THE CENTRAL LIMIT THEOREM FOR WEIGHTED MINIMAL SPANNING TREES ON RANDOM POINTS

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1 The Aals of Applied Probability 996, Vol. 6, No. 2, THE CENTRAL LIMIT THEOREM FOR WEIGHTED MINIMAL SPANNING TREES ON RANDOM POINTS By Harry Keste ad Sugchul Lee Corell Uiversity ad Natioal Uiversity of Sigapore Let X i i < be i.i.d. with uiform distributio o 0 d ad let M X X α be mi e T e α T a spaig tree o X X. The we show that for α > 0, M X X α EM X X α d 2α /2d N 0 σα 2 d i distributio for some σ 2 α d > 0.. Itroductio. Let X X be a fiite subset of d, d 2. For give α > 0, a miimal spaig tree (MST) o X X with weight fuctio ψ α y = y α is a coected graph T such that M X X = M X X α = e α e T { } = mi e α T a spaig tree o X 0 X e T where e = X i X j is the Euclidea legth of the edge e = X i X j. Whe X i i < are i.i.d. with commo distributio µ, which has compact support i d, d 2, Steele (988) showed that for 0 < α < d, (.) d α /d M X X α c α d f x d α /d dx a.s. d where f is the desity of the absolutely cotiuous part of µ ad c α d is a strictly positive but fiite costat which depeds oly o the power α ad the dimesio d. Moreover, if X i i < are i.i.d. with uiform distributio o 0 d, the Aldous ad Steele (992) showed that (.2) M X i X d c d d i L 2 I this paper we prove a cetral limit theorem for miimal spaig trees o uiformly distributed poits i 0 d. Ramey (982) already demostrated oe approach to this problem. He argued that a certai property of cotiuum percolatio would imply a cetral limit theorem for miimal spaig trees for d = 2, α =. However, he did ot prove this property of cotiuum percolatio, ad oly very recetly did Alexader (996) prove a cetral limit theorem by Received February 995; revised September 995. AMS 99 subject classificatios. 60D05, 60F05. Key words ad phrases. Miimal spaig tree, cetral limit theorem. 495

2 496 H. KESTEN AND S. LEE Ramey s approach for (the Poissoized versio of) miimal spaig trees for d = 2, α =. Here we use a completely differet approach to the problem. We represet M X X α EM X X α as the sum of martigale differeces [see (4.2) below] ad apply Lévy s martigale cetral limit theorem [see Lévy (937), Theorem 67.2]. I this way the proof of the cetral limit theorem for miimal spaig trees is reduced to a kid of weak law of large umbers estimate for certai coditioal variaces. Eve though a weak law of large umbers is much easier to obtai, i geeral, tha a cetral limit theorem, it still requires some idepedece. The required idepedece will be obtaied from a (determiistic) mootoicity property for miimal spaig trees which allows us to approximate the coditioal variaces by quatities which deped oly locally o the X i. This takes a surprisig amout of work. This is somewhat aoyig because the mootoicity ad the approximatio which we prove i Propositios 3 ad 4 seems much stroger tha eeded for the weak law of large umbers statemet (4.9). Aother drawback of our approach is that it is ot quatitative. Further ideas are eeded to obtai a error estimate i our cetral limit theorem. The ext theorem is our pricipal result. Theorem. Let X i i be i.i.d. with uiform distributio o 0 d d 2. The for ay give α > 0 (.3) M X X α EM X X α d 2α /2d N 0 σ 2 α d for some σ 2 α d > 0. It is more atural for the proof to rescale the variables, so that the desity of the poit set X X remais costat. I our proof we shall therefore work with i.i.d. poits which are uiform i 0 /d d. This also allows us to work with more geeral weight fuctios tha ψ α y = y α. Theorem is the a simple rescaled versio of the followig result. Theorem 2. that (.4) Let ψ: 0 0 be a strictly icreasig fuctio such log ψ y 0 y as y Let X X 2 X be i.i.d. ad uiform o 0 /d d ad defie (.5) M X X = M X X ψ { = mi ψ e : T a spaig tree o X e T X }

3 The (.6) M X for some σ 2 ψ d > 0. LIMIT THEOREM FOR SPANNING TREES 497 X ψ EM X X ψ N 0 σψ 2 d /2 To avoid a few techicalities, oe step i the proof (Lemma 0) will be carried out oly for ψ = ψ α. Details for geeral strictly icreasig ψ are available from the authors. A byproduct of our proof cocers a liearity property of α/d EM X, X α. Alexader (994) ad Redmod ad Yukich (994) proved i the case α = with i.i.d. uiform X i o 0 d that for a suitable costat 0 < c d <, (.7) EM X X c d d /d = O d 2 /d The followig is a corollary to the proof of Lemma 3. Corollary. Uder the hypotheses of Theorem, there exists for each α > 0 a costat ρ α d > 0 such that (.8) ad (.9) lim α/d E M X X α = ρ α d lim + α/d E M X X + α α/d E M X X α = ρ α d Remark. We would like to metio here that Jaso (995) has prove a cetral limit theorem for a miimal spaig tree o a large umber of poits X X uder the assumptio that all the distaces betwee all the pairs of poits are i.i.d. uiform o 0. This setup is sometimes called the complete graph case, whereas our situatio is described as the Euclidea case. I Sectio 2 we collect all determiistic properties of MST which we eed. I particular we prove the mootoicity of D [see (2.5) for the defiitio of D ]. I Sectio 3 we estimate various momets of D. I Sectio 4 we prove Theorem 2 i a Poissoized versio first ad the the origial versio of Theorem 2. I this paper there are lots of strictly positive but fiite costats whose specific values are ot of iterest to us. We deote them by C i, C q or D d. For ay radom variable X ad probability distributio µ, we write µ X for the expectatio of X with respect to µ. 2. Mootoicity of D A B. I this sectio we study two algorithms for a MST: Kruskal s greedy algorithm ad the add ad delete algorithm. Usig these algorithms we establish the mootoicity of D.

4 498 H. KESTEN AND S. LEE Kruskal s greedy algorithm. Let G = V E w be a coected weighted graph with vertex set V, edge set E ad weight fuctio w: E 0. Assume that the cardialities V, E of V, E, respectively, are fiite. A miimal spaig tree o G w is a tree T with vertex set V (which makes it a spaig tree) ad edge set cotaied i E (which we express by T E for brevity) ad such that w T = { } w e = mi w e : T E T a spaig tree e T e T A miimal spaig tree T o G ca be costructed by the followig greedy algorithm due to Kruskal (956). Step. Let T 0 = φ ad E 0 = E \ e E: e is a circuit. Step 2. Oce T i ad E i have bee determied, choose a edge e i+ E i such that w e i+ = mi e Ei w e ad costruct T i+ by addig the edge e i+ to T i, that is, take T i+ = T i e i+. The take E i+ = e E i \ e i+ : T i+ e does ot cotai a circuit If there exists o such edge e i+, that is, if E i = φ, the let T = T i ad stop. Step 3. Replace i by i +. Retur to Step 2. Propositio. Let G = V E w be a coected weighted graph with V <, E < ad let T E be a tree obtaied by Kruskal s greedy algorithm. The T is a MST o G w. See Theorem 3A of Chartrad ad Lesiak (986) for the proof. Add ad delete algorithm. Let G = V E w ad G = V E w be two coected weighted graphs with V <, E < ad such that V = V or V = V v for a sigle vertex v / V ad E = E e e for a fiite umber of edges e i / E. Note that the coectedess of G requires that if v is preset, at least oe edge e i coects v to some v V. We assume that w e = w e for all e E so that w is a extesio of w. We shall therefore drop the prime from the w i the sequel ad deote the weight fuctio o G also by w. Now let T be a MST o G ad assume that we wat to costruct a MST T o G. We may, of course, costruct a MST T by directly applyig the greedy algorithm to the graph G. However, the greedy algorithm is ot effective i this case, because it does ot use the fact that T is already a MST o G. We propose the followig add ad delete algorithm to costruct a MST T o G from T. Step. Let E \ E = e e 2 e i some order ad take T 0 = T. Step 2. Whe T i is give, add e i+ to T i, that is, form T i e i+. The first time we add a edge to T i which is icidet to v / V there will be o circuit i T i e i+ ad we take T i+ = T i e i+. I all other cases, T i e i+ cotais a uique circuit C i+. Choose a edge f i+ C i+ such that w f i+ = max e C w e. Delete f i+ i+ from T i e i+. That is, defie

