Zebra-percolation on Cayley trees

0.35 MB·English

by  D. Gandolfo

ZEBRA-PERCOLATION ON CAYLEY TREES
D. GANDOLFO, U. A. ROZIKOV, J. RUIZ
Abstract. WeconsiderBernoulli(bond)percolationwithparameterpontheCayley
tree of order k. We introduce the notion of zebra-percolation that is percolation by
3 paths of alternating open and closed edges. In contrast with standard percolation
1 with critical threshold at p = 1/k, we show that zebra-percolation occurs between
c
0
two critical values p and p (explicitly given). We provide the specific formula of
c,1 c,2
2
zebra-percolation function.

n
a
J
Mathematics Subject Classifications (2010). 60K35, 82B43
7
Key words. Cayley tree, percolation, zebra-percolation, percolation function.

]
R 1. Introduction and definitions
P
. Percolation on trees still remains the subject of many open problems. The purpose
h
t of this paper is to study the percolation phenomenon by paths of alternating open and
a
closed bonds. Such paths are called zebra-paths.
m
We consider the Cayley tree Γk = (V,L) where each vertex has k+1 neighbors with
[
V being the set of vertices and L the set of bonds. Bonds are independently open with

1 probability p (and closed with probability 1−p). We let P be corresponding probability
p
v
measure.
5
6 On this tree we fix a given vertex e (the root) and consider the following event
1
E = {An infinite zebra-path contains the root}. (1.1)
1
.
1 By path we mean a collection of consecutive bonds (appearing only once) sharing a
0 common endpoint. The zebra-percolation function is defined by
3
1 ζ (p) = P (E). (1.2)
k p
:
v
i The paper is organized as follows. In Section 2 we show that zebra-percolation occurs
X
in the range p ∈ (p ,p ). This holds as soon as k ≥ 3 and the two critical values are
c,1 c,2
r
a explicitly given. Section 3 is devoted to standard percolation. In Section 4 we give a
relation between standard percolation and zebra-percolation. The last section is devoted
to some discussions and open problems.
2. Two critical values
The existence of two critical values is a consequence of the following dichotomy
Theorem 1. The zebra percolation function satisfies
1) If k2p(1−p) < 1, then ζ (p) = 0.
k
1
2 D.GANDOLFO,U.A.ROZIKOV,J.RUIZ
2) If k2p(1−p) > 1, then ζ (p) > 0.
k
Proof. 1) Consider on the tree Γk all paths of length n starting from the root. We will
denote hereafter by W the set of endpoints of these paths (excluding the root). Let F
n n
be the event that there is a zebra-path of length n. The probability P for such an event
n
is
(cid:40)
2(p(1−p))n/2, if n is even
P = (2.1)
n
(p(1−p))(n−1)/2, if n is odd.
The number of paths is at most |W | = (k+1)kn−1. This implies that
n
P (F ) ≤ 2(k+1)kn−1(p(1−p))[n/2],
p n
which, under the condition k2p(1−p) < 1, goes to 0 as n → ∞. Hereafter [·] denotes
the integer part. We then get ζ (p) = 0.
k
2) We shall show that if k2p(1−p) > 1, then the root zebra-percolates with positive
