Random simplicial complexes - around the phase transition
RRandom Simplicial Complexes - Around the Phase Transition
Nathan Linial ∗ Yuval Peled † In memory of Jiˇr´ı Matouˇsek
Abstract
This article surveys some of the work done in recent years on random simplicial complexes. We mostly considerhigher-dimensional analogs of the well known phase transition in G ( n, p ) theory that occurs at p = n . Our mainobjective is to provide a more streamlined and unified perspective of some of the papers in this area. There are at least two different perspectives from which our subject can be viewed. We survey some recentdevelopments in the emerging field of high-dimensional combinatorics . However, these results can be viewed aswell as part of an ongoing effort to apply the probabilistic method in topology . The systematic study of randomgraphs was started by Erd˝os and R´enyi in the early 1960’s and had a major impact on discrete mathematics,computer science and engineering. Since graphs are one-dimensional simplicial complexes, why not developan analogous theory of d -dimensional random simplicial complexes for all d ≥ ? To this end, an analogof Erd˝os and R´enyi’s G ( n, p ) model, called Y d ( n, p ) , was introduced in [12]. Such a simplicial complex Y is d -dimensional, it has n vertices and has a full ( d − -dimensional skeleton. Each d -face is placed in Y independently with probability p . Note that Y ( n, p ) is identical with G ( n, p ) .One of the most natural questions to ask in any model of random graphs concerns graph connectivity . AsErd˝os and R´enyi famously showed, the threshold for graph connectivity in G ( n, p ) is p = ln nn . To draw the anal-ogy from a topological perspective, one should seek the threshold for the vanishing of the ( d − -st homology.This indeed was the motivating problem in [12]. As that paper showed, and together with subsequent work [17]this threshold in Y d ( n, p ) is p = d ln nn . Here the coefficients can come from any fixed finite Abelian group. Thesame question for integral ( d − -st homology has attracted considerable attention and the answer is believedto be the same. This was recently confirmed for d = 2 [15], and is not yet fully resolved for higher dimensions(but see [10]). The threshold for the vanishing of the fundamental group of Y ( n, p ) is fairly well (but still notperfectly) understood [7, 11].Since we tend to work by analogy with the G ( n, p ) theory, it is a very challenging problem to seek ahigh-dimensional counterpart to the phase transition that occurs at p = n . It is here that the random graphasymptotically-almost-surely (a.a.s.) acquires cycles. Namely, for every > c > there is a > q = q ( c ) > such that a graph in G ( n, cn ) is a forest with probability q + o n (1) , but for p ≥ n , a G ( n, p ) graph has, a.a.s., atleast one cycle. These notions have natural analogs in higher-dimensional complexes that suggest what is beingsought. However, even more famously, a giant connected component with Ω( n ) vertices emerges at p = n . ∗ Department of Computer Science, Hebrew University, Jerusalem 91904, Israel. e-mail: [email protected] . Supported by ERCgrant 339096 ”High-dimensional combinatorics”. † Department of Computer Science, Hebrew University, Jerusalem 91904, Israel. e-mail: [email protected] . Yuval Peled isgrateful to the Azrieli Foundation for the award of an Azrieli Fellowship. a r X i v : . [ m a t h . C O ] S e p γ d .
455 3 .
089 3 .
509 3 .
822 4 .
749 7 .
555 10 . c d .
754 3 .
907 4 .
962 5 .
984 11 − − . − − . − − . Table 1: The critical constants γ d and c d .Since there is no natural notion of connected components at dimensions d > , it is not even clear what to ask.Finding the correct framework for asking this question and discovering the answer is indeed one of the mainaccomplishments of the research that we survey here.Another reason that makes the high-dimensional scenario more complicated than the graph-theoretic pictureis that there are several natural analogs for acyclicity. A ( d − -face τ in a d -complex Y is free if it is containedin exactly one d -dimensional face σ of Y . In the corresponding elementary collapse step , τ and σ are removedfrom Y . We say that Y is d -collapsible if it is possible to eliminate all its d -faces by a series of elementarycollapses. Otherwise, the maximal subcomplex of Y in which all ( d − -faces are contained in at least d -facesis called the core of Y . Clearly a graph (i.e., a -dimensional complex) is -collapsible if and only if it is acyclic,i.e., a forest.A d -complex Y is said to be d -acyclic if its d -th homology group vanishes. Namely, if the d -dimensionalboundary matrix ∂ d ( Y ) has a trivial right kernel. Unless otherwise stated, we consider this matrix over the reals.The real d -Betti number of Y is β d ( Y ; R ) := dim H d ( Y ; R ) = dim(ker ∂ d ( Y )) .Whereas acyclicity and -collapsibility are equivalent for graphs, this is no longer the case for d -dimensionalcomplexes . Clearly, a d -collapsible simplicial complex has a trivial d -th homology, but the reverse implicationdoes not hold in dimension d ≥ .In this view, there are now two potentially separate thresholds to determine in Y d ( n, p ) : For d -collapsibilityand for the vanishing of the d -th homology. These questions were answered and the respective thresholds weredetermined in a series of four papers. A lower bound on the threshold for d -collapsibility was found in [6] anda matching upper bound was proved in [4] . An upper bound on the threshold for the vanishing of the d -thhomology was found in [5], with a recent matching lower bound for real homology [14]. We conjecture that thesame bound holds for all coefficient rings, but this remains open at present.The purpose of this paper is to survey these results and present the main ingredients of the proofs. Inparticular, we highlight the key role of the local structure of random complexes in all these proofs.Both thresholds are of the form p = cn . Namely there is a constant c = γ d corresponding to the d -collapsibility threshold and c = c d for acyclicity. As functions of the dimension d , the constants γ d and c d differ substantially. Our results allow us to numerically compute them to desirable accuracy (See Table 1).Before stating the theorems, a small technical remark is in order. An obvious obstacle for a d -complex Y tobe either d -collapsible or d -acyclic is that it contains the boundary of a ( d + 1) -simplex ∂ ∆ d +1 , i.e. all the d + 2 d -faces that are spanned by some d + 2 vertices. In the random complex Y d (cid:0) n, cn (cid:1) , these objects appear withprobability bounded away from both zero and one, and it is easy to see that their number is Poisson distributedwith a constant expectation. In particular, Y d (cid:0) n, cn (cid:1) is ∂ ∆ d +1 -free with positive probability. There are severalways to go around this technical difficulty. In [6] a model of random complexes conditioned on being ∂ ∆ d +1 -free was considered, which allowed a cleaner form for the theorems. Here we work with the simple binomialmodel, and consequently must mention these simplices.We turn to the main theorems. Let d ≥ be an integer, c > real and denote the core of Y = Y d (cid:0) n, cn (cid:1) by ˜ Y . We define γ d as the minimum of the function ψ ( x ) := − ln x (1 − x ) d , < x < . Furthermore, we let x ∗ be theunique root in (0 , of ( d + 1)(1 − x ) + (1 + dx ) ln x = 0 , and c d := ψ ( x ∗ ) . Recall that an elementary collapse is a homotopy equivalence. We briefly refer to a d -dimensional complexes as a d -complex In the present article we note an error in that paper, but also indicate how to overcome it. In doing so we also derive some additionalinformation on critical complexes.
2n addition, for an integer k and λ > real, we let Ψ k ( λ ) := Pr[ Poi ( λ ) ≥ k ] , and t = t ( c, d ) be the smallestpositive root of t = e − c (1 − t ) d , or equivalently − t = Ψ ( c (1 − t ) d ) . Theorem 1.1.
