WEIGHTED PERSISTENT HOMOLOGY SUMS OF RANDOM ČECH COMPLEXES

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1 WEIGHTED PERSISTENT HOMOLOGY SUMS OF RANDOM ČECH COMPLEXES BENJAMIN SCHWEINHART Abstract. We study the asyptotic behavior of rando variables of the for Eα i x,..., x n = d b α b,d PH ix,...,x n where { x j are i.i.d. saples fro a probability easure on a triangulable }j N etric space, and PH i x,..., x n denotes the i-diensional reduced persistent hoology of the Čech coplex of {x,..., x n }. These quantities are a higherdiensional generalization of the α-weighted su of a inial spanning tree; we seek to prove analogues of the theores of Steele [6] and Aldous and Steele [2] in this context. As a special case of our ain theore, we show that if { x j are distributed }j N independently and uniforly on the -diensional Euclidean sphere, α <, and 0 i < n, then there are real nubers γ and Γ so that α γ li n E α i x,..., x n Γ in probability. More generally, we prove results about the asyptotics of the expectation of Eα i for points sapled fro a locally bounded probability easure on a space that is the bi-lipschitz iage of an diensional Euclidean siplicial coplex.. Introduction We are interested in rando variables of the for Eα i x,..., x n = b,d PH i x,...,x n d b α where { x j are independent saples drawn fro a probability easure on a }j N triangulable etric space, and PH i x,..., x n denotes the i-diensional reduced persistent hoology of the Čech coplex of {x,..., x n }. The special case i = 0 is, Date: July 208.

2 2 BENJAMIN SCHWEINHART under a different guise, already the subject of an expansive literature in probabilistic cobinatorics; Eα 0 x gives the α-weight of the inial spanning tree on a finite subset of a etric space x, T x : Eα 0 x = 2 α e α In 988, Steele [6] showed the following: e T x Theore Steele. Let µ is a copactly supported probability distribution on R, and let {x n } n N be i.i.d. saples fro µ. If α <, α li n E 0 α x,..., x n c α, f x α/ R d with probability one, where f x is the probability density of the absolutely continuous part of µ, and c α, is a positive constant that depends only on α and. In 992, Aldous and Steele [2] showed that if {x i } i N sapled independently fro the unifor distribution on the unit cube in R, then li E α x,..., x n c d, d in the L 2 sense. Under the sae hypotheses, Kesten and Lee proved the following central liit theore in 996 [2]: Eα 0 X,..., X n E Eα 0 X,..., X n N 0, σ 2 n 2α α,d 2d in distribution, for any α > 0. Here, we take the first step toward a higher-diensional generalization of these celebrated results. Another special case of Eα i x α = gives the total lifetie persistence of x. Rando variables of the for E i x have been investigated by Hiraoka and Shirai [] in the context of Linial Meshula processes. They showed that if X is sapled fro the -Linial Meshula process then E E X O n which is a higher-diensional generalization of Frieze s ζ 3-theore for Erdós Rényi rando graphs [0]. Also, Adas et al. [] studied the behavior of the lifetie persistence of rando easures on Euclidean space, perforing coputational experients and conjecturing the existence of a liit function capturing finer properties of the persistent hoology.

3 WEIGHTED PERSISTENT HOMOLOGY SUMS OF RANDOM ČECH COMPLEXES 3 The properties of E i α x for general i and n have until now, as far as we know, not been studied in a probabilistic context see the note at the end of the introduction. However, soe work has been done in the extreal context. In 200, Cohen-Steiner et al. [6] showed that if M is the bi-lipschitz iage of an -diensional siplicial coplex and α >, then E α i X is uniforly bounded for X M. We use their results to prove the upper bounds in Section 2. Furtherore, in our previous paper [3] we related the upper box diension of a subset X of a etric space to the behavior of E i α Y for extreal subsets Y X. We will say ore about the relation of this to the present work in Section.2... Our Results. The following are special cases of our ain theore: Theore 2. Let { x j }j N be be distributed independently and uniforly on the Sn. If α <, 0 i < n, and persistent hoology is taken with respect to the intrinsic etric on S n, γ li n α E α i x,..., x n Γ in probability, where γ and Γ are constants that depend on µ and α. Furtherore, there exists a D R so that in probability. li log n E i x,..., x n D Theore 3. Let { x j be be distributed independently and uniforly on an - }j N diensional Euclidean ball. If α <, 0 i < n, γ li n α E E α i x,..., x n Γ where γ and Γ are constants that depend on µ and α. In fact, the lower bound holds in probability. Furtherore, there exists a D R so that li log n E E i x,..., x n D We show a stronger result for copactly supported probability easures on R 2 that are locally bounded:

