Simplicial Homology of Random Configurations

SIMPLICIAL HOMOLOGY OF R ANDOM CONFIGURA TIONS L. DECREUSEFOND, E. FERRAZ, H. RANDRIAMBOLOLONA, AND A. VER GNE Abstract. Giv en a Poisson process on a d -dimensional toru s, its random geo- metric si mplicial complex is the complex whose ve rtices are the points of the Po isson process and simplices are giv en by the ˘ Cec h complex asso ciated to the co v erage of eac h point. By means of Malliavin calculus, w e compute explic- itly the three first order momen ts of the n umber of k -s i mplices, and pro vide a wa y to compute higher order moment s. The n, w e deriv e the mean and the v ar iance of the Euler c hara cteristic. Using the Stein metho d, we estimate the speed of con v ergence of the num ber of occurrences of any connect ed subcom- plex conv e rges to w ards the Gaussian law when the in tensit y of the Poisson point process tends to infinit y . W e use a concent ration inequality for Poisson processes to find bounds f or the tail distribution of the Betti n um ber of first order and the Euler cha racter istic in such si mplicial complexes. 1. Motiv a tion Algebraic to p olo gy is the domain of mathematics in w hich the top olo gical pro p- erties of a set are a nalyzed thr o ugh algebr aic to ols. Initially dev eloped f or the classification of manifolds, it is by now heavily use d in image pro cessing and geo - metric data analysis. More rece ntly , applications to senso r net works were developed in [7, 11]. Ima gine that we are given a b ounded domain in the plane and s ensors which can detect in truders within a fixed dista nce. The so- called cov erage problem consists in determining whether the do ma in is fully co vered, i.e. whether there is any pa rt of the domain in which an intrusion can o ccur without being detected. The mathematical set which is to b e analyzed here is the union of the balls cen- tered on each sensor. If this set has no hole then the domain is fully cov ered. It turns out that algebra ic topolog y provides a computationally e ffective pro cedure to determine whether this prop erty holds. In view of the rapid developmen t of the 2010 Mathematics Subje ct Classific ation. 60G55,60 H07,55U10. Key wor ds and phr ases. ˘ Cec h complex, Concentr ation inequality , Homology , Malliavin cal- culus, point processes, Rips-Vietoris complex. L. Decreusefond w as partially supp orted by A NR Masterie. The authors would like to thank the anony mous referee for his/her thoroug h reading which helped us to impro v e the presen tation of the curren t article. 1 2 L. DECREUSEFOND, E. FERRAZ, H. RANDRIAMBOLOLONA, AND A. VER GNE tech nology of sens or netw orks [17, 18, 2 4], which are small and cheap devices with limited capacity of a utonomy and communications, dev oted to measure some lo c a l ph ysical quantit y (temperature, humidit y , intrusion, etc.), this k ind o f question is likely to become recurrent. The cov erage problem, via homology techniques, for a set of sensors was first addressed in the pap ers [7, 11]. The metho d co ns ists in calculating fr o m the geo- metric data, a combinatorial ob ject known as a simplicial complex which is a list of po in ts, edges, triangles, tetrahedron, etc. satisfying some compatibilit y conditions: all the f aces of a k - simplex ( k = 0 mea ns p oints, k = 1 means edges, etc.) of the complex must b elong to the set of ( k − 1) - simplices of the complex. Then, a n algebraic structure on these lists and linear maps, known as b oundary op e r ators, are constructed. Some of the top ologica l pr op erties (like co nnectivit y and cov er- age) are giv en b y the s o-called Betti num bers whic h mathematically sp eak ing are dimension of some quotien t v ector spaces (see b elow). Another key parameter is the alternated sum of the Betti n um b ers, known as Euler characteristic whic h gives some infor mation on the glo bal top olo gy o f the studied set. Persistence homology [5, 10] is b oth a w ay to co mpute the Betti n um b ers av o iding a (frequen t) com bi- natorial explosio n and a wa y to detect the robustness of the topolo gical prop erties of a set with resp ect to so me pa rameter: F or instance, in the in trusion detectio n setting, how the connectivity o f the covering domain is altered b y v ariations of the detection distance. When p o int s (i.e. sensors) ar e randomly lo cated in the ambien t spa ce, may it be R d or a ma nifold, it is natural to ask ab out the statistical prop erties of the Betti n umbers a nd the Euler characteristic. W e completely solved the problem in o ne dimension (see [8]) by basic metho ds inspired by queuing theo r y , without using the forthcoming sophisticated to ols of alge br aic topolo gy . Since w e cannot order p oints in R d , it is not p ossible to g eneralize the results obtained in this earlier work to higher dimension. A very few pa per s deal with the prop erties of random simplicial co mplexes. In [15] and [22], f or Binomial point pr o cesses whose n umber of points are going to infinity , the a symptotic regimes of the mean v alue o f the Betti num ber s and simplices num bers are inv estigated. In [16], these results are re fined by providin g P oisson and Gaus s ian a pproximations of the Betti nu mbers in asymptotic regimes. As will b e appar e n t b elow, for the Rips-Vietoris simplicial complex, the num ber of k -simplices bo ils do wn to the n um ber of ( k + 1) -cliques o f the underlying graph. SIMPLICIAL HOM OLOGY OF RANDOM CONFIGURA TIONS 3 As we mainly analyze this kind of complex, our w ork has thus strong links with the pioneering work of P enrose [22] and with [3, 23] as well. In [3], for P oisson input, the limiting reg imes of the n um b e r of k -simplices on a square of size a , a re inv e s tigated through limit theore ms of U-statistics. The size of the square is growing to infinit y with a cons