u(i, j, p) lies on exactly one path in C(e), namely on ␲, by virtue of 共2.30兲 and
共2.34兲. In fact u(i, j, p) is the endpoint of ␲ and we did not include the points within distance 2
from this endpoint in ⳵␲. A similar argument leads to a contradiction when 兩 u 2ki⫹r ⫺u(i, j,p) 兩
⫽1 and necessarily u 2ki⫹r ⫺u(i, j, p)⫽0. If u 2ki⫹r ⫽w⫾ ␰ a , then this leads again to w⫽u 2ki ,
which we already excluded. On the other hand, u 2ki⫹r ⫽w⫾ ␰ a ⫾ ␰ b is impossible. Indeed
兩 u 2ki⫹r ⫺u(i, j, p) 兩 ⫽1 can only be if u 2ki⫹r is an odd vertex, while w is an even vertex. Since we
assumed u 2ki⫹r ⫺u(i, j, p)⫽0 this takes care of all possibilities, so that 共2.37兲 is indeed implied by
共2.36兲.
It follows from the preceding paragraph that if we restrict ourselves to extensions satisfying
共2.36兲, then the extensions will not go through points v whose state X( v ) has been determined
earlier. For the next part of the construction we shall, therefore, only consider paths with steps in
Wª 兵 v 苸Zd : v共 l 兲 ⫽0 for l苸⌳ 其 .
Of course, W is isomorphic to Zd⫺ 兩 ⌳ 兩 , where 兩⌳兩 denotes the cardinality of ⌳. We shall write d̄ for
d⫺ 兩 ⌳ 兩 . We further write
W 2 ⫽V 2 艚W,
for the set of even vertices in W. In order to decide whether to declare the edge e from 共i,j兲 to
(i⫹1,j⫺1) open or not we are now going to look for suitable paths from the u(i, j,l), 1⭐l
⭐m, to B(i⫹1,j⫺1). When 共i,j兲 is open 关and the edge from (i, j⫺2) to (i⫹1,j⫺1) is closed兴,
then we shall go through a specific procedure to construct a random forest M, that is, a subgraph
of Zd without circuits, with the following properties 共2.39兲–共2.43兲:
If f is an edge in M with endpoints a and b, then b⫺a苸W.
There exist 2m distinct even vertices u(i⫹1,j⫺1,l), 1⭐l⭐2m,
in B(i⫹1,j⫺1) such that each u(i⫹1,j⫺1,l) is connected by
an AB-path of length 2k to one of the m special vertices u(i, j,p),
1⭐p⭐m. Moreover, 兩 u 共 i⫹1,j⫺1,l 兲 ⫺u 共 i⫹1,j⫺1,l ⬘ 兲 兩 ⭓4,l⫽l ⬘ .
The total number of even vertices in M is at most k 2 .
共2.39兲
共2.40兲
共2.41兲
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1312
J. Math. Phys., Vol. 41, No. 3, March 2000
H. Kesten and Z. Su
All paths in M start at some u(i, j,p),1⭐ p⭐m,
They are AB-paths and have length ⭐2k.
共2.42兲
The forest M is determined by only the X( v ) with v 苸M
ˆ )艚W, where ␲
ˆ is any path in M which starts
or with v 苸( ⳵␲
at one of the u 共 i, j, p 兲 ,1⭐ p⭐m,
共2.43兲
and ⳵␲
ˆ is as in 共2.33兲. The construction of M may or may not succeed. e is declared open 共closed兲
when the construction succeeds 共or fails, respectively兲. We have two tasks left. First, for the next
edge e ⬘ to be examined we must define a new C(e ⬘ ) with the properties 共2.29兲–共2.34兲 when e is
replaced by e ⬘ . Second, we must prove 共2.27兲, which amounts to showing that the conditional
probability, given the information on the previous steps, of the construction of M succeeding is at
least 1⫺5 ⑀ . We work on the second task first.
To construct M we need a further process 兵 Z n 其 n⭓0 on W 2 . This will be a branching random
walk, albeit not in the strict sense, because the displacements of the children of the same parent are
not independent. Z n will count certain particles, which we refer to as nth generation particles. The
displacement of a particle from its parent will always be a vector in W 2 . The locations of the
particles of the nth generation will be denoted by v n,1 ,..., v n,Z n 共in our fixed order for vertices of
Zd 兲. To construct 兵 Z n 其 we need new random variables X (r) (u), u苸Zd , r⭓1. For different 共u,r兲
these random variables are assumed independent. They will take the values A, B, C, or D and their
distribution is specified by
P 兵 X 共 r 兲 共 u 兲 ⫽A 其 ⫽ p A ⫽1⫺ P 兵 X 共 r 兲 共 u 兲 ⫽C 其 if u is even,
P 兵 X 共 r 兲 共 u 兲 ⫽B 其 ⫽p B ⫽1⫺ P 兵 X 共 r 兲 共 u 兲 ⫽D 其 if u is odd.
