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The random connection model in high dimensions
STATISTICS& PROBABILrlrY LETTERS ELSEVIER
Statistics & Probability Letters 35 (1997) 145-153
The random connection model in high dimensions Ronald Meester a, ,, Mat_hew D. Penrose b, Anish Sarkar
c
a Department of Mathematics, University of Utrecht, P.O. Box 80.010, 3508 TA Utrecht, Netherlands b Department of Mathematical Sciences, University of Durham, South Road, Durham DH1 3LE, UK c Indian Statistical Institute, 7 S.J.S. Sansanwal Marg, New Delhi 110016, India
Abstract
Consider a continuum percolation model in which each pair of points of a d-dimensional Poisson process of intensity 2 is connected with a probability which is a function g of the distance between them. We show that under a mild regularity condition on g, the critical value of 2, above which an infinite cluster exists a.s., is asymptotic to (fad g(lxl)dx) -~ as d---, cx~. (~) 1997 Elsevier Science B.V. Keywords: Continuum percolation; Critical value; High dimensions
1. Introduction and statement of results
In percolation models, the values of the critical point and the percolation probability are of great physical significance. It is rarely possible to evaluate these quantities exactly. However, simple exact asymptotic expressions are possible in certain limiting regimes, notably the high-dimensional limit (Kesten, 1990) and the "spread-out" limit (Penrose, 1993). In this paper we prove high-dimensional asymptotics for a model of continuum percolation with long-range interactions. The random connection model in Euclidean space R d with connection function g can be described as follows: let X I = {xl,x2 .... } be a homogeneous Poisson point process (PPP) with intensity 2 on R a and g:(0,c<~)---, [0, 1] be a nonincreasing function that is not identically zero. Fix x0 = 0 , the origin, and let X = {x0,xl .... }. Given two points xi and x1 of X (with i > j ) , connect them by an edge with probability g(Ixi-xjl) independently of all other pairs of points and the process X. Let the resulting random graph be denoted G. The cluster o f the origin, C, is the vertex set of the connected component of G which contains the origin. This model has been studied by Penrose (1991), Meester (1995) and others. For a general account, see Meester and Roy (1996). Define the percolation probability 0d(2) by 0d(2) = P(card(C) = c~),
* Corresponding author. 0167-7152/97/$17.00 (~) 1997 Elsevier Science B.V. All rights reserved PI1 S0167-7152(97)00008-4
R. Meester et al. I Statistics & Probability Letters 35 (1997) 145-153
146
where card denotes cardinality. Define the critical intensity by 2~d)(g) = inf{2 > 0: 0d(2) > 0}. Define the integral
Id(g) = fa~ g(Ixl)dx. If Id(g) is finite, then the critical intensity is nontrivial, i.e. 0 < 2 ~ ) ( g ) < c ~ , for d~>2. Our main result generalises a theorem of Penrose (1996), who showed that in the case g ( t ) = l{t~
tdg(t)dt < oe for all d~>O,
(1)
where := denotes definition. In particular, connection functions of the form g(t) = t - r for some r > 0 are not interesting from this point of view. Set ad := #d/#d--1. We require the connection function g to satisfy a further technical condition, namely sup(O~d+l/~d ) < 00.
(2)
d~>l
The significance of these quantities is that if X is a d-dimensional random vector with density function proportional to g(]. ]), then ad =E(]X]), while ad+l/ad =E(IXI2)/E(IXI) 2 We can now state our main result, which says roughly that under the above conditions the cluster at the origin behaves like a simple (Galton-Watson) branching process with offspring distribution that is Poisson with mean # := 2Id(g). Let ~O(#) denote the extinction probability for such a branching process. Theorem 1.
For a nonincreasing function g:(0, o o ) ~ [0, 1] satisfying (1) and (2), we have
).(¢d)(g)Id(g)--+ l
as d--*oc,
(3)
and lim
sup (Od(#/Id(g))-- 1 + ~k(#)) = 0.
d---*c~ #E(0, go)
(4)
Condition (2) does not seem to be very restrictive: first, if g has bounded support, (2) is satisfied. As we already remarked, polynomial decay is excluded because of (1), and all "exponential type" functions satisfy this criterion: if g ( t ) = e x p ( - t/~) for some fl > 0, then #d = F((d + 1)/[3)/[3. Thus, aa+l/~a = F((d+2)/fl)F(d/fl)/ [F((d+l)/fl)] 2 --~ I as d -~ ~z~. Remark. It is not difficult to show that there exists #0 such that 2 ( # - 1)~< 1 -~k(#)~< 6 ( # - 1) for all # E [1, #o]-
Thus, for fixed fll,fl2 with 0 < fll < f12 < # o - 1, using (3) and (4) we can find do such that for all f i e [fll,fl2]
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and all d>~do, we have fl~Od(2(cd)(1 + fl))~<7fl. This comes close to the so-called mean field behaviour of the percolation function in high dimensions and gives some information about the percolation probability. For example, if there is a jump at the critical point, then the jump sizes must tend to zero as the dimension goes to infinity. In view of results of Hara and Slade (1994) for lattice percolation, we expect a stronger result to be true, namely that Od(') is continuous at 2~ ) for d greater than some dl.
