Contemporary Mathematics

Topology of random simplicial complexes: a survey

Matthew Kahle

1. Introduction

This expository article is based on a lecture from the Stanford Symposium onAlgebraic Topology: Application and New Directions, held in honor of GunnarCarlsson, Ralph Cohen, and Ib Madsen in July 2012.

“I predict a new subject of statistical topology. Rather thancount the number of holes, Betti numbers, etc., one will bemore interested in the distribution of such objects on noncom-pact manifolds as one goes out to infinity.”— Isadore Singer, 2004 [61]

1.1. Motivation.1.1.1. Randomness models the natural world.(1) The field of topological data analysis has received a lot of attention over

the past several years — see, for example, the survey articles of Carlsson[16] and Ghrist [30]. In order to quantify the statistical significance oftopological features detected with these methods, it will be helpful to havefirmer probabilistic foundations.

(2) Certain areas of physics seem to require random topology. John Wheelersuggested in the 1960’s, for example that inside a black hole, one wouldneed a topological and geometric theory to account for relativity, andthat near the presumed singularity one would also require a quantumtheory, necessarily stochastic. He reasoned that inside a black hole thetopology of space-time itself may best be described as a quantum foam, oras a probability distribution over shapes rather than any particular fixedshape.

(3) We might also want to understand better why certain mathematical phe-nomena are so ubiquitous. For example, a folklore theorem is that “almostall groups are hyperbolic”. This turns out to be true under a variety ofdifferent measures — see, for example, Ollivier’s survey on random groups[56].

The author gratefully acknowledges support from DARPA grant # N66001-12-1-4226.

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It is known that many simplicial complexes and posets found in thewild are homotopy equivalent to bouquets of spheres. See for exampleForman’s comments in Section 1 of his discrete Morse theory notes [28].He points out that Morse theory gives a convenient way to prove such the-orems, but goes on to say, “However, that does not explain why so manysimplicial complexes that arise in combinatorics are homotopy equiva-lent to a wedge of spheres. I have often wondered if perhaps there issome deeper explanation for this.” From the point of view of this article,one might hope to make mathematical sense of such questions measure-theoretically.

(4) Random topology might even provide tractable toy models for difficult tounderstand number-theoretic settings.

The following family of simplicial complexes was apparently intro-duced and first studied topologically by Björner [11]: ∆n has primes lessthan n as its vertices, and its faces correspond to square-free numbersi with 1 ≤ i ≤ n. He pointed out that the Euler characteristic χ(∆n)coincides with the Mertens function

M(n) =

n∑k=1

µ(k),

where µ(k) is the Möbius function. The Riemann hypothesis is equivalentto the statement that M(n) satisfies

|M(n)| = O(n1/2+ε)

for every ε > 0, suggesting that studying the topology of ∆n might bequite interesting.

Björner proved these complexes are all homotopy equivalent to wedgesof spheres (not necessarily of the same dimension), hence homologyH∗(∆n)is torsion-free. He also provided some estimates for Betti numbers, show-ing that

βk ≈n

2 log n

(log log n)k

k!

for k ≥ 0 fixed, and also that∑k≥0

βk (∆n) =2n

π2+O

(nθ)

for all θ > 17/54.Pakianathan and Winfree [57] studied a fairly general framework

“quota complexes”, and gave natural topological formulations of the primenumber theorem, twin prime conjecture, Goldbach’s conjecture, and theexistence of odd perfect numbers, among others.

Unfortunately, these attractive papers do not seem to get us any closerto proving the Riemann hypothesis, but these kinds of complexes meritfurther study, and might be interesting to model probabilistically. Ofcourse primes are not random, but they are pseudorandom, and for manypurposes behave as if they were a random subset of the integers withdensity predicted by the prime number theorem— the Green–Tao theoremis a celebrated example [31].

TOPOLOGY OF RANDOM SIMPLICIAL COMPLEXES: A SURVEY 3

1.1.2. The probabilistic method provides existence proofs. This is a complemen-tary point of view. Random objects often have desirable properties, and in somecases it is difficult to construct explicit examples. This has been one of the mostinfluential ideas in discrete mathematics of the past several decades — for a broadoverview of the subject, see Alon and Spencer’s book [3].

(1) In extremal graph theory, for example, the probabilistic method has provedto be an extremely powerful tool. Almost all graphs are known to havestrong Ramsey properties (i.e. no large cliques or independent sets), butafter several decades of research no one knows how to construct exam-ples. This somewhat paradoxical situation is sometimes referred to as theproblem of “finding hay in a haystack.”

(2) Since early work of Pinsker [58], and even earlier work of Barzdin andKolmogorov [9], it has been known that in various senses almost all graphsare expanders. For an extensive survey of expander graphs and theirmany applications in mathematics and computer science, see [38]. Someof the work surveyed in this article may be viewed as higher-dimensionalanalogues of this paradigm. One of the goals of this article is to describeexpander-like qualities of random simplicial complexes.