共2.44兲
We identify X (1) (u) with X(u). For the zeroth generation of the process 兵 Z n 其 we take m particles,
one each located at u(i, j,l), 1⭐l⭐m. Thus all the particles in the Z-process will have positions
in
m
艛 关 u 共 i, j,l 兲 ⫹W 2 兴 .
l⫽1
For the time being the precise location of the initial particles is unimportant and we shall denote
them by v 0,l . The only important aspect of these locations is that
兩 v 0,l ⫺ v 0,l ⬘ 兩 ⭓4,
l⫽l ⬘ ,
共2.45兲
which is a consequence of the requirements in 共2.34兲. By our choice for the zeroth generation we
also have
v 0,l 苸V 2 艚B 共 i, j 兲 ,
1⭐l⭐Z 0 .
共2.46兲
Assume that each particle of the zeroth generation has state A, that is, X( v 0,l )⫽A for 1⭐l
⭐Z 0 . If 共i,j兲 is open, then this actually is the case, because each u(i, j,l) is the endpoint of an
AB-path starting at an even u(0,0,p); see 共2.34兲. Our construction will be such that all particles of
the Z-process have state A. During the construction we shall check various of the X (r) (u). At the
start we only know the X( v 0,l ).
We shall only construct the first k⫺1 generations and part of the kth generation of the
Z-process. Later generations are not needed. Now assume that we have determined the size of the
nth generation and the locations and states of its particles. First consider n⫹1⬍k. To form the
(n⫹1)th generation we then check successively for l⫽1,2,...,Z n a new state for each of the
W 2 -neighbors of v n,l . More specifically, the W 2 -neighbors of a vertex v are the vertices v ⬘ with
v ⬘ ⫺ v 苸W 2 and 兩 v ⫺ v ⬘ 兩 ⫽2. Let the neighbors of a given v n,l be w 1 ,...,w q . Assume that during
the construction so far the highest r-value for which X (r) (w j ) was examined is r j . We then check
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J. Math. Phys., Vol. 41, No. 3, March 2000
Some remarks on AB-Percolation in high dimensions
1313
the value of X (r j ⫹1) (w j ). We include in the (n⫹1)th generation a particle at those w j for which
X (r j ⫹1) (w j )⫽A. These new particles are called the children of the nth generation particle at v n,l .
Note that more than one (n⫹1)th generation particle may be born at a given vertex w, because
several v n,l may have w as a neighbor and give birth to a child at w. When n⫹1⫽k we use
essentially the same procedure to construct the kth generation, except that we stop considering
neighbors of the particles in the (k⫺1)th generation as soon as we found 2m kth generation
particles. Of course we may not stop; the whole kth generation may contain fewer than 2m
particles. Next we apply the map L to the positions of the particles in the Z-process. We obtain a
branching random walk on L. The nth generation of this new branching random walk has Z n
particles, located at L( v n,l ), 1⭐l⭐Z n . Let us call the process on L the L-process. We remind the
reader that the dimension of W equals d̄⫽d⫺ 兩 ⌳ 兩 . In addition we introduce
d 1 ª 兩 兵 1,2,...,d/2其 \ ⌳ 兩 ,
d 2 ª 兩 兵 d/2⫹1,...,d 其 \ ⌳ 兩 .
Clearly d/2⫺ 兩 ⌳ 兩 ⭐d 1 , d 2 ⭐d/2, so that d i /d→1/2. It is straightforward to check that for a particle
at (s,t)苸L, the expected number of its children at (s⫾ ␣ ,t⫾ ␤ ) in the L-process equals
冦
d2 A
d 1d 2 p ⬃ p
4
A
if 共 ␣ , ␤ 兲 ⫽ 共 ⫾1,⫾1 兲
1
d2
d 1 共 d 1 ⫺1 兲 p A ⫹d 1 p A ⬃ p A
2
8
1
d2
d 2 共 d 2 ⫺1 兲 p A ⫹d 2 p A ⬃ p A
2
8
关 d 1 共 d 1 ⫺1 兲 ⫹d 2 共 d 2 ⫺1 兲兴 p A ⬃
if 共 ␣ , ␤ 兲 ⫽ 共 ⫾2,0兲
.
