0 (2012) no 0 1ndash22ISSN DOI 101214vVOL-PID

Percolation and convergence properties of graphsrelated to minimal spanning forests

Christian Hirsch Tim Breretondagger Volker Schmidtdagger

Abstract

Lyons Peres and Schramm have shown that minimal spanning forests on randomlyweighted lattices exhibit a critical geometry in the sense that adding or deletingonly a small number of edges results in a radical change of percolation propertiesWe show that these results can be extended to a Euclidean setting by consideringfamilies of stationary super- and subgraphs that approximate the Euclidean minimalspanning forest arbitrarily closely but whose percolation properties differ decisivelyfrom those of the minimal spanning forest Since these families can be seen as gen-eralizations of the relative neighborhood graph and the nearest-neighbor graph re-spectively our results provide a new perspective on known percolation results fromliterature We argue that the rates at which the approximating families converge tothe minimal spanning forest are closely related to geometric characteristics of clus-ters in critical continuum percolation and we show that convergence occurs at apolynomial rate

Keywords Euclidean minimal spanning forest percolation nearest-neighbor graph relativeneighborhood graph rate of convergenceAMS MSC 2010 Primary 60D05 Secondary 82B43Submitted to on FIXME final version accepted on FIXME

1 Introduction

Euclidean minimal spanning forests were introduced by Aldous and Steele in orderto study the first-order asymptotics of certain functionals of the minimal spanning treeon n independent points that are uniformly distributed in the unit cube [minus12 12]d inRd d ge 2 see [3] Subsequently this idea has been successfully refined to derive centrallimit theorems [5] and conduct perturbation analysis [1] Although initially introducedas auxiliary objects for the investigation of large-scale minimal spanning trees mini-mal spanning forests themselves exhibit a rich geometric structure on a global scaleand have therefore attracted further research interest [4 23] In a discrete settingLyons Peres and Schramm [23] showed that for independent edge weights the mini-mal spanning forest exhibits a critical percolation behavior perturbing the forest byremoving only an arbitrarily small proportion of edges results in a graph with no in-finite connected components whereas adding only an arbitrarily small proportion ofedges results in a graph with a unique infinite connected component ndash in fact the en-tire graph is connected Returning to the Euclidean origins of the minimal spanningforest in the present paper we extend the percolation analysis to minimal spanningforests based on point processes in Rd by considering a family of perturbations that are

Mathematisches Institut Ludwig-Maximilians-Universitaumlt Muumlnchen Theresienstraszlige 39 80333Muumlnchen Germany E-mail hirschmathlmude

daggerInstitute of Stochastics Ulm University Helmholtzstr 18 89069 Ulm Germany

Percolation of graphs related to minimal spanning forests

inspired by graphs in computational geometry First this provides a new perspectiveon percolation results for the relative neighborhood graph and the nearest-neighborgraph Second our analysis gives rise to a non-trivial example of the locality of the crit-ical threshold for Bernoulli percolation on stationary random geometric graphs Thirdwe show that the perturbations converge to the minimal spanning forest at polynomialvelocity thereby providing a connection to the order of shortest-path scaling in criticalcontinuum percolation

As perturbations we consider two families of graphs that approximate the Euclideanminimal spanning forest arbitrarily closely but exhibit decisively different percolationbehaviors We consider the family of creek-crossing graphs Gnnge2 (see [20]) thatapproximate the spanning forest from above and introduce a new family of graphsthe minimal-separator graphs Hnnge1 that approximate the minimal spanning for-est from below Both families are constructed from an underlying stationary pointprocess using deterministic connection rules and are closely related to well-studiedgraphs in computational geometry including the relative neighborhood graph [28] andthe nearest-neighbor graph [13] Here both the relative neighborhood graph and thenearest-neighbor graph are geometric graphs on a locally finite subset of the Euclideanspace In the nearest-neighbor graph each node is connected by an edge to the nodethat is closest in Euclidean distance In the relative neighborhood graph two nodes xy are connected by an edge if there does not exist a third node z whose distance to bothx and y is smaller than the distance between x and y ie max|xminus z| |z minus y| lt |xminus y|with | middot | denoting the Euclidean norm

We compare percolation on the Euclidean minimal spanning forest to percolation onthe approximating families of creek-crossing and minimal-separator graphs First weshow that although every connected component is unbounded there is no Bernoullipercolation on the minimal spanning forest constructed on stationary and ergodic pointprocesses satisfying a weak condition on void probabilities In contrast the criticalthresholds for Bernoulli percolation on Poisson-based creek-crossing graphs are strictlysmaller than 1 As special cases our general result implies non-triviality of Bernoullipercolation on the Gabriel graph and the relative neighborhood graph which have beenconsidered separately in literature [9 10] In the Gabriel graph two nodes x y areconnected by an edge if there does not exist a third node z such that |z minus (x + y)2| lt|xminus y|2 Hence the relative neighborhood graph is a sub-graph of the Gabriel graph

Although the creek-crossing graphs do exhibit non-trivial Bernoulli percolation thisbecomes increasingly difficult with growing n in the sense that the critical percolationthresholds tend to 1 the percolation threshold of the minimal spanning forest In par-ticular this result provides a point-process based example for the locality of critical per-colation thresholds In the setting of discrete transitive graphs the problem of findinga non-trivial example for locality with limiting threshold equal to 1 has been advertisedas an open problem [24] Recently Beringer Pete and Timaacuter have derived a localitycriterion for the class of uniformly good unimodular random graphs [8] However theminimum spanning tree does not appear to be uniformly good so that the techniquesdeveloped in [8] do not apply in the present setting Second we show that the expectedcluster sizes of minimal-separator graphs on a Poisson point process are finite This re-sult generalizes and provides a new perspective on the classical result of the absenceof percolation in the nearest-neighbor graph [18] Nevertheless with growing n theexpected cluster sizes tend to infinity thereby reflecting that in the minimal spanningforest every connected component is unbounded

In the final part of the paper we investigate the relationship between the ratesat which the approximating graphs converge to the minimal spanning forest and thefailure of the minimal spanning forest to inherit geometric properties from its approxi-

0 (2012) paper 0Page 222

Percolation of graphs related to minimal spanning forests

mations For instance if the creek-crossing graphs converged to the minimal spanningforest too quickly not only would the minimal spanning forest be connected but it wouldalso have cycles As the minimal spanning forest does not contain cycles with probabil-ity 1 this yields an upper bound on the rate of convergence We argue that the rates ofconvergence are related to the tail behavior of chemical distances and cluster sizes incritical continuum percolation Thus one would expect these rates to be of polynomialorder We show that the convergence rates have a polynomial lower bound in dimension2 and polynomial upper bounds in all dimension

The paper is organized as follows In Section 2 we present our main results on min-imal spanning forests and their approximations see Theorems 25ndash29 In Sections 3 4and 5 we provide proofs of Theorems 25 26 and 27 where we investigate percolationproperties of spanning forests creek-crossing graphs and minimal-separator graphsrespectively Finally in Section 6 we prove Theorems 28 and 29 where we derivepolynomial upper bounds (Section 61) and polynomial lower bounds (Section 62) onthe rates at which the approximating families converge to the minimal spanning forest

2 Definitions and main results

21 Approximating Euclidean minimal spanning forests

On any finite subset ϕ sub Rd we can define a minimal spanning tree as a tree withvertex set ϕ and minimal total edge length There is a unique minimal spanning treeprovided that ϕ is ambiguity free in the sense that there do not exist x1 y1 x2 y2 isin ϕwith |x1 minus y1| = |x2 minus y2| gt 0 and x1 y1 6= x2 y2 see eg [4]

Minimal spanning forests are analogues of minimal spanning trees in cases whereϕ sub Rd is locally finite rather than finite There are a number of equivalent definitionsof minimal spanning trees that can be easily extended to locally finite ϕ sub Rd Howeverin the locally finite case these definitions cease to be equivalent This means that thereis more than one possible definition of a minimal spanning forest Additionally theresulting graphs may not be connected so they are forests rather than trees