(3) The probabilistic method has found applications in other areas of math-ematics as well. Gromov asked, “What does a random group look like?As we shall see the answer is most satisfactory: nothing like we have everseen before,” [32], and then later fulfilled his own prediction by provingthe existence of a finitely generated group Γ whose Cayley graph admitsno uniform embedding in the Hilbert space [33]. Rather than exhibit sucha group explicitly, he makes a very delicate measure-theoretic argument.

1.2. Earlier work. Although this article will focus on random simplicial com-plexes, we first put this in a larger context of random topology.

(1) One of the earliest references to random topology of which I am awareis Milnor’s 1964 paper “Most knots are wild” [53]. Milnor showed that“most” knots are wild, in the sense of wild embeddings being Baire dense inthe space of all embeddings. In the concluding remarks he points out thatquestions about embeddings being wild or knotted with probability oneare of a very different kind. In particular, he noted that with probabilityone, Brownian motion in 4-space does not self intersect and asked if isknotted. Kendall [46] and independently Weinberger [66] showed thatthe answer is “no.”

(2) Random triangulated surfaces were studied by Pippenger and Schleich[59]. Their model is randomly gluing together n oriented triangles, uni-formly over all such glueings, and they compute the expected genus E[gn]of the resulting oriented surface as n → ∞. Part of the motivation dis-cussed comes from physics — such random surfaces arise in 2-dimensionalquantum gravity and as world-sheets in string theory.

(3) Dunfield and W. Thurston [24] simultaneously studied this model forrandom surfaces, and they pointed out that in general one can not makea random 3-manifold by gluing together tetrahedra in an analogous way,as the probability that a gluing results in a manifold tends to 0 as thenumber of tetrahedra n→∞. They introduced a new model for random

4 MATTHEW KAHLE

3-manifolds M where one takes a random walk on the mapping classgroup, resulting in a random gluing of a two handlebodies.

(4) Farber and Kappeler [27] introduced random planar linkages, were thelengths of the links are random. The configuration space for such a link-age is a smooth manifold with probability one. They let the number oflinks (and hence the dimension of the manifold) tend to infinity, and giveasymptotic formula for the expectations of the Betti numbers.

(5) Gaussian random functions on manifolds were studied by Adler and Tay-lor, and in particular they discovered the kinematic formula for the ex-pected Euler characteristic of the sub-level sets. See for example, Chapter12 of their book [2]. In this setting, giving a formula for the expectationof the Betti numbers seems to be a difficult open problem. For a surveyof this area see [1].

(6) Random 2-dimensional simplicial complexes were first studied by Linialand Meshulam [49], and the k-dimensional version by Meshulam andWallach [52]. These are natural higher-dimensional analogues of ran-dom graphs — the main results of [49], [52], and [41] are cohomologicalanalogues of the Erdős–Rényi theorem which characterizes the thresholdfor connectivity of a random graph. All of these theorems describe sharptopological phase transitions where cohomology passes to vanishing withhigh probability, within a very short window of parameter.

Since the influential papers [49] and [52], random complexes and their topo-logical properties have continued to be explored by several teams of researchers.Random flag complexes [39, 41] are another generalization of Erdős–Rényi ran-dom graphs to higher dimensions, putting a measure on a wide range of possibletopologies.

2. Random graphs

Random graphs are 1-dimensional random simplicial complexes. The ran-dom graph G(n, p), sometimes called the Erdős–Rényi model, has vertex set [n] ={1, 2, . . . , n} and every possible edge appears independently with probability p. Theclosely related random graph G(n,m) is selected uniformly among all

((n2)m

)graphs

on n vertices with exactly m edges.It is also sometimes useful to consider a closely related “random graph process”;

see Figure 1. In the process

{G(n,m)}(n2)m=1,

the mth edge is selected uniformly randomly from the remaining(n

2

)− (m− 1)

edges.In random graph theory, one is usually interested in the asymptotic behavior

of such graphs as n → ∞, and p = p(n). We say that an event happens with highprobability (abbreviated w.h.p.) if the probability approaches one as n → ∞. Acelebrated theorem about the topology of random graphs is the following [26].

TOPOLOGY OF RANDOM SIMPLICIAL COMPLEXES: A SURVEY 5

Figure 1. The beginning of a random graph process on n = 12 vertices.

Theorem 2.1 (Erdős–Rényi, 1959). Let ε > 0 be fixed, and G ∼ G(n, p). If

p ≥ (1 + ε) log n

n,

then G is connected w.h.p., and if

p ≤ (1− ε) log n

n,

then G is disconnected w.h.p.

(Although this is generally attributed to Erdős and Rényi, it seems that Erdősand Rényi actually proved the analogous statement for G(n,m), and Stepanov firstproved the G(n, p) statement in 1969 [65].)

The strongest possible statement of the Erdős–Rényi theorem is slightly sharperthan this, as follows. Let

β̃0(G) = # of connected components of G− 1,

i.e. to a topologist, the reduced 0th Betti number of G.