Let d ≥ be an integer, c > real, and Y = Y d (cid:0) n, cn (cid:1) .(I) The collapsible regime: If c < γ d then a.a.s. either Y is d -collapsible or its core is comprised of O d (1) vertex disjoint ∂ ∆ d +1 ’s.(II) The intermediate regime: If γ d < c < c d then a.a.s.(a) Y is not d -collapsible. Moreover, its core contains a constant fraction of the ( d − -faces: f d − ( ˜ Y ) = Ψ ( c (1 − t ) d ) (cid:18) nd (cid:19) (1 + o (1)) , f d ( ˜ Y ) = cn (cid:18) nd + 1 (cid:19) (1 − t ) d +1 (1 + o (1)) . (1) In particular, f d ( ˜ Y ) < f d − ( ˜ Y ) .(b) Either Y is d -acyclic or H d ( Y ; R ) is generated by O d (1) vertex disjoint ∂ ∆ d +1 ’s.(III) The cyclic regime: If c > c d then a.a.s. H d ( Y ; R ) is non-trivial. Furthermore, f d − ( ˜ Y ) and f d ( ˜ Y ) stillsatisfy equation (1), but in this regime f d − ( ˜ Y ) < f d ( ˜ Y ) and β d ( Y ) = (cid:18) cd + 1 (1 − t ) d +1 − (1 − t ) + ct (1 − t ) d (cid:19) (cid:18) nd (cid:19) (1 + o (1)) = (cid:0) f d ( ˜ Y ) − f d − ( ˜ Y ) (cid:1) (1 + o (1)) . f f β c - c γ Figure 1: Illustration of Theorem 1.1 for d = 2 . Here f , f are the face numbers of ˜ Y and β is the secondBetti number. The functions are normalized by (cid:0) n (cid:1) , and n → ∞ .It is not hard to determine the asymptotic behaviour (in d ) of these expressions, namely, c d = ( d + 1)(1 − e − ( d +1) ) + O d ( d e − d ) and γ d = (1 + o d (1)) ln d. Consequently, there is a wide range of the parameter p = p ( d, n ) for which almost all the complexes in Y d ( n, p ) are acyclic and non-collapsible. There are strong indications that complexes from this range have interestingand unexpected properties. 3ote that f d ( ˜ Y ) − f d − ( ˜ Y ) > implies that H d ( Y ; R ) (cid:54) = 0 . Moreover, this difference of the face numbersis a lower bound for the d -th Betti number. Theorem 1.1 shows that for all ≤ p ≤ , and up to the appearanceof ∂ ∆ d +1 ’s these two conditions are typically equivalent and the lower bound is asymptotically tight. This isclearly a probabilistic statement which does not hold in general.We turn to deal with the emergence of the giant component, a subject on which there exists an extensive bodyof literature. As mentioned above, there is no obvious high-dimensional counterpart to the notion of connectedcomponents, and we need a conceptual idea in order to even get started. The notion of shadows , introduced in[13], offers a way around this difficulty. The idea is to tie connected components with cycles, which do havenatural high dimensional counterparts. The shadow of a graph G is the set of those edges that are not in G ,whose addition to G creates a new cycle. It turns out that the giant component emerges exactly when the shadowof the evolving random graph acquires positive density. In particular, for c > the shadow of G ( n, cn ) hasdensity (1 − t ) + o (1) , where t is the unique root in (0 , of t = e − c (1 − t ) (See Figure 2a).This suggests how we should define SH R ( Y ) , the shadow of Y , a d -dimensional complex with full ( d − -skeleton. Namely, it is the following set of d -faces:SH R ( Y ) = { σ / ∈ Y : H d ( Y ; R ) is a proper subspace of H d ( Y ∪ { σ } ; R ) } . In words, a d -face belongs to SH R ( Y ) if it is not in Y and its addition to Y creates a new d -cycle.We are considering throughout the vanishing of the d -th homology and d -collapsibility. These two notionscapture acyclicity from an algebraic resp. combinatorial perspective. For d = 1 the two coincide, but they differwidely for d ≥ . This dual perspective carries over to two notions of shadows. A d -face σ that does not belongto a d -complex Y is in Y ’s R -shadow if its addition to Y increases the d -homology. It is in the C-shadow of Y if its addition to Y increases the core. Again, these notions coincide for d = 1 , and the R -shadow is alwayscontained in the C-shadow, but for d ≥ they may differ.The notion of shadows lets us compare the phase transitions of random graphs and random complexes ofhigher dimensions, and a substantial qualitative difference reveals itself. While the density of the shadow of G ( n, p ) undergoes a smooth transition around p = 1 /n , when d ≥ both the C-shadow and the R -shadow of Y d (cid:0) n, cn (cid:1) undergo discontinuous first-order phase transitions at the critical points γ d and c d respectively. Theorem 1.2.
Let d ≥ be an integer, c > real, and Y = Y d (cid:0) n, cn (cid:1) .(I) The collapsible regime: If c < γ d then a.a.s. | SH R ( Y ) | ≤ | SH C ( Y ) | = Θ( n ) . (II) The intermediate regime: If γ d < c < c d then a.a.s. | SH R ( Y ) | = Θ( n ) , and | SH C ( Y ) | = (cid:18) nd + 1 (cid:19) ((1 − t ) d +1 + o (1)) . (III) The cyclic regime: If c > c d then a.a.s. the size of both SH R ( Y ) and SH C ( Y ) is (cid:0) nd +1 (cid:1) ((1 − t ) d +1 + o (1)) . An essential idea that is common to all these results is that in the range p = Θ( n ) many of the interestingproperties of Y d ( n, p ) can be revealed by studying its local structure. Initially, this seemed as merely a usefultool in studying the threshold for d -collapsibility, and in establishing an upper bound on the threshold of thevanishing of the d -th homology. However, in obtaining a lower bound on this threshold, it became apparentthat this idea should be viewed in the wider context of local weak limits . This framework was introduced byBenjamini and Schramm [8] and Aldous and Steele [3]. In recent years, this approach was used in deriving newasymptotic results in various fields of mathematics (e.g. [1, 16]).The study of d -collapsibility in random complexes was significantly influenced by work on k -cores in ran-dom hypergraphs and specifically the works by Molloy [18] and Riordan [20]. Also, the proof of Theorem 1.1makes substantial use of tools from the paper of Bordenave, Lelarge and Salez [9] on the rank of the adjacencymatrix of random graphs. 4 c (a) Density of the shadow of G ( n, cn ) . C - Shadow ℝ - Shadow c c γ (b) Density of the C-shadow and R -shadow of Y (cid:0) n, cn (cid:1) . Figure 2: Illustration of Theorem 1.2 for d = 2 , and comparison to the density of the shadow of a random graph.The rest of the paper is organized as follows. Section 2 gives some necessary background material aboutsimplicial complexes. In Section 3 we introduce the concept of a Poisson d -tree which is the local weak limitof random simplicial complexes. The main ingredients of the proofs of the main theorems are presented inSections 4 and 5, that respectively address the subjects of collapsibility and acyclicity. Concluding remarks andopen questions are presented in Section 6. A simplicial complex Y is a collection of subsets of its vertex set V that is closed under taking subsets. Namely, σ ∈ Y and τ ⊆ σ imply that τ ∈ Y as well. Members of Y are called faces or simplices . The dimension of the simplex σ ∈ Y is defined as | σ | − , and dim( Y ) is defined as max dim( A ) over all faces A ∈ Y . A d -dimensional simplex is also called a d -simplex or a d -face, and a d -dimensional simplicial complex is alsoreferred to as a d -complex. The set of j -faces in Y is denoted by Y j , and the face numbers by f j ( Y ) := | Y j | .For t < dim( Y ) , the t - skeleton of Y is the simplicial complex that consists of all faces of dimension ≤ t in Y ,and Y is said to have a full t -dimensional skeleton if its t -skeleton contains all the t -faces of V . In this paper, the degree d Y ( τ ) of a face τ in a complex Y is the number of dim( Y ) -faces that contain it. A face of degree zerois said to be exposed . Although we are directly interested only in finite complexes, infinite ones do play a rolehere, but we consider only locally-finite complexes in which every face has a finite degree. We occasionally usethe bipartite incidence