Following [23] we consider separately two versions of the minimal spanning forestintroduced in [3] the free minimal spanning forest and the wired minimal spanningforest Both graphs have the same vertex set but their edges may be different Theconnection rule for the free minimal spanning forest is based on the concept of a creek-crossing path

Definition 21 Let ϕ be a locally finite subset of Rd The free minimal spanning foreston ϕ FMSF(ϕ) is a geometric graph with vertex set ϕ and edge set defined by drawingan edge between x y isin ϕ if and only if there is no creek-crossing path connecting x

and y That is there does not exist an integer n ge 2 and pairwise distinct verticesx = x0 x1 xn = y such that maxiisin0nminus1 |xi minus xi+1| le |xminus y|

The connection rule for wired minimal spanning forests is based on the notion ofminimal separators We say that x y forms a minimal separator of ϕ and ψ disjointlocally finite subsets of Rd if x isin ϕ y isin ψ and |xminus y| lt inf(xprimeyprime)isin(ϕtimesψ)(xy) |xprime minus yprime| Inparticular x y is the unique minimizer of distances between ϕ and ψ On the otherhand if ϕ and ψ are both infinite then x y can be the unique minimizer even if theinequality does not hold

Definition 22 Let ϕ be a locally finite subset of Rd The wired minimal spanningforest on ϕ WMSF(ϕ) is a geometric graph with vertex set ϕ and edge set determinedas follows Two points x y isin ϕ are connected by an edge in WMSF(ϕ) if and only if thereexists a finite ψ sub ϕ such that x y forms a minimal separator of ψ and ϕ ψ

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Percolation of graphs related to minimal spanning forests

In general WMSF(ϕ) sub FMSF(ϕ) but the free minimal spanning forest and wiredminimal spanning forest need not coincide

In this paper we consider approximations of both the FMSF and the WMSF Thecreek-crossing graphs Gnnge2 approximate the free minimal spanning forest fromabove in the sense that FMSF(ϕ) =

⋂nge2Gn(ϕ) if ϕ is ambiguity-free The minimal-

separator graphs Hnnge1 approximate the wired minimal spanning forest from belowwith WMSF(ϕ) =

⋃nge1Hn(ϕ)

These graphs are defined using lsquofinitersquo analogues of the connection rules for the freeand wired minimal spanning forests as follows

Definition 23 Let ϕ be a locally finite subset of Rd The creek-crossing graphsGn(ϕ)nge2 are a family of graphs with vertex set ϕ and the connection rule thatx y isin ϕ are connected by an edge in Gn(ϕ) if and only if there do not exist m le n

and x = x0 x1 xm = y isin ϕ such that maxiisin0mminus1 |xi minus xi+1| lt |xminus y|

Figure 1 Creek-crossing graphs Gn(ϕ) for n isin 2 5 10 (from left to right)

Definition 24 Let ϕ be a locally finite subset of Rd The minimal separator graphsHnnge1 are a family of graphs with vertex set ϕ and the connection rule that x y isin ϕare connected by an edge in Hn(ϕ) if there exists ψ sub ϕ with ψ le n such that x yforms a minimal separator of ψ and ϕ ψ where ψ denotes cardinality of the set ψ

As mentioned in the introduction these families of graphs are closely related tographs arising in computational geometry In particular G2(ϕ) is the relative neigh-borhood graph on ϕ and H1(ϕ) is the nearest-neighbor graph on ϕ Figures 1 and 2illustrate the graphs Gn(ϕ) and Hn(ϕ) for a variety of values of n

Figure 2 Minimal-separator graphs Hn(ϕ) for n isin 1 10 50 (from left to right)

22 Percolation

First we consider percolation properties of Gnnge2 and Hnnge1 and analyze howthese properties behave when passing to the limiting objects In general it is unclear

0 (2012) paper 0Page 422

Percolation of graphs related to minimal spanning forests

how global properties behave under local graph limits For instance it was shownin [20] that if X is a homogeneous Poisson point process in Rd then the graphsGn(X)nge2 are as connected regardless of the dimension d In contrast this connect-edness property is not expected to hold for the minimal spanning forest in sufficientlyhigh dimensions [23 Question 68]

In this paper we investigate Bernoulli percolation on Euclidean minimal spanningforests and their approximations Recall that in Bernoulli percolation edges are re-moved independently with a certain fixed probability That is we consider the familyof graphs defined as follows Let G sub Rd denote a stationary random geometric graphwith vertex set given by a stationary point process X We attach to each x isin X an iidsequence Uxiige1 of random variables that are uniformly distributed in [0 1] Nowconsider x y isin X such that x is lexicographically smaller than y We say that the linkx y is p-open if Uxi le p where i is chosen such that in the set X the point y is theith closest point to x Then Gp denotes the graph on X whose edge set consists ofthose pairs of points that are both p-open and form an edge in G Finally we say thatthe graph Gp percolates if there exists an infinite self-avoiding path in Gp The criticalpercolation probability of the graph G is given by

pc(G) = infp isin [0 1] P (Gp percolates) gt 0

Similar to the lattice setting [23] we show that all connected components of WMSF(X)

are infinite under general assumptions on the underlying point process However ifeven an arbitrarily small proportion of edges is removed then all components are finiteThis continues to be true when passing from WMSF(X) to the larger graph FMSF(X)

Theorem 25 Let X be a stationary point process with positive and finite intensity

(i) If P(X cap [minus s2 s2 ]d = empty) isin O(sminus2d) then pc(FMSF(X)) = 1

(ii) If X is ambiguity free then all connected components of WMSF(X) are infinite

Part (ii) of Theorem 25 is a direct consequence of the definition see also [3 Lemma1] For part (i) an immediate adaptation of the arguments in [23 Theorem 12] doesnot seem possible If X is ambiguity-free then part (i) remains valid even withoutassumptions on the void probabilities see [4 Theorem 25 (i)]

From now on we assume that X is a Poisson point process where throughoutthe manuscript a Poisson point process in Rd is always assumed to be homogeneouswith positive and finite intensity In this important special case we have FMSF(X) =

WMSF(X) see [4 Proposition 21] Although free minimal spanning forests do not ad-mit Bernoulli percolation in the sense that pc(FMSF(X)) = 1 the following result showsthat this changes when FMSF(X) is replaced by any of the approximating creek-crossinggraphs

Theorem 26 Let X be a Poisson point process Then

(i) pc(Gn(X)) lt 1 for all n ge 2 and

(ii) limnrarrinfin pc(Gn(X)) = 1

In other words Theorem 26 shows that the critical probability for Bernoulli perco-lation is strictly smaller than 1 in any of the creek-crossing graphs Gn but as n rarr infinthe critical probabilities approach 1

Part (i) of Theorem 26 yields an example of a class of supergraphs whose criticalpercolation probability is strictly less than that of the original graph Lattice modelswith this property are discussed in [16 Sections 32 33] and the references given

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Percolation of graphs related to minimal spanning forests

there Related results in point-process based percolation are given in [14] Theorem 26generalizes the results for Bernoulli percolation on the Gabriel graph obtained in [9]and the results on the relative neighborhood graph announced in [10]

Part (ii) of Theorem 26 provides evidence to the heuristic that for a large class ofgraphs the critical probability for Bernoulli percolation should be local in the sense thatit is continuous with respect to local weak convergence of the underlying graphs Forinstance in the setting of (discrete) transitive graphs this heuristic is made precise bya conjecture of Schramm see [7 Conjecture 12] Schrammrsquos conjecture has so far onlybeen verified for specific classes of graphs such as Cayley graphs of Abelian groups [24]including as a special case the celebrated result of Grimmett and Marstrand [17] Afinite analogue of the locality is shown for expander graphs in [7 Theorem 13] To thebest of the authorsrsquo knowledge parts (i) and (ii) provide the first example of a family oflocally weakly convergent stationary random geometric graphs satisfying pc(Gn) lt 1 forevery n ge 2 but supnge2 pc(Gn) = 1 According to the remark following [24 Conjecture11] it is an open problem whether this is possible for discrete transitive graphs