共2.47兲
if 共 ␣ , ␤ 兲 ⫽ 共 0,⫾2 兲
d2 A
p
2
if 共 ␣ , ␤ 兲 ⫽ 共 0,0兲
Note that the expected number of children of an individual in this branching random walk on L is
共 4d 1 d 2 ⫹2d 21 ⫹2d 22 兲 p A ⫽2d̄ 2 p A ⭐2 ␥ .
共2.48兲
It is also straightforward that the second moment of the number of children of a given individual
is bounded by some constant C 1 , independent of d 关use 共2.26兲兴.
Let v 0,1 ,..., v 0,m 苸B(i, j). It will be helpful to introduce the event
A 1 共 k,⫾ 兲 ⫽A 1 共 k,⫾, v 0,1 ,..., v 0,m ,i, j 兲 ª 兵 there are at least 2m particles in B 共 i⫹1,j
⫾1 兲 which are descendants in generation k of the Z-process of the m
particles at v 0,1 ,..., v 0,m 其 .
We claim that if 共1.12兲 and 共2.26兲 prevail and d is so large that the expected number of children
per individual given by 共2.48兲 is at least (1⫹ ␥ )/2⬎1, then there exist integers k⭓4 and m such
that for any (i, j)苸L and any m zeroth generation particles in the Z-process at positions
v 0,1 ,..., v 0,m it holds that
P p 兵 A 1 共 k,⫾ 兲 其 ⭓1⫺ ⑀ .
共2.49兲
This holds for each of the choices of the sign in B(i⫹1,j⫾1) and uniformly in the v 0,l . It is
important that one pair k, m can be chosen so that 共2.49兲 holds for all large d. 共2.49兲 with this
almost uniformity in d can be proven as in lemmas 1 and 2 of Penrose 共1993兲.16 These rely only
on the local central limit theorem for the offspring distribution in the L-process and simple
moment estimates for Galton–Watson processes. Even stronger results have been proven repeatedly in the branching process literature; see for instance Asmussen and Kaplan 共1976兲.22 It is only
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1314
J. Math. Phys., Vol. 41, No. 3, March 2000
H. Kesten and Z. Su
to obtain this uniform estimate that the L-process has been introduced. For the remainder, k and m
are fixed such that 共2.49兲 holds. We repeat that if A 1 occurs we do not construct the full kth
generation. Only the first 2m particles of this generation are determined.
We need to intersect A 1 with the event
A 2 共 k 兲 ⫽A 2 共 k, v 0,1 ,..., v 0,m ,i, j 兲 ª 兵 the total number of descendants in generations
0,1,...,k⫺1 of the m particles at v 0,1 ,..., v 0,m is at most k 2 ⫺2m 其 ,
for a suitable k 2 . When 共2.26兲 holds, we can find an integer k 2 such that
P p 兵 A 1 共 k 兲 艚A 2 共 k 兲 其 ⭓1⫺2 ⑀ .
共2.50兲
Again k 2 can be chosen independent of d. Indeed, since the expected number of children per
individual in the Z-process or in the L-process is 2d̄ 2 p A ⭐2 ␥ for large d 关under 共2.26兲兴, the
k⫺1
(2 ␥ ) l . Thus
expected total number of individuals in the generations 0,1,...,k⫺1 is at most m 兺 l⫽0
共2.50兲 follows from 共2.49兲 and Markov’s inequality. For the remainder of this proof we fix a k 2 for
which 共2.50兲 holds.
We need a further restriction on the positions of the particles in the first k generations of the
Z-process. We do not want any pair of particles other than a parent and its child to be adjacent on
W 2 or to have the same position. More formally, define
A 3 ⫽ 兵 No pair of particles other than a parent-child pair, in the first k generations of the
Z-process have positions u 1 ,u 2 with 兩 u 1 ⫺u 2 兩 ⭐2 其 .
We claim that 关under 共2.26兲兴
P 兵 A 2 ⶿A 3 其 →0 as d→⬁.