graph between ( d − -faces and d -faces of a d -complex Y . This allows us, in particular,to speak about distances among such faces.The permutations on the vertices of a face σ are split in two orientations , according to the permutation’s sign.The boundary operator ∂ = ∂ d maps an oriented d -simplex σ = ( v , ..., v d ) to the formal sum (cid:80) di =0 ( − i ( σ i ) ,where σ i = ( v , ...v i − , v i +1 , ..., v d ) is an oriented ( d − -simplex. We fix some commutative ring R andlinearly extend the boundary operator to free R -sums of simplices. We denote by ∂ d ( Y ) the d -dimensionalboundary operator of a d -complex Y .When Y is finite, we consider the f d − ( Y ) × f d ( Y ) matrix form of ∂ d by choosing arbitrary orientationsfor ( d − -simplices and d -simplices. Note that changing the orientation of a d -simplex (resp. d − -simplex)results in multiplying the corresponding column (resp. row) by − .The d -th homology group H d ( Y ; R ) (or vector space if R is a field) of a d -complex Y is the (right) kernelof its boundary operator ∂ d . Most of the homology groups in this paper are considered over R . An elementin H d ( Y ; R ) is called a d -cycle , and the whole group is called the d -cycle space of Y . The d -th Betti number β d ( Y ; R ) of a complex Y is defined to be the dimension of H d ( Y ; R ) .Recall the concept of elementary collapse as defined in the introduction. A d -collapse phase is a procedure inwhich all the possible elementary d -collapses take place at once. In case more than one ( d − -face can collapse5ome d -face, one of them is chosen arbitrarily. Given a d -complex Y , the complex R k ( Y ) is the complex that isobtained from Y after k phases of d -collapse. Similarly, R ∞ ( Y ) is obtained after all possible d -collapse stepsare carried out. A d -core (or core, for brevity) is a d -complex in which all the ( d − -faces are of degree ≥ .The d -core of Y is the maximal d -core subcomplex of Y . Note that the d -core of Y is obtained from R ∞ ( Y ) byremoving the exposed ( d − -faces. d -tree The concept of a Poisson d -tree process was introduced in [6] and turned out to be extremely useful in the studyof random simplicial complexes. It can be viewed as a high-dimensional counterpart of the Poisson Galton-Watson process which plays a key role in the study of the giant component in G ( n, p ) graphs.A rooted d -tree is a pair ( T, o ) where T is a d -complex and o is some ( d − -face of T . A d -tree is generatedby the following process. Initially the complex consists of the ( d − -face o . At every step k ≥ , every ( d − -face τ of distance k from o picks a non-negative number m = m τ of new vertices v , ..., v m , and adds the d -faces v τ, ..., v m τ to T .We use some self-explanatory terminology in our study of d -trees. A leaf is a ( d − -face with no descendant d -faces. A ( d − -face τ is an ancestor of a ( d − -face τ (cid:48) if τ (cid:48) belongs to the subtree rooted at τ . If ( T, o ) isa rooted d -tree and T (cid:48) is a subtree of T which contains the ( d − -face o , we refer to ( T (cid:48) , o ) as a rooted subtree of ( T, o ) . The depth of a ( d − -face is its distance from the root, and the depth of the d -tree is the maximaldepth of any of its ( d − -faces.A Poisson d -tree with parameter c , denoted by T d ( c ) , is a rooted d -tree in which all the numbers m τ through-out this generative process are i.i.d. P oi ( c ) -distributed. The rooted subtree of T d ( c ) that consists of the first k generations of this process is denoted T d,k ( c ) .The most important fact about T d ( c ) in this context is that it approximates the local neighborhood of a ( d − -face in Y d (cid:0) n, cn (cid:1) . Lemma 3.1. [6] For every fixed integer k > , the k -neighborhood of a fixed ( d − -face τ in Y = Y d (cid:0) n, cn (cid:1) converges in distribution to the k -neighborhood of the root of T d ( c ) as n → ∞ .Proof. First, we observe that the degree of every ( d − -face in Y is Bin (cid:0) n − d, cn (cid:1) -distributed. Chernoff’sInequality implies that a.a.s. no degree exceeds ( d + 1) log n . We claim that for every fixed k ≥ , the k -neighborhood of τ in Y is a.a.s. a d -tree. The cases k = 0 and k = 1 are trivial. Assume, by induction, thatthe k -neighborhood N k ( τ ) of τ is a d -tree. The tree structure will be violated in the ( k + 1) -th layer if and onlyif there is some ( d − -face η of distance k from τ , and some vertex v ∈ N k ( τ ) such that ηv ∈ Y . However,by the bound on the degrees of the ( d − -faces, there only O (log k n ) such ( d − -faces η and O (log k n ) such vertices v . Therefore, the probability that a face of the form ηv belongs to Y is negligible. In addition,conditioned on the ( k + 1) -neighborhood being a d -tree, the number m η of new vertices that η adds to the d -treeis Bin (cid:0) n − o ( n ) , cn (cid:1) -distributed, which tends to Poisson with parameter c as n → ∞ .This lemma easily implies convergence of Y d (cid:0) n, cn (cid:1) → T d ( c ) in the sense of local weak convergenceintroduced by Benjamini and Schramm [8] and Aldous and Steele [3]. Here is a brief explanation of thisconcept. A rooted d -complex is a pair ( Y, τ ) of a d -complex and some ( d − -face in it. We denote by ( Y, τ ) k the τ -rooted subcomplex of ( Y, τ ) comprised of all the d -faces of distance at most k from τ and their subfaces.Let Y d be the set of all (isomorphism types of) rooted d -complexes, equipped with the metricdist (( Y, τ ) , ( Y (cid:48) , τ (cid:48) )) = inf (cid:26) t + 1 : ( Y, τ ) k ∼ = ( Y (cid:48) , τ (cid:48) ) k (cid:27) . It can be easily verified that ( Y d , dist ) is a separable and complete metric space, which comes, as usual, equippedwith its Borel σ -algebra (See [2]). The fact that Y d (cid:0) n, cn (cid:1) converges to T d ( c ) means that for every bounded and6ontinuous function f : Y d → R , E Y ∼ Y d ( n, cn )[ f ( Y, τ )] −−−→ n →∞ E T ∼ T d ( c ) [ f ( T, o )] . As we explain below, this fact will be applied directly to a function of particular interest in this context, namely,the degree of the root τ after k phases of τ -rooted collapse. In addition, it will be used in combination with thespectral theorem to bound the Betti numbers of Y d (cid:0) n, cn (cid:1) with the spectral measure of the Poisson d -tree. Let Y be a d -complex and τ some ( d − -face of Y . A τ -rooted collapse of Y is a d -collapse process in whichwe forbid to collapse τ . Let k be a non-negative integer. The complex obtained from Y after k phases in the τ -rooted collapse process is denoted by R k ( Y, τ ) .In the case Y = Y d (cid:0) n, cn (cid:1) , the degree d R k ( Y,τ ) ( τ ) turns out to be relevant to several different questions. Weapproximate it using δ k := d R k ( T,o ) ( o ) , where T = T d ( c ) , the Poisson d -tree with root o . Lemma 3.2.
With the above notations E [ d R k ( Y,τ ) ( τ )] −−−→ n →∞ E [ δ k ] . Note that this is not a direct corollary of the local weak convergence. Even though the function d R k ( Y,τ ) ( τ ) is continuous, being dependent only on some fixed neighborhood of the root, it is not bounded. Nevertheless,we allow ourselves to omit the proof, since this difficulty can be bypassed by a simple calculus trick. Namely,by considering the function min { d R k ( Y,τ ) ( τ ) , A } , where A is a sufficiently large constant. Lemma 3.3.
Let c > and ( t k ) k ≥− a sequence of real numbers defined by t − = 0 , t k +1 = e − c (1 − t k ) d , ∀ k ≥ . Then, δ k is Poisson distributed with parameter c (1 − t k − ) d , for every k ≥ . We refer throughout the paper to the sequences t k of real numbers and δ k of random variables that are definedhere without denoting the underlying parameter c > that is clear from the context. Proof.