4 4 BENJAMIN SCHWEINHART Definition. A probability easure µ on R is locally bounded if there is a A R with positive volue and real nubers a a 0 > 0 so that for all Borel sets B A. a 0 vol B µ B a vol B Theore 4. Let µ is a copactly supported, locally bounded probability easure on R 2, and let {x n } n N be i.i.d. saples fro µ. If α <, γ li n α E α x,..., x n Γ in probability. In fact, the upper bound holds with probability one. Furtherore, there exists a constant D so that with probability one li log n E 2 x,..., x n D More generally, we prove results for locally bounded probability easures on spaces that are the bi-lipschitz iage of a copact, -diensional Euclidean siplicial coplex: Definition 2. Let M be the bi-lipschitz iage of a copact -diensional Euclidean siplicial coplex M under a ap φ M. A probability easure µ on M is locally bounded if there exists a subset A M with positive -diensional volue, and real nubers a a 0 > 0 so that a 0 for all Borel sets B A. vol B vol M µ φ vol B M B a vol M For exaple, a the unifor easure on a -diensional Rieannian anifold is locally bounded, as is any easure that is locally bounded with respect to the Rieannian volue. While there exist etric spaces M with point sets { x j }j N so that PH i x,..., x n O n this is thought to be soewhat pathological behavior [3].

5 WEIGHTED PERSISTENT HOMOLOGY SUMS OF RANDOM ČECH COMPLEXES 5 Definition 3. A probability easure µ on a triangulable etric space has linear PH i expectation if E PH i {x,..., x n } O n Siilarly, µ has linear PH i variance if E PH i {x,..., x n } E PH i {x,..., x n } 2 O n For exaple, the unifor easure on a Euclidean ball [8] and any positive, continuous probability density on the Euclidean n-sphere [7] has linear PH i expectation. It is ore difficult to prove that a probability easure has linear PH i variance. As far as we are aware, this is only known for probability easures on R 2 and the unifor easure on the n-diensional Euclidean sphere [7] see Equation and Proposition 3. Theore 5. Let M be the bi-lipschitz iage of an -diensional Euclidean siplicial coplex, and 0 i <. If µ is a locally bounded probability easure on M, there are real nubers 0 < γ < Γ so that γn α E E i α x,..., x n Γ E PH i {x,..., x n } α for all sufficiently large n. In particular, if µ has linear PH i expectation, there is a real nuber Γ 0 so that γ li n α E E α i x,..., x n Γ 0 The lower bound holds in probability, and the upper bound does if µ has linear PH i variance. Furtherore, there exists a real nuber D so that E En i x,..., x n D log E PH i x,..., x n where analogously sharper stateents hold if µ has linear PH i expectation or variance. We prove the upper bound in Proposition 2 and the lower bound in Proposition 5. After copletion of this anuscript, we becae aware that Divol and Polonik [7] independently and concurrently proved a sharper result for the persistent hoology of points sapled fro bounded, absolutely continuous probability densities on [0, ]. We believe this anuscript is still useful in that the proofs are largely self-contained, and the ethods are applicable to other situations. In a later paper [4], we use the