probability. Let X denote the number of vertices belonging to W and zebra-connected
n n
to the root. We will apply the method of second moment to the random variable X
n
(see, e.g. [6]). We have
E[X ]2
n
P(X > 0) ≥ . (2.2)
n E[X2]
n
By linearity, we have that E(X ) = |W |P . If we can show that for some constant
n n n
M and for all n,
E(X2) ≤ ME(X )2, (2.3)
n n
we would then have that P (X > 0) ≥ 1 for all n. The events {X > 0} are decreasing
p n M n
and so countable additivity yields P (X > 0, ∀n) ≥ 1 . But the latter event is the
p n M
sameastheeventthattherootispercolatingandoneisdone. Wenowboundthesecond
moment in order to establish (2.3). Letting U be the event that both v and w are
v,w
zebra-connected to the root, we have that
(cid:88)
E(X2) = P (U ). (2.4)
n p v,w
v,w∈Wn
Now P (U ) = P2P−1 , where m is the level at which paths from e to v and to w
p v,w n mv,w v,w
split. For a given v and m, the number of w with m being m is at most |W |/|W |.
v,w n m
Hence
n n
(cid:88) (cid:88) 1
E(X2) ≤ |W | P2P−1|W |/|W | = E(X )2 . (2.5)
n n n m n m n P |W |
m m
m=0 m=0
If (cid:80)∞ 1 < ∞, then we would have (2.3). If k2p(1−p) > 1, then using formula
m=0 Pm|Wm|
(2.1) one can see that 1 decays exponentially like (k2p(1 − p))−m/2 giving the
Pm|Wm|
desired convergence. (cid:3)
ZEBRA-PERCOLATION ON TREES 3
This theorem gives two critical values for the zebra-percolation which are solutions to
k2p(1−p) = 1:
√ √
k− k2−4 k+ k2−4
p (k) = , p (k) = .
c,1 c,2
2k 2k
Note that if k ≥ 3, 0 < p (k) < 1 < 1 < p (k) < 1. Moreover p (k)+p (k) = 1.
c,1 k 2 c,2 c,1 c,2
This tells that p and p are symmetric with respect to 1/2. When k = 2, p (k) =
c,1 c,2 c,1
p (k) = 1/2 so that no zebra-percolation occurs.
c,2
3. On percolation function
Consider standard percolation model on a Cayley tree. Denote by θ (p) the standard
k
percolationfunction,thatistheprobabilitywithrespecttoP thatthereexistsaninfinite
p
cluster of open edges containing the root. We refer the reader to [2], [3], [4], [5].
Proposition 1. The function θ (p) satisfies
k
(cid:40) 0, if p ≤ 1
k
θ (p) =
k
θˆ (p), if p > 1,
k k
where θˆ (p) is a unique solution to the following functional equation
k
(cid:16) (cid:17)k 1
θˆ (p) = 1− 1−pθˆ (p) , p > . (3.1)
k k
k
Proof. Let e be the root of the Cayley tree, and S(e) the set of direct successors of the
root. Denote by A the event that vertex i ∈ S(e) is in an infinite component, which is
i
not connected to e. Then by self-similarity we get
P (A ) = θ (p), for any i ∈ S(e).
p i k
Let B be the event that the edge (cid:104)e,i(cid:105) is open and A holds. Then
i i
P (B ) = pθ (p), for any i ∈ S(e).
p i k
Since B , B , ..., B are independent, using inclusion-exclusion principle, we get
1 2 k
(cid:32) k (cid:33)
(cid:91)
θ (p) = P B =
k p i
i=1
k (cid:32) k (cid:33)
(cid:88) (cid:88) (cid:88) (cid:92)
P (B )− P (B ∩B )+ P (B ∩B ∩B )−···+(−1)k−1P B =