tan t mean n um be r of p oints p er unit of surface. In [ 2 3], there is a n extension of the latter result to non-linea r manifolds and n o n-uniform distributions of the po int s. In the ab ov e cited pap ers, so phisticated combinatorial a r guments are at the ro ot of the arguments of the conv ergence theorems. W e her e replace this line of though t b y a functional analytic appro ach which transfers the difficulty to the computation of a (p oss ibly in volv ed) deterministic in tegral. By doing so, w e can, in principle, o btain a CL T for the n um b e r of o ccurrence s of an y co nnected sub-complex and no t only for cliques. Figure 1. A sub-complex whic h is not a clique One of our main con tributions are exact formulas for the fir st thr e e momen ts of the num ber of simplices for both the Poisson and the B inomial pro cesses at the price of working on a square bo unded domain, which w e em bed into a torus in order to a void side effects. The rationale behind this simplificatio n is that when the size o f the cov ering balls is sma ll compar ed to the s ize of the square, the top ology of the tw o sets (the union of balls in the square a nd th e union of balls in the corresp onding tor us) must b e simila r . Our method could be generalized to compute the moment s of any order but the computations b ecome more and more tricky as the or der incr eases. W e a lso inv e s tigate the prop erties of the momen ts of the Euler characteristic. Moreov er, b y using Malliavin calculus, w e g o further than the previously cited works since we can ev aluate the spe e d of conv ergence in the CL T. W e also g ive a concentration inequality to bound the distr ibution tail o f the first Betti num ber. During the final preparatio n o f this pap er, we lear ned that such an approach was used indep e ndently in [26] for the analysis of U -statistics (functionals of a fixed 4 L. DECREUSEFOND, E. FERRAZ, H. RANDRIAMBOLOLONA, AND A. VER GNE chaos in our vocabular y) of P oisson pro cesses. B oth approaches r ely on the ideas which app ear ed in [9, 20]. Our metho d go es as follows: W e wr ite the n um be r s of k -s implices (i.e. po int s, edges, tr iangles, tetrahedron, etc) as iterated in tegrals with resp e ct to the under- lying P oisson pro cess. Then, the co mputation of the means is reduced to the com- putation of deterministic iterated integrals thanks to Ca mpbell formula. By using the definition of the Euler characteristic as an alternating sum of the num bers of simplices, we find its expec tation. The p oint is that even if the summing index go es to infinity , the expectatio n of χ dep ends o nly on the d -th power of the int ensity of the P oisson pro cess where d is the dimension of the underlying space. By de- po issonization, we obtain the exact v alues o f the mean num ber of simplices of an y order and then the mean v a lue of the Euler characteristic for Binomial pro cesses. Using the multiplication formula of iterated integrals, one c a n repro duce the s a me line of thought for higher order momen ts to the price of a n incr eased co mplexity in the computations. W e obtain clo sed for m formulas for the v ar iance of the num ber of k -simplices and of the Euler characteristic and a series ex pa nsion for third order moment s. O ur metho d is a prio ri suitable for an y higher order momen ts but the computations become muc h inv olved. Using Stein’s metho d mixed with Malliavin calculus, w e generalize the results of [22] b y proving a precis e (i.e. with sp eed of conv ergence) CL T for sub-complexes count. As exp ected, the speed of con vergence is of the order of λ − 1 / 2 . The paper is o rganized a s follows: Sections 2 and 3 are primers on algebraic top o logy and Malliavin calculus. In Section 4, the av erage num ber of simplices and the mean of the Euler characteristic are computed. This is sufficient to bound the t ail di stribution of β 0 using c o ncent ration inequality . Section 5 applies the Malliavin calculus in order to find the explicit expression of second order momen ts of the n um ber of k -s implices and the Euler characteristic. Using the same strategy , in Section 6, w e find the expres sion for the third o r der moment of the n um be r of simplices. In Section 7, we prov e a c e n tral limit theorem for the num b er of o ccurrences of a finite simplex into a Poisson random g eometric complex. 2. Algebraic Topology F or further rea ding on algebraic top olo gy , see [1, 13, 19]. Gra phs can be gener - alized to more generic co m binatorial ob jects kno wn a s s implicia l co mplexe s . While graphs model binary relations, simplicial complexes represent higher order relations. SIMPLICIAL HOM OLOGY OF RANDOM CONFIGURA TIONS 5 Given a set o f vertices V , a k -simplex is an unordered subset { v 0 , v 1 , . . . , v k } where v i ∈ V and v i 6 = v j for a ll i 6 = j . The faces of the k - simplex { v 0 , v 1 , . . . , v k } a re defined as all the ( k − 1) -simplices of the form { v 0 , . . . , v j − 1 , v j +1 , . . . , v k } with 0 ≤ j ≤ k . A simplicial complex C is a collection of simplices which is closed with resp ect to the inclusio n of faces, i.e. if { v 0 , v 1 , . . . , v k } is a k -simplex then all its faces are in the set o f ( k − 1) -simplices. One can define an orientation on simplices b y defining an order on vertices and with the c onv e n tion that: [ v 0 , . . . , v i , . . . , v j , . . . , v k ] = − [ v 0 , . . . , v j , . . . , v i , . . . , v k ] , for 0 ≤ i, j ≤ k . F or eac h integer k , C k is the vector spa c e spanned by t he set of oriented k - simplices of V . F or any int eger k , the b oundary map ∂ k is the linear transformation ∂ k : C k → C k − 1 which ac ts on basis elements [ v 0 , . . . , v k ] as ∂ k [ v 0 , . . . , v k ] = k X i =0 ( − 1) i [ v 0 , . . . , v i − 1 , v i +1 , . . . , v k ] , and ∂ 0 is the n ull