Algebraic and combinatorial expansion in random simplicial complexes

In this paper we consider the expansion properties and the spectrum of the combinatorial Laplace operator of a $d$-dimensional Linial-Meshulam random simplicial complex, above the cohomological connectivity threshold. We consider the spectral gap of the Laplace operator and the Cheeger constant as this was introduced by Parzanchevski, Rosenthal and Tessler ($Combinatorica$ 36, 2016). We show that with high probability the spectral gap of the random simplicial complex as well as the Cheeger constant are both concentrated around the minimum co-degree of among all $d-1$-faces. Furthermore, we consider a generalisation of a random walk on such a complex and show that the associated conductance is with high probability bounded away from 0.


Introduction
In this paper will consider the expansion properties of a random binomial simplicial complex past the threshold for the cohomological connectivity. This model was introduced by Linial and Meshulam [27] and it is a generalisation of the binomial random graph G(n, p). Let Y (n, p; d) denote the random d-dimensional simplicial complex on [n] := {1, . . . , n} where all possible faces of dimension up to d − 1 are present but each subset of [n] of size d + 1 becomes a face with probability p = p(n) ∈ [0, 1], independently of every other subset of size d + 1. When d = 1, the model reduces to the binomial random graph on [n] with edge probability equal to p.
In their seminal paper, Linial and Meshulam [27] considered the cohomological connectivity of Y (n, p; 2), that is whether the cohomology group H 1 (Y (n, p; d); where ω is as above. Linial and Meshulam [27] asked whether this can be extended to Z (for random 2-complexes). For d = 2, Łuczak and Peled [29] proved a hitting time version of this result considering the generalisation of the random graph process. This is a random process in which one constructs a random simplicial complex on n vertices with complete skeleton, where in each step a new 2-dimensional face is added, selected uniformly at random. They showed that w.h.p. the 1-homology group over Z becomes trivial the very moment all edges (1-faces) lie in at least one 2-face. This was proved for Z 2 by Kahle and Pittel [25]. These results generalise the classic result of Bollobás and Thomason [6] on the w.h.p. coincidence of the hitting times of connectivity with that of having minimum degree at least 1. For d ≥ 3, Hoffman et al. [22] provided a partial answer showing that the 1-statement holds for H d−1 (Y (n, p; d); Z) provided that np ≥ 80d log n.
Furthermore, Gundert and Wagner [21] showed that H d−1 (Y (n, p; d); R) is trivial provided that np ≥ C log n where C is a sufficiently large constant. Their approach was extended by Hoffman, Kahle and Paquette [23] which extended this to p such that np ≥ (1 + ε)d log n. Very recently, Cooley, del Guidice, Kang and Sprüssel [11] considered the cohomological connectivity of a generalised version of the Linial-Meshulam model in which random selection of faces takes place at all levels and not merely at the top level.
In this paper, we will study the expansion properties of Y (n, p; d) for p as in the supercritical regime of the Linial-Meshulam-Wallach theorem. To be more precise, we will consider the case np = (1 + ε)d log n, for an arbitrary fixed ε > 0, and we will deduce sharp concentration results about the spectral gap of the (combinatorial) Laplace operator as well as the Cheeger constant of Y (n, p; d).
For a sequence of events (E n ) n∈N , where E n is an event in the probability space of Y (n, p; d), we say that they occur with high probability (w.h.p.), if P (E n ) → 1 as n → ∞. (We will use the same term for events in the probability space of G(n, p).) If X n is a random variable defined on the probability space of Y (n, p; d) and c ∈ R, we write X n = c(1 + o p (1)), if P (|X n − c| > ε) → 0 as n → ∞ -so loosely speaking X n → c in probability as n → ∞.