2. The branching random walk approach We shall prove our results by comparison with a branching random walk (BRW). Given an integrable function h : R d ~ [0, co), the B R W with intensity function h is a sequence of finite sets Zn C Ed, with Z0 = {0}, defined as follows. Given Zn, for each y E Z, take an independent inhomogeneous PPP with intensity function h ( y - . ) (the "offspring" of that individual) and let the (n+ 1)th generation Zn+l be the superposition of these processes. The sequence (card(Z,), n~>0) forms a simple branching process whose offspring distribution is Poisson with mean fa~ h(x)dx. Given x E Ed and A = {Yl . . . . . Yr} C Ed, define the function r
0x;A(y) --
o(ly-xl)
H (1 -
o(ly- yil)).
i=1
This is the probability that a point at y is connected to a point at x but not to any point of A; in particular, 9x;0(Y) = 9 ( l y - x ] ) • The cluster C may be generated sequentially as follows. Start with the set {0}. Then add the points of X connected to {0}; these points form a PPP with intensity function 290;~('), enumerated as {Yl . . . . . yr} say. Then add the points connected to yl but not to 0; conditioned on {yl . . . . . yr}, these also form a PPP with intensity function 2gy,;{0}, as we shall see in the appendix. Then add the points connected to y2 but not to yl or to 0 (a PPP with intensity function 29yMO, y,}), and so on. For more details, see the appendix. Note that a PPP with intensity function 2gx;.~ can be obtained by starting with a PPP with intensity function 29~;0, then thinning this by removing points independently with probability 1 - 1-IweA( 1 - 0 ( ] ' - w ] )). We now define a procedure which we denote the pruned BR W construction, based on the BRW with intensity 29(-). Let T be the total number of points in all generations of this BRW, and list the points as o l , o 2 . . . . . Or (or as Ol,O)2 .... if T = o ¢ ) , in such a way that each point is preceded in the list by all of its ancestors (aside from this constraint, the choice of order is arbitrary). Consider each of the points in order, and discard point ~oi, along with all its descendants, with probability 1 - I-[ ( 1 - 9(loci-coil)), the product being over all undiscarded coj which occur in the list before the parent of coi. Let Td be the cardinality of the resulting set of undiscarded points. We claim that the pruned BRW construction is equivalent to the sequential construction of C described above, and therefore Td has the same distribution as card(C). In the appendix we shall give a proof of this claim; for now, we assume that we can construct C by running the pruned BRW as described above. Since I'd <. T, and since the underlying branching process has a Poisson offspring distribution with mean ~:=2Id(9), we have the following (cf. Penrose, 1993, Theorem 3): 0d(,~) = p ( r d = ~ ) -,
(5)
Since ~b(p) = 1 for/2 ~< 1, we have the well-known lower bound
2~)(g)Ia(g) >~ 1.
(6)
These estimates (which hold for all d/> 1) show that to prove Theorem 1, it suffices to prove pointwise convergence in (4), i.e. O,t(Ia/Ia(g))-~ 1 -qJ(#) for all #. This will give us (3) because qJ(/~) < 1 for # > 1.
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Also, the uniform convergence in (4) follows from pointwise convergence because ~k(.) is continuous and bounded, and Od is monotone for all d. By (5), it suffices to prove the pointwise convergence for It > 1. For the remainder of the paper we reparametrise the model by fixing It and g, setting 2 =#lid(9), and taking d ~ cx~ (so 2 varies with d; we sometimes write Pu to indicate that It is fixed). Let T u denote the total number of progeny of a simple branching process with Poisson(It) offspring distribution. We first prove that the distribution function of card(C) converges to that of T u at all finite values, a result of some independent interest.
Proposition 1. Let g be any nonincreasing function satisfying (1). For any It > 0 and any finite k >>.1, we have lim Pu(card( C) = k) = P( T u = k ).
d--* c~
(7)
Proposition 1 is a generalisation of Proposition l(b) of Penrose (1996). We shall prove (7) using the following preliminary result.
Lemma 1. (a) I f g is a nonincreasing function of unbounded support, then lim sup fR~g(Ixl)g(ly-xl)dx = 0 .
f.~g(Ixl)dx
d"*C~ yCRd
(8)
(b) I f g is supported by the interval [0, R], and X ( d ) is a random vector with probability density function 9(1" [)lid(g), then IX(d)l in probability as d ~ c ~ , and also lim
sup
P(lx + X(d)[ <~R) = 0.
(9)
d---*c~ xERa: Ixl ~>(3/4)R
Proof. (a) By Cauchy-Schwarz, the numerator in (8) is bounded above by fRa g(Ixl) 2 dx. Let e > 0. By the hypothesis on g, we can (and do) choose r < s with 0 < g(s)<<.o(r)<~e. Since g(t)<~ 1 for all t<~r and g(t)<<,e for t > r, we have faa g(lx[)2 dx<~ndrd +e flxt>rg(lxl)dx, where nd is the volume of the unit ball. Also, since g(t)>>,g(s ) for t<~s, we have
g(Ixl) dx/> 7Zdsag(s). Combining these inequalities, we obtain
g(Ixl) fR g(Ixl)dx
-z0-
(10)
Therefore, the left-hand side of (10) converges to zero, proving (8). The proof of the first assertion in (b) is straightforward, while the proof of the second assertion (9) is similar to that of Lemma 3 of Penrose (1996). We omit the details. []
Proof of Proposition 1. First assume g has unbounded support. In the pruned BRW construction, let Gi be the event that at least one child of ~oi is discarded. Let Y~ be the set of all undiscarded points in {o91..... oi-~). Given ~oi and Y/, the set of discarded children of tni forms a PPP with intensity function uniformly bounded
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by (p/Ia(g))g(Iogi-" l ) ~ - { g ( l o j - " I)" If Z is Poisson with mean 0~, then P(Z > 0)~<~, and so i--I
`1o(IoJ,-xl) ~ o(I~,j -xl)dx
P(GiloJi, Yi) <~ ~
j=l
( i - 1 ) p sup ~ g(lxl)g(ly-xl)dx. Id(g) yEWt ,t This bound is uniform o v e r ((Di, Yi) , and so holds for P(Gi). Thus P(Gi)---+O by Lemma 1. Hence, Pu(Td<~ k < T)-+ 0 for each k, and (7) follows. When g has bounded support, one can prove P(Gi)-~ 0 using part (b) of Lemma 1. The argument is similar to the corresponding proof in Penrose (1996), and we do not give it here. []
3. Proof of Theorem 1
By the remarks following (6), it suffices to show that for any fixed/~ > 1 we have Od(#/Id(g)) ~ 1 -- ~9(Ia). To compare different dimensions, define the dimension-dependent linear projection function L : R d ~ R 2 by L(yl, Y2..... Yd) = --~ (Yl, Y2), O~d
where ~d was defined in the first section. Let K and p be fixed positive constants, to be chosen later. We shall restrict our attention to connections {xi,xj } satisfying xi - xj E Bd, where Bd = Bd(K, p ) C ~d is defined by Ba:=
{
x6•a:
cca/K<~[xl<~aK,~a L
<~p .