So far we have seen that adding an arbitrarily small proportion of edges is sufficientto turn the minimal spanning forest into a graph exhibiting non-trivial Bernoulli perco-lation On the other hand removing only a small proportion of edges in the minimalspanning forest immediately destroys all of the infinite connected components Moreprecisely writing CnH(X) for the connected component of Hn(X cup o) at the originwe show that ECnH(X) is finite for every n but tends to infinite as nrarrinfin

Theorem 27 Let X be a Poisson point process Then

(i) ECnH(X) ltinfin for all n ge 1 and

(ii) limnrarrinfinECnH(X) =infin

23 Rates of convergence

The approximating families Gnnge2 and Hnnge1 can get arbitrarily close to thefree and wired minimal spanning forests However we have not yet discussed therates at which they converge The convergence is quantified using the expected totaldifference of degrees for vertices inside the unit cube More precisely

a(n) = E

(x y)isin(X cap [minus 1

2 12 ]d)timesX x y is an edge in Gn(X) but not in FMSF(X)

and

b(n) = E

(x y)isin(X cap [minus 1

2 12 ]d)timesX x y is an edge in WMSF(X) but not in Hn(X)

We note that the rates of convergence are linked to distributional properties of con-nected components in continuum percolation For instance the rate of convergence ofthe creek-crossing graphs Gn is related to the scaling behavior of the chemical distancein continuum percolation where the chemical distance between two points is the mini-mal number of edges in a path connecting them if the points are in the same connectedcomponent and infin otherwise More precisely the link between continuum percolationand Euclidean minimal spanning forests is based on the following observation Recallthat for r gt 0 the Gilbert graph G(ϕ r) on a locally finite vertex set ϕ sub Rd is definedby imposing that x y isin ϕ are connected by an edge if and only if |x minus y| lt r Thenthe pair x y is an edge in Gn(X) and not in FMSF(X) if and only if x and y can beconnected by a path in G(X |x minus y|) but any such path consists of more than n edgesThat is the chemical distance between x and y in G(X |xminus y|) must be finite but largerthan n Likewise the rate of convergence of the minimal-separator graphs Hn depends

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Percolation of graphs related to minimal spanning forests

on the size of the connected component This is because if x y forms an edge inWMSF(X) but not in Hn(X) then the connected components of G(X |xminus y|) containingx and y must be disjoint and both consist of more than n vertices

When X is a Poisson point process and the Euclidean distance between two pointsx y isin X is close to the critical distance in continuum percolation then the tail behaviorof chemical distance between x and y should be of polynomial order provided that thedeviation of |x minus y| from the critical distance is small when compared to the inverse ofthe parameter n Numerical evidence is provided in [11 15] The same should holdfor the tail behavior of the sizes of the connected component containing x and y Thissuggests that a(n) and b(n) should also be of polynomial order We make this rigorous byshowing that a(n) and b(n) lie between polynomial lower and upper bounds Our proofof the lower bound is based on the Russo-Seymour-Welsh (RSW) theorem in continuumpercolation and is therefore only derived in dimension d = 2

Theorem 28 If X is a Poisson point process then

lim supnrarrinfin

minus log mina(n) b(n)log n

le 2d2 + 6d

Theorem 29 If X is a Poisson point process in R2 then

lim infnrarrinfin

minus log maxa(n) b(n)log n

gt 0

3 Proof of Theorem 25

First we note that part (ii) of Theorem 25 follows as in [23] For any finite set ofpoints the minimal separator to the complementary set of the point process is an edgein the wired minimal spanning forest In the rest of this section we prove part (i) ofTheorem 25 ie the absence of Bernoulli percolation

31 Absence of Bernoulli percolation on Euclidean minimal spanning forests

To begin with we provide an auxiliary result on the maximal length of edges of thegraph G2(X) in a bounded sampling window that is used frequently throughout themanuscript Let ms(G) be the length of the longest edge in the geometric graph G

having at least one vertex in the cube Qs(o) = [minuss2 s2]d That is

ms(G) = max |xminus y| x isin Qs(o) x y is an edge in G

If G does not contain any vertices in Qs(o) we put ms(G) = 0

We show that for G2(X) and by extension for FMSF(X) ms(G) grows much slowerthan s with high probability (whp) That is with a probability tending to 1 as srarrinfin

Lemma 31 Let α isin (0 2) and X be a stationary point process Then for srarrinfin

P(ms(G2(X)) gt sα) isin o(sdP(X capQ2sα2(o) = empty))

Proof The idea of proof is to show that long edges can exist only if there are large areasnot containing any points of X First assume that ms(G2(X)) gt sα Then there existx isin X capQs(o) and y isin X such that |xminus y| ge sα and x y is an edge in G2(X) Now putsprime = sds1minusα2e and subdivide Q3s(o) into k(s) = 3d(ssprime)d subcubes Q1 Qk(s) of sidelength sprime Then we define P = x+

radicdsprime yminusx|yminusx| and let Qi denote the subcube containing

P By construction every point P prime isin Qi satisfies max|xminusP prime| |P primeminus y| lt |xminus y| so that

0 (2012) paper 0Page 722

Percolation of graphs related to minimal spanning forests

the definition of G2(X) gives that Qi capX = empty Therefore

P(ms(G2(X)) gt sα) le P(X capQi = empty for some i isin 1 k(s))

lek(s)sumi=1

P(X capQi = empty)

= k(s)P(X capQsprime(o) = empty)

Since k(s) isin o(sd) we conclude the proof

For s gt 0 and ϕ sub Rd locally finite Fs(ϕ) denotes the set of edges x y of G2(ϕ)

having at least one endpoint in Qs(o) We show that under suitable assumptions on thepoint process X the size of Fs(X) grows at most polynomially in s with high probability

Lemma 32 Let X be a stationary point process with positive and finite intensity suchthat P(X capQs(o) = empty) isin O(sminus2d) Then limsrarrinfinP(Fs(X) ge s2d+2) = 0

Proof Lemma 31 shows that

limsrarrinfin

P(Fs(X) le ((X capQ3s(o)))

2)ge limsrarrinfin

P(ms(G2(X)) le s) = 1

and the Markov inequality implies that limsrarrinfinP((X capQ3s(o)) ge sd+1

)= 0

The proof of Theorem 25 is based on the observation that due to the absence ofcycles the number of paths in FMSF(X) starting close to the origin and leaving a largecube centered at the origin grows polynomially in the size of the cube whereas theprobability that any such path is open decays at an exponential rate

Proof of Theorem 25 Let p isin (0 1) be arbitrary and consider Bernoulli bond percola-tion on FMSF(X) In the following we fix a labeling En of the edges of FMSF(X)Let Os denote the family of open self-avoiding paths in FMSF(X) starting in Q1(o) andleaving Qs(o) By stationarity it suffices to show that limsrarrinfinP(Os gt 0) = 0

Let Ts denote the set of all self-avoiding paths of the form Γ = (En1 Enk) inFMSF(X) such that

1 at least one endpoint of En1is contained in Q1(o)

2 precisely one endpoint of Enk is contained in Rd Qs(o)

3 all other endpoints of the edges En1 Enk are contained in Qs(o)

Note that Os consists of all self-avoiding paths in Ts of the form Γ = (En1 Enk) suchthat each edge in Γ is p-open Thus

Os =sum

Γ=(En1Enk )isinTs

kprodj=1

1Enj is p-open

where 1A denotes the indicator of the event A If ms(G2(X)) le s34 and s is sufficientlylarge then every Γ isin Ts consists of at least s18 edges Therefore

P(Os gt 0) le P(ms(G2(X)) ge s34) + P(Ts ge s2d+3)

+ P( sum

ΓisinTs

1ms(G2(X))les341Tsles2d+3

prodnge1EnisinΓ

1En is p-open gt 0)

le P(ms(G2(X)) ge s34) + P(Ts ge s2d+3) + s2d+3ps18

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Percolation of graphs related to minimal spanning forests

By Lemma 31 P(ms(G2(X)) ge s34) tends to 0 as s rarr infin Thus it is sufficient to showthat P(Ts ge s2d+3) tends to 0 as s rarr infin Because FMSF(X) does not contain cyclesTs is bounded from above by the product of (X capQ1(o)) and the number of edges ofFMSF(X) leaving Qs(o) In other words Ts le (X capQ1(o)) middotFs(X) so that

