Berg, J. (Rob) van den

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J. Appl. Proh. 33, 420-426 (1996) Printed in Israel © Applied Probability Trust 1996 A NOTE ON DISJOINT-OCCURRENCE INEQUALITIES FOR MARKED POISSON POINT PROCESSES J. VAN DEN BERG,* CW/ Abstract For (marked) Poisson point processes we give, for increasing events, a new proof of the analog of the BK inequality. In contrast to other proofs, which use weak-convergence arguments. our proof is 'direct' and requires no extra topological conditions on the events. Apart from some well-known properties of Poisson point processes. the proof is self-contained. DISJOINT OCCURRENCE: CONTINUUM PERCOLATION; POISSON POINT PROCESSES AMS 1991 SUBJECT CLASSIF!CATION: PRIMARY 611EI5; 611055; 60K35 1. Introduction The BK inequality for product measures on {0, l }"has become a basic tool in lattice percolation theory (see, e.g., Grimmett (1989), Ch. 2). This inequality says that the probability that two increasing events 'occur disjointly' is smaller than or equal to the product of the two individual probabilities. We will refer to this inequality as the 'standard BK inequality' to distinguish it from generalizations which we will now discuss. Firstly, van den Berg and Kesten (1985) conjectured that the inequality not only holds for increasing events but for all events (see also van den Berg and Fiebig (1987)). This conjecture remained open for about ten years, until it was recently proved by Reimer (1994). Further, for several years continuum percolation models have been the subject of attention (see, e.g., Menshikov and Sidorenko (1987), Roy (1990), Penrose (1991) and Meester and Roy (1994 )). In these models the points are no longer the vertices of a regular lattice, but the points in a stationary point process, often a Poisson point process. This development has led to a need to obtain BK-like inequalities for (marked) Poisson point processes. This need also came from a different but somewhat related field, namely, interacting particle systems. In this field many relevant problems can be formulated as percolation-like problems involving the so-called space-time diagram of the process. Bezuidenhout and Grimmett (1991, Section 2) pointed out that this generalization to Poisson point processes can be done quite easily by general weak-convergence arguments. It seems that these arguments are sufficiently flexible to obtain also some Poisson-point-Received 13 January 1995; revision received 18 April 1995. *Postal address: CW!, Kruislaan 413, 1098 SJ Amsterdam, The Netherlands. 420 Disjoint-occurrence inequalities for marked Poisson point processes 421 process analog of Reimer's extension mentioned above. However, it also seems inevitable in such arguments that an extra topological condition on the events is required (namely, that the boundaries of the events, with respect to a certain topology on the space of all realizations of the point process, have probability 0). Independently of Bezuidenhout and Grimmett, Roy and Sarkar (1992) and Sarkar (1994) also obtained a generalization for Poisson point processes. Their proof is given only for certain connection events (a special class of increasing events; see Remark (iii) in Section 2 below) and requires a topological condition of the same type as mentioned above. In the present paper we give, for the class of increasing events (which is one of the most important classes of events in the fields of application mentioned above), a more direct proof of the inequality, which requires no extra technical condition on the events. 2. Definitions and statement of results Consider a, possibly inhomogeneous, marked Poisson point process on IR'1• Intuitively, a realization of this process is obtained as follows. First generate a realization of the point process. Then assign to each point, independent of the others, a ·mark', which is drawn from the mark space according to a certain distribution (which may depend on the position of the point). A realization w of the process can be (and will be in this paper) represented as a set {(x;, S;): i EN l of pairs, where the x, form a discrete subset of IR'1 (here discrete means that for every element of this subset there exists an i; > 0 such that the distance to every other element of the subset is larger than t:) and each s, is the mark assigned to x,. The set of all realizations is denoted by Q. For a formal construction of the process, see e.g. Daley and Vere-Jones ( 1988). Here we only remark that the sigma-algebra for the process is generated by the collection of all subsets of Q of the form { w: the number of points in V with a mark in M equals n}, where Vis a bounded Borel subset of IR", n a non-negative integer, and M a measurable subset of the mark space. Before we state our theorem we need more definitions and notation. By a 'region' we will always denote a Borel set in IR". The distribution of the process will be denoted