5 LIMIT THEOREM FOR SPANNING TREES 499 T i+ = T i e i+ \ f i+. If i = ad there exists o more edges e i+, the let T = T i ad stop. Step 3. Replace i by i +. Retur to Step 2. I order to prove that a tree T o V obtaied by the add ad delete algorithm is a MST o G, we eed the followig criterio for a spaig tree to be a MST. Lemma. Let G = V E w be a coected weighted graph with V <, E <, ad let T E be a spaig tree o V. The T is a MST o V if ad oly if for each edge e E \ T (2.) w e max w f : f C \ e where C is the uique circuit i T e. Proof. Let T be a MST o V. If there exists a edge e E \ T such that (2.) fails, the there exists a edge f C \ e such that w e < w f. Note that f T. Let T = T e \ f. The T is a spaig tree o V such that w g = w e w f < 0 g T w g g T This cotradicts the fact that T is a MST o V, so for each e E \ T (2.) holds. Coversely, let T be a spaig tree o V such that for each e E \ T, (2.) holds. Assume that T is ot a MST o V. For the momet assume that w f w f 2 for all f f 2. Let T g = e e be a MST o V obtaied by the greedy algorithm, where e i is the ith edge obtaied by the algorithm. Let e i0 be the first edge i T g such that e i0 / T, that is, e i T for i < i 0 but e i0 / T (ote that ecessarily T T g if T is ot a MST). Add e i0 to T; the T e i0 cotais a circuit C. This circuit C must cotai a edge e / T g because T g does ot cotai a circuit. By (2.) ad by the assumptio that w f w f 2 for f f 2, w e < w e i0. Now e / T g ad e e i0 e do ot form a circuit, because all those edges occur i T. Moreover, w e < w e i0. Therefore, e i0 caot be the i 0 th edge obtaied by the algorithm for T g. This cotradictio shows that T is a MST o V. If w f = w f 2 for some f f 2, the we apply the precedig argumet to a MST T g = e e obtaiable by the greedy algorithm ad with the followig additioal property: τ T g T = max τ T g T : T g a MST obtaiable by the greedy algorithm where, for T g = e e 2 e, τ T g T = max j + : e T e 2 T e j T Let e i0 be the first edge i T g such that e i0 / T, that is, τ T g T = i 0. The tree T e i0 cotais a circuit C, ad C cotais a edge e / T g. By (2.), w e w e i0. If we ca fid such a e with w e < w e i0, the we reach a cotradictio as before. Assume the that w e = w e i0 ad costruct a MST

6 500 H. KESTEN AND S. LEE T g by the followig greedy algorithm. For i i 0 pick e i as the ith edge of T g. At stage i 0 pick e istead of e i0 as the i 0 th edge of T g. The complete T g by the greedy algorithm. This costructio leads to τ T g T τ T g T + which is agai a cotradictio, so T is a MST o V. Propositio 2. Let G = V E w be a coected weighted graph with V <, E <, ad let T be a MST o V. Let G = V E w be a ew coected weighted graph obtaied by addig at most oe vertex v / V to V ad a fiite umber of weighted edges e / E to E ad with w a extesio of w. Let T be a spaig tree of V obtaied from T by the add ad delete algorithm. The T is a MST o V. Proof. Let e / T. We will show that (2.) holds, that is, w e max w f : f C \ e, where C is the uique circuit i T e. The by Lemma, T is a MST o V. Let the edges added to E be e e ad let T i be the ith tree obtaied i the add ad delete algorithm as described above. Thus T i+ = T i e i+ \ f i+ except for the first i for which e i+ is icidet to v ; i this case T i+ = T i e i+. If e E \ T, the, sice T is a MST o V, by Lemma there exists a circuit C 0 i T 0 e = T e with w e max w f : f C 0 \ e. If e T, but e / T, the e T 0 but e / T so that there is some i with e T i but e / T i, that is, e = f i ad e belogs to a circuit C i i T i e i ad (2.2) w e = max w f : f C i Such a C i also exists if e is oe of the edges e e which does ot belog to the fial T. Now let e / T ad C i as just described. If C i \ e T, the, by virtue of (2.2), (2.) holds. If ot, let T j, j > i, be the first spaig tree after T i i our sequece such that C i \ e T j. That is, C i \ e T j f j but C i \ e T j. The f j C i \ e. By choice of f j, T j f j cotais a circuit C j, f j C j, ad w f j max w f : f C j \ f j. Therefore (see Figure ), T j e cotais a circuit C j C i \ f j C j \ f j such that w e max w f : f C i \ e = max w f : f C i \ e f j w f j max w f : f C i \ e f j max w f : f C j \ f j max w f : f C j \ e Therefore, if C i \ e T, the for some j > i there exists a circuit C j i T j e with w e max w f : f C j\ e. We ca ow repeat the argumet with C i replaced by C j. Applyig this argumet fiitely may times, we see that there exists a circuit C i T e with w e max w f : f C \ e, that is, (2.) holds.

7 LIMIT THEOREM FOR SPANNING TREES 50 Fig.. The circuit C j i C i \ f j C j \ f j. Circuit C j cosists of the boldly draw parts. Lemma 2. Let G = V E w be a coected weighted graph with V < ad E <. If there exists a path π = e e i G from v V to v 2 V such that w e i λ i the i ay MST T o G there exists a path π = e e m i T from v to v 2 such that w e j λ, j m. Proof. Let T be a MST o G. The either e i is a edge i T or, if e i is ot a edge i T, by Lemma, T e i cotais a circuit C i such that for all edges f i C i we have w f w e i λ. Thus the edpoits of e i ca be coected i T by a path with edges e with w e λ. This is true for each i, so also v ad v 2 are coected i T by a path with edges e with w e λ. Let ψ: 0 0 be a give fuctio. From ow o we shall take our weight fuctio to be of the followig form: for the edge x y betwee x d ad y d, (2.3) w x y = ψ x y where x y is the Euclidea legth of the lie segmet from x to y. Accordigly, we make the followig defiitio for a fiite subset of d : (2.4) M = M ψ { = mi x y T } ψ x y : T a spaig tree o Note that M ψ is just the weight of a miimal spaig tree for G = E w whe w is give by (2.3) ad E cosists of all edges betwee ay two poits of. Defie further, for two disjoit fiite subsets of d, (2.5) D = D ψ = M M Let Q be the cube Q = 0 /d d