共2.51兲
Of course we use 共2.45兲 here; no two particles in the zeroth generation may be adjacent for 共2.51兲
to hold. To see 共2.51兲, let Y 1 ,Y 2 ,... be independent, identically distributed random variables
whose distribution is the uniform distribution on the 2d̄ 2 vectors of l 1 -norm 2 in W 2 . In addition,
n
Y i . Then
let S n ⫽ 兺 i⫽1
P 兵 S 2 ⫽0 其 ⫽
1
2d̄ 2
sup P 兵 S 2 ⫽w 其 ⭐
w: 兩 w 兩 ⫽2
,
C 2 d̄
d̄ 4
⫽
C2
d̄ 3
,
because Y 1 ,Y 2 must contain the two unit vectors whose sum equals w plus another unit vector and
its opposite. Similarly, for n⭓3
sup P 兵 S n ⫽w 其 ⭐sup P 兵 S 3 ⫽w 其 ⭐
w
w
C2
d̄ 3
,
for a suitable constant C 2 . Let S n⬘ and S ⬙n be two independent copies of S n . Then these inequalities
show that the probability for a given particle in the n 0 th generation of the Z-process to have two
different children, one of which has a descendant in the n 1 th generation at some u 1 and the other
of which has a descendant in the n 2 th generation at some u 2 , such that 兩 u 1 ⫺u 2 兩 ⭐2 and n 1
⫺n 0 ⫹n 2 ⫺n 0 ⭓2 共but n 1 ⫽n 0 or n 2 ⫽n 0 allowed兲, is at most
共 2 ␥ 兲 n 1 ⫺n 0 ⫹n 2 ⫺n 0 P 兵 兩 S n⬘ 1 ⫺n 0 ⫺S n⬙ 2 ⫺n 0 兩 ⭐2 其 ⫽ 共 2 ␥ 兲 n 1 ⫹n 2 ⫺2n 0 P 兵 兩 S n 1 ⫹n 2 ⫺2n 0 兩 ⭐2 其 ⭐
C3
d̄
.
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J. Math. Phys., Vol. 41, No. 3, March 2000
Some remarks on AB-Percolation in high dimensions
1315
Equation 共2.51兲 follows from this; compare also the estimate for E p6 on p. 265 in Penrose
共1993兲.16
Now each particle in the Z-process has associated to it a path in v 0,l ⫹W 2 for some 1⭐l
⭐m. This is the path giving the locations of all its ancestors in order, starting at v 0,l . Such a path
is of the form v 0,l 0 , v 1,l 1 ,... with 兩 v j,l j ⫺ v j⫹1,l j⫹1 兩 ⫽2. In the construction of the Z-process we
check at any stage only the state of the W 2 -neighbors of one of the particles already included in
the family trees at an earlier stage. Thus if A 3 occurs, then the only way for a path associated to
one of the particles to visit a vertex whose state has already been determined at an earlier stage, is
by a so-called ‘‘immediate reversal,’’ that is by having v j,l j ⫽ v j⫹2,l j⫹2 for some j. This can happen
if a particle at some vertex v has a child at some w and, in turn, this child has a child at v . Until
the first immediate reversal in the construction, a particle is placed at a vertex v only because the
original X( v )⫽X (1) ( v )⫽A. Until then we do not use X (r) (u) with r⭓2 for any vertex u to get to
the particle at v . Therefore, if we exclude the occurrence of immediate reversals, then on A 3 all
the vertices v which contain a particle of the first k generations of the Z-process genuinely have
state A and the associated path is an A-path on
艛 兵 v 0,l ⫹W 2 其 .
共2.52兲
1⭐l⭐m
Further, an immediate reversal can occur only if the following event A 4 fails:
A 4 ª 兵 each vertex v which is reachable from some v 0,l , 1⭐l⭐m,
by an A-path in the set 共 2.52兲 of no more than k steps has X 共 2 兲 共 v 兲 ⫽C 其 .
But the probability that A 4 fails is at most
兺
1⭐l⭐m
k
E p 兵 number of A-paths on v 0,l ⫹W 2 of length ⭐k 其 p ⭐m
A
兺 共 2 ␥ 兲l p A⭐ ⑀ ,
l⫽0
共2.53兲
when d is sufficiently large.
The construction of the Z-process only involved vertices in some v 0,l ⫹W 2 , and on the event
A 1 艚A 2 艚A 3 艚A 4 we have found A-paths on the set in 共2.52兲 from 兵 v 0,1 ,..., v 0,m 其 to the positions
of the particles in the branching random walk. We would like to make sure that there actually
exists an AB-path with steps in W from 兵 v 0,1 ,..., v 0,m 其 to each of these vertices. This will actually
be the case if also the following event A 5 occurs:
A 5 ⫽ 兵 for each particle in the branching random walk there is a vertex u
苸W which is adjacent to the particle and to its parent and with X 共 u 兲 ⫽B 其 .