6 6 BENJAMIN SCHWEINHART to study the behavior of E i α x,..., x n for i.i.d. points sapled fro a easure supported on a set of fractional diension..2. PH -diension. In [3], we defined a faily of persistent hoology diensions for a subset X of a etric space M in ters of the extreal behavior of Eα i Y for subsets x of X: { } di i PH X = inf α : Eα i x < C x X That is, E i α x is uniforly bounded for all α > di i PH X, but not for α < di i PH X. Note that the persistent hoology is taken with respect M. Our results were the first rigorously relating persistent hoology to a classically defined fractal diension, the upper box diension, but the definition is difficult to copute with in practice. Here, we define a siilar notion of fractal diension for easures on a etric space that ay be ore coputable in practice: Definition 4. The PH i -diension of a probability easure on a a triangulable etric space is { } di i PH µ = sup α : li sup E Eα i x,..., x n = Clearly, di i PH µ di i PH supp µ. As a corollary to our ain theore, we show: Theore 6. Let M be the bi-lipschitz iage of a copact -diensional Euclidean siplicial coplex, and 0 i <. If µ is a locally bounded probability easure on M, di i PH µ =.3. Persistent Hoology. If X is a bounded subset of a triangulable etric space M, let X ɛ denote the ɛ-neighborhood of X : X ɛ = {x M : d x, X < ɛ} Also, let H i X be the reduced hoology of X, with coefficients in a field k. The persistent hoology of X is the product ɛ>0 H i X ɛ, together with the inclusion aps i ɛ0,ɛ : H i X ɛ0 H i X ɛ for ɛ 0 < ɛ. The structure of persistent hoology is captured by a set of intervals, which we refer to as PH i X [8]. These intervals represent how the topology of X ɛ changes as ɛ increases. Under certain finiteness hypotheses which are satisfied if X is a finite point set PH i X is the unique

7 WEIGHTED PERSISTENT HOMOLOGY SUMS OF RANDOM ČECH COMPLEXES 7 set of intervals so that the rank of i ɛ0,ɛ equals the nuber of intervals containing ɛ 0, ɛ [5]. If X is finite PH i X is the sae as the persistent hoology of the Čech coplex of X. Note that this depends on the abient etric space. Here, if µ is a probability easure on M and { x j }j N are sapled fro µ, then PH i x,..., x n is the persistent hoology with abient etric space M. All questions we study here would also be interesting in the context of the Vietoris Rips Coplex..4. Notation. In the following, an -space will be the bi-lipschitz iage of a copact -diensional Euclidean siplicial coplex. Also, if the easure µ is obvious fro the context, { x j will denote a collection of independent rando }j N variables with coon distribution µ. Also, x n will be shorthand for {x,..., x n } and x will denote a finite point set. 2. Upper Bounds Our strategy to prove an upper bound for the asyptotics of E i α {x,..., x n } will be to bound the nuber and length of the persistent hoology intervals in ters of the nuber of siplices in a triangulation of the abient etric space. The approach is siilar to that in our earlier paper [3]. 2.. Preliinaries. We require the following result, which is proven by bounding the nuber of persistent hoology intervals of a triangulable etric space of length greater than δ in ters of the nuber of siplices in a triangulation of esh δ: Proposition. Cohen-Steiner, Edelsbrunner, Harer, and Mileyko [6] Let M be an -space. There exists a real nuber C 0 so that for any 0 i <, X M, and δ > 0, {b, d PH i X : d b > δ} C 0 δ We use this result to bound E i α x in ters of the nuber of PH i intervals of x: Lea. Let M be an -space, α <, and i N. There exists a real nuber C > 0 so that Eα i X C PH i X α for all X M. Furtherore, there exists a real nuber D > 0 so that E i X D log PH i X

8 8 BENJAMIN SCHWEINHART for all X M. Proof. Dilating M by a factor r ultiplies E i α X by r α, so we ay assue without loss of generality that the diaeter of M is less than one. Let n = PH i X and I k = { b, d PH i X : Also, let C 0 be as in Proposition so I k C 0 2 k 2 < d b } k+ 2 k The largest C 0 intervals of PH i X each have length less than or equal to 2 0, the next largest C 0 2 intervals have length less than or equal to 2, and so on. It follows that if then log2 2n/C 0 l = n l C 0 2 k k=0 and E i α X l α C 0 2 k 2 k k=0 If α =, the previous inequality becoes as desired. E i α X C 0 l = O log n