function. Examples o f such o per ations are given in T a ble 1. v 0 v 1 v 2 v 0 − v 2 + [ v 0 , v 1 ] + [ v 1 , v 2 ] ∂ − → [ v 2 ] − [ v 0 ] v 0 v 1 v 2 v 0 v 1 v 2 [ v 0 , v 1 , v 2 ] ∂ − → [ v 1 , v 2 ] − [ v 0 , v 2 ] +[ v 0 , v 1 ] v 0 v 1 v 2 v 3 Filled Empty v 0 v 1 v 2 v 3 [ v 0 , v 1 , v 2 , v 3 ] ∂ − → +[ v 1 , v 2 , v 3 ] − [ v 0 , v 2 , v 3 ] +[ v 0 , v 1 , v 3 ] − [ v 0 , v 1 , v 2 ] a) b) c) T able 1. Examples o f b oundar y maps. F rom left to right. An application o ver 1-simplices. Over a 2 -simplex. Over a 3-simplex, turning a filled tetrahedron to a n empt y o ne. These maps give rise to a chain complex: a sequence of vector s paces and linear transformations : · · · ∂ k +2 − → C k +1 ∂ k +1 − → C k ∂ k − → · · · ∂ 2 − → C 1 ∂ 1 − → C 0 . A standard result then ass e r ts that for any in teger k , ∂ k ◦ ∂ k +1 = 0 . If o ne defines Z k = k er ∂ k and B k = im ∂ k +1 , this induces that B k ⊂ Z k . 6 L. DECREUSEFOND, E. FERRAZ, H. RANDRIAMBOLOLONA, AND A. VER GNE 0 0 0 C k Z k B k C k +1 C k − 1 ∂ − → ∂ − → Figure 2. A c hain complex showing the sets C k , Z k and B k . The k -th homolog y g roup of C , deno ted b y H k , is the quotien t vector space, H k = Z k /B k and the k -th Betti n um ber of C is its dimension: β k = dim H k = dim Z k − dim B k . The simplicial complexes we consider are of a sp ecial t ype since they are built on top ologica l rules. Definition 1. Give n U = ( U v , v ∈ ω ) a c ol le ction of op en sets of some t op olo gic al sp ac e X , the ˘ Ce ch c omplex of U denote d by C ( U ) , is the abstr act simplicial c omplex whose k -simpli c es c orr esp ond to ( k + 1) -tuples of distinct elements of U that have n on empty interse ction, so { v 0 , v 1 , . . . , v k } is a k - simplex if and only if T k i =0 U v i 6 = ∅ . F or the a pplica tions we have in mind, the set U v is meant to b e the cov ered zone b y the s e ns or located at v . F o r the s a ke of tractabilit y , it is supp osed to b e a ball cent ered at v with a fixed radius. As said earlier , in or der to avoid side effects, we work on the d -dimensio nal torus of length a , deno ted by T d a and to simplify the computations, we consider the l ∞ distance. Namely , the tor us is defined as the quotient of the action o f the group of translations a Z d on R d , i.e. T d a = R d /a Z d . The space X = [0 , a ) d can b e embedded in T d a as a fundamental domain of this action. If we equip X with the distance ρ d ( x, y ) = inf k ∈ Z d k x − y + k a k ∞ where k x k ∞ is the l ∞ -norm in R d , then the embedding of X into the tor us is a bi- jectiv e isometry . One can thus identify these tw o spaces and use the represent ation which is the most co n venien t according to the s itua tio n. Definition 2. Given ω a finite set of p oints on the torus T d a . F or ǫ > 0 , we define U ǫ ( ω ) = { B ρ d ( v , ǫ ) , v ∈ ω } and C ǫ ( ω ) = C ( U ǫ ( ω )) , wher e B ρ d ( x, ǫ ) = { y ∈ T d a , ρ d ( x, y ) < ǫ } . SIMPLICIAL HOM OLOGY OF RANDOM CONFIGURA TIONS 7 The follo wing r esult is known for R d , there is a slight mo dification of the pro o f for the to rus. Theorem 1. Supp ose ǫ < a/ 4 . T hen C ǫ ( ω ) has the same homolo gy ve ctor sp ac es as U ǫ ( ω ) . In p articular they have the same Betti numb ers. Pr o of. This will follow from the so-called nerve lemma of Leray , as s tated in [27, Theorem 7.2 6] or [4 , Theor em 10 .7]. One only needs to chec k that a n y non-empty in tersection of s ets B ρ d ( v , ǫ ) is co n tractible. Consider such a non-empt y intersection, a nd let x b e a p oint contained in it. Then, since ǫ < a/ 4 , the ball B ρ d ( x, 4 ǫ ) can b e ident ified with a cub e in the Euclidean spa ce. Then each B ρ d ( v , ǫ ) containing x is contained in B ρ d ( x, 4 ǫ ) , hence also b ecomes a cube with this identification, hence co nvex. T hen the intersection of these co nv ex sets is conv ex, hence contractible.  F or any finite sets of points ω of the tor us, acco r ding to the geometrica l definition of the ˘ Cech complex, the Betti num b ers hav e a geometrica l meaning: β 0 ( ω ) (with obvious notatio ns ) is the num ber o f co nnected co mp onents a nd for k ≥ 1 , β k ( ω ) is the num ber of k -dimensional ho les of U ǫ ( ω ) . F o r k = 0 and k = 1 , a n intuitiv e explanation can b e g iven. By definition, β 0 ( ω ) is the n um be r of p oints minus the n umber of indep enden t edges. Eac h time there exists a cycle w ith n p oints, we can remov e an edge without alter ing β 0 ( ω ) since there a re n − 1 independent edges in such a cycle. Doing this rep eatidly , one can reduce the orig i nal gr aph to as man y linear chains of edges as there are connected comp onents. A linea r c hain of edges which con tains n p oints has n − 1 edges, hence a β 0 ( ω ) equal to 1 . Thus β 0 ( ω ) counts the n um ber of connected comp onents. As to β 1 ( ω ) , w e remark that ker ∂ 1 is comp osed b y the cycles and that B 1 is the set of linear combinations of edges forming triangles, hence β 1 ( ω ) is the n um be r of cycles which are not triangles, hence represents the n um b e r of “coverage” holes. The well known top olo g ical in v ariant named Euler characteristic for U ǫ ( ω ) , deno ted by χ ( ω ) , is an integer defined b y: χ ( ω ) = ∞ X i =0 ( − 1) i β i ( ω ) . Let s k ( ω ) be the n um b e r o f k -simplices in the simplicial complex C ǫ ( ω ) . A w ell known theorem states that this is also given by: χ ( ω ) = ∞ X i =0 ( − 1) i s i ( ω ) . 