Note that the u’s, which are required here, must be odd vertices and that we have not examined
X(u) for any odd vertex yet. Moreover, if A 2 occurs, then we need at most k 2 vertices to be in
state B. Thus, given a realization of the first k generations of the Z-process on which A 2 occurs, the
conditional probability of A 5 is at least (1⫺ p B ) k 2 , and we may assume d so large that this is at
least 1⫺ ⑀ 关see 共1.12兲兴. Combining this with 共2.50兲, 共2.51兲, and 共2.53兲 we have that for large d
P p 兵 A 1 共 k 兲 艚A 2 共 k 兲 艚A 3 艚A 4 艚A 5 其 ⭓1⫺5 ⑀ .
共2.54兲
Finally we come to the forest M. When
A 1 共 k,⫺,u 共 i, j,1兲 ,...,u 共 i, j,m 兲 ,i, j 兲 艚A 2 共 k,u 共 i, j,1兲 ,...,u 共 i, j,m 兲 ,i, j 兲 艚A 3 艚A 4 艚A 5
occurs, then we take for M the union of the edges in the AB-paths which we found in the last
paragraph, from the vertices u(i, j,l), 1⭐l⭐m, to any of the particles in the first k generations of
the Z-process which we investigated 共this means that the kth generation may be truncated at 2m
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1316
J. Math. Phys., Vol. 41, No. 3, March 2000
H. Kesten and Z. Su
particles兲. This is indeed a forest, because A 3 occurs. For the u(i⫹1,j⫺1,l) required in 共2.40兲 we
take the location of the 2m particles in the kth generation of the Z-process. They satisfy the last
condition in 共2.40兲 because A 3 occurs 共when two even particles have l 1 -distance⬎2, then the
distance between them is at least 4兲. The other conditions in 共2.39兲–共2.43兲 hold by construction. In
this case, when A 1 ⫺A 5 occur, we say that the construction of M succeeded and declare the edge
e from 共i,j兲 to (i⫹1,j⫺1) to be open.
The construction of the Z-process only investigates the states of vertices in
m
艛 关 u 共 i, j,l 兲 ⫹W 兴 .
l⫽1
By condition 共2.33兲, and the fact that 共2.36兲 implies 共2.37兲, none of these vertices had its state
investigated before we came to the determination of the state of e. Thus the conditional probability
in 共2.27兲 is at least equal to the left hand side of 共2.54兲. This shows that 共2.27兲 holds.
The last step of the proof is the definition of C(e ⬘ ) for the edge e ⬘ which succeeds e in the
order of the edges of L. If the vertex 共i,j兲 is closed, or the edge from (i, j⫺2) to (i⫹1,j⫺1) is
open, then we did not investigate any vertices in order to decide on the state of e. Accordingly we
do not change C(e), that is we take C(e ⬘ )⫽C(e). If 共i,j兲 is open and the edge from (i, j⫺2) to
(i⫹1,j⫺1) is closed, then we go through the construction described in this proof and roughly
speaking take C(e ⬘ ) to be the union of C(e) and the forest M. Since M is not a collection of paths
we have to be more precise. Formally, C(e ⬘ ) is the union of C(e) and the collection of paths ␲
˜ of
ˆ , where ␲ 苸C(e) is a path from one of the u(0,0,l), 1⭐l⭐2m, to some
the form ␲ followed by ␲
u(i, j,p), 1⭐ p⭐m, and ␲
ˆ is a path in M starting at u(i, j,p).