p i p i j p i j q p i
i=1 i,j: i,j,q: i=1
i<j i<j<q
(cid:18) (cid:19) (cid:18) (cid:19)
k k
kpθ (p)− (pθ (p))2+ (pθ (p))3−···+(−1)k−1(pθ (p))k =
k k k k
2 3
1−(1−pθ (p))k.
k
Hence θ (p) is a fixed point of the function
k
f(x) = 1−(1−px)k, x ∈ [0,1].
The proof is then completed by using the following lemma: (cid:3)
4 D.GANDOLFO,U.A.ROZIKOV,J.RUIZ
Lemma 1. The function f satisfies
i. If p ≤ 1 then the function f(x) has a unique fixed point 0.
k
ii. If p > 1 then the function f(x) has two fixed points 0 and θˆ.
k
Proof. Note that 0 is a fixed point of f. On the other hand, f(1) = 1−(1−p)k and
f(cid:48)(x) = kp(1−px)k−1 ≥ 0, f(cid:48)(cid:48)(x) = −k(k−1)p2(1−px)k−2 ≤ 0, x ∈ [0,1].
Hence f is increasing and concave. It is easy to see that f has a unique fixed point
θˆ∈ (0,1] when f(cid:48)(0) = kp > 1 and no fixed point when f(cid:48)(0) = kp ≤ 1. This completes
the proof. (cid:3)
Simple computations show that
(cid:40) 0, if p ≤ 1
2
θ (p) =
2 2p−1, if p > 1.
p2 2
and
 0, if p ≤ 1
 3
θ (p) =
3 2(√3p−1) , if p > 1.
 p(3p+ p(4−3p)) 3
The general solution is given through the inverse function
Proposition 2. The function θˆ (p), p > 1/k, k ≥ 2 is invertible with inverse
k
√
1− k 1−p
θˆ−1(p) = . (3.2)
k p
Proof. First we shall prove that θˆ (p) is one-to-one. For p ,p ∈ (1/k,1), we get from
k 1 2
equation (3.1)
(cid:104) (cid:16) (cid:17)(cid:105)
θˆ (p )−θˆ (p ) = (p −p )θˆ (p )+p θˆ (p )−θˆ (p ) ·U, (3.3)
k 1 k 2 1 2 k 1 2 k 1 k 2
where U = (cid:80)k−1(1−p θˆ (p ))k−1−i(1−p θˆ (p ))i > 0.
i=0 1 k 1 2 k 2
Since θˆ (p) > 0 for any p > 1/k, if θˆ (p ) = θˆ (p ) then from equality (3.3) we get
k k 1 k 2
p = p . Hence θˆ (p) is one-to-one, i.e. invertible.
1 2 k
Solving the equation x = 1 − (1 − px)k with respect to p for x ∈ [0,1], we get
√
p = g(x) = x−1(1− k 1−x). Now by (3.1) we have p = g(θˆ (p)) for any p > 1. Hence
k k
g is the inverse function of θˆ (p). (cid:3)
k
Note that the function θ (p) has following properties:
k
(1) θ (p) is nondecreasing in p
k
(2) θ (1/k) = 0, θ (1) = 1, θ (p) (cid:54)= 1 for any p < 1
k k k
(3) θ (p) is differentiable for any p (cid:54)= 1/k.
k
ZEBRA-PERCOLATION ON TREES 5
4. Relation between standard and zebra percolation
Starting from the Cayley tree Γk = (V,L), we construct a new tree Γˆk = (Vˆ,Lˆ) as
follows (see Fig. 1)
∞ ∞
(cid:91) (cid:91)
Vˆ = W , Lˆ = {(x,z) : x ∈ W ,z ∈ S(y),y ∈ S(x)},
2m 2m
m=0 m=0
where S(x) denotes the set of direct successors of x.
It is easy to see that Γˆk is a regular tree of order k2 (except on the root).
We denote by l an edge in L and by λ and edge in Lˆ. Note that any edge λ ∈ Lˆ can
be represented by two edges l ,l ∈ L, which have a common endpoint. We write this
1 2
as λ = (l ,l ), moreover l is the closer to the root of the Cayley tree.
1 2 1
Now for a given configuration σ ∈ Ω = {0,1}L we define a configuration φ ∈ Φ =
{−1,0,+1}Lˆ as the following (see Fig. 1)
 −1, if σ(l ) = 0, σ(l ) = 1
1 2