A key property of this set is that IL(x)l<<.gp for XEBd. Also, it can be arranged that Bd supports most of the measure with density g(l" 1), uniformly in d, as we now show. Lemma 2. Suppose g satisfies conditions (1) and (2). Given 1 < # l
liminf fs`1 g([x[) dx
#1
d--,o~ f•`1O([xl)dx > --.~
(II)
Proof. Define a random vector X ( d ) on ~d with density given by g(l" I)/Id(g) • Set R d [X(d)l and Od = X(d)/Rd. Then Rd and Od are independent. The ratio of the integrals in (11) reduces to P ( X ( d ) E Bd), which factorises as P(Rd/O~dE [K-1,K])P(~d]L(Od)[ ~
P(K -1 <~Rd/ad <~K) = 1 - P[Rd < O~d/K]--P[Rd > C~dK] >~ 1
eaE(1/Ra) K
E(Ra) ~eK
= 1 - K - l ( ( e a / e a _ l ) + 1).
(12)
By condition (2), this lower bound can be made arbitrarily close to 1, uniformly in d, by taking K to be suitably large.
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To estimate the second factor, note that 6}d is uniform on the surface of the unit sphere in R d. Let N be a standard bivariate normal random vector. By a straightforward modification of the proof of Lemma 4 of Penrose (1995),
OCdL(Od)~ N,
(13)
so for any 6 > 0 we can find p so that l i m i n f d ~ P ( ~ d l L ( 6 ) d ) l ~p)>~ 1--6. This, combined with the estimate (12) on the first factor, finishes the proof. [] Consider the random connection model (RCM) with intensity (#lid(g)) and connection function gd('):= we denote the restricted RCM. The corresponding BRW has intensity function (#/Ia(o))ga, and is denoted the restricted BR W. Its image under L is a two-dimensional BRW, which we denote the projected BRW. Let Znd denote the nth generation of this projected BRW; for A C R 2, let Zff(A) be the number of members of Zff in the set A. A final technical result is a stability property for the distributions of the steps of this BRW.
g(l" [)18,, which
Lemma 3. Let DK := {x C ~d: Ctd/K <~Ix[ <~O~dK}. Let X x ( d ) be a random vector on Dr with density function g([" [)/ fDx g([x[) dx. Then L(XK(d) ) has the same distribution as ~dNd, where ~d is a one-dimensional random
variable taking values in [1/K,K], and Nd o N ~d and Nd are independent.
where N is a standard bivariate normal random vector, and
Proof. Let Ud be a random variable on [eta/K, ctaK] with density function g(t)t d-1 /(f~a/K CtdK g(u)ud-1 du), and let Od be an independent random vector that is uniform on the surface of the unit sphere in Ed. Then X r ( d ) has the same distribution as UdOd. Set ~d = CtdlUd and Nd = ~dL(Od). Then ~d takes values in [1/K,K], and N d ~ N by (13). [] For integer i and j, define Ai, j to be the square in ~2 of side 1, centred at (i,j). The next result says that the projected BRW starting in Ai, j is likely to produce many nearby descendants in finitely many generations, provided the sequence ~d converges in distribution. Lemma 4. Assume ~d ~ ~ as d ~ 0% for some 4. Let 1 < #1 < P, and assume K and p are chosen so that
(11 ) holds, i.e. #ld(ga)/la(g) > #1 for large enough d. Given 6 > O, we can find integers m and No such that for d large enough, P(Zdo(Al,o)>~m) > 1 - 6
(14)
whenever Zod (Ao,o) ) >~m, and P(ZdNo(AI,o)>~m,ZdNo(Ao,1)~m ) > 1 --ff(#l)--6
(15)
whenever Zod(Ao,o)----{0}. Proof. By Lemma 3, the steps of the projected BRW have the same distribution as that of ~aNd conditioned on INdl <.P, with conditional density function denoted ha. This distribution converges weakly to that of I N given INI ~
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Let Z ~ denote the nth generation of the BRW in ~2 with intensity function #lh. By arguing as in Lemmas 2 and 3 of Penrose (1993), one can arrange for (14) and (15) to hold with Zff replaced by Z ~ ; by convergence of the distribution of steps, one can also arrange that for large d, (14) and (15) also hold for Z~d n , and hence for Zft. []
Proof of Theorem 1. It suffices to prove that Od(#/Id(9))~ 1 --@(#) for each # > 1. By (5) and the continuity of @(.), it suffices to prove that for 1 < #1 < # and e > 0, lim inf Od(#/Id(9)) > 1 -- ~b(#1)-- e. d--*oo
(16)
First, assume that 4d =~ 4, sO that we can use Lemma 4. Now (16) can be proved by a similar argument to the proof of Theorem 1 of Penrose (1996), and we sketch it here. The argument is based on a comparison with oriented site percolation on the lattice Z+ × Z+, with site (0,0) open with probability 1 - ~ k ( p l ) - 2 6 and all other sites independently open with probability 1 - 2 6 . By a contour (Peierls) argument, we can (and do) choose 6 > 0 so that there is an infinite open oriented path from (0, 0) with probability exceeding 1 - ~0(#1) - e. Choose K, p, m and No as in Lemma 4, and consider the pruned BRW, or equivalently the sequential construction of the cluster C, for the restricted RCM. By making the list of points of the BRW in an appropriate order, this construction can be made equivalent to the running of a series of restricted BRWs indexed by (i,j) on the lattice Z+ × Z+, taken in the order (0,0), (1,0), (0, 1), (2,0), (1, 1), (0,2), (3,0) . . . . . with each of these BRWs