Measures of expansion: the spectral gap and the Cheeger constant
The definition and the use of the discrete Laplace operator in quantifying expansion properties of graphs dates back to Alon and Milman [3]. For a graph G = (V, E), the (combinatorial) Laplace operator ∆ + G is defined as the difference D G − A G , where D G = diag(deg(v)) v∈V and A G is the adjacency matrix of G. In [3], Alon and Milman showed how the smallest positive eigenvalue of the Laplace operator is linked to the structure of the graph as a metric space and, in particular, to the distribution of distances between disjoint sets and the diameter of the graph. The Laplace operator had been considered in graph theory earlier [4,5,16] in relation to the number of spanning trees, the girth and connectivity of a graph.
Let λ(G) denote the smallest positive eigenvalue of ∆ + G also known as the spectral gap of ∆ + G . It is a consequence of a more general result in [3] that λ(G) is bounded from above (up to some multiplicative constant) by the edge expansion of G. This is defined as where e(A, V \ A) denotes the number of edges with one endpoint in A and the other in V \ A; it is called the Cheeger constant of G. Lemma 2.1 in [3] implies that for any non-empty (proper) subset A ⊂ V we have If |A| ≤ |V |/2, then the above is at most 2c(G). This is the discrete analogue of an inequality proved by Cheeger in [8]. Setting h(G) = min A : 0<|A|≤|V |/2 h(A; G) one can complete the above inequality with a lower bound (proved by Dodziuk [12]) and get where d max (G) is the maximum degree of G. Another consequence of this result is that if d min (G) denotes the minimum degree of G, then λ(G) ≤ |V | |V |−1 δ(G) (this was also proved in [16]). Moreover, if G is disconnected, then λ(G) = 0. Thus, sometimes λ(G) is called the algebraic connectivity of G. Further properties of expander graphs in relation to the smallest positive eigenvalue of the Laplace operator were obtained by Alon in [1].
More generally, the spectrum of the Laplace operator of a graph also determines how the edges between subsets of vertices are distributed. This is expressed through the well-known Expander-Mixing Lemma. Roughly speaking, it states that if the entire non-trivial spectrum of the Laplace operator of a graph G = (V, E) is close to d, then the density of edges between any two non-empty subsets A, B ⊂ V is about d/n. Such an estimate about the number of edges within any given subset of A ⊂ V was proved by Alon and Chung [2], in the case where G is a d-regular graph. It was generalised by Friedman and Pippenger in [18].
One of the results of our paper is to generalise this result to higher dimensions in the context of the Linial-Meshulam random simplicial complex Y (n, p; d).

High dimensional Laplace operators
Let Y be a d-dimensional simplicial complex on a set V with |V | < ∞. We let Y (j) denote the set of j-dimensional faces in Y , that is, the faces containing exactly j + 1 vertices, where −1 ≤ j ≤ d. It is customary to set Y (−1) = {∅}. Also, note that Y (0) = V . For abbreviation, we will be calling a j-dimensional face a j-face. If all possible j-faces are present in Y , for j ≤ d − 1, then Y is said to have complete skeleton.
In our context a j-face for j ≥ 2 has two orientations which are the two equivalence classes of all permutations of its vertices which have the same sign. In other words, two permutations correspond to the same orientation if we can derive one from the other applying an even number of transpositions. If σ is an oriented j-face, we denote byσ the opposite orientation of it, and by Y (j) ± we denote the set of oriented j-faces. Finally for an oriented j- The space of j-forms, which we denote by Ω (j) (Y ; R) is the vector space over R of all skew-symmetric functions on oriented j-faces. In other words, for j ≥ 2 we define whereas Ω (0) (Y ; R) is just the set of all real-valued functions on V and Ω (−1) (Y ; R) is defined as the set of all functions from Y (−1) = {∅} to R, which can be identified with R. The space Ω (j) (Y ; R) is endowed with the inner product: It is well-known and easy to verify that for j ≥ 1, we have ∂ j ∂ j+1 = 0, whereby Im∂ j+1 ⊆ Ker∂ j . We set Z j (Y ) = Ker∂ j and B j (Y ) = Im∂ j+1 . The set Z j (Y ) is the set of j-cycles, whereas B j (Y ) is the set of j-boundaries.
Note that both Z j , B j ⊆ Ω (j) (Y ; R) and furthermore (Ω (j) (Y ; R), ∂ j ) d j=1 is a chain complex. The group H j (Y ; R) = Z j (Y )/B j (Y ) is the jth homology group over R.
A straightforward calculation shows that δ j−1 is the adjoint operator of ∂ j : for