φ(λ) = φ (λ) = 0, if σ(l ) = σ(l )
σ 1 2


1, if σ(l ) = 1, σ(l ) = 0.
1 2
e
11
0 1
2 3
1 0 1 0
(cid:45)1 0 0 1
44 55 66 77
0 1 0 0 1 0 1 1
8 9 10 11 12 13 14 15
0 1 0 0 1 1 0 1 0 0 0 1 0 0 0 0
0 (cid:45)1 1 1 (cid:45)1 (cid:45)1 0 (cid:45)1 1 1 0 (cid:45)1 1 1 1 1
1166 1177 1188 1199 2200 2211 2222 2233 2244 2255 2266 2277 2288 2299 3300 3311
Fig. 1. Correspondence between configurations σ on Γ2 (solid lines) and φ on Γˆ2 (dotted lines).
A given configuration φ divides the set Lˆ into clusters of (+) and (−) bonds.
6 D.GANDOLFO,U.A.ROZIKOV,J.RUIZ
We speak of the edge λ ∈ Lˆ as being open with probability q (in φ) if φ(λ) (cid:54)= 0 and
as being closed if φ(λ) = 0. Let µ be corresponding product measure. Denote
q
θ (q) = µ (|Cˆ| = ∞). (4.1)
k2 q
By our construction the following is obvious
Proposition 3. The functions ζ (p) and θ (p) are related by
k k
ζ (p) = θ (p(1−p)). (4.2)
k k2
This proposition provides an alternative proof of Theorem 1. By properties of θ (p)
k2
we get ζ (p) = 0 iff p(1−p) ≤ 1/k2 and ζ (p) > 0 iff p(1−p) > 1/k2.
k k
The two critical values p and p are the solutions of p(1−p) = 1/k2.
c,1 c,2
By Proposition 3 we get
Theorem 2. The function ζ (p) has the following properties:
k
a. ζ (p) is increasing in p ∈ [0,1/2], and deacreasing in p ∈ [1/2,1].
k
b. ζ (p ) = ζ (p ) = 0, max ζ (p) = ζ (1/2) = θ (1/4).
k c,1 k c,2 p k k k2
b. ζ (p) is differentiable on [0,1]\{p ,p }.
k c,1 c,2
c. there is no zebra-percolation for k = 2.
The graphs of functions θ (p), θ (p) and ζ (p) are presented for k = 3 in Fig.2.
k k2 k
1
0.898
0
0 1pc,1 1 1 1 pc,2 1
9 4 3 2
Fig. 2. Graphs of θ (p) (dashed line), θ (p) (dotted line), and ζ (p) (solid line).
3 9 3
ZEBRA-PERCOLATION ON TREES 7
5. Open problems
An interesting problem in percolation theory is to study the distribution of the num-
ber of vertices in clusters and geometric properties of open clusters when p is close to the
critical value p . It is believed that some of these properties are universal, i.e., depend
c
only on the dimension of the graph. Some open problems are in order.
Problem 1. Study distribution of the number of vertices and geometric properties
of the zebra-connected clusters (made of zebra paths) when p is close to p or p .
c,1 c,2
It is known that Zd for large d behaves in many respects like a regular tree.
Problem 2. Define a notion of zebra-connected component on Zd. Find the critical
value(s) for zebra-percolation on Zd.
When an infinite cluster exists, it is natural to ask how many there are (see e.g. [1]).
Problem 3. How many infinite cluster exist for zebra-percolation ?
Acknowledgements
U.Rozikov thanks CNRS for financial support and the Centre de Physique Th´eorique
- Marseille for kind hospitality during his visit (September-December 2012).
8 D.GANDOLFO,U.A.ROZIKOV,J.RUIZ
References
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[2] Grimmett, G.: Percolation, 2nd ed. Springer, Berlin. (1999)
[3] vanderHofstad,R.: Percolationandrandomgraphs.New perspectives in stochastic geometry,173–
247, Oxford Univ. Press, Oxford, (2010).
[4] Lyons, R.: Phase transitions on nonamenable graphs. J. Math. Phys. 41, 1099–1126 (2000)
[5] Peres,Y.: Probabilityontrees: anintroductoryclimb.Lecturesonprobabilitytheoryandstatistics
(Saint-Flour, 1997), 193–280, Lecture Notes in Math., 1717, Springer, Berlin, (1999)
[6] Steif, J. E.: A mini course on percolation theory. http://www.math.chalmers.se/∼steif/perc.pdf
D. Gandolfo and J.Ruiz, Centre de Physique The´orique, UMR 6207,Universite´s Aix-
Marseille et Sud Toulon-Var, Luminy Case 907, 13288 Marseille, France.
E-mail address: [email protected] [email protected]
U. A. Rozikov, Institute of mathematics, 29, Do’rmon Yo’li str., 100125, Tashkent,
Uzbekistan.
E-mail address: [email protected]