8 L. DECREUSEFOND, E. FERRAZ, H. RANDRIAMBOLOLONA, AND A. VER GNE Definition 3. L et ω b e a finite set of p oints in T d a . F or any ǫ > 0 , the R ips- Vietoris c omplex of ω , R ǫ ( ω ) , is the abstr act simplicial c omplex whose k -simplic es c orr esp ond to unor der e d ( k + 1) -tuples of p oints in ω which ar e p airwise within distanc e less than ǫ of e ach other. Lemma 2. F or the torus T d a e quipp e d with the pr o duct distanc e ρ d , the Rips- Vietoris c omplex R 2 ǫ ( ω ) has the same Betti numb ers as the ˘ Ce ch c omplex C ǫ ( ω ) . The pr o of is g iven in [1 1] in a slightly different con text, but it is easy to chec k tha t it w orks here as w ell. It m ust b e p ointed out that ˘ Cech and Rips-Vietoris simplicial complexes ca n b e defined similarly for any distance on T d a but it is only for the pro duct distance that the homology vector spaces of bo th complexes co incide. Prop osition 3. L et ω ∈ T d a b e a set of p oints, gener ating the simplicia l c omplex C ǫ ( ω ) . Th en, if i > d , β i ( ω ) = 0 . Pr o of. By Theorem 1, C ǫ ( ω ) has the same ho mology as U ǫ ( ω ) . But U ǫ ( ω ) is an op en manifold o f dimensio n d , so its Betti num b ers β i ( ω ) v anish for i > d , s ee for example [12, Theorem 22.24].  Prop osition 4. L et ω ∈ T d a b e a set of p oints, gener ating the simplicia l c omplex C ǫ ( ω ) . Th er e ar e only two p ossible values for the d -th Betti numb er of C ǫ ( ω ) : i) β d ( ω ) = 0 , or ii) β d ( ω ) = 1 . If the latter c ondition holds, then we also have χ ( ω ) = 0 . Pr o of. By Theorem 1, C ǫ ( ω ) has the same homology as U ǫ ( ω ) . Now, U ǫ ( ω ) is an op en sub-manifold of the torus, s o there are o nly tw o pos sibilities: i) U ǫ ( ω ) is a strict o pen sub-manifold, hence non-compact ii) U ǫ ( ω ) = T d a . In the first case, β d ( ω ) = 0 by [1 2, Corollary 22.25]. In the second case C ǫ ( ω ) has same homolo gy as the torus, hence β d ( ω ) = 1 and χ ( ω ) = 0 .  3. Poisson point process and Mallia vin calculus The s pa ce o f configurations on X = [0 , a ) d , is the set of lo c a lly finite s imple po in t measur es (see [6, 2 5] for details): Ω X = ( ω = n X k =1 δ ( x k ) : ( x k ) k = n k =1 ⊂ X, n ∈ N ∪ {∞} ) , SIMPLICIAL HOM OLOGY OF RANDOM CONFIGURA TIONS 9 where δ ( x ) deno tes the Dirac measure at x ∈ X . It is often con venien t to identify an element ω of Ω X with the se t corresp onding to its support, i.e. P n k =1 δ ( x k ) is iden tified with the unordered set { x 1 , . . . , x n } . F or A ∈ B ( X ) , we ha v e δ ( x )( A ) = 1 A ( x ) , so that ω ( A ) = X x ∈ ω 1 A ( x ) , counts the n umber of atoms in A . Simple measur e means that ω ( { x } ) ≤ 1 for any x ∈ X . Lo cally finite means that ω ( K ) < ∞ for any compact K of X . The configuration space Ω X is endow ed with the v ague top ology and its a sso ciated σ - algebra denoted b y F X . T o c haracterize the randomness of the system, we co nsider that the set of p oints is repr esented by a Poisson p oint pro cess ω with in tensit y measure d Λ( x ) = λ d x in X . The para meter λ is called the intensit y of the P oisson pro cess. Since ω is a Poisson po in t pro cess o f in tensity measure Λ : i) F o r any co mpact A , ω ( A ) is a r andom v a riable of pa rameter Λ( A ) : P ( ω ( A ) = k ) = e − Λ( A ) Λ( A ) k k ! · ii) F o r any disjoint sets A, A ′ ∈ B ( X ) , the ra ndom v ariables ω ( A ) and ω ( A ′ ) are independent. Along this pa per , w e r efer E Λ [ F ] a s the mean of some function F depe nding on ω giv en that the in tensit y measure o f this proc e s s is Λ . The notations V ar Λ [ F ] and Cov Λ [ F, G ] are defined accordingly . As sa id ab ov e, a configuration ω can b e viewed as a mea sure on X . It also induces a mea sure on an y X n , called the factorial measure asso cia ted to ω of order n , defined b y ω ( n ) ( C ) = X ( x 1 , ··· , x n ) ∈ ω x i 6 = x j 1 C ( x 1 , · · · , x n ) , for any C ∈ X n , with the conv en tion that ω ( n ) is the null measure if ω has les s than n p oints. Let f ∈ L 1 (Λ ⊗ n ) and let F be a random v ariable given by F ( ω ) = X x i ∈ ω x i 6 = x j f ( x 1 , . . . , x n ) = Z f ( x 1 , . . . , x n ) d ω ( n ) ( x 1 , · · · , x n ) . The Campb ell-Meck e formula for P oisson p oint pro cesses states that E Λ [ F ] = Z X n f ( x 1 , . . . , x n ) d Λ( x 1 ) . . . d Λ( x n ) . In view of this result, it is natural to in tro duce the compens ated factoria l measures defined by : dω (1) Λ ( x ) = dω ( x ) − d Λ( x ) 10 L. DECREUSEFOND, E. FERRAZ, H. RANDRIAMBOLOLONA, AND A. VER GNE and for n ≥ 2 , for any f ∈ L 1 (Λ ⊗ n ) , Z f ( x 1 , · · · , x n ) dω ( n ) Λ ( x 1 , · · · , x n ) = Z   Z f ( x 1 , · · · , x n )( d ( ω − n − 1 X j =1 δ ( x j ))( x n ) − d Λ( x n ))   dω ( n − 1) Λ ( x 1 , · · · , x n − 1 ) . A rea l-v alued function f : X n → R is ca lled symmetric if f ( x σ (1) , . . . , x σ ( n ) ) = f ( x 1 , . . . , x n ) for all p ermutations σ of S n . Then the space of square integrable symmetric functions o f n v ar iables is denoted by L 2 ( X, Λ) ◦ n . F or f ∈ L 2 ( X, Λ) ◦ n , the m ultiple P oisson sto chastic in tegral I n ( f n ) is then defined a s I n ( f n )( ω ) = Z f n ( x 1 , . . . , x n ) d ω ( n ) Λ ( x 1 , · · · , x n ) . It is known that I n ( f n ) ∈ L 2 (Ω X , P ) . Moreov er, if f n ∈ L 2 ( X, Λ) ◦ n and g m ∈ L 2 ( X, Λ) ◦ m , the isometry formula (1) E Λ [ I n ( f n ) I m ( g m )] = n ! 1 m ( n ) h f n , g m i L 2 ( X, Λ) ◦ n holds true. F urthermore, we hav e: Theorem 5. Eve ry r andom variabl e F ∈ L 2 (Ω X , P ) admits a unique Wiener- Poisson de c omp osition of the typ e F = E Λ [ F ] + ∞ X n =1 I n ( f n ) , wher e the series c onver ges in L 2 (Ω X , P ) and, for e ach n ≥ 1 , the kernel f n is an element of L 2 ( X, Λ) ◦ n . Mor e over, by definition V ar Λ [ F ] = k F − E Λ [ F ] k 2 L 2 (Ω X , P ) then we have the isometry (2) V ar Λ [ F ] = ∞ X n =1 n ! k f n k 2 L 2 ( X, Λ) ◦ n . F or f n ∈ L 2 ( X, Λ) ◦ n and g m ∈ L 2 ( X, Λ) ◦ m , w e define f n ⊗ l k g m , 0 ≤ l ≤ k , to be the function: (3) ( y l +1 , . . . , y n , x k +1 , . . . , x m ) 7− → Z X l f n ( y 1 , . . . , y n ) g m ( y 1 , . . . , y k , x k +1 , . . . , x m ) d Λ( y 1 ) . . . d Λ( y l ) . W e denote by f n ◦ l k g m the symmetrization in n + m − k − l v ariables of f n ⊗ l k g m , 0 ≤ l ≤ k . This leads us to the next prop os itio n (see [25] for a pro of ): SIMPLICIAL HOM OLOGY OF RANDOM CONFIGURA TIONS 11 Prop osition 6. F or f n ∈ L 2 ( X, Λ) ◦ n and g m ∈ L 2 ( X, Λ) ◦ m , we have I n ( f n ) I m ( g m ) = 2( n ∧ m ) X s =0 I n + m − s ( h n,m,s ) , wher e h n,m,s = X s ≤ 2 i ≤ 2( s ∧ n ∧ m ) i !  