The Laplace operator and the spectral gap
The Laplace operator associated with Y is the operator ∆ : Note that (4) implies that Ker∆ The partial Laplace operators decompose the space Ω (d−1) (Y ; R): The subspace H d−1 (Y ) = Ker∆ is the called the space of harmonic d − 1-forms. Note that for any f ∈ H d−1 (Y ), the fact that δ j−1 is the adjoint of ∂ j implies that The discrete Hodge decomposition is due to Eckmann [13]: It can be shown (see [34] p. 203) that A quantity that is of interest is the spectral gap λ(Y ) of a d-dimensional complex Y . This is defined as the minimal eigenvalue of the Laplacian or the upper Laplacian over Z d−1 (the set of d − 1-cycles). (Note that the two operators ∆ + and ∆ coincide on Horak and Jost [24] developed the theory of the Laplace operator for general weight functions. We will focus on special weighting schemes that give rise to generalisations of the well-studied combinatorial Laplace operator as well as the normalised Laplace operator.

The combinatorial Laplace operator and the Cheeger constant
In the case where w(σ) = 1 for all σ ∈ Y , the operator ∆ + is called the combinatorial (upper) Laplace operator associated with Y . An algebraic manipulation can give explicitly the combinatorial Laplace operator: If Y is a finite simplicial complex with |Y (0) | = n vertices, we define where the minimum is taken over all partitions of Y (0) into d + 1 non-empty parts A 0 , . . . , A d and F (A 0 , . . . , A d ) is the set of d-faces with exactly one vertex in each one of the parts.

Theorem 1. For a finite complex Y with a complete skeleton,
Furthermore, the authors also derive an expander mixing lemma for complexes with complete skeleton. This assumption was removed by Parzanchevski [32].
In [34], the authors discuss the existence of a lower bound in the spirit of the lower bound in (2). They observe (cf. Section 4.2 in [34]), that a bound of the form , for some C, m > 0 cannot hold, providing as counterexample the minimal triangulation of a Möbius strip, which has λ(Y ) = 0 but h(Y ) > 0.
Parzanchevski et al. [34] conjecture that an inequality of the form C · h(Y ) 2 − c ≤ λ(Y ) should hold, where C, c > depend on the maximum degree of any d − 1-face of Y as well as on the dimension of Y .
Furthermore, they showed [34] Our results strengthen the latter, showing that if np = (1 + ε)d log n and ε > 0 is fixed, the upper bound λ(Y (n, p; d)) ≤ h(Y (n, p; d)) which follows from Theorem 1 becomes tight in that λ(Y (n, p; d)) = h(Y (n, p; d))(1 + o p (1)). Furthermore, we show that λ(Y (n, p; d))/np converges in probability as n → ∞ to a certain constant which depends on ε and d. Recall that δ(Y (n, p; d)) denotes the minimum co-degree among all d − 1-dimensional faces of Y (n, p; d).
The above theorem not only strengthens the results of Parzanchevski et al. [34] as far as the range of p is concerned, but it also gives more precise asymptotics for large ε. Remark 1.2 in [26] states that as ε → ∞, a(ε) The proof of the above theorem has three parts. We start with the result on δ(Y (n, p; d)) in Section 2 (cf. Lemma 2). Thereafter, in Section 3 we show that h(Y (n, p; d)) ≤ δ(Y (n, p; d)) (cf. Theorem 4). Hence, the upper bound on λ(Y (n, p; d)) follows from Theorem 1.
For the lower bound on λ(Y (n, p; d)) in Section 4 we follow an approach similar to that of Gundert and Wagner [21]. The lower bound is derived through a decomposition, essentially due to Garland [19], of the Laplace operator ∆ + of a simplicial complex Y into the sum of the (combinatorial) graph Laplace operators of the link graphs defined by the d − 2-faces of Y . We show that the positive eigenvalues of these are bounded from below by δ(Y ) and hence the lower bound in Theorem 2.
However, this approximation incurs a term which involves the adjacency matrix of these link graphs. As we will see in Section 4, in the case where Y is Y (n, p; d) these graphs are distributed as G(n − d + 1, p). At this point we use sharp results of Feige and Ofek [15] to show that this term has no essential contribution.