Theorem 2.2 (Erdős–Rényi, 1959). Let G = G(n, p) where

p =log n+ c

n,

and c ∈ R is fixed. Then as n → ∞, β̃0(G) is asymptotically Poisson distributedwith mean e−c. In particular,

P[G is connected]→ e−e−c

Corollary 2.3. Let ω →∞ arbitrarily slowly as n→∞.If

p ≥ log n+ ω

n,

6 MATTHEW KAHLE

then G is connected w.h.p., and if

p ≤ log n− ωn

,

then G is disconnected w.h.p.

Is there any way we could guess that p = log n/n the right threshold, if wedid not already know that? To give an intuition for this, we set p = (log n + c)/nwith c ∈ R fixed, and ask what is the expected number of isolated vertices. Theprobability that a vertex is isolated is (1−p)n−1, by independence. Then by linearityof expectation, the expected number of isolated vertices X0 is given by

E[X0] = n(1− p)n−1.

Since p → 0 as n → ∞, it is reasonable to approximate 1 − p by e−p, and then itfollows easily that E[X0] → e−c as n → ∞. With a little more work, for exampleby computing all the higher moments, one can show that in fact X0 converges inlaw to a Poisson distribution with mean e−c. See, for example, Chapter 8 in [3] foran overview of limit theorems in random graph theory.

To finish the proof of Theorem 2.2, one also needs a structure theorem, namelythat for p is in this range, w.h.p. G ∼ G(n, p) consists of only two kinds of con-nected components: a unique “giant component,” and isolated vertices. Given thisstructure, G is connected if and only if there are no isolated vertices. See Chapter7 of [12] for a complete proof.

Corollary 2.3 shows that p = log n/n is a sharp threshold for connectivityof G(n, p), meaning that the phase transition from probability 0 to probability 1happens within a very narrow window. More precisely, a function f is said to bea sharp threshold for a graph property P if there exists a function g = o(f) suchthat

P[G(n, p) ∈ P]→

1 : p ≥ f + g

0 : p ≤ f − gExactly which monotone graph properties have sharp thresholds is a question

which has been extensively studied. See for example the paper of Friedghut withappendix by Bourgain [29].

Corollary 2.3 can be summarized as saying that the threshold function for “Gis connected” is the same as the threshold function for “G has no isolated vertices.”The following result of Bollobás and Thomasson takes this idea all the way to itslogical conclusion [13].

Theorem 2.4. For a random graph process {G(n,m)}(n2)m=1, with high probability

min{M : G(n,M) has no isolated vertices} = min{M : G(n,M) is connected}.

We leave it to readers to convince themselves that Theorem 2.4 is even sharperthan Corollary 2.3.

There is another topological phase transition for G ∼ G(n, p), namely wherecycles first appear, or to a topologist, where H1(G) first becomes nontrivial. SeePittel [60] for a proof of the following characterization of the appearance of cycles.

TOPOLOGY OF RANDOM SIMPLICIAL COMPLEXES: A SURVEY 7

Figure 2. The beginning of a random 2-complex process on n =12 vertices.

Theorem 2.5. Let G ∼ G(n, p), where p = c/n and c > 0 is constant.

P[H1(G) 6= 0]→

1 : c ≥ 1

√1− c exp(c/2 + c2/4) : c < 1

Note that in contrast to the connectivity threshold, this threshold is sharp onone side but not on the other.

3. Random 2-complexes

Linial and Meshulam initiated the topological study of random 2-dimensionalsimplicial complexes Y (n, p) in [49]. This model random simplicial complex isdefined to have vertex set [n], edge set

([n]2

)(i.e. the underlying graph is a complete

graph), and each of the(n3

)possible triangle faces is included with probability

p, independently. The beginning of a random 2-complex process is illustrated inFigure 2.

The main result of [49] is a cohomological analogue of Theorem 2.1.

Theorem 3.1 (Linial–Meshulam, 2006). Let ε > 0 be fixed and Y ∼ Y (n, p).If

p ≥ (2 + ε) log n/n,

then w.h.p. H1(Y,Z/2) = 0, and if

p ≤ (2− ε) log n/n,

then w.h.p. H1(Y,Z/2) 6= 0.

Although Theorem 3.1 is analogous to Theorem 2.1, the proof is much harder.Linial and Meshulam used a combination of co-isoperimetric inequalities (analogousto Cheeger constant of a graph) and intricate cocycle-counting arguments.

A few comments might be in order.(1) The Linial–Meshulam theorem is sharper than this, analogous to Corollary

2.3, but in this survey article we sometimes trade the strongest result fora simpler statement.

8 MATTHEW KAHLE

(2) The threshold p = 2 log n/n is just what is necessary in order to ensurethat there are no isolated edges. The analogue of the “stopping time”Theorem 2.4 for the random 2-complex process was recently establishedin [43].

(3) Theorem 3.1 is stated for cohomology, but universal coefficients for ho-mology and cohomology give the analogous result for homology.