running for at most No generations. The 0th generation of the (i,j)th BRW is the set {0} in the case (i,j) = (0, 0), and is otherwise a subset of size m of the N0th generation of the ( i - 1,j)th or ( i , j - 1 ) t h BRW, with all m points required to be in L-I(Ai, j), if this set exists. We call the (i,j)th BRW "successful" if it is started at all (i.e. there is a suitable 0th generation) and its N0th generation contains at least m points in L-I(Ai+I,j) and also at least m points in L-I(Ai,j+I ). If the different BRWs did not interact, it would follow from Lemma 4 that the (0, 0)th BRW has probability at least 1 - ~b(#1) - t5 of being successful, and all other BRWs probability at least 1 - 6. Next, consider possible interactions between the different BRWs. Since step sizes in the projected BRW are bounded by pK, for I ( i ' , f ) - ( i , j ) l > 2NopK+2 the (il,j')th BRW cannot "interfere" with the (i,j)th BRW, i.e. cause particles to be thrown away. Also, "interference" due to a particular (il,j')th BRW has low probability (for d large) by an argument using Lemma 1 as in the proof of Proposition 1. These observations imply that for ( i , j ) ~ (0,0), the probability that the (i,j)th BRW is successful, given the outcomes of all previous BRWs in the above ordering and given that it starts at all, exceeds 1 - 26 for d large. Now view the indices of the successful BRWs as open sites of Z+ × Z+; a comparison with independent oriented percolation shows that the number of successful BRWs is infinite with probability exceeding 1--~k(#l)--s, and (16) follows. Next we drop the assumption that ~d =~ 4 for some 4. We claim that (16) still holds. Indeed, if it did not there would exist an increasing sequence (d(k), k >i-1) with Od(k)(g/Ia(k)(9))~< 1--~k(#l ) - e for all k. By the tightness of the set of probability measures on [1/K,K], one could then take a subsequence (k') and a random variable 4, such that ~d~k')=~ 4, leading to a contradiction of the case of (16) established above. []
Appendix We give more details here on the sequential construction of the cluster at the origin C. Given a realisation of the point process X and the random graph G, define a sequence of subsets Co C C1 C Cz C • • • of X, and certain subsets ai C Ci, by the following iterative procedure. Initially let Co = {0}, and tr0 = 0. Inductively,
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R. Meester et al. / Statistics & Probability Letters 35 (1997) 145-153
assume Co. . . . . Ci and tro. . . . . ai have been defined, and define
R i = { x E X : {x,y}~_G for all y E a i } , where {x, y} E G means x and y are connected by an edge. Then perform the following two operations: (1) If Ci ~ tri, select Xi+l E Ci\ai and set o'i+ 1 -----tri (2 {Xi+l}. If not, terminate the algorithm and set J = i, the "termination time". When there is a choice, the selection ofxi+l should be determined by Co..... Ci according to some pre-specified rule. (2) Set W/+I : {x ERi: {x, xi} E G}, and set Ci+l = Ci tA Wi+l. J C i, where J : = cxD if the algorithm never terminates. Then Coo C C, and Coo = C if J Define Co~ : = [-Ji=0 is finite. Therefore c a r d ( C ~ ) and card(C) have the same distribution. The following result on conditional distributions shows that it is possible to view the successive sets W1, W2.... as a series o f inhomogeneous Poisson processes, as described informally in the sequential construction of C in Section 2; it then follows that the variable Td given by the pruned B R W construction has the same distribution as card(Co,~), and so has the same distribution as card(C). This was the claim made in Section 2.
Proposition 2. Let k >,O. The conditional joint distribution, given the outcomes of Co, W1,..., 14~, of the point processes Wk+l and Rk+l, is that of independent PPPs with intensities 2g([. - x k + l I) H (1 - g ( l " - xj])) j<~k
and
]-I (1-g(I.-xjl)), j<~k+l
respectively. Given also the outcomes of Wk+l and Rk+l, and given that J > i + 1, the points of Rk+l are independently connected to xk+2 with probability g(I. - xk+21). Proof. We sketch a proof by induction. The result is clear for k = 0, and assume now that it is true for k = 0, 1. . . . . n - I. Then, given W1. . . . . Wn_ 1, the point processes W~ and Rn are independent PPPs with intensities 2 g(l' - xn I) H j ~
Acknowledgements We wish to thank Richard Gill for suggesting the proof in the appendix, which simplifies our original argument considerably.
References H/iggstr6m, O. and R. Meester (1996), Nearest neighbour and hard sphere models in continuum percolation, Random Structures and Algorithms 9, 295-315. Hara, T. and G. Slade (1994), Mean field behavior and the lace expansion, in: G, GrimmeR, ed., Probability Theory and Phase Transitions, Nato ASI series C, Vol. 420 (Kluwer, Dordrecht). Kesten, H. (1991), Asymptotics in high dimensions for percolation, in: G.R. Grimmer and D.J.A. Welsh, eds., Disorder in Physical Systems." a Volume in Honour of J.M. Hammersley (Clarendon Press, Oxford).