The normalised Laplace operator
Under a weighting scheme where the operator ∆ + is called the normalised Laplace operator associated with Y . An explicit calculation as above (cf. (2.6) in [33]) shows that for f ∈ Ω (d−1) (Y ; R) and For graphs, the normalised Laplace operator acts on functions on the vertex set of a graph G = (V, E) and is defined as where I is the identity operator. However, note that for d = 1 the definition of ∆ + yields the operator ∆ This has the same spectrum as L G , provided that deg(σ) > 0, for all σ ∈ Y (0) . Furthermore, the constant function on V is an eigenfunction corresponding to eigenvalue 0, whereas all other eigenvalues are positive.
Gundert and Wagner [21] showed that w.h.p. the non-trivial eigenvalues of the normalised Laplacian of Y (n, p; d) are close to 1, for p such that np ≥ C log n. This implies that H d−1 (Y (n, p; d); R) is trivial for such p. Hoffman, Kahle and Paquette [23] extended this argument for p such that np ≥ (1/2 + δ) log n, showing that w.h.p. all non-trivial eigenvalues are within C/ √ np from 1.
The argument of Gundert and Wagner [21] relies on proving the sharp concentration of the non-trivial eigenvalues of the normalised Laplacian of G(n, p) around 1. Hence, the sharpening of Hoffman et al. [23] follows from their main result about the eigenvalues of the Laplacian of G(n, p) for any p such that np ≥ (1/2 + δ) log n, for arbitrary fixed δ > 0. For the denser regime where np = Ω(log 2 n), this was proved by Chung, Lu and Vu [9]. However, for sparser regimes (np bounded) this fact has been proved by Coja-Oghlan [10] for the Laplace operator restricted on core of G(n, p), although for G(n, p) itself the spectral gap is o p (1).

Random walks on Y (n, p; d) and expansion
This part of the paper is motivated by the notion of a random walk on Y introduced by Parzanchevski and Rosenthal [33]. This is in fact a random walk on Y (d−1) ± and, more precisely, on the graph (Y One may consider the projection of such a walk on If (X 0 , X 1 , . . .) denotes this Markov chain, then for any n ≥ 1 the transition probabilities are P ( In a more general setting, one may consider a γ-lazy version of this random walk, for In this Markov chain, the stationary distribution on We consider the mixing of such a random walk in the case where Y is Y (n, p; d) with np ≥ (1 + ε)d log n. In particular, we will consider the conductance of this Markov chain which we denote by Φ Y . First for any non-empty The first author and Reed [17] showed the analogous result for largest connected component of G(n, p) when np = Ω(log n).
We prove Theorem 3 in Section 5. Its proof is based on a double counting argument that is facilitated by a weak version of the Kruskal-Katona theorem (cf. Theorem 10).