by tt. The restriction of a realization w to a region U is denoted by we. More formally (recall that we consider w as a set of pairs), (I) w1 : = { (x, s) : (x, s) E w and x E U}. This leads naturally to definitions of QL' and tlc (the marginal distribution of the marked point process restricted to U). We also define the cylinder event [we] : = { w' : w; = w1 } . We say that w ~ w' if uJ is contained in uJ'. An event A is called increasing if w E A and w~w' implies o/ EA. For an event A and a region U we say that 'A lives on U' if w EA implies [wu] c A. Note that for increasing A this is equivalent to saying that w EA implies Wu E A. Let A and B be increasing events living on a bounded region. Analogously to the 'standard' case (i.e. the case of Bernoulli random variables) introduced by van den Berg and Kesten (1985), we say that A and B 'occur disjointly' if there exist two disjoint 422 J. VAN DEN BERG regions such that by looking only at the first (second) we are convinced that the realization is in A (B). More formally, (2) A 0 B: = {w: 3 disjoint regions Kand L with [oh] CA and [wd CB} (3) ={co: 3 disjoint regions Kand L with Oh EA and w1. E B}, where the equality of (2) and (3) follows easily from the fact that A and Bare increasing. Remarks. (i) Since A and Bare increasing, it is not difficult to see that, in the above definition, we can restrict to sets K and L which are finite unions of (hyper- )cubes with rational coordinates. This guarantees the measurability of A lJ B. (ii) In the remainder of this paper we always work with (3) as the definition of A 0 B. Note that for general (i.e. not necessarily increasing) events (2) and (3) are usually not the same. (iii) Another equivalent definition, explicitly using the definition of w as a set, is A 0 B= {w: 3w' Cw with w' EA and w\(I/ E BJ. Roy and Sarbar have used this form for the special case mentioned in the introduction. In that case each mark is a non-negative real number, interpreted as the (random) radius of a ball centered at the corre-sponding point, and A and B are of the form "there exists a path of overlapping balls from one region to another'. More precisely, A= {w: 3(x 1, r 1),-··, (x,,, r,,) E w with x 1 E V1, x,, E ~Vi and Xy··, x,,_ 1 EU, and , 1 - x, II < r, ~ 1 + r, , i = I , .. ·, n - I } , and B is similar, but with V1 and W1 replaced by U2 and W2 respectively. Here U is a bounded region, and Jli, 11-'1 , V2 and W, are regions contained in li. Our main result is the following theorem, which will be proved in Section 3. Theorem. Let A and B he incrca.1·ing aents /it,ing 011 11 hounded region. IVe luwe (4) p(A 0 B);£p(A)/t(B). We will finish this section by giving some background for Section 3. The proofs by Bezuidenhout and Grimmett and by Roy and Sarkar consist. roughly speaking, of partitioning space in small cells, applying the standard BK inequality to a collection of independent random variables indexed by these cells, and then taking limits as the cell size tends to zero. We too will partition space in cells, but instead of applying the standard BK inequality, we modify the method in one of the proofs which van den Berg and Kesten had obtained for the standard BK inequality (see e.g. van den Berg and Fiebig (1987, Sect. 4.4 and 5)). In this so-called splitting method the event Bis changed step by step and eventually becomes an independent copy of the original B. Moreover, at each step the probability of A 0 B increases (or remains the same). In the Poisson point process case we can no longer prove this directly, but it appears that Disjoint-occurrence inequulities for marked Poisson point process<'.\' 423 the 'total error' over all steps goes to 0 with the cell size, which makes the method still work. 3. Proof of the theorem First we assume that the process is homogeneous (that is, the Poisson process has constant density i. and the mark distribution does not depend on location). The key of the proof is the proposition below, for which we need some more definitions. Let x E lR" and V be a bounded region. Define x + V: = {x + y : y E V}. For a realization w, define by T~'( w) the realization obtained by exchanging the configurations on V and x + V. More precisely, T~ ( w) is defined by (T;(w)L+1 :=((x+y,s):(y,s)E<1)1l: ( T; ( w) )i : = { ( )' - x, s) : ( y, s) E Wn,.}: Finally, define for an event £, T'. (£) :· = { T'..(w) : w E £}. Proposition. Let £ 1 and £ 2 be increasing events living on a bounded region M. Let VC Mand xEIR'1 be such that Mn(x+ V)=0. Then where 1101· I is the size (i.e. number of elements) of w1 • Remark. Note that without the last two terms in the r.h.s of (5) we would immediately have the theorem. Namely, with A and B as in the formulation of the theorem and U a bounded region on which A and B live, take V and M equal to U, £ 1 =A and £2 = B. Then, since A and T; (B) live on disjoint regions, the first term in the r.h.s of (5) is Jl(A )JL( T,t(B)) which, by homogeneity, equals Jl(A )p(B). Proof id' the proposition. For Y. E !21111 let Analogously define Ec{ct.). We need the following lemma. Lemma. (a) If w E £ 1 D £ 2, then Wv E £ 1(w,1,\I') or wr E E2(W.H11·) or lw1· I ;;::; 2. (b) If W1· E E1(wM11·) or w,+1· E T;'(E2(wM11 · )), then uJ E £1 D I".'