9 WEIGHTED PERSISTENT HOMOLOGY SUMS OF RANDOM ČECH COMPLEXES 9 Otherwise, if α <, E i α X l C 0 2 k α k=0 2 αl+ = C 0 2 α C 0 2 α 2 αl+ C 0 2 α 2 = C n α log 2 2n/C 0 +2 α where C = C 04 α 2 α The Upper Bound. The upper bound in our ain theore now follows iediately fro Jensen s inequality, as the function f x = x α is concave for 0 < α : Proposition 2. Let M be an -space, let i be a natural nuber less than, and let µ be a locally bounded probability easure on M. For all 0 < α < there exists a real nuber C > 0 so that E Eα i x,..., x n C E PH i x,..., x n α In particular, if µ has linear PH i expectation and linear PH i variance then there is a C > 0 so that α li n E i α x,..., x n C in probability. Furtherore, there exists a real nuber D so that E E i x,..., x n D log PH i x,..., x n In particular, if µ has linear PH i expectation and linear PH i variance then there is a D > 0 so that li log n Ei x,..., x n D in probability.

10 0 BENJAMIN SCHWEINHART Proof. Let x be a finite subset of B, and let C be as in Lea. If α <, E Eα i x E C PH i x α by Lea C E PH i x α by Jensen s inequality as desired. If µ has linear PH i expectation and linear PH i -variance, Chebyshev s Inequality iplies that li PH i x,..., x n /n C 2 in probability, for soe C 2 > 0, and the desired stateent follows fro Lea. The proof for the case α = is siilar Sharper Upper Bounds. Our sharper upper bounds in Theores 2 and 3 follow fro the fact that if {x,..., x n } is a finite subset of R of S in general position then PH i x,..., x n DT x,..., x n where DT x,..., x n is the nuber of siplices of the Delaunay triangulation on {x,..., x n }. In fact, the Alpha coplex is a filtration on the siplices of the Delaunay triangulation that is hootopy equivalent to the ɛ-neighborhood filtration of the points {x,..., x n } [9]. This construction is usually defined for points in Euclidean space, but easily extends to points on the -sphere, in which case the Delaunay triangulation is the spherical convex hull of the points. Proposition 3. If B be a bounded subset of R Eα i x,..., x n = O n + 2 α for any general position point set {x,..., x n } contained in B. Proof. The Upper Bound Theore [5] iplies that if X R then DT x,..., x n = O n + 2 The desired stateent follows iediately fro Lea and Equation

11 WEIGHTED PERSISTENT HOMOLOGY SUMS OF RANDOM ČECH COMPLEXES 3. Lower Bounds Our strategy to prove lower bounds for the asyptotics of weighted PH -sus is to study collections of sets whose persistent hoology obeys a super-additivity property. We define certain cubical occupancy events giving rise to such collections, and prove that they occur with positive probability for sets of i.i.d. points drawn fro a locally bounded probability easure on an -space. We bootstrap these results by subdividing a subset of an -diensional cube into any sall sub-cubes. This bootstrapping arguent is siilar to the one we used to prove a lower bound for PH i diension in our previous paper [3]. In the following, fix 0 i <. 3.. Super-additivity for Persistent Hoology. Persistent hoology does not in general obey a super-additivity property, but we can define a subclass of sets whose persistent hoology does. If X and T are subsets of a triangulable etric space and b < d, let M X,T b, d be the rank of the hooorphis on hoology induced by the inclusion X b X d X d T d where X ɛ denotes the ɛ-neighborhood of X. Note that M X,C b, d N X b, d where N X b, d is the nuber of intervals of PH i X with birth ties less than b and death ties greater than d. We will show that if C is an -diensional cube and X C, then quantities of the for M X, C d, b obey a super-additivity property. Lea 2. Let {C,..., C n } be -diensional cubes in R so that C j C k C j If X j C j for j =,..., n for any 0 b < d. N j X j b, d M j X j, j C j b, d j, k {,..., n} : j k n M Xj, C j b, d Proof. Let k {,..., n}, S = k j= X j, T = n j= C j, X = X k, and C = C k. See Figure. j=