(4) It was shown by Meshulam and Wallach [52] that the same result holdswith (Z/`)-coefficients for every fixed ` (and for an analogous d-dimensionalmodel).

(5) The threshold for homology with Z-coefficients is still unknown. It mightseem that this would follow from Meshulam and Wallach’s work, but theproblem is that there could be `-torsion, where ` is growing with n.

Linial and Meshulam asked in [49] for the threshold for simple connectivity,and this was eventually shown to be much larger [7].

Theorem 3.2 (Babson et al., 2011). Let ε > 0 be fixed and Y ∼ Y (n, p). If

p ≥ nε√n,

then w.h.p. π1(Y ) = 0, and if

p ≤ n−ε√n,

then w.h.p. π1(Y ) is a nontrivial group, hyperbolic in the sense of Gromov.

The study of fundamental groups of random 2-complexes is continued in [37].A group G is said to have Kazhdan’s property (T) if the trivial representation isan isolated point in the unitary dual of G equipped with the Fell topology. Moreintuitively, Property (T) is an “expander-like” property of groups; in fact the firstexplicit examples of expanders were constructed by Margulis from Cayley graphsof quotients of (T) groups such as SL(3,Z) [51]. For a comprehensive overview ofProperty (T) see the monograph [10].

The following theorem shows that the threshold for π1(Y ) to have Kazhdan’sProperty (T) is the same as the vanishing threshold for H1(Y,Z/2) [37].

Theorem 3.3 (Hoffman et al., 2012). Let ε > 0 be fixed and Y ∼ Y (n, p).Then as n→∞,

P[π1(Y ) is Kazhdan]→

1 : p ≥ (2 + ε) log n/n

0 : p ≤ (2− ε) log n/n

The proof that Y (n, p) is Kazhdan when p ≥ (2+ε) log n/n utilizes the followingtheorem of Żuk [67].

Theorem 3.4 (Żuk). If X is a pure 2-dimensional locally-finite simplicial com-plex so that for every vertex v, the vertex link lkv(X) is connected and the normalizedgraph Laplacian L = L(lkv(X)) has smallest positive eigenvalue λ2(L) > 1/2, thenπ1(X) has property (T).

The link of a vertex in the random 2-complex Y (n, p) has the same probabilitydistribution as a random graph G(n − 1, p). So Żuk’s theorem reduces the proofof Theorem 3.3 to a question about Laplacians of random graphs. However, new

TOPOLOGY OF RANDOM SIMPLICIAL COMPLEXES: A SURVEY 9

results about such Laplacians are still required in order to prove Theorem 3.3.Establishing the following comprises most of the work in [37].

Theorem 3.5 (Hoffman et al., 2012). Fix k ≥ 0 and ε > 0. Let 0 = λ1 ≤ λ2 ≤· · · ≤ λn ≤ 2 be the eigenvalues of the normalized Laplacian of the random graphG(n, p). There is a constant C = C(k) so that when

p ≥ (k + 1) log n+ C√

log n log log n

nis satisfied, then

λ2 > 1− ε,with probability at least 1− o(n−k).

The proof of Theorem 3.3 only requires Theorem 3.5 with k = 1 and ε = 1/2.By Żuk’s theorem, the proof is almost immediate once Theorem 3.5 is established:There are n vertex links, and for each one, the probability that its spectral gap issmaller than 1/2 is o(n−1). So the probability that there is at least one vertex linkwith spectral gap too small is o(1) by a union bound — the probability that atleast one bad event occurs is never more than the sum of the probabilities of theindividual bad events.

The k = 0 case of Theorem 3.5 is also of particular interest, as this gets veryclose to the connectivity threshold for G(n, p). It would be interesting to see justhow close one can get. Consider a random graph process, for example, addingone edge at a time: is the graph already an expander (smallest positive eigenvaluebounded away from zero) at the moment of connectivity? We discuss applicationsof Theorem 3.5 with other values of k in Section 4.

Both Theorem 3.1 and Theorem 3.3 have the following corollary.

Corollary 3.6. Let ε > 0 be fixed and Y ∼ Y (n, p). If

p ≥ (2 + ε) log n/n,

then w.h.p. H1(Y,Q) = 0, and if

p ≤ (2− ε) log n/n,

then w.h.p. H1(Y,Q 6= 0.

This follows from Theorem 3.1 by the universal coefficient theorem, and followsfrom Theorem 3.3, since a Kazhdan group has finite abelianization.

Costa and Farber described two phase transitions for the cohomological dimen-sion cdπ1(Y ) [19, 20].

Theorem 3.7 (Costa–Farber). Let Y ∼ Y (n, p), and set p = n−α.(1) If α > 1 then w.h.p. cdπ1(Y ) = 1 ,(2) if 3/5 < α < 1 then w.h.p. cdπ1(Y ) = 2, and(3) if 1/2 < α < 3/5 then w.h.p. cdπ1(Y ) =∞.