8 502 H. KESTEN AND S. LEE For x = x x d, y = y y d d ad L, l > 0, we write B x y = C x = A B x y l L = d x i y i i= d x i x i + i= d d x i l L y i + l + L \ x i l y i + l i= We shall call B x y, C x ad A B x y l L a rectagle, a cube ad a aulus, respectively. Defiitio. Let δ > 0. A separatig set of width δ for the rectagle B x y is a fiite set i the aulus A B x y δ with the property (see Figure 2): (2.6) Each lie segmet from x i y i + to x i δ y i + + δ passes withi distace /3 of some poit of. Most of the time we shall cosider situatios i which all vertices of iterest lie i Q = 0 /d d. For weight fuctios w of the form w e = ψ e with ψ strictly icreasig, it is reasoable to thik of the edge x y as the straight lie segmet betwee x ad y. Thus the edges of iterest will also lie i Q. I this situatio it is useful to relax (2.6) to the followig property: (2.7) Each straight lie segmet from a poit i x i y i + Q to aother poit i x i δ y i + + δ Q passes withi distace /3 of some poit Q. i= Fig. 2. A separatig set for the rectagle B x y. The asterisk deotes a poit of withi distace /3 of the dashed lie segmet.

9 LIMIT THEOREM FOR SPANNING TREES 503 We call a set Q A B x y δ with this property a -separatig set of width δ for the rectagle B x y. The reader should ote that if x i δ y i + + δ lies outside Q, the = φ satisfies (2.7) so that φ is a -separatig set. The term separatig set is justified by the ext lemma. Lemma 3. Let ψ be strictly icreasig. If A B x y δ is a separatig set for the rectagle B x y the i ay MST T (with weight fuctio ψ) o ay fiite subset of d cotaiig (with all edges betwee a pair of vertices i allowed), there are o edges betwee vertices i B x y ad vertices i d \ x i 2 δ y i δ. If Q the the same result holds for a -separatig set. Proof. We oly deal with the first case, whe is urestricted; the case whe all vertices ad are restricted to Q is essetially the same. Assume T cotais a edge v v 2 betwee v B x y ad v 2 d \ x i 2 δ y i δ. This edge v v 2 cotais a lie segmet from x i y i + to x i δ y i + + δ, ad hece there exists a s withi distace /3 of this edge. Removig the edge v v 2 breaks up T ito two compoets. If v ad s lie i the same compoet, the coect v 2 to s. If v 2 ad s lie i the same compoet, the coect v to s. I each case we obtai a ew spaig tree T. We claim that (2.8) ψ e ψ e ψ v v 2 max ψ v s ψ v 2 s > 0 e T e T This leads to a cotradictio. To prove our claim, ote that v s ad v 2 s because B x y has distace to x i y i + ad d \ x i 2 δ y i δ has distace to x i δ y i + + δ. Let t be a poit o the lie segmet v v 2 such that t s /3. The v v 2 = v t + v 2 t v s + v 2 s 2 s t = max v s v 2 s + mi v s v 2 s 2 s t max v s v 2 s > max v s v 2 s The last iequality i (2.8) ow follows from the strict mootoicity of ψ. The ext propositio is about the mootoicity of D. Propositio 3. Let ψ be strictly icreasig ad let A B x y δ be a separatig set for the rectagle B x y. Let ad be fiite subsets of d such that B x y ad = φ. The for ay fiite subset of d such that = φ ad (2.9) x i 2 δ y i δ = φ

10 504 H. KESTEN AND S. LEE we have (2.0) Proof. D D If = v v k, the it suffices to prove D v v r D v v r v r+ for r = 0 k. Thus we may restrict ourselves to the case whe we add oly oe poit to, say Also, if = u u 2 u l, the ad D = M M = D = = v l M u u r M u u r r= l M u u r M u u r r= We therefore oly have to prove M u u r M u u r M u u r M u u r for r = 2 l. Thus it suffices to prove the propositio for the case = u, = v. Let f f m be the edges from v to all the vertices of i order of icreasig legth; m is the cardiality of the set. Let E be the set of edges betwee all pairs of. Let T be a MST for E. The by the add ad delete algorithm costruct a MST T v o v. Next let h h m be the edges from u to agai i order of icreasig legth. Agai apply the add ad delete algorithm to T to costruct a MST T u o u. Also costruct a MST T vu o u v by applyig the add ad delete algorithm to T v. Sice cotais a separatig set for B x y, by Lemma 3, T vu does ot cotai the edge u v (recall (2.9) ad u x y ). So whe we apply the add ad delete algorithm to T v i order to costruct T vu, we do ot eed to add the edge u v. Thus whe carryig out the add ad delete algorithm to form T vu from T v, we oly have to add the same set of edges h h m from u to, as whe formig T u from T. Let T i be the tree obtaied from T after addig f f i ad deletig appropriate edges g 2 g i, that is, T i = T f f i \ g 2 g i

11 LIMIT THEOREM FOR SPANNING TREES 505 By Propositio 2, T i is a MST o v E f f i. Also, let T i j be the tree obtaied from T i after addig h h j ad deletig appropriate edges k i 2 k i j. We shall prove (2.) ψ e ψ e ψ e ψ e e T i j+ e T i j e T i+ j+ e T i+ j for i = m ad j = m. This will give (by summig over j) ψ e ψ e ψ e ψ e e T i m e T i e T i+ m e T i+ ad, by iteratio, (2.2) ψ e ψ e ψ e ψ e e T m e T e T m m e T m Note that T m is the tree obtaied by addig f ad the addig h h m ad deletig k 2 k m. As log as we oly add the sigle edge f from v to, v is a leaf ad f is ever deleted, so that T m is the same as T u plus the sigle edge f. Therefore, e T m ψ e = M u + ψ f Similarly e T ψ e = M + ψ f + ψ h e T m m ψ e = M u v e T m ψ e = M v + ψ h Thus (2.2) says M u M M u v M v This is the propositio for the case = u, = v. To prove the propositio, it therefore suffices to prove (2.). To prove (2.) we costruct T i+ j ad T i j+. By Propositio 2, T i j, which is obtaied by first addig f f i to T ad the addig h h j to T, is a MST o u v E f f i h h j. To costruct T i+ j we should first add f f i+ ad the h h j. By Propositio 2, T i+ j is the a MST o u v E f f i+ h h j. Cosider istead of this the followig costructio. First costruct T i j, the add f i+ ad delete a edge of maximal weight i the circuit formed i T i j f i+. Let this circuit be Ɣ i j ad the deleted edge γ i j ad let the resultig tree be T i+ j. By Propositio 2, T i+ j is also a MST o u v E f f i+ h h j. We may therefore replace e T i+ j ψ e i (2.) by e T i+ j ψ e. Similarly, we