12 2 BENJAMIN SCHWEINHART Figure. The setup in the proof of Lea 2. We consider the cases i = and i < separately. If i =, Alexander Duality iplies that N S b, d is the nuber of bounded coponents of the copleent of S b that intersect non-trivially with the copleent of S d. Siilarly, M X,C b, d is the nuber of bounded coponents of the copleent of X b that intersect non-trivially with X d C d c. Note that all bounded coponents of Xb c are contained within the interior of C, because C is convex and separates R into two coponents. Let Y be a coponent of the copleent of X b that intersects non-trivially with Xd C d c, and let y Y Xd C d c. C separates R into two coponents so Therefore, d y, S d y, S T = d y, X C > d Y S d c Y S d T d c = Y X d C d c Applying the sae arguent to each X j and counting coponents of the copleent yields the desired inequalities. Otherwise, assue that i. We will show that M S X,T b, d M S,T b, d + M X, C b, d and the desired result will follow by induction. Note that X ɛ S ɛ X ɛ S ɛ T ɛ C ɛ for any ɛ > 0. Consider the following coutative diagra of inclusion hooorphiss and Mayer-Vietoris sequences:

13 WEIGHTED PERSISTENT HOMOLOGY SUMS OF RANDOM ČECH COMPLEXES 3 α b +β b H i X b S b H i X b H i S b H i X b S b 0 = H i Cd φ ψ ζ α d +β d H i Xd C d Hi S d T d H i X d S d T d Observe that M X, C b, d = rank φ, M S,T b, d = rank ψ, and M X S,T b, d = rank ζ. It follows that M X S,T b, d = rank ζ rank α d + β d φ ψ = rank φ ψ because H i Cd = 0 = rank φ + rank ψ = M X, C b, d + M S,T b, d k M Xj, C j b, d j= by induction 3.2. Occupancy Events. If B is a subset of an -space, define the occupancy event { 0 x B = 0 δ B, x = x B > 0 Also, if {A i } r i= and { } s B j, are collections of subsets of M, let j= ξ x, {A i }, { B j } = { δ A i, x = 0 and δ B j, x = i, j 0 otherwise Lea 3. Let µ be a locally bounded probability easure on an -space M. There exists a real nuber V 0 > 0 so for any r, s N there there exists a real nuber γ 0 > 0 so that for any collections of disjoint, congruent cubes { } { } A k i and Bj k, for i {,..., r}, j {,..., s}, and k {,..., n} for a total of r + s n cubes with volue

14 4 BENJAMIN SCHWEINHART Figure 2. The setup in the proof of Lea 4. then for all sufficiently large n. vol A k i = vol Bj k = V 0 /n i, j, k n P ξ k= { } { x, A k i, } Bj k s γ 0 Proof. The proof is nearly identical to that of Lea 3 in [4]. Lea 4. Let 0 < b < d < /6, and V 0 > 0. There exists a λ 0 > 0 so that if C R is an -diensional cube of width R and λ > λ 0, there exist disjoint, congruent cubes { } A j and {Bk } of width R V 0 /λ so that ξ x, { } A j, {Bk } = = M x, C Rb, Rd > 0 Proof. We ay assue without loss of generality that R = and C is centered at the origin. Let S i R be an i diensional sphere of diaeter /3 centered at the origin; note that PH i S i consists of a single interval 0, /6. Let κ = in b, /6 d and 0 = κ/. li δ /δ = δ 0