P(Ts(X) ge s2d+3) le P((X capQ1(o)) ge s) + P(Fs(X) ge s2d+2)

By Lemma 32 the right-hand side tends to 0 as srarrinfin

4 Proof of Theorem 26

41 Bernoulli percolation on creek-crossing graphs

First we prove part (i) of Theorem 26 ie we show that if X is a homogeneousPoisson point process in Rd then pc(Gn(X)) lt 1 for every n ge 2 Without loss ofgenerality the intensity of X is assumed to be 1 The idea of proof is to define adiscretized site-percolation model which exhibits finite range of dependence and whosepercolation implies existence of an infinite open path in the graph Gn(X)

Now the proof of Theorem 26 proceeds roughly as follows For z isin Zd we callthe site z open if for all zprime isin plusmne1 plusmned the points q(sz) and q(s(z + zprime)) can beconnected by a p-open path in the graph Gpn(X) where for i isin 1 d we denote by eithe ith standard unit vector in Rd and for x isin Rd we write q(x) for the closest elementof the graph Gn(X) to the point x In order to apply standard results from percolationtheory we need to ensure that this bond percolation process is m-dependent in thesense that events defined on lattice regions of dinfin-distance at least m from one anotherare independent

As an important tool in the proof of Theorem 26 we use the fact that shortest-pathlengths in the graphs Gn(X) n ge 2 grow at most linearly in the Euclidean distanceexcept for events of rapidly decaying probability More precisely we make use of thefollowing strengthening of the standard concept of convergence with high probability

Definition 41 A family of events Assge1 occurs with very high probability (wvhp) ifthere exist c1 c2 gt 0 such that 1minus P(As) le c1exp(minussc2) for all s ge 1

To quantify the growth of shortest-path length for x y isin Rd we write `(x y) forthe minimum Euclidean length amongst all paths in Gn(X) connecting q(x) and q(y)In [19 Theorem 1] a linear growth result is shown under assumptions that are verifiedin [19 Section 31] for creek-crossing graphs on the homogeneous Poisson point pro-cess Hence in the setting of the present paper we have the following linear growthresult

Proposition 42 Let n ge 2 and consider paths in the graph Gn(X) Then there existsu0 ge 1 depending only on d n and the intensity of the Poisson point process X suchthat the events `(o se1) le u0s occur wvhp

Proposition 42 is an extension of earlier growth and shape results for planar graphssee [2 Theorem 1] The proof is based on a renormalization argument showing thatregions of good connectivity dominate a supercritical percolation process with the con-struction of [6] used to extract macroscopic paths of at most linearly growing length

Now we leverage Proposition 42 to prove Theorem 26

Proof of Theorem 26 part (i) The proof has two major parts In the first step we showthat percolation in the discretized site-percolation model implies existence of an infiniteopen path in the graph Gn(X) In the second step we show that the site-percolationprocess is m-dependent This allows us to use a result from [22 Theorem 13] to infer

0 (2012) paper 0Page 922

Percolation of graphs related to minimal spanning forests

percolation We begin by formally defining the approximating site-percolation processFor s gt 0 and p isin (0 1) we say that z isin Zd is (p s)-good if and only if the followingconditions are satisfied

1 X capQradics(sz) 6= empty and (X capQ4u0s(sz)) le sd+1

2 `(sz s(z + zprime)) le u0s for all zprime isin plusmne1 plusmned

3 every edge Em in Gn(X) with Em capQ3u0s(sz) 6= empty is p-open and

4 Gn(X) cap Q3u0s(sz) = Gn((X cap Q4u0s(sz)) cup ψ) cap Q3u0s(sz) for all locally finite ψ subQd Q4u0s(sz)

The first step of the proof is straightforward as conditions 1 2 and 3 imply that if the(p s)-good sites percolate then Bernoulli bond percolation occurs on the graph Gn(X)

at parameter p That is if ziige1 forms an infinite self-avoiding path of (p s)-good sitesthen there exists an infinite open path in Gn(X) connecting the points q(szi)ige1 Inorder to complete the second step the percolation of (p s)-good sites we apply thestandard m-dependent percolation technique from [22 Theorem 13] Observe thatcondition 4 implies that the site-percolation process of (p s)-good sites exhibits finiterange of dependence and additionally the range of dependence can be bounded fromabove by quantities that do not depend on p and s We can choose both s gt 0 andp isin (0 1) sufficiently large so that the probability that conditions 1 2 3 and 4 aresatisfied becomes arbitrarily close to one For condition 1 this follows from elementaryproperties of the Poisson distribution Condition 2 follows from Proposition 42 andcondition 4 follows from [19 Lemma 4] Using conditions 1 and 4 we deduce thatcondition 3 is satisfied with probability at least ps

2d+2

Therefore by first choosing s gt 0

and then p isin (0 1) large the probability of (p s)-goodness can be chosen arbitrarilyclose to 1 so that an application of [22 Theorem 13] completes the proof

42 Locality of the critical probability in Bernoulli percolation

In this section we prove part (ii) of Theorem 26 That is if X is a homogeneousPoisson point process then limnrarrinfin pc(Gn(X)) = 1 For this purpose we fix p isin (0 1)

in the entire section and show that for n = n(p) sufficiently large there is no Bernoullipercolation in Gn(X) at the level p The proof is based on a renormalization argumentFirst we choose a discretization of Rd into cubes such that whp when constructed onthe local configuration in the cube the graph Gn agrees with the minimal spanningforest By a monotonicity argument any p-open path of Gn(X) crossing such a cube isalso a p-open paths in the graph constructed on the local configuration In particularwe obtain a large number of cubes exhibiting long p-open path in the minimal spanningforest This will lead to a contradiction to the behavior identified in Theorem 25

First we show that if the approximation is sufficiently fine then whp we do not seepercolation in the cubes used in the renormalization Additionally whp there are novery long edges crossing such cubes

Lemma 43 Let Cn denote the event that

1 there does not exist a p-open path in Gnd+1(X capQ5n(o)) from partQn(o) to partQ3n(o)

2 |xminusy| le n2 holds for every locally finite ϕ sub QdQ5n(o) and x y isin (XcapQ5n(o))cupϕsuch that x y forms an edge in Gnd+1((X capQ5n(o)) cup ϕ) intersecting Q3n(o)

Then limnrarrinfinP(X capQ5n(o) isin Cn) = 1

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Percolation of graphs related to minimal spanning forests

Proof To begin with we proceed similarly as in Lemma 31 to verify the second re-quirement Subdivide Q5n(o) into subcubes Q1 Q(20d)d of side length n(4d) As-sume that there is a locally finite ϕ sub Qd Q5n(o) and x y isin (X cap Q5n(o)) cup ϕ suchthat x y forms an edge of length at least n2 in G2((X cap Q5n(o)) cup ϕ) intersectingQ3n(o) Let P0 isin Q3n(o) cap [x y] be arbitrary and put z = (x + y)2 Then we define

P = P0 +radicdn

4dzminusP0

|zminusP0| and let Qi denote the subcube containing P By construction ev-

ery point P prime isin Qi satisfies max|x minus P prime| |P prime minus y| lt |x minus y| so that the definition ofG2((X cap Q5n(o)) cup ϕ) gives that Qi cap X = empty Now we conclude as in Lemma 31 Itremains to verify absence of long p-open path We observe that the relation

G(XcapQ5n(o))(X capQ5n(o)) = FMSF(X capQ5n(o))

implies that the probability that Gnd+1(XcapQ5n(o)) and FMSF(XcapQ5n(o)) coincide tendsto 1 as n rarr infin Hence it suffices to show that whp there are no p-open paths inFMSF(XcapQ5n(o)) from partQn(o) to partQ3n(o) This can be achieved using similar argumentsas in the proof of Theorem 25 More precisely since FMSF(X cap Q5n(o)) is a tree thenumber of edges in FMSF(X capQ5n(o)) is given by (X capQ5n(o))minus 1 so that whp thereexist at most 6dnd edges intersecting partQn(o) or partQ3n(o) Again using the tree propertywe conclude that whp the number of paths between partQn(o) and partQ3n(o) is at most62dn2d and that whp each of these paths consists of at least 1