12 506 H. KESTEN AND S. LEE may form a tree T i j+ by addig h j+ to T i j ad deletig a edge δ i j of maximal weight from the circuit i j i T i j h j+. We may the replace e T i j+ ψ e i (2.) by e T i j+ ψ e. Fially we ca form a tree T i+ j+ by addig first f i+ to T i j ad deletig γ i j ; this gives T i+ j. The add h j+ ad delete the edge δ i j of maximal weight i the circuit i j of T i+ j h j+. Agai e T i+ j+ ψ e = e T i+ j+ ψ e. This gives ψ e ψ e = ψ h j+ ψ δ i j e T i j+ e T i j ψ e ψ e = ψ h j+ ψ δ i j e T i+ j+ e T i+ j Thus (2.) is equivalet to (2.3) ψ δ i j ψ δ i j This, however, is easy. If γ i j / i j, the i j is preset i T i+ j h j+ = T i j f i+ h j+ \ γ i j. Cosequetly, i j = i j, δ i j = δ i j ad (2.3) is trivial. If γ i j i j, the situatio is as i Figure 3. Circuit i j is formed from a piece of i j ad a piece of Ɣ i j, but the by the choice of δ i j, ψ δ i j max e i j ψ e γ i j i j ad the choice of γ i j i Ɣ i j further imply that ψ δ i j ψ γ i j max e Ɣ i j ψ e Fig. 3. The trees T i+ j = T i j f i+ \ γ i j, T i j+ = T i j h j+ \ δ i j ad T i+ j+ = T i+ j h j+ \ δ i j. Edges γ i j, δ i j ad δ i j are the logest edges i the circuits Ɣ i j, i j, ad i j, respectively; i j is the boldly draw circuit ad i j is the dashed circuit.

13 LIMIT THEOREM FOR SPANNING TREES 507 Therefore ψ δ i j max ψ e max ψ e = ψ δ i j e i j Ɣ i j e i j Thus (2.3) holds. [I fact this argumet shows that if δ i j i j \ Ɣ i j, the ψ δ i j = ψ δ i j, but we do ot eed this fact.] We close this sectio by statig that the degree of a MST is uiformly bouded. Lemma 4. Let ψ be strictly icreasig. The there is a fiite costat D d which depeds oly o d such that for ay MST T o a fiite subset of d T has maximum vertex degree bouded by D d. The same proof works for all strictly icreasig weight fuctios. See Lemma 4 of Aldous ad Steele (992) or Talagrad [(995), proof of Lemma.3.]. 3. Momet estimates. I Sectio 4 we represet M X X EM X X as the sum of martigale differeces ad the we approximate (i the sese of Propositio 4) the martigale differeces by some radom variables which deped oly o the local cofiguratio of X X. This ca be justified by the estimates of momets i this sectio. We deote the (radom) set of a Poisso poit process of desity λ i d by λ ad the correspodig probability measure by µ P λ. If we are iterested i the process oly o Q, we deote the Poisso poits ad the correspodig probability measure by λ ad µ P λ, respectively. Thus λ = λ Q. I the case λ = we simplify the otatios, µ P, ad µ P to, µ P, ad µ P, respectively. Lemma 5. There exist costats 0 < C C 2 < such that for all rectagles B x y ad for /2 λ 2 (3.) µ P λ there is o separatig set of width δ of Poisso poits for the rectagle B x y i the aulus A B x y δ C γ 2 d exp C 2 δ where γ = [ yi x i δ 2] /2 is the diameter of the rectagle xi δ y i + + δ. Moreover for B x y Q φ, (3.2) µ P λ there is o -separatig set of width δ of λ -poits for the rectagle B x y i the aulus A B x y δ C γ 2 d exp C 2 δ

14 508 H. KESTEN AND S. LEE Proof. Partitio x i y i + ito at most C 3 γ 2δ d d dimesioal cubes of edge legth at most / 3 d /2. Similarly partitio x i δ y i + + δ ito C 3 γ d such cubes. Assume there exists a lie segmet from x i y i + to x i δ y i + + δ such that o poit from the Poisso process i the aulus x i δ x i ++δ \ x i y i + lies withi distace /3 of the segmet. Let the segmet ru from a poit p i a d -dimesioal cube F o x i y i + to a poit p 2 i a d dimesioal cube F 2 o x i δ y i ++δ. Let f i be the ceter of F i ad ote that f i p i /6. The o Poisso poit π i the aulus lies withi distace /6 from the lie segmet coectig f F to f 2 F 2, because π αp α p 2 π αf α f 2 + /6, 0 α. For each f ad f 2, µ P λ there is o Poisso poit i the aulus withi distace /6 of the segmet f f 2 C 4 exp C 2 δ The umber of choices for each of F ad F 2 is at most C 5 γ d so that the probability i (3.) is at most C γ 2 d exp C 2 δ This proves (3.) ad the proof of (3.2) is essetially the same. We shall also choose m vertices uiformly i the cube Q ad idepedetly of each other. We deote these radom poits by X X m ad deote the set of these radom poits by m. The correspodig probability measure goverig m is deoted by µ U m. I the case m = we simplify the otatios m ad µ U m to ad µ U, respectively. Lemma 6. There exist costats 0 < C C 2 C 6 C 7 < such that for all rectagles B x y with B x y Q φ ad for all 3/4 m 5/4 (3.3) µ U m there is o -separatig set m for the rectagle B x y i the aulus A B x y δ C γ 2 d exp C 2 δ + C 6 exp C 7 where as i Lemma 5 γ = [ yi x i δ 2] /2 is the diameter of the rectagle x i δ y i + + δ. Proof. Choose poits i Q accordig to a Poisso process with desity /2. If the Poisso process has l poits i Q with l m, the add m l idepedet poits, chose uiformly i Q. If l > m, the choose m ew idepedet poits uiformly i Q. It is easy to see that the resultig set of m poits i Q has the distributio µ U m. Therefore, sice icreasig the

15 LIMIT THEOREM FOR SPANNING TREES 509 umber of poits ca oly create more separatig sets, the left-had side of (3.3) is at most (3.4) µ P /2 l > 3/4 + µ P /2 there is o -separatig set of /2 -poits for the rectagle B x y i the aulus A B x y δ Now (3.5) µ P /2 l > 3/4 C 6 exp C 7 So (3.3) follows from (3.4), (3.5) ad (3.2). We shall apply Propositio 3 whe is the set of Poisso poits i C x Q ad is the set of Poisso poits i Q \ C x. I this case we shall wat to approximate D C x Q λ Q \ C x λ by D C x Q λ B λ, where (for a suitable L ) B = x i L x i + + L Q \ C x. We shall also wat to approximate D x Q \ x λ by D x B λ, where B = x i L x i + L Q \ x. For these situatios it is coveiet to have some abbreviated otatio. For a locally fiite set of poits i d we defie D L x = D C x x i L x i + + L \ C x D L x = D x x i L x i + L \ x The followig lemma provides a simple estimate for the various fuctios D. Lemma 7. Let ψ be strictly icreasig. Let be a fiite subset of the rectagle B x y with cardiality ad let be a fiite subset of d which cotais a separatig set for B x y i the aulus A B x y δ. The (3.6) D \ C 8 ψ γ + 2d /2 where γ is the diameter of x i δ y i + + δ. Similarly if is a locally fiite set of poits i d which cotais a separatig set for C x i the aulus A C x δ the (3.7) D L x C 8 C x ψ 2δ + 5 2L + d /2 ad if cotais a separatig set for x i the aulus A x δ the (3.8) D L x C 8 ψ 2δ + 4 2L d /2 Fially if Q A C x δ the cotais a -separatig set for C x i the aulus (3.9) D C x Q Q \ C x C 8 C x Q ψ 2δ + 5 /d d /2