Tools: concentration inequalities
In our proofs, we make use of the following variant of the Chernoff bounds (see [31,Chapter 4]). and 2 The minimum (co)degree of Y (n, p; d) The following lemma builds very strongly on the work of Kolokolnikov, Osting, and Von Brecht [26], who obtained very sharp bounds on the minimum vertex degree in G(n, p), and its relation to the spectral gap of the graph just above the connectivity threshold, in particular, when p = (1 + ε) log n n for ε > 0. (More specifically, see Lemmas 3.3. and 3.4 in [26].) Lemma 2. Let p = (1 + ε) d log n n , and let a = a(ε) denote the solution to Let Y = Y (n, p; d) and let Proof. For a random variable X following the binomial distribution Bin(n, p) and c > 0, let Observe that by (12) and (13) we have that (1 + ε)H(a(ε)) = −1.
It can be shown (see Lemma 3.3 in [26]) that for p = Θ(log n/n) there exist constants c 1 , c 2 > 0 such that Recall that for a set σ ⊂ [n] of d vertices, deg(σ) denotes the number of ddimensional faces in Y containing σ. Clearly, deg(σ) follows the binomial distribution Bin(n − d, p). The application of (14) to the random variable deg(σ) yields: Furthermore, Since a(ε) ∈ (0, 1) for all ε > 0 (see Remark 1.3 in [26]), by taking we can use (15) to see that Since both H(c) and H ′ (c) = − log c are continuous and positive on (0, 1), we have that We start by showing that w.h.p. we have δ(Y ) ≥ anp − √ np. For a fixed subset σ of size d we have that Therefore the expected number of d-element sets σ with deg(σ) ≤ anp − √ np is at most To bound δ(Y ) from above, we first find an upper bound on deg(σ). Using that (1 + ε)H(a) = −1, we have as n → ∞. By Chebyshev's inequality we then have The co-degrees of two subsets σ, σ ′ are independent whenever |σ ∩ σ ′ | = d − 1. Thus the variance of N 0 satisfies we focus on the value of P (X σ = X σ ′ = 1). Let deg \σ ′ (σ) denote the number of ddimensional faces that contain a d-subset σ but do not contain the d-subset σ ′ . Using the law of total probability, conditioning on the presence or absence of the unique face that contains both σ and σ ′ , and since deg \σ ′ (σ) and deg \σ (σ ′ ) are identically distributed, we have In particular, the random variable deg \σ ′ (σ) follows the binomial distribution Bin(n − d − 1, p) and, thereby, it is stochastically dominated by Bin(n − d, p). So For the second term in (19), we have Hence for n large enough we have Thus by (17), (18), and (20), we obtain since we have µ → ∞. Thus with high probability we have that the minimum co-degree of a d-element set is at most anp + √ np and the lemma holds.
For k ∈ Z, let W k (x) be the k-th branch of the Lambert W function, defined as We now discuss some further properties of the function a(ε) defined in (12). This lemma shows that for small ε the function a(ε) is bounded by a linear function on ε. We will use this bound in the next section.
First, using the property that for any branch of the Lambert W function and any z ∈ (−e −1 , 0) we have W ′ (z) = W (z) z(1+W (z)) , and that e W (z) = z/W (z), we obtain .

Cheeger constant
In this section we consider the measure of expansion of a simplicial complex which is called its Cheeger constant and was defined in (8). As the main result in this section we prove the following theorem.  ; d) and let a = a(ε) be as in (12). There exists a positive constant C > 0 such that w.h.p. we have where the last equality follows from |A d−1 | ≥ n/(log n) 1/2 . Since there are at most (d + 1) n partitions of V into d + 1 disjoint sets, by the union bound we see that with where the last inequality follows from a(ε) < 1. Now, assume that C(d, ε) ≤ |A d−1 | < n/(log n) 1/2 for some constant C(d, ε) > 0 to be determined later. Note that by the assumption that (1) We will show that in this case with probability 1 − o n −(d+|A 0 |+|A 1 |+...+|A d−1 |) we have Since We note that, since d ≥ 2, if ε > 3.01 then for n large enough we can bound On the other hand, if ε ∈ (0, 3.01] then we have that As a polynomial in ε, this is positive on (0, 3.04). So for any ε ∈ (0, 3.01], if n is large enough then we also have d− (1+o(1)) min{1,0.33ε} Hence, the union bound completes the proof for this case. The final remaining case is when |A d−1 | < C(d, ε) for C(d, ε) > 0 constant. In this case we can use Lemma 2. Since we know that w.h.p. the minimum co-degree of a d-element set is at least (1 + ε)a(ε)d log n − C √ log n, it follows that w.h.p. for any selection of elements a 0 ∈ A 0 , . . . , This completes the proof of Theorem 4.