2

radicn hops Therefore the

expected number of p-open paths connecting partQn(o) and partQ3n(o) is at most 62dn2dpradicn2

Since this expression tends to 0 as nrarrinfin we conclude the proof

Now we complete the proof of Theorem 26 using m-dependent percolation theory

Proof of Theorem 26 part (ii) We start by defining a renormalized site percolationprocess of good sites in Zd where a site z isin Zd is good if (X minus nz) cap Q5n(o) isin CnIn particular the process of good sites is a 5-dependent site percolation process and byLemma 43 the probability that a given site is good becomes arbitrarily close to 1 if n ischosen sufficiently large In particular by [22 Theorem 13] the percolation process ofbad sites is stochastically dominated by a subcritical Bernoulli site percolation processeven if we allow bonds of dinfin-distance 1 Finally we establish the connection betweenbad sites and Bernoulli bond percolation on Gnd+1(X) by showing that if the p-openbonds in Gnd+1(X) percolated then so would the process of bad sites If Γ is an infinitep-open path in Gnd+1(X) and Qn(nz1) Qn(nz2) denotes the sequence of n-cubesintersected by Γ then we claim that z1 z2 is a path of bad sites such that thedinfin-distance of successive sites equals 1 This will result in the desired contradiction tothe subcriticality of the bad sites To prove the claim we note that if zi was good then(X minus nzi) cap Q5n(o) isin Cn so that every edge in Γ intersecting Q3n(nzi) is of length atmost n2 Moreover by monotonicity every such edge in Gnd+1(X) is also an edge inGnd+1(X cap Q5n(nzi)) In particular taking a suitable subpath of Γ we obtain a p-openpath in Gnd+1(X cap Q5n(nzi)) connecting partQn(nzi) and partQ3n(nzi) But this is possibleonly if zi is bad

5 Proof of Theorem 27

In this section we prove Theorem 27 That is we show the absence of percolationin minimal-separator graphs We make extensive use of generalized descending chainswhich were introduced in [20] as a modification of the concept of descending chainsconsidered in [12 18]

Definition 51 Let b gt 0 and ϕ be a locally finite subset of Rd A (possibly finite)sequence x1 x2 isin ϕ forms a b-bounded generalized descending chain if there existsan ordered set I = i1 i2 sub 1 2 with the following properties

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Percolation of graphs related to minimal spanning forests

1 |ij+1 minus ij | le 2 for all j ge 0

2 0 lt |xi minus xi+1| le b for all i ge 1

3 |xij+1 minus xij | lt |xijminus1+1 minus xijminus1| for all j ge 2

where we use the convention i0 = 0

Definition 51 is illustrated in Figure 3 where segments corresponding to elementsof I are drawn thicker

le b

le b

Figure 3 Generalized descending chain

The following deterministic result highlights an essential connection between exis-tence of long paths in Hn(ϕ) and occurrence of long generalized descending chains inϕ Informally speaking Lemma 52 allows us to produce long generalized descendingchains from long paths in Hn(ϕ) We recall that ms(Hn(ϕ)) denotes the length of thelongest edge in Hn(ϕ) having at least one vertex in Qs(o)

Lemma 52 Let n ge 1 s gt 64 ϕ sub Rd be a locally finite set containing o and as-sume 128nms(Hn(ϕ)) le s If CnH(ϕ) 6sub Qs(o) then there is an nms(Hn(ϕ))-boundedgeneralized descending chain starting at x isin ϕ capQs(o) and leaving Qs16(x)

Proof Let γ = (x1 xl) be a self-avoiding path in Hn(ϕ) consisting of l ge n + 2 hopsand satisfying x1 = o x2 xlminus1 isin Qs(o) and xl 6isin Qs(o) We say that xi forms a peakin γ if i ge n+ 2 and maxjisiniminusniminus1 |xjminus1 minus xj | le |ximinus1 minus xi| see Figure 4

i2 3 4 5 6

|ximinus1 minus xi|

Figure 4 Peak at i = 6 when n = 3 Vertical axis shows hop lengths

We claim that for every peak xi with i+n lt l there exists an index j isin i+1 i+nsuch that xj also constitutes a peak Let Gprime be the graph obtained by removing theedge ximinus1 xi from G(ϕ |ximinus1minus xi|) and write C(x) for the connected component of Gprime

containing x isin ϕ Since ximinus1 xi forms an edge in Hn(ϕ) there exist x isin ximinus1 xi anda finite ψ sub ϕ such that x isin ψ ψ le n and ximinus1 xi is a minimal separator between ψand ϕ ψ As ximinus1 xi is a minimal separator we obtain that C(x) sub ψ In particular

C(x) le ψ le n (51)

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Percolation of graphs related to minimal spanning forests

This forces x = xi since the contrary would imply

ximinusnminus1 ximinusn ximinus1 isin C(x)

contradicting (51) If none of xi+1 xi+n is a peak then maxjisini+1i+n |xjminusxjminus1| le|xi minus ximinus1| so that xj isin C(xi) for all j isin i i+ n contradicting (51) again Henceif there exists a peak xi with i isin n + 2 j0 then there exists a sequence xi =

xi1 xi2 xik of peaks with i1 lt i2 lt middot middot middot lt ik |ij minus ijminus1| le n and such that |ik minus l| le nwhere j0 = mini isin 1 l xi 6isin Qs4(o) is the index of the first node in the path γ

that does not lie in Qs4(o) In particular xik xikminus1 xi1 xi1minus1 forms an nms(Hn(ϕ))-bounded generalized descending chain that starts in xik and leaves Qs4(xik)

It remains to consider the case where xi does not form a peak for all i isin n +

2 j0 Define f 1 j0 rarr 1 j0 by i 7rarr argmax jisiniminusniminus1 |xj minus xjminus1| fori ge n+2 and i 7rarr 1 for i isin 1 n+1 Due to our assumption that none of xn+2 xj0forms a peak we conclude that |xj minus xjminus1| le |xf(j) minus xf(j)minus1| for all j isin n + 2 j0We set k = infmge1xf(m)(j0) isin Qs16(o) where f (m) denotes the m-fold composition of

f Observe that due to the assumption nms(Hn(ϕ)) le s128 we have f (k)(j0) ge n + 2Therefore xf(k)(j0)minus1 xf(k)(j0) xj0minus1 xj0 forms an nms(Hn(ϕ))-bounded generalizeddescending chain starting in xf(k)(j0)minus1 and leaving Qs16(xf(k)(j0)minus1)

In order to prove the absence of percolation with the help of Lemma 52 it is usefulto consider bounds on the probability for the existence of long generalized descendingchains We make use of the following result from [19 Lemma 5]

Lemma 53 Let X be a homogeneous Poisson point process in Rd For each s gt 1consider the event that there is no s-bounded generalized descending chain in X cup ostarting at o and leaving Q8ds2d+3(o) These events occur wvhp

For the convenience of the reader we provide a brief sketch of proof for Lemma 53referring to [19 Lemma 5] for details We consider the sequence of decreasing segmentlengths embedded in a generalized descending chain and note that the occurrence ofa long descending chain means that for some sub-interval of [0 s] of length sminus2d thereare a large number of consecutive segment lengths falling inside this subinterval Viaappropriate renormalization this event produces a long open path in a suitable subcrit-ical m-dependent percolation process Similar to the proof of Theorem 26 it is thenpossible to conclude via the finite-dependence approach from [22 Theorem 13]

Now we can proceed with the proof of Theorem 27

Proof of Theorem 27 Applying the monotone convergence theorem the second part ofTheorem 27 becomes an immediate consequence to part (ii) of Theorem 25 In orderto prove part (i) of Theorem 27 note that

ECnH(X) =

int infin0

P(CnH(X) gt s)ds

leint infin

0

P((X capQs1(2d)(o)) gt s) + P(CnH(X) 6sub Qs1(2d)(o))ds

where in the last line we use Lemma 52 Hence it suffices to show that the eventCnH(X) sub Qs(o) occurs with very high probability Put A(1)

s = ms(Hn(X cup o)) ges1(4d+8) Then Lemma 31 implies that the complements of the events A