We briefly check that C(e ⬘ ) satisfies 共2.29兲–共2.34兲 with e replaced by e ⬘ 关and the required
small modification in 共2.34兲兴. 共2.29兲 is obvious from the definition of C(e ⬘ ). 共2.30兲 follows from
the fact that no circuits can be formed by means of the paths which have been added to C(e). In
turn, this is so because 共2.36兲 implies 共2.37兲 and because the edges in M ‘move in directions of
W only, i.e., 共2.39兲. Next 共2.31兲 also follows from the nature of the paths which have been added
ˆ which have been added have length ⭐2k. Moreover,
to C(e). To see this note that the parts ␲
apart from their initial point, they can contain only one special point, namely their endpoint. The
ˆ is a special point if and only if ␲
ˆ has length equal to 2k. In addition, the initial
endpoint of ␲
point of any such ␲
ˆ is a special point in B(i, j) and is therefore the 2kith point and endpoint of
some path ␲ 苸C(e) 关see 共2.31兲 and 共2.34兲兴. As for conditions 共2.32兲–共2.34兲 for C(e ⬘ ), these are
immediate from the same conditions for C(e) and 共2.40兲–共2.43兲. Thus if we start with C(e 0 )⫽0”
for e 0 ⫽the edge from 共0,0兲 to 共1,⫺1兲, and increase C according to the procedure just outlined at
each step, then the resulting C(e) will have the properties 共2.29兲–共2.34兲 at each stage.
This completes the recursive step when we have to decide on the state of the edge from 共i,j兲
to (i⫹1,j⫺1). When we want to decide on the state of the edge from 共i,j兲 to (i⫹1,j⫹1) no
essential changes need to be made. Only, in this case the role of the u(i, j,l) with 1⭐l⭐m is taken
over by the u(i, j,l) with m⫹1⭐l⭐2m. Thus, we try to connect the vertices u(i, j,l) with m
⫹1⭐l⭐2m to B(i⫹1,j⫹1), and ignore the u(i, j,l) with 1⭐l⭐m.
䊏
III. PROOF TO THEOREM 2
First we discuss 共1.8兲. We are now back to the case when P 兵 X( v )⫽A 其 ⫽ p⫽1⫺ P 兵 X( v )
⫽B 其 . In the case of AB-percolation on Zd the result can be proven by the ideas of Section 15 in
Penrose 共1993兲.16 The probability of an infinite AB-path starting at the origin is at most P 兵 X(0)
⫽A 其 ⭐ p⫽O(d ⫺2 ). More precisely, this upper bound equals p times the probability that a certain
Galton–Watson process 兵 Ẑ n 其 does not die out. The offspring distribution of this Galton–Watson
process is a binomial distribution corresponding to 2d 2 trials with success parameter p. The
Ẑ-process starts with one individual. Intuitively, Ẑ n counts the number of particle alive in the nth
generation of the Z-process of the preceding proof, except that we now drop the restriction that the
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Some remarks on AB-Percolation in high dimensions
J. Math. Phys., Vol. 41, No. 3, March 2000
1317
displacement of a child from its parent has to lie in W 2 . The limsup of the survival probability of
the Ẑ-process 共as d→⬁兲 is at most equal to the survival probability of a branching process whose
offspring distribution is Poisson with mean ␥. This is so, because the distribution of the number of
offspring of one particle in the Ẑ-process converges to a Poisson distribution with mean ␥ when
p⫽ ␥ /(2d 2 ). The survival probability of such a process is the y( ␥ ) of 共1.6兲. This result implies
that the probability of a BA-path starting at the origin is in the limit at most the expected number
of neighbors of 0 in Zd which have state A times y( ␥ ), that is
2d
␥
␥y共␥兲
y 共 ␥ 兲⫽
.
2d 2
d
This provides an upper bound for the left-hand-side of 共1.8兲.
To obtain a lower bound, we merely have to show that we can get started in the preceding
proof. That is, we have to find 2m vertices u(0,0,l), 1⭐l⭐2m, in B(0,0) which are connected to
the origin by AB-paths of length⭐C 4 . When these paths are fixed, they use at most 2mC 4 edges
which involve at most 2mC 4 coordinates. This set of coordinates plays the same role as ⌳ in the
preceding proof. We can then obtain a continuation to infinity of at least one of these AB-paths by
considering only paths which take no steps in these 2mC 4 directions. The probability that such a
continuation exists can be made as close to one as desired by taking ⑀ small 共and hence k,m large兲
in the preceding proof. Thus, a lower bound for the left hand side of 共1.8兲 is essentially given by
the probability that there exist 2m even points in B(0,0) which satisfy 共2.28兲 and which are
connected to a neighbor of the origin by an AB-path. It is not hard to show that an asymptotic
lower bound for this is again the probability that at least one neighbor of 0 has state A times the
survival probability y( ␥ ) of a Galton–Watson process with a mean ␥ Poisson offspring distribution. Since the probability of 0 having at least one neighbor in state A is asymptotically equivalent
to ␥ /d, this proves 共1.8兲.