To give some intuition for where the exponent 3/5 comes from: the smallesttriangulation of the projective plane has 6 vertices and 10 faces, and these appear assubcomplexes once p� n−6/10 = n−3/5. Costa and Farber show that the inducedmap on fundamental group is injective, hence for p in this regime there is 2-torsionin π1(Y ).

10 MATTHEW KAHLE

The random fundamental groups are related to other models random finitelypresented groups, as in the survey article [56], especially the “triangular” modelstudied earlier by Żuk, but the geometric and topological considerations in therandom fundamental group setting are somewhat more complicated.

As an aside, very different models of random groups based on random graphs,with large but finite cohomological dimension, for example, have recently beenstudied by Charney–Farber [17], and by Davis–Kahle [21].

The two-dimensional analogue of Theorem 2.5 describing the first appearanceof cycles has also been studied, in a series of papers by Kozlov [47], Cohen et al.[18], and Aronshtam–Linial [4]. In this last paper, the following strong estimate isestablished.

Theorem 3.8 (Aronshtam–Linial, 2012). Let Y ∼ Y (n, p) where p = c/n andc > 0 is constant. If c > 2.75381 . . . then w.h.p. H2(Y ) 6= 0.

Cohen et al. had already noticed that this theorem holds with c > 3, essentiallyfor linear algebraic reasons. It is surprisingly difficult to improve the constant past3. The number 2.75381 . . . is the unique positive solution of a fairly complicatedequation described in the paper, and Aronshtam and Linial conjecture that thisconstant is best possible.

Homology vanishing, simple connectivity, and Property (T) are all monotoneproperties for random 2-complexes. This is in contrast to what we will see inSection 4, where each homology group Hk(X),k ≥ 1, passes through two distinctphase transitions: one where it first appears, and a later one where it vanishes.

4. Random flag complexes

The flag complex X(H) of a graph H is the maximal simplicial complex com-patible with H as its 1-skeleton; in other words, the i-dimensional faces of X(H)correspond to the cliques of order i + 1 in H. (Such complexes have apparentlyarisen independently several times, and X(H) is also sometimes called the cliquecomplex or the Vietoris–Rips complex of H.)

We define the random flag complexX(n, p) to be the flag complex of the randomgraph G(n, p). Every simplicial complex is homeomorphic to a flag complex, e.g.by taking the barycentric subdivision. So X(n, p) puts a measure on a wide varietyof topologies as n→∞.

One can also consider a random flag complex process, which is the same prob-ability space as the random graph process, edges being added one at a time. SeeFigure 3. The Betti numbers of an instance of such a process on n = 100 verticesand roughly 3000 steps are illustrated in Figure 4.

We see immediately that homology no longer behaves in a monotone way withrespect to the underlying parameter.

4.1. Vanishing homology. First of all, we check that, as suggested by Figure4, there is a range of p outside of which Hk = 0 with high probability [39].

For the sake of simplicity, we state theorems in the next few sections withassuming p = n−α.

Theorem 4.1. Let k ≥ 1 and α > 0 be fixed, p = n−α, and X ∼ X(n, p).(1) If α > 1/k then w.h.p. Hk(X,Z) = 0.

TOPOLOGY OF RANDOM SIMPLICIAL COMPLEXES: A SURVEY 11

Figure 3. The beginning of a random flag complex process.

0

200

400

600

800

1000

1200

1400

1600

1800

2000

0 0.1 0.2 0.3 0.4 0.5 0.6

homo

logy

edge probability p

!0!1!2!3!4!5

Figure 4. The Betti numbers for the beginning of a random flagcomplex process on n = 100 vertices. (Computation and image cour-tesy of Afra Zomorodian.)

(2) If α < 1/(2k + 1), then w.h.p. Hk(X,Z) = 0.

The proof of (1) is based on showing first that homology is supported on cyclesof small support (bounded in size as n → ∞), and then showing that every suchcycle is a boundary.

The key observation for (2) is that link of a vertex in a random flag complex is arandom flag complex with shifted parameter. Indeed, even intersecting the links ofseveral vertex links results in another random flag complex. Then the Nerve Lemmaallows one to bootstrap local information about connectivity of a large number ofrandom graphs into global information about cohomology vanishing. This argumentshows something stronger topologically: that if p = n−α where α < 1/(2k+1) thenw.h.p. X is k-connected, i.e. πi(X) = 0 for i ≤ k. Recent work of Babson showsthat this exponent 1/3 is tight when k = 1 [6].

12 MATTHEW KAHLE

In Section 4.4 we will see that the exponent in (2) can be improved, with aspectral gap argument, if one relaxes to cohomology with rational coefficients.

4.2. Nontrivial homology and cohomology. It is also known that for ev-ery k ≥ 0 there is a range of p = pk(n) for which Hk(X(n, p)) 6= 0 with highprobability. Here are three ideas for how one might try to prove this.

(1) Linear algebra. Let fi denote the number of i-dimensional faces of X.Then if fk > fk−1 + fk+1, we already have that Hk 6= 0 for dimensionalreasons, i.e.