By induction on k . The case k = 0 is trivial since δ is Poisson distributed with parameter c . For theinduction step, let us consider the distribution of δ k . A d -face σ that contains the root o survives k phases ofrooted collapse if and only if each of its ( d − -faces τ other than the root (there are d such τ ’s) is contained in a d -face other then σ after k − phases. This occurs if and only if τ has a positive degree in the subtree of T rootedat τ after k − phases of τ -rooted collapse. Since this subtree is also a Poisson d -tree with parameter c , thisoccurs, by the induction hypothesis, with probability Pr[ δ k − >
0] = 1 − t k − . Moreover, different branches ofthe tree are independent, so these events for different τ ’s and σ ’s are independent. Namely, the distribution of δ k is a Binomial distribution with δ ∼ Poi(c) trials and success probability (1 − t k − ) d . By a standard computationin probability theory, this implies that δ k is Poisson distributed with parameter c (1 − t k − ) d .We say that a rooted d -tree is collapsible if its root gets exposed in the rooted collapse process. For instance,the previous lemma shows that the probability that T d ( c ) is collapsed after k phases is t k , and the probabilitythat T d ( c ) is collapsible is t = t ( c, d ) . 7 d -collapsibility A simple calculus exercise tells us that the behavior of the sequence ( t k ) of Lemma 3.3 changes quite substan-tially when c = γ d . Namely, lim k →∞ t k = (cid:26) c < γ d t c > γ d , where t = t ( c, d ) is as defined just before Theorem 1.1. In other words, if c < γ d then the root of T d ( c ) getsexposed after k collapse phases with probability − o k (1) . Moreover, the expected degree of the root is o k (1) .On the other hand, if c > γ d then, for arbitrarily large k , with probability bounded away from zero some d -facesthat contain the root will survive the collapse process.How do these facts reflect on the behavior of the random simplicial complex Y = Y d (cid:0) n, cn (cid:1) under d -collapsephases? Many parameters of R k ( Y ) can be understood almost directly from the τ -rooted collapse of Y , where τ is a typical ( d − -face. Moreover, the Poisson d -tree plays a key role here since k phases of τ -rooted collapsedepend only on the k -neighborhood of τ . Consequently, as k grows, almost all ( d − -faces of Y will eithercollapse or become exposed in R k ( Y ) if c < γ d . On the other hand, if c > γ d , a constant fraction of the ( d − -faces survive k phases of d -collapse, but only very few of them remain with degree , giving the collapse processa slim chance to continue much further. In fact, the fraction of the ( d − -faces that are contained in Y ’s coreis asymptotically approximated by the probability that the root of T d ( c ) has degree ≥ after infinitely manycollapse phases.While the transition from the Poisson d -tree to the random simplicial complex is straightforward in thesubcritical regime, in the supercritical regime we follow an involved argument of Riordan [20] for k -cores ofrandom graphs. The reader is encouraged to read the introduction of Riordan’s paper for an intuitive discussionon the proof method. Let Y = Y d (cid:0) n, cn (cid:1) , c < γ d , and let τ be some ( d − -face in Y . E [ f d ( R ∞ ( Y ))] ≤ E [ f d ( R k ( Y ))]= 1 d + 1 E (cid:88) τ ∈ Y d − τ ∈ R k ( Y ) · d R k ( Y ) ( τ ) = 1 d + 1 (cid:18) nd (cid:19) E (cid:2) τ ∈ R k ( Y ) · d R k ( Y ) ( τ ) (cid:3) (2) ≤ d + 1 (cid:18) nd (cid:19) E (cid:2) d R k ( Y,τ ) ( τ ) (cid:3) (3) = 1 d + 1 (cid:18) nd (cid:19) (1 + o (1)) E [ δ k ]= 1 d + 1 (cid:18) nd (cid:19) (1 + o (1)) c (1 − t k − ) d . Identity (2) is obtained by considering some fixed ( d − -face τ , using linearity of expectation and symmetry.The subsequent inequality (3) is due to the fact that in the τ -rooted collapse process fewer collapses occur thanin d -collapse phases, whence an inequality d R k ( Y ) ( τ ) ≤ d R k ( Y,τ ) ( τ ) between τ ’s degrees after k phases in either d -collapse processes. The following equations are straightforward applications of the lemmas in Section 3.Consequently, f d ( R ∞ ( Y )) = o ( n d ) a.a.s. for every c < γ d .The argument which completes the proof says that for every c > , the complex Y d (cid:0) n, cn (cid:1) has no coresubcomplex with o ( n d ) d -faces, other than vertex disjoint ∂ ∆ d +1 ’s. This is proved in Theorem 4.1 of [6],8oncerning inclusion-minimal core complexes. It turns out that a slight modification of that proof yields a moregeneral conclusion.A d -complex whose d -faces are comprised of a vertex-disjoint union of boundaries of ( d + 1) -simplices iscalled here a d - gravel . Lemma 4.1.
For every integer d ≥ and real c > real there is α > such that a.a.s. the following holds. Let Y = Y d (cid:0) n, cn (cid:1) , then either R ∞ ( Y ) is a d -gravel or f d ( R ∞ ( Y )) > αn d .Proof. Let m := ( d log n ) d . Our first goal is to show that every core d -subcomplex C of Y with f d ( C ) ≤ m is a d -gravel. A simple first moment argument yields that Y cannot contain two intersecting copies of ∂ ∆ d +1 ,nor can it contain more than log log n copies of ∂ ∆ d +1 . A core d -complex C that is comprised of exactly l vertex disjoint ∂ ∆ d +1 ’s and m additional d -faces is said to have type ( l, m ) .Let C have type ( l, m ) . We partition its vertex set into S ˙ ∪ T , where S is the set of the vertices in some ∂ ∆ d +1 of C . Let T (cid:48) ⊆ T be those vertices in T of degree d + 1 in C (i.e., such a vertex is in exactly d + 1 of C ’s d -faces). Since C is a core, the degree of every vertex in C is at least d + 1 .In addition, every d -face of C with two or more vertices of degree d + 1 is included in a ∂ ∆ d +1 . Recallthe notion of a link of a vertex v in a simplicial complex Y . Namely, lk Y ( v ) = { τ ∈ Y : v ˙ ∪ τ ∈ Y } .In particular, the link of a vertex in a d -core is a ( d − -core. Therefore, if a vertex has degree d + 1 in C then its link is a ∂ ∆ d . Suppose that the vertices v, u ∈ C have degree d + 1 in C and are contained ina common d -face σ = ( u, v, x , ..., x d − ) . Their links are ∂ ∆ d ’s so there exist vertices u (cid:48) , v (cid:48) / ∈ σ such thatlk C ( u ) = ∂ ( u (cid:48) , v, x , ..., x d − ) and lk C ( v ) = ∂ ( u, v (cid:48) , x , ..., x d − ) . This can occur only if u (cid:48) = v (cid:48) and theclaim follows.As a result, every non-gravel d -face contains at most one vertex of T (cid:48) , so that m ≥ | T (cid:48) | ( d + 1) . Countingincidences of vertices and d -faces in C yields (cid:0) m + l ( d + 2) (cid:1) · ( d + 1) ≥ ( | S | + | T | )( d + 1) + ( | T | − | T (cid:48) | ) . But | S | = l ( d + 2) , and a simple manipulation of these inequalities gives | T | ≤ m · d +3 d +4 . We can assumew.l.o.g. that m ≤ m , l ≤ log log n and derive the following upper bound on the number of type ( l, m ) core d -complexes with at most n vertices n l ( d +2)+ d +3 d +4 m · ( | S | + | T | ) ( d +1) m = n l ( d +2) · (cid:104) n d +3 d +4 · O (log d ( d +1) n ) (cid:105) m . The first term counts the choices for
S, T , and the second the choice of non-gravel d -faces. We conclude thata.a.s. Y contains no core of type ( l, m ) with l < log log n and < m ≤ m . This is because any subcomplexof type ( l, m ) appears in Y with probability ( c/n ) l ( d +2)+ m .The proof of Theorem 4.1 in [6] yields a constant α = α ( c, d ) such that a.a.s. Y d (cid:0) n, cn (cid:1) has no inclusion-minimal subcomplex that is a core with m ≤ m ≤ αn d d -faces. In fact, their argument only uses the factthat a minimal core C is connected in the sense that between every two ( d − -faces τ, τ (cid:48) in C there is a pathalternating between ( d − -faces and d -faces of C with an inclusion relation. However, since every core is aunion of connected cores, this means that there are no cores of size m in Y . It follows that the only possiblecores that Y can contain have type ( l, , i.e., it is a d -gravel. Y d (cid:0) n, cn (cid:1) - Theorem 1.1 (II.a) The proof of this theorem closely follows the argument of Riordan [20]. We fix the dimension d and refer to r ( c ) := 1 − t ( c, d ) as a function of c . For brevity we denote r + ( c ) := Ψ ( cr ( c ) d ) . Note that both r ( c ) and r + ( c ) are continuous, bounded away from and increasing when c > γ d . Our main goal is to show that forevery ˜ c > γ d and ε > , f d − ( ˜ Y ) > ( r + (˜ c ) − ε ) (cid:0) nd (cid:1) , where ˜ Y is the d -core of Y d (cid:0) n, ˜ cn (cid:1) .This is motivated by the fact that r + (˜ c ) is the probability that the root’s degree is ≥ after every finitenumber of rooted collapse phases in T d (˜ c ) . In other words, this is the probability that the root survives the9on-rooted collapse process. Although this argument is simple and appealing, the actual proof is substantiallymore involved. Our strategy is to define some carefully crafted property A of d -trees of depth log log n (cid:28) S = S ( n ) (cid:28) log n such that the following two statements hold a.a.s. First, the subset A ⊂ Y d − of ( d − -faces τ such that Y contains a τ -rooted d -tree with property A is of density at least ( r + (˜ c ) − ε ) . Second, for every τ ∈ A there exists a d -tree T τ ⊂ Y in which τ ’s degree is at least and every leaf also belongs to A . Consequently, no ( d − -face in A can be collapsed.We refer throughout the proof to certain properties of rooted d -trees ( T, o ) , and occasionally write T ∈ P tosay that T has property P . Every property P of rooted d -trees induces a property of ( d − -faces in a d -complex Y . Namely, we say that the ( d − -face τ has property P if Y contains a d -tree rooted on τ that has property P . Here are some relevant properties: D ≤ L means T has depth ≤ L , and D < ∞ means that it has finite depth.Let P and P be properties of finite (resp. general) d -trees. A d -tree ( T, o ) has property P ◦ P when: (i) T hasa finite subtree T (cid:48) rooted at o with property P , and (ii) For every leaf τ of T (cid:48) , the subtree of T rooted at τ hasproperty P . For example, we consider the property that T has depth k + 1 and it does not collapse in k phases,i.e., B k := { T ∈ D ≤ k +1 | δ k ( T ) > } and note that B k = B ◦ B k − . Property B means that T does not collapseat finite time.We also define for k ≥ , the properties R k which are stronger than B k as follows R = { T ∈ D ≤ | δ ( T ) ≥ } , R k = R ◦ B k − ∪ B ◦ R k − , k > . The difference between B k and R k is this: T ∈ B k means that T has depth k + 1 and every non-leaf ( d − -facehas at least one descendant d -face. In defining R k we add the requirement that along every root-to-leaf path weencounter at least one ( d − -face with ≥ descendant d -faces.Finally, we introduce a stochastic version of P , a property of finite rooted d -trees ( T, o ) . For some ≤ p ≤ we mark each leaf of T independently with probability p , and remove every d -face that contains any unmarkedleaf. We say that the event M ( P , p, T ) holds if the remaining d -tree has property P . Marking is a convenientway of capturing the following phenomenon: We let each leaf in a finite d -tree grow a Poisson d -tree and weonly ask whether or not this ”tail” has some desired property. This is expressed in the simple identity Pr[ M ( P , p, T d,k ( c ))] = Pr[ T d ( c ) ∈ P ◦ P ] (4)where d, k are integers, c > , P is a property of depth- k trees and P is a property of probability p for Poisson d -trees with parameter c . For instance, Pr[ M ( B k , r ( c ) , T d,k +1 ( c ))] = Pr[ T d ( c ) ∈ B ] = r ( c ) The following lemma can be viewed as a variation on this identity. It shows that although property R k isstronger than B k , the two are almost equally likely in a Poisson d -tree. Lemma 4.2.