Algebraic expansion
Recall from (7) that for a d-dimensional simplicial complex Y with complete skeleton, the spectral gap λ(Y ) is the minimal eigenvalue of the upper Laplacian on (d−1)-cycles, i.e., λ(Y ) = min Spec ∆ + | Z d−1 .
As our main result in this section, we prove the following theorem about the spectral gap of Y (n, p; d).
Theorem 5. Let Y = Y (n, p; d) for d ≥ 2 and p = (1+ε)d log n n , where ε > 0. There exists a constant C > 0 such that w.h.p. the spectral gap of Y satisfies Proof. The upper bound in (23) follows immediately from Theorem 4 and through Theorem 1. Recall that the latter states that λ(Y ) ≤ h(Y ). Furthermore, at the beginning of the proof of Theorem 4, we observed that h(Y ) ≤ δ(Y ). We now focus on the lower bound on λ(Y ). We will give a lower bound on ∆ + f, f for f ∈ Z d−1 (Y ) which relies on a decomposition of ∆ + into the Laplace operators of the link graphs of all d − 2-faces of Y .
For a (d− 2)-dimensional face τ in Y (d−2) , let lkτ be the link graph of face τ , i.e., the graph with vertex set ; for any f ∈ Ω (d−1) and any σ ∈ Y (d−1) we set Furthermore, for a form f ∈ Ω (d−1) and a face τ ∈ Y (d−2) , we define f τ : (lkτ ) (0) → R as f τ (v) = f (vτ ). The operator ∆ + τ has the same effect as the Laplace operator associated with lkτ (see 2. in the following theorem). This is made precise in the following result by Garland [19] (we state it as in Lemma 4.2 in [34]). Theorem 6. Let X be a d-dimensional simplicial complex and f ∈ Ω (d−1) . Then the following hold.
we seek a lower bound on ∆ + f, f . We will use the decomposition of ∆ + in terms of the local Laplace operators as in 1. of the above theorem. To make use of this, we will rewrite the operator D as a sum of localised versions of it. In particular, we write D lkτ : Ω (0) (lkτ ; R) → Ω (0) (lkτ ; R) for the operator such that for f ′ ∈ Ω (0) (lkτ ; R) and v ∈ lkτ (0) we have ( The following holds.

Claim 1. For any
Proof. Let f ∈ Ω (d−1) (Y ). We write Note that if A lkτ is the adjacency matrix of lkτ , then Hence using Theorem 6 parts 1. and 2., for any f ∈ Ω (d−1) we can write But note that Thereby, Now, we will give an upper bound on τ ∈Y (d−2) A lkτ f τ , f τ , for Y = Y (n, p; d) and f ∈ Z d−1 . As we observed above, lkτ is distributed as a G(n − d + 1, p) random graph for any τ ∈ Y (d−2) (n, p; d). Of course, these random graphs are not independent. Hence, the simplest way to bound from above this sum is to give an upper bound on A lkτ f τ , f τ that holds with probability 1 − o(n −(d−1) ) and apply the union bound over all τ ∈ Y (d−2) (n, p; d), which there are O(n d−1 ) of them.
To this end, we will use some results by Feige and Ofek [15] on the spectrum of the adjacency matrix of G(n, p).

The adjacency matrix of G(n, p) and its spectral gap
In brief, Feige and Ofek [15] showed that the second largest eignvalue of G(n, p) is O( √ np), provided that np = Ω(log n). Let A be the adjacency matrix of G(n, p). If G(n, p) were regular, then the all-1s vector 1 would span the eigenspace of the leading eigenvalue and the above result would imply that Af, f = O( √ np) f, f for any function f on the vertex set of G(n, p) such that f ⊥ 1. However, G(n, p) is not regular but almost regular in the sense that for any ε > 0 w.h.p. most vertices have degrees np(1 ± ε). Feige and Ofek [15] proved that despite this, the bound on this quadratic form still holds w.h.p.
The main theorem in [15] is as follows.
Furthermore, the authors state and prove the following claim.

Claim 2 (Claim 2.4 in [15]). Suppose that for some
for any f, f ′ ∈ T . Then for any f ∈ S, we have Af, f < c From the above two statements the following result is deduced. Deducing the lower bound on λ(Y (n, p; d)).
We apply this in our setting, recalling that for any τ ∈ Y (d−2) , the link graph lkτ is distributed as G(n−d+1, p) with p = (1+ε)d log n n . Taking c = d in Theorem 8, we deduce that for some constant c ′′ d > 0 with probability at least 1 − n −d for any f τ ∈ Z 0 (lkτ ), we have The union bound (over all τ ∈ Y (d−2) ) implies that for any f ∈ Z d−1 (Y ) we have w.h.p.
Using the lower bound of (26) and the upper bound of (27), Equation (25) yields that w.h.p. for any f ∈ Z d−1 (Y ) we have But Lemma 2 states that δ(Y ) ≥ (1+ε)ad log n−C √ log n w.h.p. for some C > 0, where a = a(ε) is the solution of (12). Hence, for some C ′ > 0 w.h.p. for any f ∈ Z d−1 (Y ) such that f = 0 we have