16 50 H. KESTEN AND S. LEE Proof. We begi with (3.6). It is trivial that M M + ψ γ because we ca coect each vertex of to a MST o by coectig it to some vertex of. Coversely, if we take a MST o, the we ca obtai a spaig tree o by deletig all the edges icidet to ay vertex i \ (by Lemma 4 we delete at most D d edges) ad by recoectig the at most D d + compoets, which by Lemma 3 all have a vertex i x i 2 δ y i δ. Note that the diameter of the rectagle x i 2 δ y i δ is at most γ + 2d /2, so M M + D d ψ γ + 2d /2 This proves (3.6). Equatios (3.7) (3.9) ca be prove i the same way if oe takes ito accout that ay pair of poits i x i L x i ++L, x i L x i +L or Q are at most a distace 2L + d /2, 2L d /2 or /d d /2, respectively, apart. Lemma 8. (3.0) Assume that ψ is strictly icreasig ad that lim log ψ x = 0 x x The for each q > 0 there exists a costat C q such that (3.) (3.2) (3.3) µ P λ D L x λ q C q µ P λ D L x λ q C q uiformly i L x /2 λ 2 µ P λ D C x Q λ Q \ C x λ q C q uiformly i x Q /2 λ 2 µ P λ D L x λ q C q uiformly i x Q L /2 λ 2 (3.4) µ U m D X m m q C q uiformly i 3/4 m 5/4 (3.5) Proof. µ U m D L X m m q C q uiformly i 3/4 m 5/4 ad L Cosider a Poisso field λ, /2 λ 2, i d. Let δ = if δ: separatig set λ for C x i the aulus A C x δ

17 LIMIT THEOREM FOR SPANNING TREES 5 The, by (3.7), (3.6) Therefore D L x λ C 8 C x λ ψ 2 δ + 5 2L + d /2 µ P λ D L x λ q = µ P λ D L 0 λ q (by traslatio ivariace) C 9 µ P λ C 0 λ 2q /2 µ P λ ψ 2 δ + 5 d /2 2q /2 C 0 µ P λ ψ 2 δ + 5 d /2 2q /2 Now by Lemma 5 ad (3.0), µ P λ ψ 2 δ + 5 d /2 2q is bouded for /2 λ 2. Thus we get the first part of (3.). Almost the same argumet works for the secod part of (3.). Similarly, for (3.2) ad (3.3) we merely have to replace δ by (3.7) δ = if δ: -separatig set λ for C x i the aulus A C x δ ad use (3.6) ad (3.7). For (3.4) ad (3.5) we use Lemma 6 istead of Lemma 5. We remid the reader that µ P Q = 0 /d d. is the distributio of = = Lemma 9. Let ψ be strictly icreasig. For all ε > 0 there exists a L = L ε such that for x L /d L d ad L L (3.8) µ P D C x Q \ C x D L x ε ε Proof. Choose δ so large that µ P there is o separatig set for C x i A C x δ ε/3 Such a δ exists by Lemma 5. Also choose N so that O the evet (3.9) we have, by (3.6), ad, by Propositio 3, µ P C x cotais more tha N Poisso poits ε 3 separatig set for C x i A C x δ C x cotais at most N Poisso poits D L x C 8 Nψ 2δ + 5 d /2 D L x is icreasig i L for L δ + 2

18 52 H. KESTEN AND S. LEE Therefore, D L x coverges to a fiite limit a.e. µ P ad oe ca choose L = L ε δ + 2 so large that (3.20) µ P{ (3.9) fails or sup L L ε D L x D } L ε x ε/2 ε Note that this estimate is uiform i x, by the traslatio ivariace of µ P. If x L /d L d, L, L L ad Q x i L x i + + L, the by Propositio 3 oe has o the evet { } (3.2) (3.9) holds ad sup D L x D L x < ε/2 L L that 0 D C x Q \ C x D L x = D C x Q \ C x D L x (3.22) D C x Q \ C x D L x = D C x Q \ C x D L x D L x D L x < ε So (3.8) follows from (3.20), (3.2) ad (3.22). We ext wat to use Propositio 3 whe is a sigle (radom) poit ad is the collectio of Poisso poits i Q. That is, we wat to see what the ifluece of addig a sigle poit is o the weight of a MST. Choose λ, the set of Poisso poits i d of desity λ, ad the choose a poit Y, idepedet of λ, uiformly i Q. We deote the correspodig probability by µ P λ Y. Similarly, we deote by µ P λ Y the joit distributio of λ = λ Q ad Y. Lemma 0. Let ψ be strictly icreasig. For all ε > 0 there exists a K = K ε ad a = ε such that uiformly for /2 λ 2 (3.23) ad (3.24) µ P λ Y D Y Q \ Y λ D K Y λ ε = µ P λ Y D Y Q \ Y λ D K Y λ ε ε µ P λ Y there is o separatig set for Y i A Y K P λ ε/4 Proof. We give the proof oly for ψ y = ψ α y = y α. Whe λ =, the (3.23) is very similar to (3.8) except that i (3.8) the Y is replaced by C x, x L /d L d. However, the basic estimate is the same, except

19 LIMIT THEOREM FOR SPANNING TREES 53 that a extra term comes i to take care of the case whe Y is too close to Q. Specifically, choose δ, idepedet of /2 λ 2, so large that for all, µ P λ Y there is o separatig set for Y i A Y δ ε/4 This ca be doe by virtue of Lemma 5 ad the traslatio ivariace of µ P λ. The iequality (3.24) will hold if we take K δ. O the evet (3.25) separatig set for Y i A Y δ we have, by Lemma 7, D K Y λ C 8 ψ 2δ + 4 d /2 for K δ + ad, by Propositio 3, D K Y P λ is icreasig i K for K δ + 2. We claim that oe ca choose K ε δ + 2 ad ε so that uiformly for /2 λ 2,, { µ P λ Y Y / K /d K d or (3.25) fails or sup D K Y λ D K ε Y λ ε/2 K K ε } ε Ideed, sice, for ay fixed x, λ /d +x has the same distributio as λ, ad ψ λ /d y = λ α/d ψ y, we have { } µ P λ Y sup D K Y λ D K Y λ ε/2 K K { } = µ P Y sup D K 0 D Kλ /d 0 ε/2 λ α/d K Kλ /d Choose K ε 2 /d δ + 2 so that { } µ P Y sup D K 0 D K ε 2 α/d 0 ε/4 2 α/d ε/4 K K ε 2 α/d Fially, we choose ε so that for ε, P Y / K /d K d ε/4 The rest of the proof is the same as that of Lemma 9. Next we prove a aalogue of Lemma 0 whe λ is replaced by m, a cofiguratio of m i.i.d. poits, each oe uiformly distributed i Q. Let us write X X m for m such poits i Q. Lemma. Let ψ be strictly icreasig. For all ε > 0 there exists a K = K ε ad a 2 = 2 ε such that uiformly for 2 3/4 m + 3/ µ U m D X m Q \ X m m D K X m m ε ε