For every c > c > γ d there is a sufficiently large k such that Pr[ M ( R k , r ( c ) , T d,k +1 ( c ))] > r ( c ) . Proof.
Consider the following probabilistic experiment where we randomize thrice. Initially we generate thefirst k + 1 generations of T d ( c ) . Then we do the random marking that yields the d -tree T . Finally we removeeach d -face of T independently with probability c /c . We denote the component of the root by T (cid:48) . Note that T (cid:48) is distributed like a T d,k +1 ( c ) to which random r ( c ) -marking is applied. In particular, Pr[ T (cid:48) ∈ B k ] = r ( c ) ,hence we need to prove that Pr[ T ∈ R k ] > Pr[ T (cid:48) ∈ B k ] . Since T (cid:48) ∈ B k implies that T ∈ B k it suffices to showthat Pr[ W ] > Pr[ L ] , where W = [ T ∈ R k , T (cid:48) / ∈ B k ] and L = [ T ∈ B k \ R k , T (cid:48) ∈ B k ] . To this end we show that
Pr[ L ] → as k → ∞ whereas Pr[ W ] stays bounded away from .10ndeed, if T ∈ B k \ R k , then there exists some d -face of depth k whose removal violates property B k . This d -face survives in T (cid:48) with probability ( c /c ) k , so that Pr[ L ] < ( c /c ) k . On the other hand W contains the eventthat δ ( T ) = 2 , T ∈ R ◦ B k − and δ ( T (cid:48) ) = 0 whose probability is positive and independent of k .A d -tree T of depth L + 1 is ( p, η ) -rigid if Pr[ M ( R L , p, T )] > − η . Lemma 4.3.
For every c > c > γ d and η > and for a large enough integer L there holds Pr[ T d,L +1 ( c ) is ( r ( c ) , η ) -rigid ] ≥ r ( c ) . Proof.
Below we assume that k is large enough, as required in Lemma 4.2. We claim that Pr[ T d ( c ) ∈ R k ◦ B ] = Pr[ M ( R k , r ( c ) , T d,k +1 ( c ))] ≥ Pr[ M ( R k , r ( c ) , T d,k +1 ( c ))] > r ( c ) . The equality is a special case of identity (4). The next inequality follows by a simple monotonicity considerationand the fact that r ( c ) > r ( c ) . The last inequality comes from Lemma 4.2. The condition of ( p, η ) -rigidity isstronger the smaller η is, so we fix it to satisfy Pr[ T d ( c ) ∈ R k ◦ B ] − r ( c ) > η. For fixed k the conditions T d,k + l +2 ( c ) ∈ R k ◦ B l become more strict as l grows and their conjunction overall l ≥ is exactly the condition T d ( c ) ∈ R k ◦ B . Therefore, we can and will choose l large enough so that Pr[ T d,k + l +2 ( c ) ∈ R k ◦ B l ] ≤ Pr[ T d ( c ) ∈ R k ◦ B ] + η . For a d -tree T of depth L + 1 = k + l + 2 , let φ ( T ) := Pr[ M ( R k ◦ B l , r ( c ) , T )] . We denote by L the property that T is ( r ( c ) , η ) -rigid. The expectation of φ ( T ) , where T is T d,L +1 ( c ) distributed, equals theprobability Pr[ T d ( c ) ∈ R k ◦ B ] . In addition, φ ( T ) = 0 if T / ∈ R k ◦ B l and φ ( T ) ≤ − η if T / ∈ L since R k ◦ B l implies R L . Therefore Pr[ T d ( c ) ∈ R k ◦ B ] ≤ Pr[ T d,L +1 ( c ) ∈ ( R k ◦ B l ) ∩ L ] + (1 − η ) Pr[ T d,L +1 ( c ) ∈ ( R k ◦ B l ) \ L ] , whence η · Pr[ T d,L +1 ( c ) ∈ R k ◦ B l \ L ] ≤ Pr[ T d,L +1 ( c ) ∈ R k ◦ B l ] − Pr[ T d ( c ) ∈ R k ◦ B ] . Putting everything together we conclude that
Pr[ T d,L +1 ( c ) ∈ L ] ≥ Pr[ T d,L +1 ( c ) ∈ R k ◦ B l ] − η > r ( c ) ,as stated.We set all the parameters that appear in the discussion below. Recall that our goal is to show that for every ˜ c > γ d and ε > , f d − ( ˜ Y ) > ( r + (˜ c ) − ε ) (cid:0) nd (cid:1) . Let γ d < c < c < ˜ c such that r + ( c ) ≥ r + (˜ c ) − ε/ and Ψ (˜ cr ( c ) d ) > r ( c ) . Again we choose k large enough to make Lemma 4.2 hold, and we fix some < η
For every s ≥ , and every T ∈ A s , Pr[ M ( B ( k +1) s ◦ R L , r ( c ) , T )] ≥ − − s d k +1) . Proof.
By induction on s . Our definition of L and the choice of η yield the case s = 0 . Let s ≥ be an integer,and T ∈ A s +1 = R k ◦ A s . Let T (cid:48) ⊂ T be a minimal d -tree of depth k + 1 such that T (cid:48) ∈ R k and every subtreerooted at its leaves has the property A s . By a straightforward computation, T (cid:48) has exactly d k +1 leaves. Byinduction, after the marking process the subtree rooted at every leaf of T (cid:48) fails to have property B ( k +1) s ◦ R L independently with probability at most − s d k +1) . Let us refer to such a leaf as bad , and remove from T (cid:48) every d -face that contains a bad leaf. Since initially T (cid:48) had the property R k , it now has property B k unless at least d -faces were removed. But this can only occur if at least leaves are bad, an event of probability at most (cid:18) d k +1 (cid:19) (cid:18) − s d k +1) (cid:19) ≤ − s +1 d k +1) . Namely, the tree T has the property B k ◦ B ( k +1) s ◦ R L = B ( k +1)( s +1) ◦ R L with the desired probability.This leads to the following key lemma. Lemma 4.5.
For every fixed ( d − -face τ in Y = Y d (cid:0) n, ˜ cn (cid:1) , the probability that τ has property A but doesnot have P is o ( n − d ) .Proof. It is easy to show that with probability o ( n − d ) the (1 + L + 2 Q ) -neighborhood of τ consists of at most n / vertices, and we condition on this event. Suppose that τ has the property A , and consider τ ∈ T ⊂ Y suchthat T ∈ A . In particular, there exists a d -tree T (cid:48) ⊂ T rooted at τ of depth at most L + 1 such that d T (cid:48) ( τ ) = 2 and every subtree T (cid:48)(cid:48) π ⊂ T rooted at a leaf π of T (cid:48) has property A S . Denote by X ⊂ Y d − the union of theleaves of T (cid:48)(cid:48) π over all the leaves π of T (cid:48) .We now expose an additional subset of the ( Q + 1) -neighborhoods of the ( d − -faces of X with thefollowing precaution. When we reach some ( d − -face ρ and query whether a d -face that contains it belongsto Y , we only consider d -faces of the form vρ where v is a vertex that does not belong to T nor did it appear inthe exposing process upto the current query. In this manner, every ρ ∈ X is the root of a d -tree ˜ T ρ ⊂ Y in whichevery ( d − -face has at least Bin ( n − n / , ˜ cn ) descendants. Therefore, the probability that ˜ T ρ ∈ B ◦ A S is atleast Pr[ T d,Q (˜ c ) ∈ B ◦ A S ] − o (1) ≥ Ψ (˜ cr ( c ) d ) − o (1) > r ( c ) , and these events are independent over ρ ∈ X . Therefore we can consider the event ˜ T ρ ∈ B ◦A S as an alternativefor marking the leaves of d -trees T (cid:48)(cid:48) π and plug it in Claim 4.4. Since the number of leaves of T (cid:48) is bounded, theclaim implies that with probability − O (cid:16) − S (cid:17) = 1 − o ( n − d ) , after the described exposure of the additional12eighborhoods, all the subtrees rooted in these leaves have the property B ( k +1) S ◦ R L ◦ B ◦ A S . But recall that R L means that in every path from the root of the d -tree to a ( d − -face of depth L + 1 , there is a ( d − -facewith at least two descendants. In other words, R L ◦ B ◦ A S implies D ≤ L ◦ R ◦ D ≤ L ◦ A S = D ≤ L ◦ A .In particular, all the leaves of T (cid:48) have the property D < ∞ ◦ A , and since d T (cid:48) ( τ ) = 2 , it follows that τ has theproperty P .We are now ready to prove the theorem. Proof of Theorem 1.1 (II.a).