15 WEIGHTED PERSISTENT HOMOLOGY SUMS OF RANDOM ČECH COMPLEXES 5 so there is a real nuber > 0 so that δ /δ < κ for all δ <. Set λ 0 = V 0 in 0, Choose λ > λ 0, set δ = V 0 /λ, and let C be the cube of width δ /δ centered at the origin. Subdivide C into /δ sub-cubes of width δ. Call this collection of sub-cubes {C l } and let { } { } Aj = c {C l } : S i c = and {B k } = See Figure 2 for an illustration. If x C and the event ξ x, { } A j, {Bk } occurs, then d H x C, S i < κ { } c {C l } : S i c where d H is the Hausdorff distance and we used the fact that the diagonal of an -diensional cube of width δ is δ. The stability of the bottleneck distance [5] iplies that PH i x C includes an interval ˆb, ˆd so that ˆb < κ b < d /3 κ < ˆd In particular, N x C b, d > 0 By construction, 2 d x C, C \ C > 2 δ d C, C > 6 6 κ d so the ɛ-neighborhoods of x C and C \ C are disjoint for all ɛ d. It follows that the aps on hoology induced by the inclusions x C ɛ x ɛ and x ɛ x ɛ C ɛ are injective for all ɛ d. Therefore, M x, C b, d > 0, as desired Proof of the Lower Bound. In the reainder, let µ be a locally bounded probability easure on an -space M, let { x j be i.i.d. saples fro µ, and let }j N x n = {x,..., x n }. Also, let C be as in Lea 3, and rescale M if necessary so that C is a unit cube. Finally, let λ 0 be as in Lea 4.

16 6 BENJAMIN SCHWEINHART The Euclidean Case. For clarity, we first consider the special case where φ M is the identity ap, and µ is a locally bounded probability easure on a copact Euclidean siplicial coplex. The arguent for the general case contains any of the sae eleents. Lea 5. Let 0 < b 0 < d 0 < /6. If n 0 > λ 0, there is a γ > 0 so that li n N x n n0 n0 b0, d0 > γ n n in probability. Proof. Let V 0 be as in Definition 3, and let r = A i and s = Bj, where {Ai } and { Bj } are as in the previous lea. Assuing n > n 0, let ω = n 0 n. Subdivide R into cubes of width ω, and let {D l } Kn l= be the cubes that are fully contained in C. Note that K n := {D l } n/n 0 By the previous lea, there are collections of disjoint, congruent sub-cubes { A l,..., A l r and { B, l..., Bs} l of width ω V0 /n 0 contained inside each cube Dl so that { } { ξ x n, A l i, Bj} l = = M xn Dl, D ωb l 0, ωd 0 > 0 } Note that N xn ωb 0, ωd 0 K n l= K n l= M xn D l, D l ωb 0, ωd 0 by Lea 2 ξ { } { } x n, A l i, Bj l Let γ 0 be as in Lea 3 and γ < γ 0 /n 0. Set δ = + γ n 0 γ 0 2 and ɛ = δ δ

17 WEIGHTED PERSISTENT HOMOLOGY SUMS OF RANDOM ČECH COMPLEXES 7 so /2 < δ < and 0 < ɛ <. Also, find a N so that K n > δn/n 0 for all n > N. Note that 2 γn = γ 0δn δ < ɛ γ 0 K n n 0 δ for all n > N. Therefore, if n > N, P N xn ωb 0, ωd 0 > γn which converges to as n. K n P ξ l= { } { x n, A l i, Bj} l > γn P B K n, γ 0 > γn by Lea 3 P B K n, γ 0 > ɛ γ 0 K n by Equation 2 We can now prove the lower bound in the Euclidean setting: Proposition 4. There is a γ > 0 so that in probability. α li n E i α x n γ Proof. Let 0 < b < d < /6, and let n 0 > λ 0 and γ be as before. Also, let ω = n 0 n /. We have that li in probability. α n E i α x n α li n ωd ωb α N xn ωb, ωd n α/ 0 = li d bα N xn ωb, ωd n n α/ 0 d b α γ by Lea 5 := γ