βk ≥ −fk−1 + fk − fk+1.

(2) Homological argument: sphere. Try to find a subcomplex Y home-omorphic to a sphere Sk, and a simplicial map f : X → Y such thatf |Y = id. Then the homology of Y is naturally a summand of the homol-ogy of X, and in particular Hk(X) 6= 0.

(3) Cohomological argument: isolated face. If σ is a k-dimensionalface not contained in any (k+ 1)-dimensional face, then the characteristicfunction of σ represents a cocycle. If one can somehow show that thisfunction is not also a coboundary, then one has a nontrivial class.

All three of these approaches work, and in roughly the same range of param-eter. They also work equally well for with any choice of coefficients. Any of theseapproaches yields the following, for example.

Theorem 4.2. Let α > 0 be fixed, p = n−α, and X ∼ X(n, p). If 1/(k + 1) <α < 1/k, then w.h.p. Hk 6= 0.

The first two approaches are discussed in [39], and the third approach in [41].In Section 4.4 we will see that the third approach has a slight edge on the other twoapproaches at the upper end of the window of nontrivial homology. In this case,a much sharper estimate may be obtained. So the exponent 1/k in Theorems 4.1and 4.2 is sharp. The exponent 1/(k + 1) in Theorem 4.2 is also sharp, as we willsee in Section 4.4.

4.3. Limit theorems. Much more can be shown in the nontrivial regime. Notonly do we know that Hk 6= 0 w.h.p., but we can also understand the expectationof βk and its limiting distribution.

The following asymptotic formula for the expectation follows from the linearalgebra approach described above.

Theorem 4.3. Let α > 0 be fixed, p = n−α, and X ∼ X(n, p). If 1/(k + 1) <α < 1/k, then

E[βk](nk+1

)p(

k+12 )→ 1,

as n→∞.

Here E[βk] denotes the expectation of βk. A similar formula can be given forthe asymptotic variance Var[βk].

The following central limit theorem characterizes the limiting distribution [42].

TOPOLOGY OF RANDOM SIMPLICIAL COMPLEXES: A SURVEY 13

Theorem 4.4 (Kahle–Meckes). Let α > 0 be fixed, p = n−α, and X ∼ X(n, p).If 1/(k + 1) < α < 1/k, then

βk − E[βk]√Var[βk]

→ N (0, 1)

as n→∞.

Here N (0, 1) is the standard normal distribution with mean 0 and variance 1,and the convergence is in distribution.

4.4. Sharp thresholds for rational cohomology. The following gives asharp upper threshold for rational cohomology [41] of random flag complexes. TheErdős-Renyi theorem corresponds to the k = 0 case.

Theorem 4.5. Let k ≥ 1 and ε > 0 be fixed, and X = X(n, p).(1) If

p ≥(

(k/2 + 1 + ε) log n

n

)1/(k+1)

,

thenP[Hk(X,Q) = 0]→ 1,

(2) and if

n−1/k+ε ≤ p ≤(

(k/2 + 1− ε) log n

n

)1/(k+1)

,

thenP[Hk(X,Q) = 0]→ 0,

as n→∞.

The main tools used to prove Theorem 4.5 this are Theorem 3.5 above whichgives a concentration result for the spectral gap, together with the following Theo-rem 4.6.

Theorem 4.6 (Garland, Ballman–Świątkowski). Let ∆ be a pure D-dimensionalfinite simplicial complex such that for every (D − 2)-dimensional face σ, the linklk∆(σ) is connected and has spectral gap is at least λ2[lk∆(σ)] > 1 − 1/D. ThenHD−1(∆,Q) = 0.

Theorem 4.6 is a special case of Theorem 2.5 of Ballman–Świątkowski [8], whichin turn based on earlier work of Garland. For a deeper discussion of Garland’smethod, see A. Borel’s account in Séminaire Bourbaki [14]. It is worth noting thatKazhdan had already proved certain cases of Serre’s conjecture in 1967 [45], andthat this is also the paper in which he introduced Property (T).

As a corollary, many random flag complexes have nontrivial rational homologyonly in middle degree.

Corollary 4.7. Let d ≥ 1 and 1/(d+ 1) < α < 1/d be fixed. If p = n−α andX ∼ X(n, p), then

H̃i(X,Q) = 0 unless i = d

It is conceivable that Theorem 4.5 could be sharpened to the following.

14 MATTHEW KAHLE

Conjecture 4.8. If

p =

((k/2 + 1) log n+ (k/2) log log n+ c

n

)1/(k+1)

,

where c ∈ R is constant, then the dimension of kth cohomology βk approaches aPoisson distribution with mean

µ =(k/2 + 1)k/2

(k + 1)!e−c.

In particular,

P[Hk(X,Q) = 0]→ exp

[− (k/2 + 1)k/2

(k + 1)!e−c],

as n→∞.