Let N A denote the number of ( d − -faces in Y = Y d (cid:0) n, ˜ cn (cid:1) that have property A .By the previous discussion, we know that f d − ( ˜ Y ) ≥ N A . We approximate the expectation of N A by Pr ˜ c [ A ] ,the probability that T d (˜ c ) has a rooted subtree that satisfies A , E [ N A ] = ( Pr ˜ c [ A ] + o (1)) (cid:18) nd (cid:19) ≥ r + ( c ) (cid:18) nd (cid:19) ≥ ( r + (˜ c ) − ε/ (cid:18) nd (cid:19) by Equation (5). Since A depends only on the O ( S ) -neighborhood of the ( d − -face, and two ( d − -faceshave non-disjoint neighborhoods with negligible probability, it follows that E [ N A ] = E [ N A ] (1 + o (1)) . By thesecond moment method
Pr[ N A < ( r + (˜ c ) − ε ) (cid:0) nd (cid:1) ] = o (1) . The upper bound is much simpler. Let N k denote thenumber of ( d − -faces that survive (= did not collapse nor become free) the first k phases of collapse. Clearly, f d − ( ˜ Y ) ≤ N k for every k . Similarly to N A , this property depends on the k -neighborhood of a ( d − -faceand by the same argument as before, N k is concentrated around its expectation. The expectation of N k can bebounded by the Poisson d -tree as follows. E [ N k ] ≤ ( Pr ˜ c [ δ k − ≥
2] + o (1)) (cid:18) nd (cid:19) = (Ψ (˜ c (1 − t k − ) d ) + o (1)) (cid:18) nd (cid:19) , and since t k → t , we obtain that Ψ (˜ c (1 − t k − ) d ) tends to r + (˜ c ) as k → ∞ .We turn to prove that a.a.s. the number of d -faces in the core f d ( ˜ Y ) = r (˜ c ) d +1 ˜ cn (cid:0) nd +1 (cid:1) (1 + o (1)) . Let M A denote the number of d -faces in Y all of whose ( d − -faces have property A . Clearly, f d ( ˜ Y ) ≥ M A sinceno ( d − -face with property A is collapsed. In addition, since this is a local property it suffices, as before, tocompute the expectation of M A . The probability that a d -simplex σ belongs to Y is ˜ cn , and we can expose asubset T of its neighborhood in the same careful fashion as done in the proof of Lemma 4.5. The probabilitythat all the d -trees growing from σ ’s ( d − -faces have property A S is at least Pr ˜ c [ A S ] d +1 − o (1) > r ( c ) d +1 .If this occurs, then every ( d − -face τ ⊂ σ has property A by letting τ be the root of the d -tree T ∪ { σ } .Consequently E [ M A ] ≥ r ( c ) d +1 ˜ cn (cid:0) nd +1 (cid:1) . The upper bound is proved similarly, by showing that the probabilitythat a d -face survives the first k collapse phases tends to r (˜ c ) d +1 as k grows. Y d (cid:0) n, cn (cid:1) Here we prove the parts of Theorem 1.2 that deal with the C -shadow of Y d (cid:0) n, cn (cid:1) . Namely, we show that for c < γ d , the C-shadow of Y = Y d (cid:0) n, cn (cid:1) has size Θ( n ) , and for c > γ d its size is | SH C ( Y ) | = (cid:18) nd + 1 (cid:19) ((1 − t ) d +1 + o (1)) . Both statements follow directly from the previous proofs. Regarding the range c < γ d , a simple second momentcalculation shows that a.a.s. there are Θ( n ) sets of d + 2 vertices in Y that span all but one of the d -faces in theboundary of a ( d + 1) -simplex. The missing d -face in every such configuration is obviously in the C-shadow.On the other hand, if the C-shadow is large, viz., | SH C ( Y ) | (cid:29) n , then for every c < c (cid:48) < γ d , with probabilitybounded away from zero, the core of Y d (cid:16) n, c (cid:48) n (cid:17) contains a complex that is not the boundary of ( d + 1) -simplex.But this contradicts Theorem 1.1(I). 13e prove the supercritical case c > γ d in much the same way that we calculated the number of d -faces inthe core. Namely, for the lower bound we count d -simplices not in Y all of whose ( d − -faces have property A . For the upper bound we count d -simplices that if added into Y do not survive k phases of collapse. Asbefore, both properties are local and by a second moment argument are concentrated around their means, whichare computed by Poisson d -tree approximations. d -acyclicity In the previous section we saw that the threshold γ d for d -collapsibility in Y d (cid:0) n, cn (cid:1) coincides with the thresholdin which rooted collapsibility in T d ( c ) almost surely eliminates all the d -faces containing the root. In the caseof d -acyclicity, the correspondence is similar but more intricate. In fact, the threshold c d for d -acyclicity coin-cides with two seemingly separate thresholds of T d ( c ) ’s parameters. These will used to bound the d -acyclicitythreshold from below and above respectively. Since both occur at c = c d it follows that these bounds are tight.Furthermore, if c > c d , these two parameters yield upper and lower bounds for β d ( Y, R ) which are tight uptosmall order error terms. Finally, the tight estimation for β d ( Y, R ) allows us to compute the density of the shadow of Y .If a d -complex Y has more d -faces than ( d − -faces, then β d ( Y, R ) ≥ f d ( Y ) − f d − ( Y ) > . For Y = Y d (cid:0) n, cn (cid:1) , this happens only when c > d + 1 , but we can say a bit more. Even though Y and its core ˜ Y have the same d -th Betti number, it turns out that there is a wider range of the parameter c for which ˜ Y hasmore d -faces than ( d − -faces. In fact, one can show that f d ( ˜ Y ) > f d − ( ˜ Y ) if and only if c > c d , usingthe expressions for these face numbers in Theorem 1.1(II.a). However, it is significantly easier to prove thesame lower bound on β d ( Y ) by analyzing T d ( c ) as follows. Let S k ( Y ) be obtained by removing all exposed ( d − -faces in R k ( Y ) . The average degree of the ( d − faces in S k ( Y ) is approximated using the conditionalexpectation E [ δ k | δ k > ∧ δ k − > . In words, this is the expected degree of the root of T d ( c ) after k phases of rooted collapses, conditioned on thefact that its degree remains strictly greater than throughout the collapse process. We claim that this capturesthe average degree of ( d − -faces in S k ( Y ) . A ( d − -face τ of Y belongs to S k ( Y ) if and only if its degreeafter k − phases of τ -rooted collapse is > and stays positive after one more phase. Indeed, as long as d τ > the τ -rooted collapse and non-rooted collapse are identical. A difference occurs when d τ = 1 , at which pointthe rooted collapse continues as usual, but the non-rooted collapse eliminates τ . If the average degree of ( d − -faces exceeds d + 1 , this yields, via a simple double-counting argument, a positive lower bound for β d ( Y ) . Asimple calculus exercise then shows that this condition holds if and only if c > c d . Namely, lim k →∞ E [ δ k | δ k > ∧ δ k − > > d + 1 ⇐⇒ c > c d . The most substantial role of local weak convergence is in proving the lower bound on the d -acyclicitythreshold. We analyze T = T d ( c ) using tools from spectral theory and functional analysis. For this reason weare still unable to resolve this question over finite fields of coefficients. As further detailed below, we define x T := π L ( T ) ,e o ( { } ) to be the measure of the atom { } according to the spectral measure π of the Laplacian L ( T ) of T with respect to the characteristic vector e o of the root o . Local weak convergence implies that x T isan upper bound on the normalized dimension of the left kernel Z of ∂ d ( Y ) . Note that if Y d (cid:0) n, cn (cid:1) is d -acyclicthen dim Z equals (1 + o (1)) (cid:0) nd (cid:1) (cid:16) − cd +1 (cid:17) , and otherwise it is greater. Indeed, the proof shows that E T ∼ T d ( c ) [ x T ] = 1 − cd + 1 ⇐⇒ c < c d , and if c > c d , this expectation is greater than − cd +1 . .1 Acyclicity beyond collapsibility - Theorem 1.1(II.b) To prove that a random complex is d -acyclic beyond the d -collapsibility threshold, we cannot restrict ourselvesto purely combinatorial arguments. It is not a-priori clear that the local weak limit of a random complex holdsenough information to prove such a