I~ Meester et al. / Statistics & Probability Letters 35 (1997.) 145-153 Meester, Meester, Penrose, Penrose, Penrose,
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R. (1995), Equality of critical densities in continuum percolation, J. AppL Probab. 32, 90-104. R. and R. Roy (1996), Continuum Percolation (Cambridge Univ. Press, Cambridge). M.D. (1991), On a continuum percolation model, Adv. Appl. Probab. 23, 536-556. M.D. (1993), On the spread-out limit for bond and continuum percolation, Ann. Appl. Probab. 3, 253-276. M.D. (1996), Continuum percolation and Euclidean minimal spanning trees in high dimensions, Annals App. Prob. 6, 528-544.
Statistics & Probability Letters 35 (1997) 145-153
The random connection model in high dimensions Ronald Meester a, ,, Mat_hew D. Penrose b, Anish Sarkar
c
a Department of Mathematics, University of Utrecht, P.O. Box 80.010, 3508 TA Utrecht, Netherlands b Department of Mathematical Sciences, University of Durham, South Road, Durham DH1 3LE, UK c Indian Statistical Institute, 7 S.J.S. Sansanwal Marg, New Delhi 110016, India
Abstract
Consider a continuum percolation model in which each pair of points of a d-dimensional Poisson process of intensity 2 is connected with a probability which is a function g of the distance between them. We show that under a mild regularity condition on g, the critical value of 2, above which an infinite cluster exists a.s., is asymptotic to (fad g(lxl)dx) -~ as d---, cx~. (~) 1997 Elsevier Science B.V. Keywords: Continuum percolation; Critical value; High dimensions
1. Introduction and statement of results
In percolation models, the values of the critical point and the percolation probability are of great physical significance. It is rarely possible to evaluate these quantities exactly. However, simple exact asymptotic expressions are possible in certain limiting regimes, notably the high-dimensional limit (Kesten, 1990) and the "spread-out" limit (Penrose, 1993). In this paper we prove high-dimensional asymptotics for a model of continuum percolation with long-range interactions. The random connection model in Euclidean space R d with connection function g can be described as follows: let X I = {xl,x2 .... } be a homogeneous Poisson point process (PPP) with intensity 2 on R a and g:(0,c<~)---, [0, 1] be a nonincreasing function that is not identically zero. Fix x0 = 0 , the origin, and let X = {x0,xl .... }. Given two points xi and x1 of X (with i > j ) , connect them by an edge with probability g(Ixi-xjl) independently of all other pairs of points and the process X. Let the resulting random graph be denoted G. The cluster o f the origin, C, is the vertex set of the connected component of G which contains the origin. This model has been studied by Penrose (1991), Meester (1995) and others. For a general account, see Meester and Roy (1996). Define the percolation probability 0d(2) by 0d(2) = P(card(C) = c~),
* Corresponding author. 0167-7152/97/$17.00 (~) 1997 Elsevier Science B.V. All rights reserved PI1 S0167-7152(97)00008-4
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146
where card denotes cardinality. Define the critical intensity by 2~d)(g) = inf{2 > 0: 0d(2) > 0}. Define the integral
Id(g) = fa~ g(Ixl)dx. If Id(g) is finite, then the critical intensity is nontrivial, i.e. 0 < 2 ~ ) ( g ) < c ~ , for d~>2. Our main result generalises a theorem of Penrose (1996), who showed that in the case g ( t ) = l{t~
tdg(t)dt < oe for all d~>O,
(1)
where := denotes definition. In particular, connection functions of the form g(t) = t - r for some r > 0 are not interesting from this point of view. Set ad := #d/#d--1. We require the connection function g to satisfy a further technical condition, namely sup(O~d+l/~d ) < 00.
(2)
d~>l
The significance of these quantities is that if X is a d-dimensional random vector with density function proportional to g(]. ]), then ad =E(]X]), while ad+l/ad =E(IXI2)/E(IXI) 2 We can now state our main result, which says roughly that under the above conditions the cluster at the origin behaves like a simple (Galton-Watson) branching process with offspring distribution that is Poisson with mean # := 2Id(g). Let ~O(#) denote the extinction probability for such a branching process. Theorem 1.
For a nonincreasing function g:(0, o o ) ~ [0, 1] satisfying (1) and (2), we have
).(¢d)(g)Id(g)--+ l
as d--*oc,
(3)
and lim
sup (Od(#/Id(g))-- 1 + ~k(#)) = 0.
d---*c~ #E(0, go)
(4)
Condition (2) does not seem to be very restrictive: first, if g has bounded support, (2) is satisfied. As we already remarked, polynomial decay is excluded because of (1), and all "exponential type" functions satisfy this criterion: if g ( t ) = e x p ( - t/~) for some fl > 0, then #d = F((d + 1)/[3)/[3. Thus, aa+l/~a = F((d+2)/fl)F(d/fl)/ [F((d+l)/fl)] 2 --~ I as d -~ ~z~. Remark. It is not difficult to show that there exists #0 such that 2 ( # - 1)~< 1 -~k(#)~< 6 ( # - 1) for all # E [1, #o]-
Thus, for fixed fll,fl2 with 0 < fll < f12 < # o - 1, using (3) and (4) we can find do such that for all f i e [fll,fl2]
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and all d>~do, we have fl~Od(2(cd)(1 + fl))~<7fl. This comes close to the so-called mean field behaviour of the percolation function in high dimensions and gives some information about the percolation probability. For example, if there is a jump at the critical point, then the jump sizes must tend to zero as the dimension goes to infinity. In view of results of Hara and Slade (1994) for lattice percolation, we expect a stronger result to be true, namely that Od(') is continuous at 2~ ) for d greater than some dl.