(1)s occur

wvhp Furthermore denote by A(2)s the event that there exists ξ isin X cap Qs(o) and

an ns1(4d+8)-bounded generalized descending chain starting in ξ and leaving Qradics(ξ)

Then Lemma 53 implies that the complements of the events A(2)s occur wvhp The

proof of Theorem 27 is completed by noting that the event CnH(X) 6sub Qs1(2d)(o)implies that at least one of the events A(1)

s or A(2)s occurs

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Percolation of graphs related to minimal spanning forests

6 Proofs of Theorems 28 and 29

In this section we prove Theorems 28 and 29 That is we show that the ratesof convergence of the families Gnnge2 and Hnnge1 to the spanning forests are ofpolynomial order The upper bound is established in Section 61 and the lower boundin Section 62

61 Polynomial upper bounds

The idea for the proof of the upper bound considered in Theorem 28 is to show that ifthe rates of convergence were faster then the minimal spanning forests would inheritcertain geometric properties from their approximations Specifically faster decay ofa(n) would imply that FMSF(X) contains cycles and faster decay of b(n) would implythat WMSF(X) does not percolate As we know that the minimal spanning forests donot possess these geometric properties we are able to obtain upper bounds on the ratesof convergence

The upper bounds for a(n) and for b(n) are established separately in Propositions 62and 63 respectively First we recall from [19 Lemma 6] that there is a close linkbetween the absence of long generalized descending chains in X and the existence ofshort connections in Gn(X)

Lemma 61 Let a gt 1 n ge 1 and ϕ sub Rd be locally finite Furthermore let η ηprime isin ϕ besuch that 4n|ηminus ηprime| le a where η ηprime are contained in different connected components ofGn(ϕ) cap Qa(η) Then there is an n|η minus ηprime|-bounded generalized descending chain in ϕ

starting at η and leaving the cube Qa2(η)

Combining Lemma 61 with the observation that long generalized descending chainsoccur only with a small probability (Lemma 53) we can now deduce an upper boundfor the decay rate of a(n)

Proposition 62 If X is a Poisson point process then

lim supnrarrinfin

minus log a(n)

log nle 2d2 + 6d

Proof For n ge 2 let A(1)n denote the event that there exists z isin Zd with |z|infin = 4n2d+4

and such that |z minus qprime(z)|infin gt n1(2d+5) where qprime(z) denotes the closest point of X to

z Since X is a Poisson point process the complements of the events A(1)n occur whp

Moreover let A(2)n denote the event that there exists x isin X capQn2d+5(o) such that there

is an n1+1(2d+4)-bounded generalized descending chain in X starting at x and leavingthe cube Qn2d+4(x) Then we conclude from Lemma 53 that the complements of the

events A(2)n occur whp If neither A(1)

n nor A(2)n occur then by Lemma 61 the points qprime(z)

and qprime(zprime) can be connected by a path in Gn(X) which is contained in [z zprime]oplusQ3n2d+4(o)

whenever z zprime isin Zd are adjacent sites with |z|infin = |zprime|infin = 4n2d+4 Next put z1 =

4n2d+4(minuse1 minus e2) z2 = 4n2d+4(e1 minus e2) z3 = 4n2d+4(e1 + e2) and z4 = 4n2d+4(minuse1 + e2)We conclude that qprime(z1) and qprime(z3) can be connected by a path in

Gn(X) cap (([z1 z2] cup [z2 z3])oplusQ3n2d+4(o))

and also by a path in

Gn(X) cap (([z1 z4] cup [z4 z3])oplusQ3n2d+4(o))

see Figure 5 Hence inside Qn2d+5(o) there exists a cycle of edges in Gn(X) SinceFMSF(X) does not contain cycles we conclude that there exist x y isin X cap Qn2d+5(o)

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Percolation of graphs related to minimal spanning forests

qprime(z1)

qprime(z3)qprime(z4)

qprime(z2)

3n2d+4

o

Figure 5 A cycle in Gn(X) capQn2d+5(o)

such that x y is an edge in Gn(X) but not in FMSF(X) Let Q1 Qn(2d+5)d

be asubdivision of Qn2d+5(o) into cubes of side length 1 Then

1minus P(A(1)n )minus P(A(2)

n ) le Esum

xisinXcapQn2d+5 (o)

sumyisinX

1xy is an edge in Gn(X) but not in FMSF(X)

=

n(2d+5)dsumj=1

Esum

xisinXcapQj

sumyisinX

1xy is an edge in Gn(X) but not in FMSF(X)

= n(2d+5)da(n)

which completes the proof of Proposition 62

The reader may have noticed that there still is some room to optimize the abovearguments in order to obtain a better exponent for the upper bounds derived in Propo-sition 62 However even after such improvements the exponent seems to be rather faraway from the true exponent in the polynomial rate and the rigorous determination (ifpossible) of its precise value will certainly need more advanced techniques Neverthe-less following the arguments and computations in [15 29] one can provide at least aheuristically motivated conjecture for the true rate in dimension d = 2 see also [11] forrelated numerical results For two points that are precisely at the critical distance ofcontinuum percolation the probability that these two points are connected by an edgein Gn(X) but not in FMSF(X) decays asymptotically as nminusmicro where micro asymp 11056 Thisevent only depends on points of the Poisson point process that are at chemical distanceat most n from the originating two points Thus even if the two originating points arenot precisely at the critical distance but still inside an interval of near-criticality of sizenminus1νt then the scaling should agree with that at criticality where 1νt asymp 06634 Thisleads to the following conjecture

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Percolation of graphs related to minimal spanning forests

Conjecture The true rate at which a(n) decays is given by micro+ 1νt asymp 1769

In order to derive the analogue of Proposition 62 for b(n) we make use of thepercolation result derived in Lemma 52

Proposition 63 If X is a Poisson point process then

lim supnrarrinfin

minus log b(n)

log nle 2d2 + 6d

Proof For n ge 1 put A(1)n = mn2d+5(G2(X)) ge n1(2d+4) Lemma 31 implies that the

complements of the events A(1)n occur whp Furthermore for n ge 1 let A(2)

n denote theevent that there is a n1+1(2d+4)-bounded generalized descending chain in X starting atx isin X and leaving the cube Qn2d+4(x) Again Lemma 53 implies that the complements

of the events A(2)n occur whp Now we apply Lemma 52 with s = 16n2d+4 and to suitably

shifted unit cubes inside Qn(o) Hence if neither A(1)n nor A(2)

n occur then there doesnot exist a path in Hn(X) starting in Qn(o) and leaving Qn2d+5(o) However since eachconnected component of WMSF(X) is unbounded we conclude that if X cap Qn(o) 6= emptythen there exist x y isin X capQn2d+5(o) such that x y forms an edge in WMSF(X) but notin Hn(X) Thus

1minus2sumi=1

P(A(i)n )minus P(X capQn(o) = empty) le E

sumxisinXcapQ

n2d+5 (o)

sumyisinX

1xy is an edge in WMSF(X) Hn(X)

=

n(2d+5)dsumj=1

Esum

xisinXcapQj

sumyisinX

1xy is an edge in WMSF(X) Hn(X)

= n(2d+5)db(n)

where as before Q1 Qn(2d+5)d

denotes a subdivision of Qn2d+5(o) into congruent sub-cubes of side length 1

62 Polynomial lower bounds

We now derive polynomial lower bounds Because the derivation of these boundsrequires more refined percolation results than the derivation of the upper bounds inSection 61 we restrict our attention to the case where X is a unit-intensity Poissonpoint process in the plane We make use of the close relationship between the minimalspanning forest and critical percolation which implies that long-range dependenciescan only occur for edges whose lengths are close to the critical radius of continuumpercolation To make this precise we let

rc = infr gt 0 P(G(X r) percolates) gt 0

denote the critical radius for continuum percolation associated with the Poisson pointprocessX Now we make use of a sophisticated result from two-dimensional continuumpercolation stating that at criticality the laws of the sizes of the occupied and vacantconnected components admit a power law decay For the convenience of the reader westate this result in a form that is most convenient for our purposes Let Eoccs denotethe event that the occupied connected component at the origin of the Boolean modelwith radius rc2 is contained in Qs(o) We also write Evacs for the corresponding eventinvolving the vacant component