Combinatorial expansion
Our lower bound on Φ Y will rely on a lower bound on the edge expansion of the graph for which there exists a d-face ρ ∈ Y (d) which contains both σ and σ ′ . We write σ ∼ σ ′ . Hence, given a subset S ⊂ Y (d−1) , we would like to bound from below the number of d-faces which contain at least one d − 1-face in S and a d − 1-face not in S. In other words, we would like to provide a lower bound on the number of d-faces which are potential exits for a random walk that starts inside S.
To express these more precisely, we introduce some relevant notation. For a d- Analogously to the oriented case, for σ ∈ Y (k) where 1 ≤ k ≤ d we define The latter is the number of d-faces that are exits out of the set S. Furthermore, Hence, In our proof, we will in fact use an upper bound on the number of d-faces in ∂ + S which are not exits. These are d-faces whose d − 1-subsets are all faces belonging to S. To bound their number from above, we will use a weak version of the Kruskal-Katona theorem. This provides an upper bound on the number of complete subgraphs on a hypergraph with a given number of hyperedges. To apply this to our context, we consider the hypergraph spanned by the d − 1-faces in S. The number of the complete d + 1-subhypergraphs of this hypergraph is an upper bound on the number of d-faces in ∂ + S which are not exits.
Proof. We shall assume that n is large enough for the estimates in the proof to hold. Given S ⊂ Y (d−1) and 1 ≤ i ≤ d + 1, let For any t ≥ 2, let K (t) t+1 be the complete t-uniform hypergraph on t + 1 vertices, and for a t-uniform hypergraph G, let K Hence, we shall use the following weak form of Kruskal-Katona theorem (see Lovász [28]).
σ ∈ S in n d many ways. We then perform a breadth-first-search on S: we first find all neighbours of σ, i.e., faces that share d − 1 vertices with σ. Exploring these faces according to the selected order, we then find all yet unexplored faces that share d − 1 vertices with consecutive neighbours of σ. Then, we move to the second neighbourhood of σ and find all of their still unexplored neighbours. Since, having picked σ, we have to discover a total of m − 1 faces and these will be first found as one of the offspring of one of m faces, we see that there are at most 2m−2 m−1 ≤ 4 m many ways to assign the numbers of offspring to consecutive faces (a collection of m non-negative integers which sum up to m − 1).
Any (d − 1)-dimensional face σ has at most dn neighbours, as we have d vertices in σ we can drop, and at most n vertices not in σ can be added to form the neighbour. Hence, for any choice of the numbers of neighbours first explored by consecutive faces, there are at most (dn) m ways to pick these neighbours. Thus, the number of tightly connected sets of size m is at most 4 m (dn) m = exp((1 + o(1))m log n).
As again the number of values of m we have to consider is at most n d , by (39) and the union bound we see that with probability 1 − o(1), (29) holds for all sets S with |S| ≤ n d (1−α) . This completes the proof of Theorem 9.
We can now easily show that the assumption that S is tightly connected can be dropped in Theorem 9 yielding a lower bound on Φ Y (n,p;d) and completing the proof of Theorem 3.

Conclusions
This paper is a study of various measures of expansion in the Linial-Meshulam random complex Y (n, p; d) past the cohomological connectivity threshold. We considered the spectral gap of the combinatorial Laplace operator and showed that w.h.p. it is very close to the the Cheeger constant associated with the simplicial complex. Furthermore, we showed that both quantities are w.h.p. very close to the minimum co-degree of the random simplicial complex. We determined explicitly the latter using the large deviations theory of the binomial distribution. Finally, we considered a random walk on the d − 1-faces of the random simplicial complex, which generalises the standard random walk on graphs. In particular, we considered the conductance of such a random walk and showed that w.h.p. it is bounded away from zero.
The above results were obtained for p such that np = (1 + ε)d log n, for any ε > 0 fixed. Our proofs seem to work when ε = ε(n) → 0 as n → ∞ slowly enough. A natural next step would be to consider these quantities for p that is closer to the threshold d log n/n. Indeed, the supercritical regime is for p such that np = d log n + ω(n), where ω(n) → ∞ as n → ∞ arbitrarily slowly. We believe it would be interesting to extend the analysis to this range of p as well. This would complete the picture of the evolution of the expansion properties of Y (n, p; d).