2. The branching random walk approach We shall prove our results by comparison with a branching random walk (BRW). Given an integrable function h : R d ~ [0, co), the B R W with intensity function h is a sequence of finite sets Zn C Ed, with Z0 = {0}, defined as follows. Given Zn, for each y E Z, take an independent inhomogeneous PPP with intensity function h ( y - . ) (the "offspring" of that individual) and let the (n+ 1)th generation Zn+l be the superposition of these processes. The sequence (card(Z,), n~>0) forms a simple branching process whose offspring distribution is Poisson with mean fa~ h(x)dx. Given x E Ed and A = {Yl . . . . . Yr} C Ed, define the function r
0x;A(y) --
o(ly-xl)
H (1 -
o(ly- yil)).
i=1
This is the probability that a point at y is connected to a point at x but not to any point of A; in particular, 9x;0(Y) = 9 ( l y - x ] ) • The cluster C may be generated sequentially as follows. Start with the set {0}. Then add the points of X connected to {0}; these points form a PPP with intensity function 290;~('), enumerated as {Yl . . . . . yr} say. Then add the points connected to yl but not to 0; conditioned on {yl . . . . . yr}, these also form a PPP with intensity function 2gy,;{0}, as we shall see in the appendix. Then add the points connected to y2 but not to yl or to 0 (a PPP with intensity function 29yMO, y,}), and so on. For more details, see the appendix. Note that a PPP with intensity function 2gx;.~ can be obtained by starting with a PPP with intensity function 29~;0, then thinning this by removing points independently with probability 1 - 1-IweA( 1 - 0 ( ] ' - w ] )). We now define a procedure which we denote the pruned BR W construction, based on the BRW with intensity 29(-). Let T be the total number of points in all generations of this BRW, and list the points as o l , o 2 . . . . . Or (or as Ol,O)2 .... if T = o ¢ ) , in such a way that each point is preceded in the list by all of its ancestors (aside from this constraint, the choice of order is arbitrary). Consider each of the points in order, and discard point ~oi, along with all its descendants, with probability 1 - I-[ ( 1 - 9(loci-coil)), the product being over all undiscarded coj which occur in the list before the parent of coi. Let Td be the cardinality of the resulting set of undiscarded points. We claim that the pruned BRW construction is equivalent to the sequential construction of C described above, and therefore Td has the same distribution as card(C). In the appendix we shall give a proof of this claim; for now, we assume that we can construct C by running the pruned BRW as described above. Since I'd <. T, and since the underlying branching process has a Poisson offspring distribution with mean ~:=2Id(9), we have the following (cf. Penrose, 1993, Theorem 3): 0d(,~) = p ( r d = ~ ) -,
(5)
Since ~b(p) = 1 for/2 ~< 1, we have the well-known lower bound
2~)(g)Ia(g) >~ 1.
(6)
These estimates (which hold for all d/> 1) show that to prove Theorem 1, it suffices to prove pointwise convergence in (4), i.e. O,t(Ia/Ia(g))-~ 1 -qJ(#) for all #. This will give us (3) because qJ(/~) < 1 for # > 1.
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Also, the uniform convergence in (4) follows from pointwise convergence because ~k(.) is continuous and bounded, and Od is monotone for all d. By (5), it suffices to prove the pointwise convergence for It > 1. For the remainder of the paper we reparametrise the model by fixing It and g, setting 2 =#lid(9), and taking d ~ cx~ (so 2 varies with d; we sometimes write Pu to indicate that It is fixed). Let T u denote the total number of progeny of a simple branching process with Poisson(It) offspring distribution. We first prove that the distribution function of card(C) converges to that of T u at all finite values, a result of some independent interest.
Proposition 1. Let g be any nonincreasing function satisfying (1). For any It > 0 and any finite k >>.1, we have lim Pu(card( C) = k) = P( T u = k ).
d--* c~
(7)
Proposition 1 is a generalisation of Proposition l(b) of Penrose (1996). We shall prove (7) using the following preliminary result.
Lemma 1. (a) I f g is a nonincreasing function of unbounded support, then lim sup fR~g(Ixl)g(ly-xl)dx = 0 .
f.~g(Ixl)dx
d"*C~ yCRd
(8)
(b) I f g is supported by the interval [0, R], and X ( d ) is a random vector with probability density function 9(1" [)lid(g), then IX(d)l in probability as d ~ c ~ , and also lim
sup
P(lx + X(d)[ <~R) = 0.