This conjecture is equivalent to showing that in this regime, cohomology isw.h.p. generated by characteristic functions on isolated k-faces. This is analogousto the fact that when c ∈ R is constant, and G ∼ G(n, p) where

p =log n+ c

n,

w.h.p. G consists of a giant component and isolated vertices.

4.5. Torsion. The question of torsion in random homology is still fairly mys-terious, but for certain models of random simplicial complex, we know that torsionwill be quite large. It may be surprising to learn, for example, that there exists a2-dimensional Q-acyclic simplicial complex S on 31 vertices with H1(S,Z) cyclic oforder

|H1(S,Z)| = 736712186612810774591.

The complex is easy to define. The vertices are the elements of the cyclic groupZ/31, the 1-skeleton is a complete graph, and a set of three vertices {x, y, z} spana 2-dimensional face if and only if

x+ y + z ≡ 1, 2, or 9 (mod 31).

This type of “sum complex” was introduced and proved to be Q-acyclic using dis-crete Fourier analysis by Linial, Meshulam, and Rosenthal [48], and I found thisparticular example choosing 1, 2, 9 randomly with a calculation in Sage [64].

Work of Kalai [44] showed that for Q-acyclic complexes, the expected size ofthe torsion group is enormous. For example, for a random 2-dimensional Q-acycliccomplex Sn on n vertices,

E [|H1(S,Z)|] ≥ ecn2

for some constant c > 0. This is in some sense the worst case scenario for torsion:all 2-dimensional Q-acyclic complexes have

|H1(S,Z)| ≤ eCn2

,

for some other constant C > 0; see for example Proposition 3 in Soulé [62].

Kalai’s result tells us about the expected size of these random torsion groups,but thirty years later, not much more seems to be known about their expectedstructure. One natural conjecture might be that the p-parts of these random torsiongroups obey approach Cohen–Lenstra distributions in the limit.

TOPOLOGY OF RANDOM SIMPLICIAL COMPLEXES: A SURVEY 15

The Cohen–Lensta distribution over finite abelian p-groups proposes that theprobability of every group is inversely proportional to the size of its automorphismgroup. These measures arise naturally in certain number-theoretic settings andhave been the subject of a lot of attention, for example in recent work of Ellenberg,Venkatesh, and Westerland [25]. As Lyons points out [50], the natural measureson Q-acyclic complexes are certain determinantal measures rather than uniformdistributions, so this question should be understood with respect to these measures.

In contrast to random Q-acyclic complexes, one might guess for random 2-complexes or random flag complexes, would be that there may be a window whenHi(X,k) = 0 for every fixed field k but such that Hi(X,Z) 6= 0, but that thiswindow is very small. The philosophy is that once homology is a finite group,it should only take a relatively small number of random relations to kill it. Inparticular I would guess the following.

Conjecture 4.9. Let d ≥ 3 and

1

d+ 1< α <

1

d

be fixed. If p = n−α and X ∼ X(n, p), then w.h.p. X is homotopy equivalent to abouquet of d-dimensional spheres.

It was shown in [39] that if α < 1/3, then w.h.p. π1(X) = 0. Since simply-connected Moore spaces are unique up to homotopy equivalence, Conjecture 4.9 isequivalent to showing that for this range of p, H∗(X) is w.h.p. torsion-free.

5. Comments

Now that several different models of random simplicial complex have beenstudied, we are starting to see a few common themes emerging.

5.1. There are at least two kinds of topological phase transitions.The “upper” phase transition where homology or cohomology vanishes seems easierto understand cohomologically. Examples of this kind of phase transition includethe Erdős–Rényi theorem, the Linial–Meshulam theorem, and Theorem 4.5 above.These thresholds tend to be sharp, happening in a very narrow window.

The “lower” phase transition where homology or cohomology passes from van-ishing to nonvanishing seems easier to understand homologically. Theorem 2.5characterizes the first appearance of cycles in G(n, p). The higher-dimensional ana-logue of this phase transition in random complexes is apparently much more subtle,but interesting recent work studying this phase transition appears in [4] and [5].These thresholds tend to be sharp on one side, not on the other.

For random graphs, the basic philosophy local properties such as “contains a tri-angle” have coarse thresholds, whereas global properties, such as “connected”, havesharp thresholds. For precise statements, see for example the paper by Friedgutwith appendix by Bourgain [29]. Since isolated faces generate cohomology nearthe upper phase transition, by the same argument, this is a global property andwe should guess the upper threshold is sharp. On the other hand, homology seemsto first appear, supported on small spheres, so the lower threshold may tend to becoarse.

16 MATTHEW KAHLE

5.2. Homology and cohomology try to be as small as possible. Withthis motto we mean something more geometric, namely that near the lower thresh-old, homology tends to be supported on small classes. For example, homologyof random geometric complexes in the sparse regime has a basis of vertex-minimalspheres [40]. This happens for both Čech and Vietoris–Rips complexes, even thoughthe minimal spheres are combinatorially different in the two cases. This alreadyaccounts for the difference in the formulas for expectation of the Betti numbers inthe sparse regime.