20 54 H. KESTEN AND S. LEE Proof. Cosider two idepedet Poisso variables N ad W with meas 2 3/4 ad 4 3/4, respectively. The N 2 = N + W is a Poisso variable with mea + 2 3/4. We ow choose i successio N 2 i.i.d. uiform poits o Q. The first N of these poits forms a realizatio of 2 /4 o Q ad the total forms a realizatio of + 2 /4 o Q. O the evet (3.27) N 3/4 < + 3/4 N 2 we further take the first m poits as our realizatio of µ U m, say, X X m (recall that 3/4 m + 3/4 ). If (3.27) is ot the case, we pick m ew i.i.d. uiform poits o Q to get a realizatio of µ U m. Fially let Y be aother uiform poit o Q, idepedet of N, W ad the X i. Let K = K ε ad = ε be such that (3.23) holds ad such that (3.24) holds eve with K replaced by K 2. Assume that for some give, (3.27) occurs ad (3.28) D Y + 2 /4 D K Y + 2 /4 < ε Assume further that (3.29) ad that (3.30) oe of the W poits umbered N N 2 fall i Y i K 2 Y i + K + 2 separatig set i 2 /4 for Y i A Y K 2 We claim that, if (3.27) (3.30) hold, the (3.3) D Y m D K Y m ε for 3/4 m + 3/4. Ideed, by (3.27) ad (3.29), for each N m N 2, (3.32) D Y m Y i K Y i + K = D Y + 2 /4 Y i K Y i + K Moreover, by (3.30) ad Propositio 3, as we keep addig poits, (3.33) D Y r icreases i r N r N 2 Therefore, if (3.27) (3.30) hold, the for, 3/4 m + 3/4, 0 D Y m D K Y m D Y + 2 /4 D K Y m [by (3.30) ad Propositio 3] [by (3.27) ad (3.33)] = D Y + 2 /4 D K Y + 2 /4 [by (3.32)] < ε

21 LIMIT THEOREM FOR SPANNING TREES 55 This proves our claim (3.3). Next we show that the probability that (3.27) (3.30) hold is close to. It is trivial that P (3.27) fails ε/4 for large, say 2. By Lemma 0, for ε/4, P (3.28) fails ε/4. Also 2K + 4 d P (3.29) fails EW = 2K + 4 d 4 /4 We assumed that (3.24) holds with K replaced by K 2 ad we may further assume that 2 is so large that, for 2, P (3.29) or (3.30) fails ε/2. The for 2 (3.34) P (3.27) (3.30) hold ε Sice the joit distributio of Y ad m is the same as that of X m ad m, (3.3) ad (3.34) together imply µ U m D X m m D K X m m ε ε The iequality (3.26) follows because D Xm Q \ X m m = D Xm m ad D K Xm m = D K X m m. 4. Cetral limit theorem for a MST. We order the vertices v i d 0 /d d i some way, say lexicographically, as v v /d + d ad defie k by ( [ k = σ i k ]) C v i ( 0 is the trivial σ-field). Also defie M ad k by (4.) { } M = mi ψ e : T a spaig tree o e T k = µ P M k µ P M k From ow o we assume that ψ is strictly icreasig, cotiuous, ad satisfies the growth coditio (3.0). Lemma 2. The quatity M is a fuctio of the idepedet radom variables = Cv l = Cv l, l = /d + d. Moreover (4.2) M E M l = k k=

22 56 H. KESTEN AND S. LEE ad (4.3) k = µ P da k da k+ da l [ ( [ ] [ D k i i<k i>k ]) ( [ a i D a k i<k ] [ i i>k a i ])] where µ P dak da l is short for µ P k da k l da l. Proof. Clearly M is a fuctio of l. I fact, with the appropriate topology o the rage of l, M is a Borel fuctio of l. Equatio (4.2) is the immediate from the fact that M is l -measurable. Moreover, by defiitio, k = µ P da k da k+ da l [ ([ ] [ ]) ([ ] [ M i a i M i a i ])] i k i>k i<k i k Now subtract ad add M ([ ] [ ]) i<k i i>k a i to the expressio i square brackets ad ote that ([ ] [ ]) ([ ] [ ]) M i a i M i a i i k i>k i<k i>k ( [ ] [ = D k i a i ]) i<k ad similarly if k is replaced by a k. i>k Now we write k L for the expressio i (4.3) whe D k ad D a k are replaced by D L v k k ad D L v k a k, respectively. That is, k L = µ P da k da k+ da l (4.4) ( [ ] [ [ D L v k i i k i>k ]) [ a i D L (v k i<k ] [ i i k a i ])] Propositio 4. For all ε > 0 there exists a L = L ε ad for each L L a 3 = 3 L such that for L L 3 (4.5) k µ P 2 k 2 k L ε

23 LIMIT THEOREM FOR SPANNING TREES 57 Proof. k k L = µ P da k da l [ { ( [ ] [ ]) (4.6) D k i a i D L i<k { ( [ D a k i<k i>k ] [ ]) i a i D L i>k ( [ v k ( v k i k ] [ i i>k a i ])} [ ] [ i a i ])}] By (4.6), (3.2) ad (3.3), k k L is uiformly [i k L ] itegrable. By (3.8), k k L 0 as L (uiformly i ad k with v k L /d L d i µ P -measure, that is, (4.7) µ P k k L 0 as L uiformly i ad k with v k L /d L d. This is close to what we wat. First we ote that the umber of k for which v k / L /d L d is at most C L d /d ad the sum of µ P 2 k 2 k L over those k is therefore at most C 2 L d /d, by (4.3), (4.4), (3.2) ad (3.3). Thus, this part of the sum i (4.5) cotributes at most C 2 L /d. For the remaiig summads we ote that (4.8) By (4.7), (4.9) 2 k 2 k L = k k L k + k L i<k 2K k k L I k K k L K + 2 k I k > K + 2 k L I k L > K 2Kµ P k k L 0 as L uiformly i ad k with v k L /d L d. Moreover, by (4.3), (4.4), (3.2) ad (3.3), i k 4 0 µ P 2 k I k > K K µp k 3 C 3 K 4 µ P 2 k L I k L > K K µp k L 3 C 3 K Now, for a give ε > 0, choose K large so that C 3 /K ε/4. The choose L large so that 2Kµ P k k L ε/4 for L L, v k L /d L d ad fially for give L L choose 3 large so that C 2 L /d ε/4 for 3. The propositio follows from (4.8) (4.), together with the fact that the boudary part of the sum i (4.5) cotributes at most C 2 L /d. Lemma Defie ρ K by ρ K = µ P D K 0

24 58 H. KESTEN AND S. LEE The 4 3 ρ = lim K ρ K exists ad ρ is fiite. Moreover for ε > 0 ad A > 0, there exists a 4 = 4 ε A such that for µ U +A /2 { M X X s M X X s ρ ε /2 for some A /2 s + A /2} + 2ε C 2 ε + 24Aε ε Proof. By the same argumet as for (3.8), as K, D K 0 is evetually bouded ad icreasig, so D K 0 coverges a.e. as K. Moreover, by (3.), D K 0 is uiformly itegrable, so (4.3) holds ad ρ is fiite. Fix K such that ρ K ρ ε/ 4 2A + ad such that (3.26), with ε replaced by ε 6 /A, holds for 2 ε 6 /A. Now take t = A /2 ad ote that 4 5 M X X s = M X X t + s p=t+ D X p p ad (4.6) sup s 2 A /2 s s 2 +A /2 p=s + [ D X p p D K X p p ] +A /2 A /2 D X p D K X p p p Itroduce K X p = D X p D K X p The, by virtue of (3.4), (3.5) ad the fact that p µ U +A /2 f = µ U p f p