(9)
d---*c~ xERa: Ixl ~>(3/4)R
Proof. (a) By Cauchy-Schwarz, the numerator in (8) is bounded above by fRa g(Ixl) 2 dx. Let e > 0. By the hypothesis on g, we can (and do) choose r < s with 0 < g(s)<<.o(r)<~e. Since g(t)<~ 1 for all t<~r and g(t)<<,e for t > r, we have faa g(lx[)2 dx<~ndrd +e flxt>rg(lxl)dx, where nd is the volume of the unit ball. Also, since g(t)>>,g(s ) for t<~s, we have
g(Ixl) dx/> 7Zdsag(s). Combining these inequalities, we obtain
g(Ixl) fR g(Ixl)dx
-z0-
(10)
Therefore, the left-hand side of (10) converges to zero, proving (8). The proof of the first assertion in (b) is straightforward, while the proof of the second assertion (9) is similar to that of Lemma 3 of Penrose (1996). We omit the details. []
Proof of Proposition 1. First assume g has unbounded support. In the pruned BRW construction, let Gi be the event that at least one child of ~oi is discarded. Let Y~ be the set of all undiscarded points in {o91..... oi-~). Given ~oi and Y/, the set of discarded children of tni forms a PPP with intensity function uniformly bounded
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by (p/Ia(g))g(Iogi-" l ) ~ - { g ( l o j - " I)" If Z is Poisson with mean 0~, then P(Z > 0)~<~, and so i--I
`1o(IoJ,-xl) ~ o(I~,j -xl)dx
P(GiloJi, Yi) <~ ~
j=l
( i - 1 ) p sup ~ g(lxl)g(ly-xl)dx. Id(g) yEWt ,t This bound is uniform o v e r ((Di, Yi) , and so holds for P(Gi). Thus P(Gi)---+O by Lemma 1. Hence, Pu(Td<~ k < T)-+ 0 for each k, and (7) follows. When g has bounded support, one can prove P(Gi)-~ 0 using part (b) of Lemma 1. The argument is similar to the corresponding proof in Penrose (1996), and we do not give it here. []
3. Proof of Theorem 1
By the remarks following (6), it suffices to show that for any fixed/~ > 1 we have Od(#/Id(g)) ~ 1 -- ~9(Ia). To compare different dimensions, define the dimension-dependent linear projection function L : R d ~ R 2 by L(yl, Y2..... Yd) = --~ (Yl, Y2), O~d
where ~d was defined in the first section. Let K and p be fixed positive constants, to be chosen later. We shall restrict our attention to connections {xi,xj } satisfying xi - xj E Bd, where Bd = Bd(K, p ) C ~d is defined by Ba:=
{
x6•a:
cca/K<~[xl<~aK,~a L
<~p .
A key property of this set is that IL(x)l<<.gp for XEBd. Also, it can be arranged that Bd supports most of the measure with density g(l" 1), uniformly in d, as we now show. Lemma 2. Suppose g satisfies conditions (1) and (2). Given 1 < # l
liminf fs`1 g([x[) dx
#1
d--,o~ f•`1O([xl)dx > --.~
(II)
Proof. Define a random vector X ( d ) on ~d with density given by g(l" I)/Id(g) • Set R d [X(d)l and Od = X(d)/Rd. Then Rd and Od are independent. The ratio of the integrals in (11) reduces to P ( X ( d ) E Bd), which factorises as P(Rd/O~dE [K-1,K])P(~d]L(Od)[ ~
P(K -1 <~Rd/ad <~K) = 1 - P[Rd < O~d/K]--P[Rd > C~dK] >~ 1
eaE(1/Ra) K
E(Ra) ~eK
= 1 - K - l ( ( e a / e a _ l ) + 1).
(12)
By condition (2), this lower bound can be made arbitrarily close to 1, uniformly in d, by taking K to be suitably large.
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To estimate the second factor, note that 6}d is uniform on the surface of the unit sphere in R d. Let N be a standard bivariate normal random vector. By a straightforward modification of the proof of Lemma 4 of Penrose (1995),
OCdL(Od)~ N,
(13)
so for any 6 > 0 we can find p so that l i m i n f d ~ P ( ~ d l L ( 6 ) d ) l ~p)>~ 1--6. This, combined with the estimate (12) on the first factor, finishes the proof. [] Consider the random connection model (RCM) with intensity (#lid(g)) and connection function gd('):= we denote the restricted RCM. The corresponding BRW has intensity function (#/Ia(o))ga, and is denoted the restricted BR W. Its image under L is a two-dimensional BRW, which we denote the projected BRW. Let Znd denote the nth generation of this projected BRW; for A C R 2, let Zff(A) be the number of members of Zff in the set A. A final technical result is a stability property for the distributions of the steps of this BRW.
g(l" [)18,, which
Lemma 3. Let DK := {x C ~d: Ctd/K <~Ix[ <~O~dK}. Let X x ( d ) be a random vector on Dr with density function g([" [)/ fDx g([x[) dx. Then L(XK(d) ) has the same distribution as ~dNd, where ~d is a one-dimensional random
variable taking values in [1/K,K], and Nd o N ~d and Nd are independent.
where N is a standard bivariate normal random vector, and
Proof. Let Ud be a random variable on [eta/K, ctaK] with density function g(t)t d-1 /(f~a/K CtdK g(u)ud-1 du), and let Od be an independent random vector that is uniform on the surface of the unit sphere in Ed. Then X r ( d ) has the same distribution as UdOd. Set ~d = CtdlUd and Nd = ~dL(Od). Then ~d takes values in [1/K,K], and N d ~ N by (13). [] For integer i and j, define Ai, j to be the square in ~2 of side 1, centred at (i,j). The next result says that the projected BRW starting in Ai, j is likely to produce many nearby descendants in finitely many generations, provided the sequence ~d converges in distribution. Lemma 4. Assume ~d ~ ~ as d ~ 0% for some 4. Let 1 < #1 < P, and assume K and p are chosen so that
(11 ) holds, i.e. #ld(ga)/la(g) > #1 for large enough d. Given 6 > O, we can find integers m and No such that for d large enough, P(Zdo(Al,o)>~m) > 1 - 6
(14)
whenever Zod (Ao,o) ) >~m, and P(ZdNo(AI,o)>~m,ZdNo(Ao,1)~m ) > 1 --ff(#l)--6
(15)
whenever Zod(Ao,o)----{0}. Proof. By Lemma 3, the steps of the projected BRW have the same distribution as that of ~aNd conditioned on INdl <.P, with conditional density function denoted ha. This distribution converges weakly to that of I N given INI ~
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151
Let Z ~ denote the nth generation of the BRW in ~2 with intensity function #lh. By arguing as in Lemmas 2 and 3 of Penrose (1993), one can arrange for (14) and (15) to hold with Zff replaced by Z ~ ; by convergence of the distribution of steps, one can also arrange that for large d, (14) and (15) also hold for Z~d n , and hence for Zft. []