We turn now to Theorem 2 in the oriented case. Presumably, 共1.7兲, can again be proven by
the method of Penrose 共1993兲,16 but we have not checked this. An alternative is to use the method
of Kesten 共1991兲.18 Because the probability of AB-percolation is not obviously monotonic in p we
cannot simply copy the proof there, but need an extra step.
It is helpful for that to consider the two parameter problem described by 共1.9兲, 共1.10兲. Write
d
␪ B 共 p A , p B ,d 兲 ª P 兵 there is a BA-path on Z⫹
from some neighbor of 0 to ⬁ 其 . 共3.1兲
d
The second moment method in the proof of Theorem 1 for Z⫹
shows that for fixed ␥ ⬎1 there
exist ␩ i ⫽ ␩ i ( ␥ )⬎0, i⫽1,2, such that for large d,
␪ B 共 p A ,p B ,d 兲 ⭓ ␩ 1 ,
共3.2兲
if
p A⬃
2␥ B
,p ⭓1⫺ ␩ 2 .
d2
共3.3兲
A closely related problem is that of A-percolation on V 2 , when V 2 is given the orientation
d
d
d
. There is an oriented edge from 兺 i⫽1
a i ␰ i to 兺 i⫽1
b i ␰ i if and only if
which it inherits from Z⫹
d
b i ⭓a i , 1⭐i⭐d, and 兺 i⫽1 (b i ⫺a i )⫽2. Even though the notation does not indicate this, V 2 will
always be oriented in this way in this section. In this model a vertex of V 2 is in state A with
probability p A . We define for this model
␪ˆ 共 p 兲 ⫽ P p 兵 there exists an oriented A-path on V 2 from some neighbor of 0 to ⬁ 其 .
We can also think of this as a situation in which all odd vertices are put in state B with probability
1, and only the even vertices have random states. Thus
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1318
J. Math. Phys., Vol. 41, No. 3, March 2000
H. Kesten and Z. Su
␪ˆ 共 p A 兲 ⫽ ␪ B 共 p A ,1,d 兲 .
The principal extra step is to show that if 共1.11兲 prevails, then
␪ B 共 p A , p B ,d 兲 ⫺ ␪ B 共 p A ,1,d 兲 ⫽ ␪ B 共 p A ,p B ,d 兲 ⫺ ␪ˆ 共 p A 兲 →0 as d→⬁.
共3.4兲
d
starting at a neighbor of the origin is an A-path
Since the restriction to V 2 of any BA-path on Z⫹
on V 2 , it is clear that
␪ B 共 p A ,p B ,d 兲 ⭐ ␪ˆ 共 p A 兲 ,
so that we need a lower bound for ␪ B (p A ,p B ,d), or equivalently, an upper bound on 1
⫺ ␪ B ( p A , p B ,d). To this end we define C to be the ‘‘BA-cluster’’ of 0, that is, C consists of 0 plus
d
starting at a neighbor of the origin; 兩C兩 will be the
all the vertices reachable by a BA-path on Z⫹
d
: 兩 z 兩 ⭐m 其 . Then
number of vertices in C. We further define ⌳ m ⫽ 兵 z苸Z⫹
1⫺ ␪ B 共 p A , p B ,d 兲 ⫽ P p 兵 兩 C兩 ⬍⬁ 其 ⫽
兺C P p 兵 C⫽C 其 ,
共3.5兲
d
where the sum over C runs over all finite possibilities for C, that is, over all finite subsets C of Z⫹
such that every point of C is reachable in C by a path from the origin. It was proved in Kesten
共1991兲18 关see lemmas 5–7, especially the estimate 共3.9兲兴 that
P 兵 C contains a vertex outside ⌳ n , but 兩 C兩 ⬍⬁ 其
can be made less than any prescribed ⑀ ⬎0 for all large d by taking n large enough. The proof in
Kesten 共1991兲18 is given for a slightly different situation, but basically the only changes needed
there are to consider only even n in lemma 5 共so that ␯ n there only counts vertices in state A兲 and
to redefine ␴ in lemma 7 as
␴ 共 n,l 兲 ⫽min兵 s:s even, s⬎n, ␯ s ⭓l 其 .
Once we have this, we can fix n so that
1⫺ ␪ B 共 p A ,p B ,d 兲 ⭐ ⑀ ⫹
兺
C傺⌳ n
P 兵 C⫽C 其 ,
共3.6兲
for all large d.