statement. Surprisingly, perhaps, this is the case when we work over R . Inthis section we describe the main ingredients of this method, which appears in [14], where complete proofs canbe found.Let Y = Y d (cid:0) n, cn (cid:1) . The primary goal in the proof is to find a tight upper bound for lim n →∞ ( nd ) E [ β d ( Y )] .It turns out more useful to work with the corresponding Laplace operator L ( Y ) = ∂ d ( Y ) ∂ d ( Y ) ∗ . We considerits kernel Z which coincides with the left kernel of ∂ d ( Y ) . Let P Z : R Y d − → Z be the orthogonal projection tothe space Z . By linear algebra, dim Z = (cid:88) τ ∈ Y d − (cid:107) P Z ( e τ ) (cid:107) , where e τ is the unit vector of τ .The spectral theorem from functional analysis offers a new perspective of (cid:107) P Z ( e τ ) (cid:107) . Associated with everyself-adjoint operator L on a Hilbert space H , and a vector ψ ∈ H is the spectral measure of L with respect to ψ .It is a real measure denoted π L,ψ which satisfies (cid:104) F ( L ) ψ, ψ (cid:105) = (cid:90) R F ( x ) dπ L,ψ ( x ) , for every measurable function F : R → C . The operator F ( L ) is uniquely defined by extending the action ofpolynomials on the operator L .If H is finite-dimensional, π L,ψ is a discrete measure supported on the spectrum of L and π L,ψ ( λ ) = (cid:107) P λ ψ (cid:107) , where P λ is the orthogonal projection to the λ -eigenspace.We use this theorem with the measure π L ( Y ) ,e τ . Here Y is a d -complex and H = (cid:96) ( Y d − ) . The self adjointoperator is the Laplacian L ( Y ) , and e τ is the characteristic vector of some ( d − -face of Y .In particular, with Y = Y d (cid:0) n, cn (cid:1) and Z as before, (cid:107) P Z ( e τ ) (cid:107) is simply the measure of the atom { } according to the spectral measure π L ( Y ) ,e τ .The difficulty with applying the spectral theorem to the Poisson d -tree is that the degrees in this tree may beunbounded. We must, therefore, consider the subtleties of the theory of unbounded operators [19]. Briefly, theLaplacian L ( T ) of an infinite d -tree T is a symmetric operator, directly defined on the dense subset of finitelysupported vectors of H = (cid:96) ( T d − ) . The symmetric densely-defined operator L ( T ) has a unique extension to H . This extension need not be self-adjoint, and when it does we say that the tree T is self-adjoint. In such casesthe spectral theorem can be applied on L ( T ) . It can be shown that a Poisson d -tree is, almost surely, self-adjoint.We employ the useful property that spectral measures are continuous with respect to local weak conver-gence. Since Y = Y d (cid:0) n, cn (cid:1) converges in local weak convergence to the Poisson d -tree T = T d ( c ) , whichis almost-surely self-adjoint, we conclude that the expected measure E Y [ π L ( Y ) ,e τ ] weakly converges to the ex-pected measure E T [ π L ( T ) ,e o ] , where o is the root of T . In particular, by measuring the closed set { } , lim sup n →∞ E (cid:2) (cid:107) P Z ( e τ ) (cid:107) (cid:3) ≤ E [ x T ] , where x T := π L ( T ) ,e o ( { } ) . Consequently, E [dim Z ] ≤ (1 + o n (1)) (cid:18) nd (cid:19) E [ x T ] . By the Rank-Nullity Theorem from linear algebra, dim Z − β d ( Y ) = f d − ( Y ) − f d ( Y ) , (cid:0) nd (cid:1) E [ β d ( Y )] ≤ E [ x T ] − cd + 1 + o n (1) . There remains the problem of bounding the expectation E T [ x T ] without directly computing the operator’skernel. This difficulty is bypassed using the recursive structure of d -trees to derive a simple recursion formulason these spectral measures, as in the following lemma.Let T be a self-adjoint d -tree with root o , and let σ , ..., σ m be the d -faces that contain the root. For ≤ j ≤ m and ≤ r ≤ d we denote by τ j,r the ( d − -faces of σ j other than the root. Also, T j,r denotes the d -treerooted at τ j,r that contains τ j,r and its branch. Lemma 5.1. x T = 0 if there exists some ≤ j ≤ m such that x T j, = ... = x T j,d = 0 . Otherwise, x T = m (cid:88) j =1 (cid:32) d (cid:88) r =1 x T j,r (cid:33) − − . Proof sketch.
Consider the following bounded family of measurable functions { H s : R → C | s ∈ R \ { }} ,H s ( x ) = isx + is . Note that H s approaches the Kronecker delta function δ x, as s → . Given a self-adjoint d -tree T with root o ,we define h T ( s ) := (cid:90) R H s ( x ) dπ L ( T ) ,e o ( x ) . In particular, x T = lim s → h T ( s ) . The proof is concluded by showing that the functions h T j,r ’s and h T satisfythe formula h T ( s ) m (cid:88) j =1 (cid:32) is + d (cid:88) r =1 h T j,r ( s ) (cid:33) − = 1 , (6)and letting s → .We turn to describe the derivation of equation (6). Denote L := L ( T ) , ˜ L := (cid:76) j,r L ( T j,r ) and M theLaplacian of the subcomplex of T which contains σ , ..., σ m and their subfaces. In particular, L = M ⊕ ˜ L . Theoperators H s ( L ) and H s ( ˜ L ) are scalar multiples of the resolvents R := ( L + is · I ) − and ˜ R := ( ˜ L + is · I ) − .In particular, h T ( s ) = is · (cid:104) Re , e (cid:105) . Simple observations about the d -tree structure yields that (i) ˜ Re o = is e o and (ii) (cid:68) ˜ Re τ j,r , e τ j (cid:48) ,r (cid:48) (cid:69) = 0 when ( j, r ) (cid:54) = ( j (cid:48) , r (cid:48) ) . In addition, we use the Second Resolvent Identity whichstates that RM ˜ R = ˜ R − R . Here the operator M has a very concrete and usable form, being the Laplacian ofa d -complex which consists of m distinct d -faces with a common ( d − -subface o . Equation (6) is obtainedusing simple algebraic manipulations by comparing terms of the form (cid:68) ( RM ˜ R ) e τ , e τ (cid:48) (cid:69) = (cid:68) ( ˜ R − R ) e τ , e τ (cid:48) (cid:69) ,when τ, τ (cid:48) are ( d − -faces of T of distance at most from the root o .The remaining step of the argument is an application of the recursive formula in Lemma 5.1 to the Poisson d -tree. Lemma 5.2.
Let T = T d ( c ) be a rooted Poisson d -tree with parameter c . Then, E [ x T ] ≤ max (cid:26) t + ct (1 − t ) d − cd + 1 (cid:16) − (1 − t ) d +1 (cid:17) | t ∈ [0 , , t = e − c (1 − t ) d (cid:27) Note that this maximum is taken over a finite set, due to the condition t = e − c (1 − t ) d .16 roof. Let T be a Poisson d -tree with root degree m and { T j,r | ≤ j ≤ m, ≤ r ≤ d } its subtrees as above.The parameters x T , { x T j,r } can be considered as random variables when T is T d ( c ) -distributed. The randomvariables { x T j,r } are i.i.d and are distributed like x T since all the subtrees T j,r are independent Poisson d -trees.In addition, these variables satisfy the equation of Lemma 5.1.These observations suggest the following equivalent description of D , the distribution of the random variable x T . First sample a P oi ( c ) -distributed integer m , and x T j,r ∼ D i.i.d for every ≤ j ≤ m and ≤ r ≤ d . Giventhese samples, the value of x T is determined by Lemma 5.1.In particular, if we let t := Pr( x T > , then t satisfies the equation t = ∞ (cid:88) m =0 e − c c m m ! (1 − (1 − t ) d ) m = e − c (1 − t ) d . (7)Let X be a D -distributed random variable, E [ X ] = E (cid:34) {∀ j ∈ [ m ] , S j > } (cid:80) mj =1 S − j (cid:35) Here S , S , . . . , S m are random variables whose distribution is that of a sum of d i.i.d. D -distributed variables.By expressing the probability Pr[ {∀ j ∈ [ m ] , S j > } ] as t , exploiting the symmetry between the different S j ’s andusing basic properties of the Poisson distribution, we are able to express this expectation in terms of t , E [ X ] = t + ct (1 − t ) d − cd + 1 (cid:16) − (1 − t ) d +1 (cid:17) , as was claimed.It requires only basic calculus to conclude that:1. For c < c d , the maximum of t + ct (1 − t ) d − cd + 1 (cid:16) − (1 − t ) d +1 (cid:17) , s.t. t = e − c (1 − t ) d is attained at t = 1 . Consequently, E [ β d ( Y )] ≤ o ( n d ) .