Proof of Theorem 1. It suffices to prove that Od(#/Id(9))~ 1 --@(#) for each # > 1. By (5) and the continuity of @(.), it suffices to prove that for 1 < #1 < # and e > 0, lim inf Od(#/Id(9)) > 1 -- ~b(#1)-- e. d--*oo
(16)
First, assume that 4d =~ 4, sO that we can use Lemma 4. Now (16) can be proved by a similar argument to the proof of Theorem 1 of Penrose (1996), and we sketch it here. The argument is based on a comparison with oriented site percolation on the lattice Z+ × Z+, with site (0,0) open with probability 1 - ~ k ( p l ) - 2 6 and all other sites independently open with probability 1 - 2 6 . By a contour (Peierls) argument, we can (and do) choose 6 > 0 so that there is an infinite open oriented path from (0, 0) with probability exceeding 1 - ~0(#1) - e. Choose K, p, m and No as in Lemma 4, and consider the pruned BRW, or equivalently the sequential construction of the cluster C, for the restricted RCM. By making the list of points of the BRW in an appropriate order, this construction can be made equivalent to the running of a series of restricted BRWs indexed by (i,j) on the lattice Z+ × Z+, taken in the order (0,0), (1,0), (0, 1), (2,0), (1, 1), (0,2), (3,0) . . . . . with each of these BRWs running for at most No generations. The 0th generation of the (i,j)th BRW is the set {0} in the case (i,j) = (0, 0), and is otherwise a subset of size m of the N0th generation of the ( i - 1,j)th or ( i , j - 1 ) t h BRW, with all m points required to be in L-I(Ai, j), if this set exists. We call the (i,j)th BRW "successful" if it is started at all (i.e. there is a suitable 0th generation) and its N0th generation contains at least m points in L-I(Ai+I,j) and also at least m points in L-I(Ai,j+I ). If the different BRWs did not interact, it would follow from Lemma 4 that the (0, 0)th BRW has probability at least 1 - ~b(#1) - t5 of being successful, and all other BRWs probability at least 1 - 6. Next, consider possible interactions between the different BRWs. Since step sizes in the projected BRW are bounded by pK, for I ( i ' , f ) - ( i , j ) l > 2NopK+2 the (il,j')th BRW cannot "interfere" with the (i,j)th BRW, i.e. cause particles to be thrown away. Also, "interference" due to a particular (il,j')th BRW has low probability (for d large) by an argument using Lemma 1 as in the proof of Proposition 1. These observations imply that for ( i , j ) ~ (0,0), the probability that the (i,j)th BRW is successful, given the outcomes of all previous BRWs in the above ordering and given that it starts at all, exceeds 1 - 26 for d large. Now view the indices of the successful BRWs as open sites of Z+ × Z+; a comparison with independent oriented percolation shows that the number of successful BRWs is infinite with probability exceeding 1--~k(#l)--s, and (16) follows. Next we drop the assumption that ~d =~ 4 for some 4. We claim that (16) still holds. Indeed, if it did not there would exist an increasing sequence (d(k), k >i-1) with Od(k)(g/Ia(k)(9))~< 1--~k(#l ) - e for all k. By the tightness of the set of probability measures on [1/K,K], one could then take a subsequence (k') and a random variable 4, such that ~d~k')=~ 4, leading to a contradiction of the case of (16) established above. []
Appendix We give more details here on the sequential construction of the cluster at the origin C. Given a realisation of the point process X and the random graph G, define a sequence of subsets Co C C1 C Cz C • • • of X, and certain subsets ai C Ci, by the following iterative procedure. Initially let Co = {0}, and tr0 = 0. Inductively,
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assume Co. . . . . Ci and tro. . . . . ai have been defined, and define
R i = { x E X : {x,y}~_G for all y E a i } , where {x, y} E G means x and y are connected by an edge. Then perform the following two operations: (1) If Ci ~ tri, select Xi+l E Ci\ai and set o'i+ 1 -----tri (2 {Xi+l}. If not, terminate the algorithm and set J = i, the "termination time". When there is a choice, the selection ofxi+l should be determined by Co..... Ci according to some pre-specified rule. (2) Set W/+I : {x ERi: {x, xi} E G}, and set Ci+l = Ci tA Wi+l. J C i, where J : = cxD if the algorithm never terminates. Then Coo C C, and Coo = C if J Define Co~ : = [-Ji=0 is finite. Therefore c a r d ( C ~ ) and card(C) have the same distribution. The following result on conditional distributions shows that it is possible to view the successive sets W1, W2.... as a series o f inhomogeneous Poisson processes, as described informally in the sequential construction of C in Section 2; it then follows that the variable Td given by the pruned B R W construction has the same distribution as card(Co,~), and so has the same distribution as card(C). This was the claim made in Section 2.
Proposition 2. Let k >,O. The conditional joint distribution, given the outcomes of Co, W1,..., 14~, of the point processes Wk+l and Rk+l, is that of independent PPPs with intensities 2g([. - x k + l I) H (1 - g ( l " - xj])) j<~k
and
]-I (1-g(I.-xjl)), j<~k+l
respectively. Given also the outcomes of Wk+l and Rk+l, and given that J > i + 1, the points of Rk+l are independently connected to xk+2 with probability g(I. - xk+21). Proof. We sketch a proof by induction. The result is clear for k = 0, and assume now that it is true for k = 0, 1. . . . . n - I. Then, given W1. . . . . Wn_ 1, the point processes W~ and Rn are independent PPPs with intensities 2 g(l' - xn I) H j ~
Acknowledgements We wish to thank Richard Gill for suggesting the proof in the appendix, which simplifies our original argument considerably.
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