Now let C2 be the set of vertices of V 2 which are connected by an A-path on V 2 to a neighbor
on V 2 of the origin. By simple path counting
共 n⫹2 兲 /2
E p A 兩 C2 艚⌳ n⫹2 兩 ⭐
兺
k⫽0
冋
册
d 共 d⫹1 兲 p A k
⭐
2
共 n⫹2 兲 /2
兺
k⫽0
共 3 ␥ 兲 k ⭐C 5 ,
for some constant C 5 and all large d. If v ⫽0 is a vertex in C2 , then there is an oriented A-path on
V 2 from the origin to v . Call the predecessor of v on this path a parent of v . We say a parent,
because it is not necessarily unique. The probability that for each v 苸C2 艚⌳ n⫹2 , v ⫽0, there is a
d
which is adjacent to v and to a parent of v and which is in state B is, for any N
vertex of Z⫹
⭓1, at least
冋
P p A 兵 兩 C2 艚⌳ n⫹2 兩 ⭐N 其 关 p B 兴 N ⭓ 1⫺
册
C5 B N
关p 兴 .
N
共3.7兲
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J. Math. Phys., Vol. 41, No. 3, March 2000
Some remarks on AB-Percolation in high dimensions
1319
By choosing N large enough and then taking d large enough 关and hence p B close to 1 共see 共1.11兲兴
we can make the right-hand-side of 共3.7兲 at least 1⫺ ⑀ . Note that if each v 苸C2 艚⌳ n⫹2 and its
parent have a common neighbor in state B, then all the vertices in v 苸C2 艚⌳ n⫹2 have an AB-path
leading to them from some neighbor of the origin. Thus in this case C2 艚⌳ n⫹2 傺C. If in addition
C傺⌳ n , this implies that
C2 艚⌳ n⫹2 傺⌳ n ,
共3.8兲
and this prevents C2 of getting out of ⌳ n , because any path in C2 to the complement of ⌳ n⫹2
would have to jump from ⌳ n to the complement of ⌳ n⫹2 by a step of size ⭓3, which is impossible. Thus 共3.8兲 forces C2 to be finite. Combined with 共3.6兲 this shows that
1⫺ ␪ B 共 p A ,1,d 兲 ⫽1⫺ ␪ˆ 共 p A 兲 ⫽ P p A 兵 兩 C2 兩 ⬍⬁ 其 ⭓ P 兵 C傺⌳ n 其 ⫺ ⑀ ⭓1⫺ ␪ B 共 p A ,p B ,d 兲 ⫺2 ⑀ .
Since ⑀ ⬎0 was arbitrary, 共3.4兲 follows.
The probability of A-percolation on V 2 with probability p for a vertex to be in state A is
increasing in p. One can, therefore, use the proof in Kesten 共1991兲18 to show that for p A
⬃(2 ␥ /d 2 ) it holds that
␪ˆ 共 p A 兲 →y 共 ␥ 兲 .
Together with 共3.4兲 this gives under 共1.11兲 that
␪ B 共 p A ,p B ,d 兲 →y 共 ␥ 兲 .
共3.9兲
Finally, by the inclusion–exclusion principle,
兺
d
P p 兵 ᭚ infinite AB-path on Z⫹
starting at x 其 ⫽
兩 x 兩 ⫽1
兺
兩 x 兩 ⫽1
p A ␪ B 共 p A ,p B ,d 兲
d
⭓ P p 兵 ᭚ infinite AB-path on Z⫹
starting at some x with 兩 x 兩 ⫽1 其 ⭓
⫺
兺
兩 x 兩 ⫽1,兩 y 兩 ⫽1,
x⫽y
兺
兩 x 兩 ⫽1
p A ␪ B 共 p A ,p B ,d 兲
P 兵 x and y have state A 其 .
Thus,
d
starting at some neighbor of 0其 ⫽d p A ␪ B 共 p A ,p B ,d 兲
P p 兵 ᭚ infinite AB-path on Z⫹
⫹O 共 d 2 关 p A 兴 2 兲 ⬃
共1.7兲 follows.
冉 冊
1
2␥y共␥兲
⫹O 2 .
d
d
䊏
ACKNOWLEDGMENTS
The work of H.K. was supported by an NSF Grant to Cornell University. Z.S. was supported
by a Fellowship from the China Scholarship Council, which he used to visit the Mathematics
Department at Cornell university for the year 1998. Much of the research for this article was
carried out during this year, Z.S. thanks the Mathematics Department at Cornell for its hospitality.
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4
5
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