25 LIMIT THEOREM FOR SPANNING TREES 59 if f depeds oly o X X p, p + A /2, oe has µ U p { ( ) } K X p { µ U ( p ) K X p } ( ) K X p ε6 A { + 2µ U p D ({ X } p ) D ({ Xp } p ) A } ε 3 p { + 2µ U ( p D K X p p ) ( D K X p p ) A } ε 3 (4.7) { + µ U ( p K X p ) K ( X p ) > ε6 A D ({ X p } p ) A ε 3 D K ( X p p ) A ε 3 } ε6 A + 2ε3 A µu p { D 2({ Xp } p )} + 2 ε3 A µu p { D 2 ( K X p p )} + ε6 2A A ε 3 ] [by (3.26) with ε replaced by ε6 ( ε 6 ) A + 4ε3 C 2 + 2ε3 A So, if 3A /2 2A /2 +, say for 4 ε A, the { s 2 } µ U +A /2 sup K X p 4 ε/2 A /2 s s 2 +A /2 p=s + A +A /2 µ U p K Xp / ( A /2 2ε C 2 ε + 24Aε ε Thus, it suffices to show { µ U +A /2 sup (4.8) ε s A /2 s t+ D K X p 4 ε/2) p ρ K } 4 ε/2 To do this we observe that the distributio of p X p K K d coverges to that of a Poisso field with mea i K K d as ad p/. I particular we have 4 9 µ U p D K Xp p ρ K = µ P D K 0

26 520 H. KESTEN AND S. LEE uiformly i A /2 p + A /2. We ca therefore further replace D K Xp p ρ K by Z p = D K X p It the suffices to show that { p ρ K µ U p D K X p ρ K µ U +A /2 sup s A /2 } s Z p 8 ε/2 ε By (3.8) (with δ = ), the Z p are bouded by C 4 K ad µ U p Z p = 0. The Z p are ot idepedet, but we have that for p p 2 the joit distributio of p X p K K d ad p 2 X p 2 K K d coverges to the distributio of two idepedet Poisso fields o K K d as. From this it follows that EZ p Z p2 0 ad t+ ( s ) 2 E Z p 0 t+ as, uiformly i A /2 s + A /2. Therefore, for fixed η > 0, /2 t+jηa/2 t+ Z p 0 i µ U +A /2 -measure, uiformly for j = 2 2/η. This proves 4 20 /2 sup i µ U +A /2 -measure because sup t+jηa /2 s t+ j+ ηa /2 s A /2 s t+ s Z p 0 t+ t+jηa /2 Z p t+ p Z p ηa /2 C 4 K Iequality (4.8) ad hece (4.4) ow follow from (4.20). Proof of Corollary. Note that µ U + M X X + µu M X X ρ = µ U + D X + ρ = µ U + K X + + D K X + ρ K + ρ K ρ It therefore follows from (4.7), (4.9) ad (4.3) that µ U + M X X + µu M X X ρ

27 LIMIT THEOREM FOR SPANNING TREES 52 I the special case whe ψ x = x α, rescalig to the uit cube shows that (with X X 2 i.i.d. ad uiform o 0 d ) 4 2 α/d E M X X + α E M X X α ρ This implies the followig asymptotic equivaleces as : 4 22 E M X X α ρk α/d ρd d α α/d provided 4 25 E M X X d ρk ρ log E M X X α ρk α/d ρd d α α/d E M X X α 0 i case α > d. Now, first, if α < d, (4.22) shows that + α/d α/d E M X ρα X + α d α if α < d if α > d which together with (4.2) proves (.9) with ρ = ρd/ d α i this case. Equatio (.8) is just (4.22). Next, if α = d, the (4.23), together with (.2) shows that ρ = 0. The, by (4.2) ad (.2), + E M X X + d E M X X d = o + E M X X + d c d d This proves (.9) whe α = d while (.8) is immediate from (.2). Fially, whe α > d, we must first prove (4.25). This follows from Lemmas 3 ad 6. Ideed, if δ = δ x is defied as i (3.7), the by Lemma 3, for ay MST T o = X X, it holds that e α d α/2 δ x + 3 α C x e T oe edpoit of e i C x Therefore, the expectatio of the left-had side with respect to µ U is at most d α/2 µ U δ x + 3 2α µ U C x 2 /2 C 5 for some C 5, idepedet of x,. Summig over x = v v 2 v /d + d proves that for ψ x = x α, µ U M X X ψ C 5 /d + d

28 522 H. KESTEN AND S. LEE Rescalig agai to the uit cube shows that E M X X α C 6 α/d Thus (4.25) ad (4.24) follow. We leave it to the reader to derive (.8) ad (.9) for α > d. The strict positivity of ρ follows i all cases by the argumet of Steele [(988), ed of Sectio 4]. Lemma 4. If alog some subsequece /2 M µ P M coverges i distributio to F with characteristic fuctio F the alog the same subsequece /2 M X X µ P M coverges i distributio to G with characteristic fuctio Ĝ θ = exp iθx dg x = F θ exp 2 θ2 ρ 2 Proof. Couple ad m by first choosig a ifiite sequece X X 2 of i.i.d. uiform radom variables o Q ad, idepedetly of the X i, a Poisso variable N = N with mea. The X X m has the distributio µ U m ad X X has the distributio µ P. I particular, M has the same distributio as M X /2 N N 0 ad, by Lemma 3, 4 26 /2 M X N X N, X M X X N ρ 0 N i probability. Thus, if /2 M µ P M is tight, the /2 M X µ P M is tight. Therefore, if alog some subsequece /2 M X µ P M coverges i distributio to F with characteristic fuctio F, the alog the same subsequece, by (4.26), /2 M X X µ P M + N ρ coverges i distributio to F. Sice N is idepedet of the X alog the same subsequece, /2 M X X µ P M coverges i distributio to G with characteristic fuctio Ĝ, where Ĝ θ exp 2 θ2 ρ 2 = F θ. Proof of Theorem 2. At first we prove the theorem i the Poissoized versio which says that /2 M µ P M coverges i distributio to a ormal distributio with mea 0 ad variace τ 2 for some τ > 0. By virtue of the represetatio (4.2) of M µ P M as a sum of martigale differeces ad Theorem 2.3 i McLeish (974), it suffices to verify the followig relatios (4.27) (4.30): k τ2 k i µ P -measure i,

29 LIMIT THEOREM FOR SPANNING TREES 523 where 4 28 τ 2 = lim L τ 2 L τ 2 L = µp[ µ P D C 0 L L + d \ C 0 µ P D C 0 L L + d \ C 0 ] 2 = σ C x : x d C x L L + d x strictly precedes 0 i the lexicographical order or x = 0 ad = σ C x : x d C x L L + d x strictly precedes 0 i the lexicographical order 4 29 /2 max k 0 k i µ P -measure ad 4 30 { µ P max k 2} is bouded i k [the existece of the limit i (4.28) is part of what eeds to be prove]. First let us deal with the easy relatios (4.29) ad (4.30). Equatio (4.29) holds, because { µ P /2 max k } { k ε µ P by (4.3) ad (3.2). For (4.30) we observe that { µ P max k 2} k k agai by (4.3) ad (3.2). Fially we prove that for fixed L, k k 3 } / ε /2 3 C 7 C 3 / ε /2 3 µ P 2 k C 7C k L τ2 L k i µp -measure We claim that (4.3) implies (4.27). To see this, ote first that (4.28) holds, as i the argumet for (4.3), by the evetual mootoicity i L of µ P D C 0, L L + d \ C 0. We may therefore choose L is so large that for all L L, 4 32 τ 2 τl 2 < ε/3

