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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 [1] Beffara, V., Sidoravicius, V.: Probability theory. Encyclopedia Math. Phys. 21–28 (2006) [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]

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