(£,). The proof of this lemma is just a precise application of the definitions. As to part (a), if w E £ 1 D £2, then there exist disjoint regions Kand L with wK E £ 1 and W1- E £;.. IC additionally, lwv I~ I, then wL n w1· =0 or wK n w1• =0. In the first case we have (taking K' =K\ Vand L' =L\ V) 424 J. VAN DEN BERG (7) (where the last equality above holds because w L n oJ1· = 0, and the result is in £ 2 since w1. E £ 2 and £ 2 lives on M), and (8) where the last assertion is true because OJi.: E £ 1 and £ 1 lives on M. By (7) and (8) we have w1 E £ 1(w 1111 ). Similarly, in the other case, we get w,. E £ 2(0>.111,, ). As to part (b) of the lemma: in the first case we have w1· E £ 1(w1111 · ). Hence there exist K' and L' outside V with (w1111 h· U w, E £1 and (w1111 le E £2. So we have (9) and ( 10) W11111. = T'. (w1rn c) = T'..( (w 1111 )i) E T'. (£2), where. in the first equality of(IO), we used MnL'n V=0 and Mnl'n(x+V)=0. Hence, since ((Mn K') U V) n (lvl n L') = 0, w E £ 1 D T'. (£2) as required. The proof for the second case is similar. This completes the proof of the lemma. We are now ready to prove the proposition. We compare the conditional prob-abilities of E1 [J £, and £1 D T'. (£,). given W1111 · By part (a) or the lemma, the first is at most ( I I ) Further, by part (b) of the lemma, we have the following: if 0 E £ 1(o; 1111 ) or 0 E T'. (£2(w1111 )), then the second conditional probability is l (and hence not smaller than the first). Moreover. again by (b), the second conditional probability is always at least (12) /lr (E1(W1111 ))+/ln1 (T'.(£c(r!J1111·)))-J11 (£1(UJ1111 )) Xfl,+r (T'.(E2(W1111))), and if the above does not hold (i.e. if 0 ef:c E 1(w.1111 ) and 0 ef:c T'. (E2(w1111 ))), this is at least (use also homogeneity) (13) Now the proposition follows immediately. Proofo(thi! thi!orl!m. Without loss of generality (use the scaling properties of Poisson point processes), we may assume that A and B live on the unit cube V=[O, I)". Take x = (2. o ... ·, 0). Fix a positive integer n and partition U in nd cubes S1, .. ·, S,,.r of the form [U1-l)/11.i1/n)x ... x[(ic1-l)/n,i,1/n), with l~i1 , .. ·,i"~n. Define the events B<kJ, k= 0, .. ·, nd - 1 as follows: JJ01 = B and B1k+ 1 l = T~·, + 1( B11' 1), k = O, ... , n" - I. Now apply the proposition n" times (the kth time with £ 1 =A, £ 2 = B11' - 11 , M =VU (x+ (S1 U ··· U S1, _ 1)), and v = s" ). Noting that, for each bounded region V, I i01· I is Poisson distributed Disjoint-occurrence inequalities fur marked Poisson point processes 425 with parameter J. I VI, so that the sum of the last two terms in (5) is at most ZA2 I V 12, we get p(A D B) = 11(A 0 B'- 01) ~ 11(A 0 Bil I)+ 2).21 s, 12 (14) ~ ... ~ p(A 0 B""" 1)+ 2J.2 IS,1 2 + .. · + 21.2 I S,,.11 2• Since B1"'11 = T~'(B) and each IS, I equals l/n'1, we get (15) 2).2 11(A 0 B)~p(A)11(B)+-.1. n This holds for any n, completing the proof of the theorem for the homogeneous case. As to the non-homogeneous case, note that neither side of the inequality in the theorem changes if we change the intensities and mark distributions outside U. So we may assume that, with x as before, for every y E U, i.,= ).x+, and the mark distribution for y is the same as for x + y. The proof now remains practically the same, the only difference being that ). in the last term in the r.h.s of ( 15) is replaced by sup, e 1, )7 . Since this is finite, the argument still works. This completes the proof of the theorem. Remark. An event A is called decreasing if w EA and w ~ w' implies w' EA. In the case of Bernoulli random variables there is complete symmetry between increasing and decreasing events. However, such symmetry is absent in Poisson point processes and we have not been able to adapt the proof of the above theorem for decreasing events. In particular, we do not know a useful analog of part (a) of the lemma in this section. Acknowledgments I thank G. R. Grimmett for enlightening me on the disjoint-occurrence result in his article with C. Bezuidenhout, M. Keane for a useful discussion on measurability matters, and R. Meester for several valuable comments on an earlier draft of this paper. References VAN DEN BERG. J. AND FIEBIG, U. (1987) On a combinatorial conjecture concerning disjoint occurrences of events. Ann Proh. 15. 354-374. VAN DEN BERG. J. AN!> KESTEN, H. (1985) Inequalities with applications to percolation and reliability. J Appl. Proh. 22. 556-569. BEZUIDENHOUT, C. AND GRIMMETT, G. R. 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A note on disjoint-occurrence inequalities for marked Poisson point processes

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