2. For c > c d , the maximum is attained at t = t ( c, d ) . Consequently, E [ β d ( Y )] ≤ (1 + o n (1)) (cid:18) nd (cid:19) (cid:18) c (1 − t ) d +1 d + 1 − t + ct (1 − t ) d (cid:19) . (8)The proof of Theorem 1.1 (II.b) is concluded by the following standard probabilistic argument. Let c < c d and ε > . The absence of small non-collapsible subcomplexes in Y (Lemma 4.1) implies that it has nosmall d -cycles except ∂ ∆ d +1 ’s. The computation above shows that the dimension of the cycle space is o ( n d ) .Therefore, removing Bin (cid:16)(cid:0) nd +1 (cid:1) , ε/n (cid:17) = Θ( n d ) random d -faces eliminates all large (non ∂ ∆ d +1 ) d -cycles withhigh probability, hence Y d (cid:0) n, c − εn (cid:1) is a.a.s. d -acyclic except ∂ ∆ d +1 ’s. The key idea of [5] for the computation of a matching upper bound for the d -acyclicity threshold uses an analysisof the collapse process. Let Y be a d -complex and k some positive integer. Recall that R k ( Y ) is the simplicialcomplex obtained from Y by k phases of d -collapse and S k ( Y ) is its subcomplex obtained by removing theexposed ( d − -faces. Clearly, β d ( Y ) = β d ( S k ( Y )) since d -collapsing and removing exposed faces doesnot affect the right kernel of ∂ d . The final ingredient of the strategy is the observation that β d ( S k ( Y )) ≥ d ( S k ( Y )) − f d − ( S k ( Y )) , and the fact that the parameter f d ( S k ( Y )) − f d − ( S k ( Y )) can be studied from thelocal weak limit.Let Y = Y d (cid:0) n, cn (cid:1) where c > c d , and Y (cid:48) := S k ( Y ) for a sufficiently large integer k . E [ β d ( Y )] ≥ E [ f d ( Y (cid:48) ) − f d − ( Y (cid:48) )]= E (cid:88) τ ∈ Y (cid:48) d − (cid:18) d Y (cid:48) ( τ ) d + 1 − (cid:19) = (cid:18) nd (cid:19) Pr[ τ ∈ Y (cid:48) ] (cid:18) E [ d Y (cid:48) ( τ ) | τ ∈ Y (cid:48) ] d + 1 − (cid:19) . (9)For the last equation we can, due to symmetry, consider a fixed τ and apply the Law of Total Expectation tothe event { τ ∈ Y (cid:48) } . As mentioned above, the ( d − -face τ of Y belongs to Y (cid:48) if and only if in the τ -rootedcollapse process of Y , the degree of τ is greater than after ( k − phases and positive after k steps. Byapproximating the τ -rooted collapse process of Y with the rooted collapse process on T d ( c ) we obtain that upto o n (1) factor, Pr[ τ ∈ Y (cid:48) ] ≥ Pr[ δ k >
1] = 1 − t k − c (1 − t k − ) d t k − . (10)Furthermore, upto o n (1) factor, E (cid:2) d Y (cid:48) ( τ ) | τ ∈ Y (cid:48) (cid:3) = E [ δ k | δ k > ∧ δ k − > ≥ ∞ (cid:88) j =2 j · Pr[ δ k = j | δ k > ∧ δ k − > ≥ ∞ (cid:88) j =2 j · Pr[ δ k = j ]Pr[ δ k − > c (1 − t k − ) d (1 − t k )1 − t k − − c (1 − t k − ) d t k − . (11)By combining Inequalities (9),(10) and (11), and letting k → ∞ , we obtain that E [ β d ( Y )] ≥ (1 + o n (1)) (cid:18) nd (cid:19) (cid:18) c (1 − t ) d +1 d + 1 − t + ct (1 − t ) d (cid:19) . This bound starts to be meaningful for c > c d , where this expression matches the upper bound from (8). Since β d is 1-Lifschitz, a straightforward application of Azuma’s inequality yields that a.a.s. β d deviates from itsexpectation by only o ( n d ) . In particular, this shows a matching upper bound for the threshold of d -acyclicity. R -Shadow of Y d (cid:0) n, cn (cid:1) The behavior of the R -shadow when c < c d is studied similarly to the C-shadow in the collapsible regime (SeeSection 4.3).We turn to the range c > c d . Here we do not give a proof, but only a general intuitive explanation. Anaccurate analysis of the measure concentration can be found in [14]. Recall that (cid:0) nd (cid:1) E [ β d ( Y ; R )] −−−→ n →∞ g d ( c ) := cd + 1 (1 − t ) d +1 − (1 − t ) + ct (1 − t ) d . A simple technical claim shows that for every c > c d , the limit function g d ( c ) is differentiable w.r.t the variable c and its derivative equals to d +1 (1 − t ) d +1 . 18t turns out to be more convenient to work here with a d -dimensional analog of the so-called evolution ofrandom graphs . Let Y d ( n, m ) be a random simplicial complex with n vertices, a complete ( d − -skeleton and m uniformly random d -faces. Y (cid:48) = Y d ( n, m + 1) can be sampled by the following procedure. First sample Y = Y d ( n, m ) and then add a random d -face which does not belong to Y . Therefore, the following equationholds in expectation, β d ( Y (cid:48) ; R ) − β d ( Y ; R ) = 1 (cid:0) nd +1 (cid:1) | SH R ( Y ) | . Letting m ∼ Bin (cid:16)(cid:0) nd +1 (cid:1) , cn (cid:17) yields (cid:18) nd (cid:19) (cid:32) g d (cid:32) c + d + 1 (cid:0) nd (cid:1) (cid:33) − g d ( c ) (cid:33) ≈ (cid:0) nd +1 (cid:1) | SH R ( Y ) | , and by letting n → ∞ , (cid:0) nd +1 (cid:1) | SH R ( Y ) | ≈ ( d + 1) g (cid:48) d ( c ) = (1 − t ) d +1 . This argument can be made rigorous by incrementing Y with εn d random d -faces at a time rather than one byone, and applying standard measure concentration inequalities. Although this article is mostly a review of previous work, it does contain several new results, e.g., Theorem1.1(I) that deals with the collapsible regime is a little stronger than the original result of [6]. Other notablenew results concern the asymptotic densities of the core and the C-shadow of Y d (cid:0) n, cn (cid:1) for c > γ d , improvingthe main theorem of [4] which says that Y d (cid:0) n, cn (cid:1) is a.a.s. non-collapsible for c > γ d . As mentioned above,although the main result in that paper is correct, there is an error in the proof, which we are able to remedy hereusing the techniques of Riordan [20].The results surveyed here can be viewed from several perspectives which suggest different problems forfuture research.From the combinatorial perspective, the phase transition in the density of the shadow of Y d (cid:0) n, cn (cid:1) is ofgreat interest. We conjecture that the R -shadow grows from linear in n to a giant (order Θ( n d +1 ) ) in a singlestep in a random evolution of simplicial complexes. This starkly contrasts with the gradual growth of the giantcomponent in random graphs. It is of particular interest to understand the structure of the critical complex.Numerical experiments suggest that its ( d − -homolgy group has torsion of size exp(Θ( n d )) , but we do notknow a proof of this yet.On the topological side, it would be very interesting to better understand the d -cycles which appear in Y d (cid:0) n, cn (cid:1) when c > c d . Provably, they consist of Θ( n d ) d -faces, but numerical experiments suggest that theylie in an unknown territory in the realm of homological d -cycles. Unlike closed manifolds, in which the degreeof all ( d − -faces equals to , it seems that in these d -cycles the average degree approaches d + 1 , which is thelargest possible for a minimal d -cycle. Namely, these d -cycles are in some sense the opposite of manifolds.In addition, the random complexes Y d (cid:0) n, cn (cid:1) , where γ d < c < c d have the nice property of being d -acyclicbut not d -collapsible. 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