On the other hand, near the upper threshold cohomology is generated by smallclasses, namely characteristic functions on isolated k-faces.

The existence of these two phase transitions already implies something aboutwhat a theory of random persistent homology will have to look like. For example,after a certain point in a random filtration, death times of whatever cycles remainshould be well approximated by an appropriate Poisson process.

5.3. Nature abhors homology. (With apologies to Aristotle.) Homologyis, after all said to measure the number of “holes” in a topological space, and holesare made out of vacuum. More seriously, it is often the case that unless there is agood reason random homology is forced to be there, then it is likely to vanish orbe “small.”

Suppose you have some measure on “pairs of random linear maps f : A→ B andg : B → C satisfying g ◦ f = 0”. What can we say about the resulting distributionon HB? Sometimes you can guess the answer from random linear algebra.

For example, one might guess that if

dimA� dimB � dimC

then there is a good chance that the map g : B → C is injective, in which caseHB = 0. Or else perhaps ker g is merely small, but then this still bounds the sizeof homology. Similarly, if

dimA� dimB � dimC,

then one might expect f : A → B is probably surjective (or nearly so) and so oneexpects that HB is small. In fact the only place the one place where we actuallyexpect to see large homology, if dimension were the only consideration, would be if

dimA� dimB � dimC.

So you might guess that dimHB ≈ max{0,− dimA+ dimB − dimC}.The random flag complex X(n, p) is an example of where this kind of argument

works surprisingly well. In a random flag complex, the theorems so far tell us thatwe should only expect to see one or two nontrivial Betti numbers at any given partin the process, and that the overlap between two nonzero Betti numbers shouldnot be too large. If we make the simplifying assumption that all of the reducedBetti numbers are zero except for one, then the nonzero Betti number equals theabsolute value of the reduced Euler characteristic. The upshot is that the expectedEuler characteristic E[χ̃] is easy to compute, as linearity of expectation gives that

E[χ̃] = −1 + n−(n

2

)p+

(n

3

)p3 −

(n

4

)p6 +

(n

5

)p10 − . . . .

See, for example, Figure 5. Using his software Perseus [54], Vidit Nanda com-puted fifty complete flag complex processes on n = 25 vertices, and the average

TOPOLOGY OF RANDOM SIMPLICIAL COMPLEXES: A SURVEY 17

Figure 5. The average of 50 complete random flag complex pro-cesses on n = 25 vertices (continuous curves), plotted against |E[χ̃]|(blue stars). As in Figure 4, the horizontal axis is the number ofedges and the vertical axis the Betti numbers. (Computation andimage courtesy of Vidit Nanda.)

is plotted against the expected Euler characteristic. It is quite striking that theprediction is so good, especially considering that all the theory is as n → ∞ andp → 0, but here the prediction seems to work reasonably well, even for relativelysmall n and large p.

5.4. Random simplicial complexes are expanders. Random simplicialcomplexes have many expander-like properties. Higher-dimensional analogues ofexpanders have attracted a lot of attention recently— see for example the discus-sion in [23], and also Gromov’s recent work on “geometric overlap” properties ofexpanders [34, 35].

For a geometric application, expander graphs are well understood to be im-possible to embed in Euclidean space with small metric distortion, by work ofBourgain [15]. Volume distortion analogues of this were recently established forrandom simplicial complexes by Newman–Rabinovich [55] and by Dotterrer [22].

We would like to better understand the higher-dimensional analogues of theCheeger–Buser inequalities relating spectral gap and the “bottleneck” expansionconstant. For recent work on higher-dimensional analogues of Cheeger–Buser, seeGundert–Wagner [36], and Steenbergen et al. [63].

5.5. A multi-parameter model. A model that deserves more attention isthe multi-parameter random complex ∆(n; p1, p2, . . . ). Here there are n vertices,the probability of an edge is p1 = p1(n), and the complex is built inductively bydimension in so that the probability of every k-dimensional simplex, conditionedthat its entire (k − 1)-dimensional boundary is already in place, is pk = pk(n),independently.

18 MATTHEW KAHLE

Several of the random simplicial complexes discussed here are special cases ofthis model; in particular, the random graph

G(n, p) = ∆(n; p1, 0, 0, . . . ),

the random 2-complex

Y (n, p) = ∆(n; 1, p2, 0, 0, . . . ),

and the random flag complex

X(n, p) = ∆(n; p1, 1, 1, . . . ).

Acknowledgements

I am grateful to IAS for hosting me for two weeks in January 2013, and forhaving a chance to present this material in two talks at the MacPherson seminarwhile the article was in preparation, and to receive feedback from a friendly audi-ence. I also thank Peter Landweber for a careful read of an earlier draft, and forseveral helpful suggestions.

I dedicate this article to Gunnar Carlsson, and thank him for his encouragementand support.

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The Ohio State University, Department of MathematicsE-mail address: mkahle@math.osu.edu