It is well-known in percolation theory (and intuitively plausible) that two events of the form ``there is an open path from $s$ to $a$' are positively correlated. We prove the (not intuitively obvious) fact that this is still true if we condition on an event of the form ``there is no open path from $s$ to $t$'

Berg, J. (Rob) van den

Kahn, J.

English

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Centrum voor Wiskunde en InformaticaREPORTRAPPORTA correlation inequality for connection events in percolationJ. van den Berg, J. KahnProbability, Networks and Algorithms (PNA)PNA-R9918 December 31, 1999Report PNA-R9918ISSN 1386-3711CWIP.O. Box 940791090 GB  AmsterdamThe NetherlandsCWI is the National Research Institute for Mathematicsand Computer Science. CWI is part of the StichtingMathematisch Centrum (SMC), the Dutch foundationfor promotion of mathematics and computer scienceand their applications.SMC is sponsored by the Netherlands Organization forScientific Research (NWO). CWI is a member ofERCIM, the European Research Consortium forInformatics and Mathematics.Copyright © Stichting Mathematisch CentrumP.O. Box 94079, 1090 GB  Amsterdam (NL)Kruislaan 413, 1098 SJ  Amsterdam (NL)Telephone +31 20 592 9333Telefax +31 20 592 4199A Correlation Inequality for Connection Events in PercolationJ. van den BergCWIP.O. Box 94079, 1090 GB AmsterdamThe NetherlandsJ. KahnDepartment of Mathematics and RUTCORRutgers University, New Brunswick, NJ 08903USAAbstractIt is well-known in percolation theory (and intuitively plausible) that two events ofthe form \there is an open path from s to a" are positively correlated. We prove the(not intuitively obvious) fact that this is still true if we condition on an event of the form\there is no open path from s to t".1991 Mathematics Subject Classication: 05C99 60C05 60K35.Key words and Phrases: Correlation inequalities, Percolation, Ahlswede-Daykin Theorem.Note: Work by the rst author carried out under CWI project PNA3.1 \Probability". Thesecond author is supported by NSF.11 Introduction and statement of resultsWe consider the usual bond percolation models on a (nite or countably innite) graphG = (V;E): each e 2 E is \open" (has value 1) with probability p(e) and \closed" (hasvalue 0) with probability 1 − p(e), independently of all other edges. We write P for thecorresponding probability distribution on Ω := f0; 1gE . For general background see [3].For s; a 2 V we write s  ! a for the event that there is an open path from s to a, ands 6 ! a for the complementary event.Positive (i.e. nonnegative) correlation of any two events s  ! a and s  ! b followsfrom Harris’ inequality [5] (Theorem 2.1 below). The correlation inequality of the title saysthat this phenomenon persists if we condition on any event s 6 ! t.Theorem 1.1 For any s; a; b; t 2 VP (s ! a; s ! b j s 6 ! t)  P (s ! a j s 6 ! t)P (s ! b j s 6 ! t):The intuition for this is not very clear. In particular it is not true if we condition ons ! t rather than s 6 ! t. (Consider the graph with vertices s; a; b; t and each of s; t joinedto each of a; b.)From now on we x s 2 V , and set, for X  V , QX = fs  ! x 8x 2 Xg andRX = fs 6 ! x 8x 2 Xg.Theorem 1.2 For any A;B;X; Y  V ,P (QARX)P (QBRY )  P (QA[BRX\Y )P (RX[Y ): (1)Remarks1. Of course we recover Theorem 1.1 from Theorem 1.2 by taking A = fag, B = fbg andX = Y = ftg. This is not generalization for its own sake: the more general form is neededfor the proof.2. The perhaps intuitively more natural statement obtained by replacing RX[Y byQA\BRX[Y in Theorem 1.2 is not true: take V (G) = fs; x; y; ag, E(G) = fsx; xa; ay; ysgand X = fxg, Y = fyg, A = B = fag.3. Note that if we replace A by A nB in Theorem 1.2, the r.h.s. of (1) remains the sameand the l.h.s. does not decrease. So Theorem 1.2 as stated above is not more general thanthe case A \B = ;.4. The original motivation for Theorem 1.1 was a conjecture we learned from the late P.W.Kasteleyn (personal communication, circa 1985), a slightly informal description of which isas follows. Let G = (V;E) be a nite graph, W some subset of V , and ~G = ( ~V ; ~E) a copyof G. For each e 2 E and v 2 V , let ~e and ~v be the corresponding edge and vertex in ~Grespectively. Now we ‘glue’ G and ~G together by identifying w with ~w for w 2 W , and onthis new graph consider any percolation model with p(~e) = p(e) for all e 2 E. The conjectureis then that, for every a; b 2 V , P (a ! b)  P (a ! ~b). There is in fact a slight concreteconnection with Theorem 1.1, in that a special case of the latter says that when jW j = 2,say W = fv;wg, one has P (a  ! bjv 6 ! w)  P (a  ! ~bjv 6 ! w). But we feel thatTheorem 1.1 is more interesting for its own sake, and believe it has potential applications inpercolation theory in general.2 BackgroundWe just recall the two correlation inequalities we will need in Section 3. For more extensivediscussions see [2].2An event A (i.e. a subset of Ω) is called increasing if A 3 !  !0 implies !0 2 A. (Here!  !0 means !e  !0e for all e 2 E). The following correlation inequality is due to Harris[5].Theorem 2.1 For any increasing A;B  Ω,P (AB)  P (A)P (B):Of course this is equivalent to saying that for any increasing A and decreasing B P (AB) P (A)P (B).There are a number of signicant extensions of Harris’ inequality, notably that of Fortuin,Kasteleyn and Ginibre [4]. Our main tool is the considerably more general Ahlswede-Daykin(or \Four Functions") Theorem [1], viz.Theorem 2.2 Let N be a nite set and let P(N) denote the set of all subsets of N Suppose; ; γ;  : P(N)! R+ satisfy(S)(T )  γ(S \ T )(S [ T ) 8S; T  N: (2)ThenP(S)P(S) P γ(S)P (S) (where the sums are over all S  N).3 Proof of Theorem 1.2We assume G is nite. (If G is countably innite, the result follows from the nite case byobvious limit arguments). The proof is by induction on the number of vertices jV j. If jV j = 1,the result is trivial. Suppose it always holds if jV j  n and consider a graph G with n + 1vertices.Set X \ Y = Z. If Z = ; then (1) follows from Harris’ inequality:P (QARX)P (QBRY )  P (QA)P (RX)P (QB)P (RY ) P (QAQB)P (RXRY )= P (QA[BRX\Y )P (RX[Y ):If Z 6= ; we proceed as follows: Set N = fy 62 Z : y  Zg (where y  Z means y isadjacent to at least one vertex of Z). Dene the (random) setS = fy 2 N : there is an open edge from y to Zg:We use S; T for possible values of S and write P (S) for P (S = S) and P (jS) for theconditional distribution given S = S. We may expandP (QARX) =XSP (S)P (QARX jS)(where the sum is over all subsets of N), and similarly for the other terms in (1). Thus if wedene(S) = P (S)P (QARX jS);(S) = P (S)P (QBRY jS);γ(S) = P (S)P (QA[BRX\Y jS);(S) = P (S)P (RX[Y jS);3then (1) becomes X(S)X(S) Xγ(S)X(S);where S runs over the subsets of N . Theorem 2.2 says that to verify this we just need toestablish (2), which, since (as one can easily check) P (S)P (T ) = P (S [ T )P (S \ T ), is thesame asP (QARX jS)P (QBRY jT )  P (QA[BRX\Y jS \ T )P (RX[Y jS [ T ): (3)Let P 0 refer to the percolation model for the graph G0, obtained from G by removing Z,with edge probabilities as in our original percolation model on G. Then it is easy to see thatfor any C;W  V n Z and S  N ,P (QCRW[Z jS) = P 0(QCRW[S): (4)Now we obtain (3) as follows: Let X 0 = X n Z and Y 0 = Y n Z. We haveP (QARX jS)P (QBRY jT ) = P 0(QARX0[S)P 0(QBRY 0[T ) P 0(QA[BR(X0[S)\(Y 0[T ))P 0(R(X0[S)[(Y 0[T )) P 0(QA[BR(S\T ))P 0(R(X0[Y 0)[(S[T ))= P (QA[BRX\Y jS \ T )P (RX[Y jS [ T );where the rst equality follows from applying (4) twice (with W = X 0 and W = Y 0 respec-tively), the rst inequality from the induction hypothesis (which says that (1) holds for G0),the second inequality from (S \T )  (X 0 [S)\ (Y 0[T ), and the second equality from againapplying (4) twice (with W = ; and W = X 0 [ Y 0 respectively). 2References[1] R. Ahlswede and D.E. Daykin, An inequality for the weights of two families of sets, theirunions and intersections, Z. Wahrscheinl. Geb. 43 (1978), 183-185.[2] B. Bollobas, Combinatorics, Cambridge Univ. Pr., Cambridge, 1986.[3] G. Grimmett, Percolation, Springer, New York, 1989.[4] C.M. Fortuin, P.W. Kasteleyn and J. Ginibre, Correlation inequalities on some partiallyordered sets, Comm. Math. Phys. 22 (1971), 89-103.[5] T.E. Harris, A lower bound for the critical probability in a certain percolation process,Proc. Cam. Phil. Soc. 56 (1960), 13-20.4

A correlation inequality for connection events in percolation

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