18 8 BENJAMIN SCHWEINHART The General Case. Before proving the lower bound in our ain theore, we require an interleaving result for the persistent hoology of iages of bi-lipschitz aps: Lea 6. Let M 0 and M be etric spaces and let ψ : M 0 M be L-bilipshitz. If X M 0 and 0 b 0 < d 0 N X b 0 /L, Ld 0 N ψx b 0, d 0 N X Lb 0, d 0 /L Proof. Fix i N, and let j ɛ0,ɛ : X ɛ0 X ɛ and k ɛ0,ɛ : φ X ɛ0 φ X ɛ denote the inclusion aps for ɛ 0 ɛ. By the definition of a bi-lipschitz ap L d M 0 x, y d M ψ x, ψ y Ld M0 x, y for all x, y M 0. In particular, we have the following inclusions: ψ X b0 /L ψ Xb0 ψ X d0 ψ X Ld0 It follows that the rank of ap on hoology induced by i b0 /L,Ld 0 is less than or equal to the rank of the ap induced by j b0,d 0 where we have used that a bi-lipschitz ap is a hoeoorphis. Therefore, N X b 0 /L, Ld 0 N ψx b 0, d 0 The arguent for the other inequality is very siilar. Proposition 5. Let µ be a locally bounded probability easure on an -space M and 0 i <. There is a γ > 0 so that in probability li α n E i α x,... x n > γ Proof. Let L be the bi-lipschitz constant of φ M, and choose b, d > 0 so that Set so L 2 b < d < /6 n 0 = ax d/l Lb, n 0 3 n 0 d/l Lb

19 WEIGHTED PERSISTENT HOMOLOGY SUMS OF RANDOM ČECH COMPLEXES 9 Let ω = n 0 Lea 5 to y n. First, n E i α x n and y n = φ x n. Our strategy is to bound E i α x n by applying ω d/l Lb α N xn ωlb, ωd/l n α/ N xn ωlb, ωd/l by Equation 3 n α/ N yn ωb, ωd by Lea 6 = n α/ N yn ωb, ωd Therefore, li α n E i α x n li n N y n ωb, ωd > γ in probability, where γ > 0 is as given in Lea 5. References [] Henry Adas, Manuchehr Ainian, Elin Farnell, Michael Kirby, Joshua Mirth, Rachel Neville, Chris Peterson, Patrick Shipan, and Clayton Shonkwiler. A fractal diension for easures via persistent hoology. Preprint, 208. [2] David Aldous and J. Michael Steele. Asyptotics for Euclidean inial spanning trees on rando points. Probability Theory and Related Fields, 992. [3] Ulrich Bauer and Florian Pausinger. Persistent Betti nubers of rando Čech coplexes. arxiv: [4] Oer Bobrowski, Matthew Kahle, and Prioz Skraba. Maxially persistent cycles in rando geoetric coplexes. The Annals of Applied Probability, 207. [5] David Cohen-Steiner, Herbert Edelsbrunner, and John Harer. Stability of persistence diagras. Discrete & Coputational Geoetry, 37:0320, [6] David Cohen-Steiner, Herbert Edelsbrunner, John Harer, and Yuriy Mileyko. Lipschitz functions have l p -stable persistence. Foundations of Coputational Matheatics, 200. [7] Vincent Divol and Wolfgang Polonik. On the choice of weight functions for linear representations of persistence diagras. arxiv: , July 208. [8] Rex Dwyer. Higher-diensional voronoi diagras in linear expected tie. Discrete and Coputational Geoetry, 99. [9] H. Edelsbrunner, D. Letscher, and A. Zoorodian. Topological persistence and siplificitaion. Discrete and Coputational Geoetry, [0] A. M. Frieze. On the value of a rando iniu spanning tree proble. Discrete Applied Matheatics, 985.

20 20 BENJAMIN SCHWEINHART [] Yasuaki Hiraoka and Tooyuki Shirai. Miniu spanning acycle and lifetie of persistent hoology in the Linial Meshula process. Rando Structures & Algoriths, 207. [2] Harry Kesten and Sungchul Lee. The central liit theore for weighted inial spanning trees on rando points. Annals of Applied Probability, 996. [3] B. Schweinhart. Persistent hoology and the upper box diension. arxiv: [4] Benjain Schweinhart. The persistent hoology of rando geoetric coplexes on fractals. arxiv: [5] Richard Stalney. The upper bound conjecture and Cohen-Macaulay rings. Studies in Applied Matheatics, 975. [6] J. Michael Steele. Growth rates of Euclidean inial spanning trees with power weighted edges. Annals of Probability, 988. [7] Johannes Steeseder. Rando polytopes with vertices on the sphere. PhD thesis, Paris-Lodron University of Salzburg, 204. [8] A. Zoorodian and G. Carlsson. Coputing persistent hoology. Discrete and Coputational Geoetry, 33: , 2005.