n i  m i  i s − i  f n ◦ s − i i g m b elongs to L 2 ( X, Λ) ◦ n + m − s , 0 ≤ s ≤ 2 ( m ∧ n ) . In what follows, given f ∈ L 2 ( X, Λ) ◦ q ( q ≥ 2) and t ∈ X , we denote by f ( ∗ , x ) the function o n X q − 1 given b y ( x 1 , . . . , x q − 1 ) 7− → f ( x 1 , . . . , x q − 1 , x ) . Definition 4. L et Dom D b e t he the set of r andom variables F ∈ L 2 (Ω X , P ) admitting a chaoti c de c omp osition such that ∞ X n =1 q q ! k f n k 2 L 2 ( X, Λ) ◦ n < ∞ . L et D b e define d by D : Dom D → L 2 (Ω X × X , P ⊗ Λ) F = E Λ [ F ] + X n ≥ 1 I n ( f n ) 7− → D x F = X n ≥ 1 nI n − 1 ( f n ( ∗ , x )) . It is known, cf. [14] , that we also hav e D x F ( ω ) = F ( ω ∪ { x } ) − F ( ω ) , P ⊗ Λ − a.e. Definition 5. The Ornstein-Uhlenb e ck op er ator L is given by LF = − ∞ X n =1 nI n ( f n ) , whenever F ∈ Dom L , given by those F ∈ L 2 (Ω X , P ) such that their chaos exp an- sion verifies ∞ X n =1 q 2 q ! k f n k 2 L 2 ( X, Λ) ◦ n < ∞ . Note that E Λ [ LF ] = 0 , by definition and (1 ) . Definition 6. F or F ∈ L 2 (Ω X , P ) such that E Λ [ F ] = 0 , we may define L − 1 by L − 1 F = − ∞ X n =1 1 n I n ( f n ) . Combining Stein’s metho d and Malliavin calculus yields the following theorem, see [21]: 12 L. DECREUSEFOND, E. FERRAZ, H. RANDRIAMBOLOLONA, AND A. VER GNE Theorem 7. L et F ∈ Dom D b e such that E Λ [ F ] = 0 and V ar ( F ) = 1 . Then, d W ( F, N (0 , 1)) ≤ E Λ      1 + Z X [ D x F × D x L − 1 F ] d Λ( x )      + Z X E Λ h | D x F | 2   D x L − 1 F   i d Λ( x ) . Another result from the Malliavin calculus used in this w ork is the following one, quoted from [25 ]: Theorem 8. L et F ∈ Dom D b e such that DF ≤ K , a.s., for some K ≥ 0 and we denote k D F k L ∞ ( L 2 ( X, Λ) , P ) := sup ω Z X | D x F ( ω ) | 2 d Λ( x ) < ∞ . Then (4) P ( F − E Λ [ F ] ≥ x ) ≤ exp  − x 2 K log  1 + xK k D F k L ∞ ( L 2 ( X, Λ) , P )  . 4. First order moments Let ω denote a generic realization of a Poisson point pro ces s on the torus T d a and C ǫ ( ω ) the a sso ciated ˘ Cech co mplex with ǫ < a/ 4 . A Poisson pro cess in R d of in tensit y λ dilated b y a factor α is a P oisson pro cess of intensit y λα − d . Hence, statistically , the homolo gical pro p er ties of a Poisson pro cess of intensit y λ , inside a torus of length a with ball size s ǫ ar e the sa me as that of a Poisson pr o cess of in tensit y λα − d , inside a torus of length αa with ball sizes αǫ . Th us there are only t wo degrees of freedom among λ , a , and ǫ . F o r ins ta nce, we can set a = 1 and the general results are obtained by a mult iplication of magnitude a d . Strictly sp eaking, Betti n um ber s, Euler characteristic a nd num ber o f k -simplices are functions of C ǫ ( ω ) but w e will skip this dep endence for the sa ke of no tations. W e also define N k as the num ber o f ( k − 1) -simplices. In this section, w e ev aluate the mean of the num b er of ( k − 1) -simplices E Λ [ N k ] and the mean of the Euler c haracteristic E Λ [ χ ] . W e intro duce some notations. Let ∆ ( d ) k = { ( v 1 , . . . , v k ) ∈ ([0 , a ) d ) k , v i 6 = v j , ∀ i 6 = j } . F or a ny integer k , we define ϕ ( d ) k as: ϕ ( d ) k : ([0 , a ) d ) k − → { 0 , 1 } ( v 1 , · · · , v k ) 7− →      Q 1 ≤ i λa d , we have P Λ ( β 0 ≥ y ) ≤ exp  − y − λa d 2 log  1 + y − λa d (2 d − 1) 2 λ  · Pr o of. β 0 is the num ber of co nnected comp onents. Since ther e are more p oints than connected comp onents, E Λ [ β 0 ] ≤ E Λ [ N 1 ] = λa d . According to the definition of D , sup x ∈ X D t β 0 is the maximum v ariation of β 0 induced by the addition of a n arbitrary po in t. If the p oint x is at a distance smaller than ǫ from ω , then D x β 0 ≤ 0 , otherwise, D x β 0 = 1 , so D x β 0 ≤ 1 for a ny x ∈ X . Besides, this added p oint can join a t most t w o connected compo nents in eac h dimension, so in d dimensions it ca n 16 L. DECREUSEFOND, E. FERRAZ, H. RANDRIAMBOLOLONA, AND A. VER GNE join a t most 2 d connected component, whic h means that D β 0 ranges from − (2 d − 1) to 1 , and then k D β 0 k L ∞ ( L 2 ( X, Λ) , P ) ≤ sup ω Z X | D x β 0 | 2 d Λ( x ) ≤ λa d (2 d − 1) 2 . Since the function f defined by f ( x, y ) = exp  − k 1 − x 2 k 2 log  1 + ( k 1 − x ) k 2 k 3 y  · is strictly increasing with res p ect to x and y for k 1 > x , it follows from Theorem 8 that: P Λ ( β 0 ≥ y ) ≤ exp  − y − λa d 2 log  1 + y − λa d (2 d − 1) 2 λa d  , for y > λa d ≥ E Λ [ β 0 ] .  5. Second order moments W e now deal with the co mputations of the second order moments. The pro ofs rely on the c haos decompo sition of the n um be r of simplices (see Lemma 14) and the multiplication for mu la fo r itera ted in tegrals (see Prop osition 6). The compu- tations are rather technical and postp oned to Appendix A. W e make the following conv en tion: F o r any in teger k , Z X 0 ϕ ( d ) k ( v 1 , · · · , v k ) d v 1 . . . d v 0 = ϕ ( d ) k ( v 1 , · · · , v k ) . Lemma 14 . W e c an re write N k as N k = 1 k ! k X i =0  k i  λ k − i I i  Z X k − i ϕ ( d ) k ( v 1 , · · · , v k ) d v 1 . . . d v k − i  . Pr o of. F or k = 1 , the r esult is immediate with the conv en tion ma de ab ov e. Once we have seen that Z ϕ ( d ) k ( v 1 , . . . , v k ) d ω ( k ) ( v 1 , · · · , v k ) = Z   Z ϕ ( d ) k ( v 1 , . . . , v k )( d ( ω − k − 1 X j =1 δ ( v j ))( v k ) − d Λ( v k ))   d ω ( k − 1) ( v 1 , · · · , v k − 1 ) + Z  Z X ϕ ( d ) k ( v 1 , . . . , v k ) d Λ( v k )  d ω ( k − 1) ( v 1 , · · · , v k − 1 ) , the result follows by induction.  SIMPLICIAL HOM OLOGY OF RANDOM CONFIGURA TIONS 17 Theorem 15. The c ova rianc e b etwe en the numb er of ( k − 1) -simplic es N k , and the numb er of ( l − 1) -simplic es, N l for l ≤ k is given by Cov Λ [ N k , N l ] = l X i =1 λa d ( λ (2 ǫ ) d ) k + l − i − 1 i !