Lemma 64 It holds that

lim infsrarrinfin

minus log maxP(Eoccs)P(Evacs)log s

gt 0

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Percolation of graphs related to minimal spanning forests

Lemma 64 can be shown using typical arguments from continuum percolation butas it is not stated explicitly in the standard textbook [25] we include a short proof atthe end of this section

As we take X to be a Poisson point process we are able to use the Slivnyak-Mecketheorem to obtain alternative representations for a(n) and b(n) For the convenience ofthe reader we restate this result and refer the reader eg to [26] for further details Wewrite N for the space of all locally finite subsets of Rd and N with the smallest σ-algebrasuch that the evaluation functions evB N rarr 0 1 ϕ 7rarr (ϕ cap B) are measurablefor every bounded Borel set B sub Rd

Proposition 65 Let X be a unit-intensity Poisson point process and f RdtimesRdtimesNrarr[0infin) be an arbitrary measurable function Then

EsumxyisinXx 6=y

f(x yX) =

intR2d

Ef(x yX cup x y) dx dy

In particular

a(n) = E(x y) isin (X capQ1(o))timesX x y is an edge in Gn(X) FMSF(X)

=

intRdP(o x is an edge in Gn(X cup o x) FMSF(X cup o x)) dx

= 2π

int infin0

rP(o re1 is an edge in Gn(X cup o re1) FMSF(X cup o re1)) dr

and

b(n) = 2π

int infin0

rP(o re1 is an edge in WMSF(X cup o re1) Hn(X cup o re1)) dr

As mentioned in Section 23 the rate of convergence of the creek-crossing graphsis closely related to the tail behavior of the chemical distance between points thatare near to one another in a Euclidean sense We introduce the random variable Rras the chemical distance between the vertices minusre12 and re12 in the graph G(X cupminusre12 re12 r) Then

a(n) = 2π

int infin0

rP(n lt Rr ltinfin) dr (61)

Likewise the convergence rates of the minimal separator graphs are related to the tailbehavior of cluster sizes We introduce the random variable Sr as the minimum of thenumbers of vertices in the connected components containing minusre12 and re12 if minusre12

and re12 are in separate components andinfin otherwise Then

b(n) = 2π

int infin0

rP(n lt Sr ltinfin) dr (62)

In order to derive lower bounds on the decay of a(n) and b(n) we subdivide the domainsof integration in (61) and (62) into several parts and consider them separately We firstgive a general bound on both P(n lt Rr lt infin) and P(n lt Sr lt infin) which we will use inthe case of large r We write Bs(x) for the ball of radius s gt 0 centered at x isin Rd

Lemma 66 Let r ge 0 be arbitrary Then

maxP(2 lt Rr)P(Sr ltinfin) le exp(minusπ4 r2)

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Percolation of graphs related to minimal spanning forests

Proof If one of the events Rr gt 2 or Sr ltinfin occurs then X capBr2(o) = empty Thus

maxP(n lt Rr ltinfin)P(n lt Sr ltinfin) le exp(minusπ4 r2)

Next we deal with the case of sub-critical r writing α for the lower limit in Lemma 64

Lemma 67 It holds that

lim infnrarrinfin

infrlerc minus log maxP(n lt Rr ltinfin)P(n lt Sr ltinfin)log n

ge α

8

Proof We consider the cases involving a(n) and b(n) separately If n lt Rr lt infin thenthere exists a path in G(X cup minusre12 re12 rc) connecting minusre12 and re12 and con-sisting of at least n+ 1 edges In particular if (X capBn14(o)) le nminus 2 then there existsxprime isin ( rc5 Z

2) cap Q5rc(o) such that the connected component of X oplus Brc2(o) at xprime leavesBn14(o) Hence by Lemma 64

P(n lt Rr ltinfin) le 54P((X capBn14(o)) ge nminus 1) + c1nminusα4

for some c1 gt 0 Since the Poisson concentration inequality [26 Lemma 12] shows thatthe probability of the event (X cap Bn14(o)) ge nminus 1 decays to 0 exponentially fast inn we have proven the assertion concerning a(n) Similarly if n lt Sr lt infin then thereexists a connected component of G(X rc) which intersects Q2rc(o) and consists of atleast n vertices Again if (X capBn14(o)) le nminus 2 then there exists xprime isin ( rc5 Z

2)capQ5rc(o)

such that the connected component ofXoplusBrc2(o) at xprime leavesBn14(o) and we concludeas before

In the remaining case r is in the supercritical regime but still not too large

Lemma 68 It holds that

lim infnrarrinfin

infrclerlenα16 minus log maxP(n lt Rr ltinfin)P(n lt Sr ltinfin)log n

ge α

4

Proof As in Lemma 67 we deal with the assertions involving a(n) and b(n) separatelyWe first consider the bound for a(n) Assuming that r lt nα16 and n lt Rr lt infin theconnected component of R2 (X oplus Brc2(o)) containing the origin leaves Bn142(o) if(X capBn14(o)) le nminus 2 Hence by Lemma 64

P(n lt Rr ltinfin) le P((X capBn14(o)) ge nminus 1) + 2minusαnminusα4

Since the events (X cap Bn14(o)) le n minus 2 occur wvhp this proves the first claimFor b(n) we claim that if n lt Sr lt infin then either (X cap Bn14(o)) le n minus 2 or theconnected component of R2 (X oplus Brc2(o)) containing the origin leaves Bn142(o)Once this claim is proven we conclude the proof as we did for a(n) To show the claimwe note that if the connected component of R2 (XoplusBrc2(o)) containing the origin lieswithin Bn142(o) then there is a closed path in G(X rc) that surrounds the origin andis contained in Bn14(o) In particular since (X capBn14(o)) le nminus 2 and the connectedcomponents of the graph G(X cup minusre12 re12 r) containing minusre12 and re12 bothconsist of at least n + 1 points we see that both connected components must intersectthe closed path However this a contradiction since Sr lt infin implies that minusre12 andre12 are contained in different connected components

Using these auxiliary results we now complete the proof of Theorem 29

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Percolation of graphs related to minimal spanning forests

Proof of Theorem 29 Since the proofs for the cases of a(n) and b(n) are essentiallyidentical we only consider a(n) The domain of integration in (61) is decomposed intothree regions (0 rc] [rc n

α16] and [nα16infin) For the first region we haveint rc

0

rP(n lt Rr ltinfin) dr le r2c suprisin[0rc]

P(n lt Rr ltinfin)

By Lemma 67 the right-hand side is of order O(nminusα4) For the second region we haveint nα16

rc

rP(n lt Rr ltinfin) dr le nα8 suprisin[rcnα16]

P(n lt Rr ltinfin)

By Lemma 68 the right-hand side is of order O(nminusα8) Finally by Lemma 66int infinnα16

rP(n lt Rr ltinfin) dr leint infinnα16

rexp(minusπ4 r2)dr = 2πminus1exp(minusπ4n

α8)

which completes the proof

It remains to provide a proof for Lemma 64 As in the classical lattice model [16Theorem 1189] the key to proving the desired polynomial rate is a RSW-theorem Sincecontinuum analogues of the classical theorem were established in [5 27] Lemma 64can be deduced using standard methods from continuum percolation theory [25] Stillfor the convenience of the reader we include a proof For a b gt 0 let Hocc(a b) denotethe event describing the existence of an occupied horizontal crossing of the rectangle[0 a]times [0 b] ie the existence of a connected component of ([0 a]times [0 b])cap(XoplusBrc2(o))

intersecting both 0 times [0 b] and a times [0 b] Furthermore Hvac(a b) denotes the eventdescribing the existence of a vacant horizontal crossing of [0 a]times[0 b] ie the existenceof a connected component of ([0 a]times [0 b]) capR2 (X oplusBrc2(o)) intersecting 0 times [0 b]

and atimes [0 b] Using this notation we now recall the following two RSW-type theoremsfor occupied and vacant percolation derived in [5 Lemma 33] and [27 Theorem 23]respectively Since it is sufficient for our purposes we only consider the case wherethe radius of the Boolean model is equal to rc2