30 524 H. KESTEN AND S. LEE Now, for give ε > 0, we ca [by (4.5)] choose L = L ε 2 L so that i additio to (4.32), for 3 L, 4 33 µ P 2 k 2 k L < ε2 k Fially, by (4.3), we ca obtai a 4 L ε 3 so that for 4, { µ P 2 k L τ2 L > ε } < ε 3 The, for 4, by (4.33) ad (4.32), µ P { k µ P } 2 k τ2 > ε { ε2 ε/3 + ε k 2 k k k 2 k L > ε } { + µ P 2 k L 3 τ2 L > ε } 3 Thus (4.3) ideed implies (4.27). To see (4.3) we merely ote that k L ad j L are idepedet as soo as v k i L v k i + L + v j i L v j i + L + = φ ad that for all v k L /d L d µ P 2 k L = τl 2 as defied above, provided we order the v k lexicographically (as we may). Sice the cotributio of the terms with v k / L /d L d to (4.3) is at most C 8 /d, (4.3) follows from a simple variace estimate ad Chebychev s iequality. As stated above, the relatios (4.27) (4.30) prove that By Lemma 4 we the have M µ P M N 0 τ 2 /2 k 4 34 M X X µ P M N 0 σ 2 /2 where τ 2 = σ 2 + ρ 2. Below we will show that σ 2 > 0. This the completes the proof of the Poissoized versio of Theorem 2. To prove the origial versio of Theorem 2, we observe first that 4 35 /2 µ P M µ U M 0 because [see (4.26)] 4 36 /2 M M N 0 ρ 2

31 LIMIT THEOREM FOR SPANNING TREES 525 ad (4.5) shows that (whe we use the couplig of Lemma 4, that is, N = N, a Poisso variable with mea ) E M M 2 { E I N = j E [ j=+ I N = j + j=0 j=+ + j p=+ p=j+ I N = j j I N = j j j=0 C 9E N 2 C 20 D X p D X p j p=+ p=j+ p 2} p E D 2 X p p ] E D 2 Xp p 2 (by Schwarz s iequality) Therefore, the left-had side of (4.35) is uiformly itegrable ad hece, by (4.36), (4.35) holds. We may therefore replace µ P M by µ U M i (4.34). Fially we show σ 2 > 0, by provig that M caot be cocetrated o a iterval of legth o /2. This is doe by a block argumet which is similar to that of Avram ad Bertsimas [(993), Propositio 5]. We just describe our block for d = 2 ad leave the case d 3 to the reader. For d = 2 we cosider a 3 3 square W Q. For each square alog the boudary of W (see Figure 4) we require that 4 37 We also require that 4 38 there is at least oe poit of i the square cotais exactly oe poit v i i each of B i i = 2 where B i are squares as i Figure 4. Fially we require that 4 39 apart from v v 2 ad the poits i the squares alog the boudary of W, there are o poits of i W. We claim that, if W has these properties, the i ay MST T o, there is oly oe edge icidet to v 2 B 2, ad this edge coects v 2 to v. Ideed, for ay v \ v v 2, the lie segmet v v 2 must itersect a square o the boudary of W. Choose a -poit w i this square. Next choose a -poit w 2 i the adjacet boudary square. Keep choosig -poits w j i successive adjacet boudary squares util we choose a - poit w m i B 3 (see Figure 4). The π = v w w m v v 2 is a path from

32 526 H. KESTEN AND S. LEE Fig. 4. If cotais this 3 3 square cofiguratio, the i ay MST o the oly edge icidet to v 2 is the edge v v 2. v to v 2 i such that v w < v v 2, w i w i < v v 2, 2 i m, w m v < v v 2 ad v v 2 < v v 2. Therefore, by Lemma 2, v v 2 caot be a edge i ay MST T o. This proves our claim that the oly edge i T icidet to v 2 is the edge v v 2. It follows from this claim that, if W satisfies (4.37) (4.39), the M = M \ v 2 + ψ v v 2 As i Avram ad Bertsimas (993), this proves that Var M C 2 for some costat C 2 > 0. However, without a proof of uiform itegrability of M µ U M 2, this does ot guaratee σ 2 > 0. For ψ x = x, this uiform itegrability is i fact cotaied i Talagrad s much more detailed ad deeper tail estimates for M [see Theorem.3.2 i Talagrad (995)]. Alteratively, we ca use the followig well kow argumet. Choose M = C 22 disjoit 3 3 squares W W M i Q. Let W i W is be the radom collectio of those squares for which W ij has the properties (4.37) (4.39) (after a suitable traslatio). Let v j ad v 2 j be the aalogues i W ij of v ad v 2, respectively. The coditioed o the idex set i i s

33 LIMIT THEOREM FOR SPANNING TREES 527 ad \ [ sj= v 2 j ], the ψ v j v 2 j are idepedet ad s s ψ v j v 2 j j= is asymptotically ormal with some mea, but with a variace ν 2 C 23 for some costat C 23 > 0. I additio, it is easy to see that there exists some C 24 > 0 such that µ U s C 24 as. Stadard argumets ow show that sup µ U a /2 M a + ε 0 a as ε 0, uiformly i. Together with (4.34) this implies σ 2 > 0. This completes the proof of Theorem 2. Ackowledgmet. The work of H.K. was supported by the NSF through a grat to Corell Uiversity. The authors wish to thak J. E. Yukich for several helpful coversatios about miimal spaig trees. REFERENCES Aldous, D. ad Steele, J. M. (992). Asymptotics for Euclidea miimal spaig trees o radom poits. Probab. Theory Related Fields Alexader, K. S. (994). Rates of covergece of meas for distace-miimizig subadditive Euclidea fuctioals. A. Appl. Probab Alexader, K. S. (996). The RSW theorem for cotiuum percolatio ad the CLT for Euclidea miimal spaig trees. A. Appl. Probab Avram, F. ad Bertsimas, D. (993). O cetral limit theorems i geometric probability. A. Appl. Probab Chartrad, G. ad Lesiak, L. (986). Graphs ad Digraphs. Wadsworth, Belmot, CA. Jaso, S. (995). The miimal spaig tree i a complete graph ad a fuctioal limit theorem for trees i a radom graph. Preprit. Kruskal, J. B. (956). O the shortest spaig subtree of a graph ad the travelig salesma problem. Proc. Amer. Math. Soc Lévy, P. (937). Théorie de l Additio des Variables Aléatoires. Gauthier Villars, Paris. McLeish, D. L. (974). Depedet cetral limit theorems ad ivariace priciples. A. Probab Ramey, D. B. (982). A o-parametric test of bimodality with applicatios to cluster aalysis. Ph.D. dissertatio, Yale Uiv. Redmod, C. ad Yukich, J. E. (994). Limit theorems ad rates of covergece for Euclidea fuctioals. A. Appl. Probab Steele, J. M. (988). Growth rates of Euclidea miimal spaig trees with power weighted edges. A. Probab Talagrad, M. (995). Cocetratio of measure ad isoperimetric iequalities i product spaces. Publ. Math. vol. 8, Ist. des Mautes Etudes Scietifiques. Departmet of Mathematics Corell Uiversity Ithaca, New York keste@math.corell.edu Departmet of Mathematics Natioal Uiversity of Sigapore Sigapore 05 matleesc@leois.us.sg