( k − i )!( l − i )!  k + l − i + 2 ( k − i )( l − i ) i + 1  d . R emark. As for the first moment it is still p ossible to find, considering the Euclidean norm, a closed-form expres sion for V ar Λ [ N k ] . W e did not find a gener al expres s ion for any dimension. How ever, when we co nsider the Rips-Vietoris complex in T 2 a , the v a riance of the n umber of 1- simplices and 2-s implices are g iven by: V ar Λ [ N 2 ] =  a 2 ǫ  2  π 2 (4 λǫ 2 ) 2 + π 2 (4 λǫ 2 ) 3  , and V ar Λ [ N 3 ] =  a 2 ǫ  2 (4 λǫ ) 3 π 6 π − 3 √ 3 4 ! + (4 λǫ 2 ) 4 π π 2 2 − 5 12 − π √ 3 2 ! +(4 λǫ 2 ) 5 π 2 4 π − 3 √ 3 4 ! 2   · Since w e ha ve an expression for the v aria nce of the n um be r o f k -simplices, it is po ssible to calculate the v ar iance of the Euler characteristic. Theorem 16 . The varianc e of t he Euler char acteristic is: V ar Λ [ χ ] = λa d ∞ X n =1 c d n ( λ (2 ǫ ) d ) n − 1 , wher e c d n = n X j = ⌈ ( n +1) / 2 ⌉   2 j X i = n − j +1 ( − 1) i + j ( n − j )!( n − i )!( i + j − n )!  n + 2( n − i )( n − j ) 1 + i + j − n  d − 1 ( n − j )! 2 (2 j − n )!  n + 2( n − j ) 2 1 + 2 j − n  d # . Theorem 17. In one dimension, the expr ession of the varianc e of the Euler char- acteristic is: V ar Λ [ χ ] = a  λe − 2 λǫ − 4 λ 2 ǫe − 4 λǫ  . Theorem 18 . If d = 2 , we have Dχ ≤ 2 and thus P ( χ − E Λ [ χ ] ≥ x ) ≤ exp  − x 4 log  1 + x 2 λa 2  · 18 L. DECREUSEFOND, E. FERRAZ, H. RANDRIAMBOLOLONA, AND A. VER GNE Pr o of. In t w o dimensions, accor ding to Prop osition 3, the Euler characteristic is given b y: χ = β 0 − β 1 + β 2 . If w e add a vertex on the torus, either the vertex is isola ted or not. In the fir s t case, it fo rms a new co nnected component increas ing β 0 b y 1 , and the n um b e r of holes, i.e. β 1 , remains the same. Other wise, a s there is no new connected comp onent, β 0 is the same, but the new vertex ca n at most fill one hole, increa s ing β 1 b y 1 . Therefore, the v ariation of β 0 − β 1 is at most 1 . F urthermore , when we add a vertex to a simplicial complex, we kno w from Prop osition 4 that D β 2 ≤ 1 hence Dχ ≤ 2 . Then, we use Eq. (4) to complete the pro of.  6. Third order moments Higher order momen ts ca n b e computed in a similar wa y but the computations bec o me trickier as the order increases. W e here restrict our computations to the third order moments to illustrate the general pro cedure. The pr o of is given in Appendix B. W e are interested in the cent ral momen t, so we in troduce the following notation for the c en tralized num b er of ( k − 1) -simplices: f N k = N k − E Λ [ N k ] . Theorem 19. The thir d c entr al moment of the numb er of ( k − 1) - simplic es is given by: E Λ h f N k 3 i = X i,j,s,t λ 3 k − i − j t ! ( k !) 3  k i  k j  k s  i t  j t  t i + j − s − t  J 3 ( k , i, j, s, t ) , with s ≥ | i − j | , and J 3 ( k , i, j, s, t ) is an inte gr al dep ending on k , i, j, s and t , define d b elow in (9) . 7. Convergence Before g oing further, we must answ er a natural question: Do we retrieve the torus homology when the in tensit y of the P oisson pro cess go es to infinit y , so that the num b er o f points b ecomes arbitrar y lar ge ? The answer is p ositive as shows the next theorem. Theorem 20. The Betti n umb ers of C ǫ ( ω ) c onver ge in pr ob ability to the Betti numb ers of the torus as λ go es to infinity: P Λ d \ i =0  β i ( ω ) =  d i  ! λ →∞ − − − − → 1 , SIMPLICIAL HOM OLOGY OF RANDOM CONFIGURA TIONS 19 wher e  d i  is the i -th Betti numb er of the d -dimensional torus, se e [13] . Pr o of. Let η < ǫ/ 2 , by compactness of the tor us , there exis ts B a finite collection of balls of radius η cov ering T d a . Since η < ǫ/ 2 , if x b elong s to some ball B ∈ B then B ⊂ B ( x, ǫ ) , hence \ B ∈ B ( ω ( B ) 6 = 0) ⊂  U ǫ ( ω ) = T d a  . Thu s, P Λ  U ǫ ( ω ) 6 = T d a  ≤ P Λ [ B ∈ B ( ω ( B ) = 0) ! ≤ K exp( − λ (2 η ) d ) λ →∞ − − − − → 0 . Moreov er, b y the nerve lemma  U ǫ ( ω ) = T d a  ⊂ d \ i =0  β i ( ω ) =  d i  , and the r esult follows.  Let Γ be an arbitrar y co nnected simplicial complex of n v ertices. The num ber of o ccurrences o f Γ in C ǫ ( ω ) is denoted as G Γ ( ω ) . It m ust b e no ted that with our construction of the simplicial complex, a complex Γ app ears in C ǫ ( ω ) a s so o n as its edges are in C ǫ ( ω ) . Th e s et of edges of Γ , denoted b y J Γ is a subset of { 1 , . . . , n } × { 1 , . . . , n } . Let us define the following function on the vertices of Γ : f h Γ ( v 1 , . . . , v n ) = 1 c Γ Y ( i,j ) ∈ J Γ 1 ρ d ( v i ,v j ) <ǫ , where c Γ is the num ber of p ermutations σ of { v 1 , . . . , v n } such that f h Γ ( v 1 , . . . , v n ) = f h Γ ( v σ (1) , . . . , v σ ( n ) ) , and let f Γ be the symmetriza tion of f h Γ . Then, we hav e: (5) G Γ ( ω ) = Z f Γ ( v 1 , . . . , v n ) d ω ( n ) ( v 1 , · · · , v n ) . Lemma 21 . The r andom variable G Γ has a chao s r epr esentation given by: G Γ = n X i =0 I i ( f Γ i ) , wher e f Γ i is the b ounde d symmetric function define d as (6) f Γ i ( v i +1 , . . . , v n ) =  n i  λ n − i Z X n − i f Γ ( v 1 , . . . , v n ) d v 1 . . . d v n − i . 20 L. DECREUSEFOND, E. FERRAZ, H. RANDRIAMBOLOLONA, AND A. VER GNE Pr o of. F rom (5), using the binomial expansion and some algebra, we obtain G Γ ( ω ) = n X i =0 Z    n i  Z X n − i f Γ ( v 1 , . . . , v n ) λ d v 1 . . . λ d v n − i   d ω ( i ) Λ ( v n − i +1 , · · · , v n ) . W e define, for any i ∈ { 1 , . . . , n } , f Γ i ( v i +1 , . . . , v n ) =  n i  λ n − i Z X n − i f Γ ( v 1 , . . . , v n ) d v 1 . . . d v n − i . T o co nclude the proo f, we note that, since X is a bounded set and h Γ is b ounded, f Γ i is b ounded.  