Lemma 69 Let ε isin (0 1363) be arbitrary Then

infsgt0

P(Hocc(s3s))geε

infsprimeisin(45rc2s3minus5rc2)

P (Hocc (3sprime sprime)) ge (Kε6)27

for some constant K gt 0

Lemma 610 Let k ge 1 and δ1 δ2 gt 0 Then there exist constants D(1)(k) andD(2)(k δ1 δ2) for which the following implication holds If `1 `2 gt 2rc are such thatP (Hvac (`1 `2)) ge δ1 and P (Hvac (`2 3`12)) ge δ2 then

P (Hvac (k`1 `2)) ge D(1)(k)D(2)(k δ1 δ2)

We also make use of the following auxiliary result from [5 Lemma 32(i)]

Lemma 611 Let δ ge 17 and ε isin (0 1363) If P(Hvac(s 3s)) ge ε then

inf`isin(rcs3)

P (Hvac(` `+ δ`)) ge ε

Finally we recall an immediate corollary to [5 Theorem 34 and 35]

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Percolation of graphs related to minimal spanning forests

Lemma 612 It holds that

lim supsrarrinfin

maxP(Hocc(3s s))P(Hvac(3s s)) lt 1

Using Lemmas 69 and 610 we now obtain the following standard corollaries whichform the basis for the proof of Lemma 64

Corollary 613 It holds that

lim infnrarrinfin

minP(Hvac(3n+1 3n))P(Hocc(3

n+1 3n)) gt 0

Proof For the vacant part Lemma 612 and P(Hvac(s 3s)) = 1minusP(Hocc(3s s)) allow usto apply Lemma 611 so that P(Hvac(s 8s7)) ge ε0 for all sufficiently large s ge 1 Inparticular applying Lemma 610 with `1 = 7 middot 3n8 `2 = 3n and k = 4 yields that

lim infnrarrinfin

P(Hvac(3n+1 3n)) ge D(1)(4)D(2)(4 ε0 ε0)

For the occupied part using Lemma 612 and P(Hocc(s 3s)) = 1minus P(Hvac(3s s)) we seethat the assumption of Lemma 69 is satisfied Thus

lim infnrarrinfin

P(Hocc(3n+1 3n)) ge (Kε6)27

Using Corollary 613 we now to complete the proof of Lemma 64

Proof of Lemma 64 We only present a proof of the first claim since the second onecan be shown using similar arguments Let g R2 rarr R2 be the rotation by π2 Forevery n ge 1 consider the event An defined as the joint occurrence of vacant horizon-tal crossings in the rectangles Rn and g2(Rn) and of vacant vertical crossings in therectangles g(Rn) and g3(Rn) where Rn = [minus32n+12 32n+12]times [minus32n+12minus32n2] Putε = lim infnrarrinfinP (An) and note that from the Harris inequality (see eg [21 Theorem14]) and Corollary 613 we obtain that

ε ge lim infnrarrinfin

P(Hvac(32n+1 32n))4 gt 0

Furthermore note that For s gt 0 let Aprimes denote the event that there exists a con-nected component of X oplus Brc2(o) intersecting both Q1(o) and R2 Qs(o) and putn(s) = b(log s log 3 minus 1)2c Since the events Anngen0 are independent provided thatn0 ge 1 is sufficiently large we arrive at

P (Aprimes) len(s)prodi=n0

P(Aci ) le (1minus ε)n(s)minusn0minus1 le (1minus ε)minusn0minus1slog(1minusε)(4 log 3)

as asserted

Acknowledgment The authors are grateful to anonymous referees for valuable com-ments and suggestions that helped to improve the presentation of the material

References

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[2] D J Aldous and J Shun Connected spatial networks over random points and a route-lengthstatistic Statistical Science 25 (2010) 275ndash288

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Percolation of graphs related to minimal spanning forests

[3] D J Aldous and J M Steele Asymptotics for Euclidean minimal spanning trees on randompoints Probability Theory and Related Fields 92 (1992) 247ndash258

[4] K S Alexander Percolation and minimal spanning forests in infinite graphs Annals ofProbability 23 (1995) 87ndash104

[5] K S Alexander The RSW theorem for continuum percolation and the CLT for Euclideanminimal spanning trees Annals of Applied Probability 6 (1996) 466ndash494

[6] P Antal and A Pisztora On the chemical distance for supercritical Bernoulli percolationAnnals of Probability 24 (1996) 1036ndash1048

[7] I Benjamini A Nachmias and Y Peres Is the critical percolation probability local Proba-bility Theory and Related Fields 149 (2011) 261ndash269

[8] D Beringer G Pete and A Timaacuter On percolation critical probabilities and unimodularrandom graphs arXiv preprint arXiv 160907043 (2016)

[9] E Bertin J-M Billiot and R Drouilhet Continuum percolation in the Gabriel graph Ad-vances in Applied Probability 34 (2002) 689ndash701

[10] JM Billiot F Corset and E Fontenas Continuum percolation in the relative neighborhoodgraph arXiv preprint arXiv10045292 (2010)

[11] T Brereton C Hirsch V Schmidt and D Kroese A critical exponent for shortest-pathscaling in continuum percolation Journal of Physics A Mathematical and Theoretical 47(2014) 505003

[12] D J Daley and G Last Descending chains the lilypond model and mutual-nearest-neighbour matching Advances in Applied Probability 37 (2005) 604ndash628

[13] D Eppstein M S Paterson and F F Yao On nearest-neighbor graphs Discrete ComputGeom 17 (1997) no 3 263ndash282

[14] M Franceschetti M D Penrose and T Rosoman Strict inequalities of critical values incontinuum percolation Journal of Statistical Physics 142 (2011) 460ndash486

[15] P Grassberger Pair connectedness and shortest-path scaling in critical percolation Journalof Physics A Mathematical and General 32 (1999) 6233ndash6238

[16] G R Grimmett Percolation second ed Springer New York 1999

[17] G R Grimmett and J M Marstrand The supercritical phase of percolation is well behavedProc Roy Soc London Ser A 430 (1990) 439ndash457

[18] O Haumlggstroumlm and R Meester Nearest neighbor and hard sphere models in continuumpercolation Random Structures amp Algorithms 9 (1996) 295ndash315

[19] C Hirsch D Neuhaumluser C Gloaguen and V Schmidt First passage percolation on randomgeometric graphs and an application to shortest-path trees Advances in Applied Probability47 (2015) 328ndash354

[20] C Hirsch D Neuhaumluser and V Schmidt Connectivity of random geometric graphs relatedto minimal spanning forests Advances in Applied Probability 45 (2013) 20ndash36

[21] G Last and M D Penrose Poisson process Fock space representation chaos expansion andcovariance inequalities Probability Theory and Related Fields 150 (2011) 663ndash690

[22] T M Liggett R H Schonmann and A M Stacey Domination by product measures Annalsof Probability 25 (1997) 71ndash95

[23] R Lyons Y Peres and O Schramm Minimal spanning forests Annals of Probability 34(2006) 1665ndash1692

[24] S Martineau and V Tassion Locality of percolation for Abelian Cayley graphs Annals ofProbability 45 (2017) 1247-1277

[25] R Meester and R Roy Continuum percolation Cambridge University Press Cambridge1996

[26] M D Penrose Random Geometric Graphs Oxford University Press Oxford 2003

[27] R Roy The Russo-Seymour-Welsh theorem and the equality of critical densities and theldquodualrdquo critical densities for continuum percolation on R2 Annals of Probability 18 (1990)1563ndash1575

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Percolation of graphs related to minimal spanning forests

[28] G T Toussaint The relative neighbourhood graph of a finite planar set Pattern Recognition12 (1980) 261ndash268

[29] R M Ziff Exact critical exponent for the shortest-path scaling function in percolation Jour-nal of Physics A Mathematical and General 32 (1999) L457ndashL459

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