Theorem 22 . Ther e exists c > 0 such that for any λ ≥ 1 , d W G Γ − E Λ [ G Γ ] p V ar Λ [ G Γ ] , N (0 , 1) ! ≤ c λ 1 / 2 · Pr o of. Let F = G Γ − E Λ [ G Γ ] p V ar Λ [ G Γ ] . Provided that Γ has n vertices, a ccording to Lemma 21, we hav e the following iden tities: D t F = 1 p V ar Λ [ G Γ ] n X i =1 iI i − 1 ( f Γ i ( ∗ , t )) , − D t L − 1 F = 1 p V ar Λ [ G Γ ] n X i =1 I i − 1 ( f Γ i ( ∗ , t )) , V ar Λ [ G Γ ] = n X i =1 i !   f Γ i   2 L 2 ( X, Λ) ⊗ i . Hence, V ar Λ [ G Γ ] is a p olyno mial of degre e 2 n − 1 with respect to λ . F ro m Prop o- sition 6 , it is tedious but straig htforward to see that h DL − 1 F, D F i L 2 ( X, Λ) is a po lynomial of degree 2 n − 2 with random co efficients dep ending on the in tegrals ov er X i of the f Γ i . According to Lemma 21, these co efficients are b ounded a lmost- surely . Hence there exists a co nstant c > 0 suc h that E Λ    1 + h DL − 1 F, D F i L 2 ( X, Λ)    ≤ c λ − 1 / 2 . The sa me kind of co mputations shows that Z X E Λ  | D x F | 2 | D x L − 1 F |  λ d x ≤ c λ − 1 / 2 . Then, the result follows fro m Theorem 7.  SIMPLICIAL HOM OLOGY OF RANDOM CONFIGURA TIONS 21 References [1] M. A. Armstrong, Basic top olo gy , Springer -V erlag, 1997. [2] E. T. Bell, Exp onential p olynom ials , Ann, Math. (193 4), no. 35, 258–27 7. [3] Rabi N. Bhatta c hary a and Jay an ta K. Ghosh, A class of U -statistics and asymptotic nor- mality of the numb er of k -clusters , J. Multiv ariate Anal. 43 ( 1992), no. 2, 300–330. [4] A. 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[18] F. L. Lewis, Wir eless sensor networks , c h. 2, John Wiley , New Y ork, 2004. [19] J. M unkres, Elements of algebr aic top olo gy , Add ison W esley , 1993. [20] I. Nourdin and G. Pec cati, Normal app r oximations with mal liavin ca lculus: F r om stein ’s metho d to universality , Cam bridge Univer sity Press, 2012. [21] G. Pec cati, J.L. Solé, M.S. T aqqu, and F. Utzet, Stein ’s metho d and normal appr oximation of p oisson functionals , Annals of Probability 38 (2010) , no. 2, 443–47 8. [22] M . Penrose , R and om ge ometric gr aphs , Oxford Studies in Probability , v ol. 5, Oxford Univer- sity Press, Oxford, 2003. [23] M . D. Pen rose and J. E. Y ukich, Limit the ory for p oint pr o c esses in manifolds , (2011) . [24] G. J. Pottie and W.J. Kaiser, Wir eless inte gr ate d network sensors , Comm unications of the ACM 43 ( 2000), 51–58. 22 L. DECREUSEFOND, E. FERRAZ, H. RANDRIAMBOLOLONA, AND A. VER GNE [25] N . Priv ault, Sto chastic analysis i n discr ete and c ontinuous settings with normal martingales , Lecture Notes in Mathematics, vol. 1982, Springe r-V erlag, Berlin, 2009 . [26] M . Reitzner and M. Sch ül te, Centr al limit the or ems for U-statistics of p oisson p oint pr o c esses , Annals of Probabilit y , to appear (2013). [27] J. J. Rotman, An intr o duction to algebr aic top olo gy (gr aduate t exts in mathematics) , Springer, July 1998. Appendix A. Proofs of the second order moments A.1. Pro of of Theorem 15. By Lemma 14, w e can rewrite the cov a riance b etw ee n N k and N l with l ≤ k : Cov Λ [ N k , N l ] = E Λ [( N k − E Λ [ N k ])( N l − E Λ [ N l ])] = E Λ   1 k ! k X i =1  k i  λ k − i I i  f k i  1 l ! l X j =1  l j  λ l − j I j  f l j    , where f k i ( v k − i +1 , . . . , v k ) = Z X k − i ϕ ( d ) k ( v 1 , . . . , v k ) d v 1 . . . d v k − i . Using the iso metry formula, giv en by Eq. (1), we hav e : Cov Λ [ N k , N l ] = 1 k ! l ! l X i =1  k i  l i  λ k + l − 2 i E Λ  I i  f k i  I i  f l i  = 1 k ! l ! l X i =1  k i  l i  λ k + l − 2 i i ! h f k i , f l i i L 2 ( X, Λ) ◦ i = l X i =1 1 i !( k − i )!( l − i )! λ k + l − 2 i h f k i , f l i i L 2 ( X, Λ) ◦ i Hence, we are reduced to compute h f k i , f l i i L 2 ( X, Λ) ◦ i = Z X i  Z X l − i ϕ ( d ) l ( v 1 , . . . , v l ) d v i +1 . . . d v l  ×  Z X k − i ϕ ( d ) k ( v 1 , . . . , v k ) d v i +1 . . . d v k  λ d v 1 . . . λ d v i . Let us denote J 2 ( m 1 , m 2 , m 12 ) the integral on tw o simplices of res p ectively m 1 + m 12 and m 2 + m 12 vertices with m 12 > 0 common vertices: J 2 ( m 1 , m 2 , m 12 ) = Z X M ϕ ( d ) m 1 + m 12 ( v 1 , . . . , v m 1 + m 12 ) ϕ ( d ) m 2 + m 12 ( v m 1 +1 , . . . , v M ) d v 1 . . . d v M , with M = m 1 + m 2 + m 12 . Then we can rewr ite: h f k i , f l i i L 2 ( X, Λ) ◦ i = λ i J 2 ( l − i, k − i, i ) , SIMPLICIAL HOM OLOGY OF RANDOM CONFIGURA TIONS 23 and it then remains to co mpute J 2 ( m 1 , m 2 , m 12 ) . First, thanks to the tensoriza tion prop erty of the max-distance, we can write: J 2 ( m 1 , m 2 , m 12 ) = Z [0 , a ) M ϕ (1) m 1 + m 12 ( x 1 , . . . , x m 1 + m 12 ) ϕ (1) m 2 + m 12 ( x m 1 +1 , . . . , x M ) d x 1 . . . d x M ! d Let us s plit the integration domain of J 2 in to tw o parts: • A 1 = { ( x 1 , . . . , x M ) ∈ ∆ (1) M , ϕ (1) M ( x 1 , . . . , x M ) = 1 } , w e r ecognize the int e- gral calculated in the pro of o f Theorem 9: Z A 1 ϕ (1) m 1 + m 12 ( x 1 , . . . , x m 1 + m 12 ) ϕ (1) m 2 + m 12 ( x m 1 +1 , . . . , x M ) d x 1 . . . d x M = M (2 ǫ ) M − 1 a. • A 2 = { ( x 1 , . . . , x M ) ∈ ∆ (1) M , ϕ (1) M ( x 1 , . . . , x M ) 6 = 1 } . As in the pro of of Theorem 9, we denote ζ ( x 1 , · · · , x M ) the index i such that x i < x j < x i + 2 ǫ or x i < x j + a < x i + 2 ǫ , which exists since ǫ < a/ 4 a nd m 12 > 0 . By symmetry , w e ca n reduce the ana lysis to the situation where ζ ( x 1 , · · · , x M ) = 1 and x 1 per tains to the first simplex of m 1 + m 12 . W e then o rder the three sets of vertices such that: x 1 < · · · < x m 1 , x m 1 +1 < · · · < x m 1 + m 12 , and x m 1 + m 12 +1 < · · · < x M . Since ( x 1 , . . . , x M ) b elongs to A 2 , we hav e x M − x 1 > 2 ǫ . Let us deno te J a ( f )( x ) = R a x f ( u ) d u and by induction J ( m ) a ( f )( x ) = Z a x J ( m − 1) a ( f )( u ) d u . Then we hav e by in v ar iance by translation of the Lebesg ue measure, Z A 2 ϕ (1) m 1 + m 12 ( x 1 , . . . , x m 1 + m 12 ) ϕ (1) m 2 + m 12 ( x m 1 +1 , . . . , x M ) d x 1 . . . d x M = 2 m 1 ! m 2 ! m 12 ! Z a 0 J ( m 1 − 1) x 1 +2 ǫ ( 1 )( x 1 ) Z x 1 +4 ǫ x 1 +2 ǫ ( − J ( m 2 − 1) x M − 2 ǫ ( 1 )( x M )) J ( m 12 ) x 1 +2 ǫ ( 1 )( x M − 2 ǫ ) d x M d x 1 . W e e a sily find that: J ( m 1 − 1) x 1 +2 ǫ ( 1 )( x 1 ) = (2 ǫ ) m 1 − 1 ( m 1 − 1)! − J ( m 2 − 1) x M − 2 ǫ ( 1 )( x M ) = (2 ǫ ) m 2 − 1 ( m 2 − 1)! , J ( m 12 ) x 1 +2 ǫ ( 1 )( x M − 2 ǫ ) = ( x 1 − x M + 4 ǫ ) m 12 m 12 ! · 24 L. DECREUSEFOND, E. FERRAZ, H. RANDRIAMBOLOLONA, AND A. VER GNE Thu s we ha ve: Z A 2 ϕ (1) m 1 + m 12 ( x 1 , . . . , x m 1 + m 12 ) ϕ (1) m 2 + m 12 ( x m 1 +1 , . . . , x M ) d x 1 . . . d x M = 2 m 1 m 2 m 12 + 1 (2 ǫ ) M − 1 a. Then, J 2 ( m 1 , m 2 , m 12 ) = ( m 1 + m 2 + m 12 + 2 m 1 m 2 m 12 + 1 ) d a d (2 ǫ ) ( m 1 + m 2 + m 12 − 1) d concluding the proo f. A.2. Pro of of Theo rem 16. The v ariance of χ is given b y: V ar Λ [ χ ] = E Λ  ( χ − E Λ [ χ ]) 2  = E Λ   ∞ X k =1 ( − 1) k ( N k − E Λ [ N k ] ! 2   = E Λ   ∞ X i =1 ∞ X j =1 ( − 1) i + j ( N i − E Λ [ N i ])( N j − E Λ [ N j ])   . W e r emark that N i ≤ N i 1 /i ! , thus E Λ   ∞ X i =1 ∞ X j =1 | ( N i − E Λ [ N i ])( N j − E Λ [ N j ]) |   ≤ E Λ   ∞ X i =1 ∞ X j =1 N i N j + E Λ [ N i ] E Λ [ N j ] + N i E Λ [ N j ] + N j E Λ [ N i ]   ≤ E Λ   ∞ X i =1 ∞ X j =1 N i + j 1 + E Λ  N i 1  E Λ h N j 1 i + N i 1 E Λ h N j 1 i + N j 1 E Λ  N i 1  i ! j !   ≤ E Λ h e 2 N 1 + e 2 E Λ [ N 1 ] + 2 e N 1 + E Λ [ N 1 ] i < ∞ . Thu s, we can write V ar Λ [ χ ] = ∞ X i =1 ( − 1) i ∞ X j =1 ( − 1) j Cov Λ [ N i , N j ] . The res ult follows by Theorem 15. SIMPLICIAL HOM OLOGY OF RANDOM CONFIGURA TIONS 25 A.3. Pro of of Theo rem 17. If d = 1 , according to Theo rem 16: V ar Λ [ χ ] = a 2 ǫ ∞ X n =1 c 1 n (2 λǫ ) n . (7) Moreov er, w e define α n = n X j = ⌈ n +1 2 ⌉   2 j X i = n − j +1 ( − 1) i + j n ( n − j )!( n − i )!( i + j − n )! − n ( n − j )! 2 (2 j − n )!   , and β n = c 1 n − α n . It is w ell known that 2 j − n X i =0 ( − 1) i  j i  = ( − 1) 2 j − n − 1  j − 1 2 j − n  , using Stiffel’s r e la tion, we obtain: α n = ( − 1) n n n ! n X j = ⌈ n +1 2 ⌉ "  n j  2 2 j − n X i =0 ( − 1) i  j i  + 2( − 1 ) n  n j  # = 1 ( n − 1)! n X j = ⌈ n +1 2 ⌉  2  n j  j − 1 n − j − 1  −  n j  j n − j  − 2( − 1 ) n  n j  = 1 ( n − 1)! n X j = ⌈ n +1 2 ⌉  n j   j − 1 n − j  −  j − 1 n − j − 1  − 2( − 1 ) n  n j  . (8) The identit y  n j  =  n n − j  allows us to write that n X j = ⌈ ( n +1) / 2 ⌉ ( − 2( − 1) n )  n j  = n X j =0  n j  = 2 n , n odd, n X j = ⌈ ( n +1) / 2 ⌉ ( − 2( − 1) n )  n j  =  n n/ 2  + n X j =0 −  n j  = − 2 n +  n n/ 2  , n even. Since  j − 1 n − j  = 0 for j <  n +1 2  , we hav e n X j = ⌈ n +1 2 ⌉  n j   j − 1 n − j  −  j − 1 n − j − 1  = n X j =1  n j   j − 1 n − j  −  j − 1 n − j − 1  −  n n/ 2  1 + ( − 1) n 2 By known formulas on h yp ergeometric functions, we hav e that: n X j = ⌈ n +1 2 ⌉  n j   j − 1 n − j  −  j − 1 n − j − 1  = ( − 1) n +1 −  n n/ 2  1 + ( − 1) n 2 Then, we substitute these la st tw o expressions in Eq. (8) to o btain α n = ( − 1) n (1 − 2 n ) 1 n ≥ 1 ( n − 1)! , 26 L. DECREUSEFOND, E. FERRAZ, H. RANDRIAMBOLOLONA, AND A. VER GNE and thus ∞ X i =0 α n x n = − xe − x + 2 xe − 2 x . Pro ceeding along the same line, β n is given b y β n = n X j = ⌈ n +1 2 ⌉   2 j X i = n − j +1 ( − 1) i + j 2( n − i )( n − j ) ( n − j )!( n − i )!( i + j − n + 1)! − 2( n − j ) 2 ( n − j )! 2 (2 j − n + 1)!  = ( − 1) n  ( − 2 + 2 n ) 1 n ≥ 1 ( n − 1)! − 2 1 n ≥ 2 ( n − 2)!  , and aga in w e can simplify the power ser ies P ∞ n =0 β n x n as ∞ X n =0 β n x n = 2 xe − x − 2( x + x 2 ) e − 2 x . Then, substituting α n and β n in Eq. (7) yields the result. Appendix B. Proof of the third order moment B.1. Pro of of Theorem 19. F ro m Lemma 14 , we k now that the chaos deco m- po sition of the nu mber of ( k − 1) -s implices is given b y f N k = k X i =1 I i ( h i ) , with h i ( v 1 , . . . , v i ) = 1 k !  k i  λ k − i Z X k − i ϕ ( d ) k ( v 1 , . . . , v k ) d v i +1 . . . d v k , and I i ( h i ) = Z X i h i ( v 1 , . . . , v i ) d ω ( i ) Λ ( v 1 , . . . , v i ) . Then, we define denoting u = i + j − s , g i,j,s,t = t !  i t  j t  t u − t  h i ◦ u − t t h j and using the c haos expa nsion (cf Prop osition 6), we have f N k 3 = ( k X i =1 I i ( h i )) 3 =   k X i =1 k X j =1 I i ( h i ) I j ( h j )   k X l =1 I l ( h l ) ! = k X i,j,l =1 i + j X s = | i − j | u ∧ i ∧ j X t = ⌈ u 2 ⌉ I s ( g i,j,s,t ) I l ( h l ) . SIMPLICIAL HOM OLOGY OF RANDOM CONFIGURA TIONS 27 A ccording to (1), w e o bta in: E Λ h f N k 3 i = E Λ   k X i,j =1 i + j ∧ k X s = | i − j |∨ 1 u ∧ i ∧ j X t = ⌈ u 2 ⌉ I s ( g i,j,s,t ) I s ( h s )   = k X i,j =1 i + j ∧ k X s = | i − j |∨ 1 u ∧ i ∧ j X t = ⌈ u 2 ⌉ Z X s g i,j,s,t h s λ s d v 1 . . . d v s = k X i,j =1 i + j ∧ k X s = | i − j |∨ 1 u ∧ i ∧ j X t = ⌈ u 2 ⌉ λ s t !  i t  j t  t u − t  Z X s ( h i ◦ u − t t h j ) h s d v 1 . . . d v s . W e denote J 3 ( k , i, j, s, t ) the following in tegral: (9) J 3 ( k , i, j, s, t ) = Z X 3 k − t − s ϕ ( d ) k ( v 1 , . . . , v k ) ϕ ( d ) k ( v 1 , . . . , v t , v k +1 , . . . , v 2 k − t ) ϕ ( d ) k ( v 1 , . . . , v 2 t − i − j + s , v t +1 , . . . , v i , v 2 k − j +1 , . . . , v 3 k − t − s ) d v 1 . . . d v 3 k − t − s , for i, j, s and t bounded a s in the previous sums. W e r e c ognize the in tegral on three ( k − 1) -simplices w ith u − t , i − t , a nd j − t co mmon v ertices to only tw o of them, and 2 t − u common v ertices to the three of them. Then, we ca n write: E Λ h f N k 3 i = k X i,j, =1 i + j ∧ k X s = | i − j |∨ 1 u ∧ i ∧ j X t = ⌈ u 2 ⌉ λ 3 k − i − j t ! ( k !) 3  k i  k j  k s  i t  j t  t u − t  J 3 ( k , i, j, s, t ) . Finally , rela xing the b ounda r ies on the sums conclude the pr o of. E-mail addr ess : laurent.decreus efond@tel ecom-paristech.fr E-mail addr ess : eduardo.ferraz@ telecom-p aristech.fr E-mail addr ess : hugues.randriam @telecom- paristech.fr E-mail addr ess : anais.vergne@te lecom-par istech.fr Institut Telecom, Telecom P arisTech, CNRS L TCI, 46, r ue Barra ul t, P a ris - 75634, France