# Limit theorems for high-dimensional Betti numbers in the multiparameter
random simplicial complexes

Takashi Owada  Department of Statistics
Purdue University
IN, 47907, USA<EMAIL_ADDRESS>and  Gennady Samorodnitsky  School of
Operations Research and Information Engineering
Cornell University
NY, 14853, USA<EMAIL_ADDRESS>
###### Abstract.

We consider the multiparameter random simplicial complex on a vertex set
$\\{1,\dots,n\\}$, which is parameterized by multiple connectivity
probabilities. Our key results concern the topology of this complex of
dimensions higher than the critical dimension. We show that the higher-
dimensional Betti numbers satisfy strong laws of large numbers and central
limit theorems. Moreover, lower tail large deviations for these Betti numbers
are also discussed. Some of our results indicate an occurrence of phase
transitions in terms of the scaling constants of the central limit theorem,
and the exponentially decaying rate of convergence of lower tail large
deviation probabilities.

###### Key words and phrases:

strong law of large numbers, central limit theorem, large deviation, Betti
number, multiparameter random simplicial complex.

###### 2010 Mathematics Subject Classification:

Primary 60F05, 60F10, 60F15. Secondary 55U05, 60C05.

Owada’s research was partially supported by the AFOSR grant FA9550-22-1-0238
at Purdue University. Samorodnitsky’s research was partially supported by the
AFOSR grant FA9550-22-1-0091 at Cornell University.

##  1\. Introduction

The main theme of this paper is topological features of large high-dimensional
complex systems. A large complex system of interest to us is a random
simplicial complex, in which minor changes in a low-dimensional and local
structure may significantly affect high-dimensional and global structure of
the entire system. The random simplicial complex considered in this paper is a
higher-dimensional analog of the Erdős-Rényi random graph on
$[n]:=\\{1,\dots,n\\}$, where edges are added with probability $p$
independently of other potential edges. The literature contains several
higher-dimensional extensions of the Erdős-Rényi random graph. One is the
_random clique complex_ , in which a set of vertices in $[n]$ forms a face (or
a simplex) if and only if these vertices form a clique in the Erdős-Rényi
random graph ([9, 21]). The _Linial–Meshulam complex_ [12] is a
$2$-dimensional random simplicial complex, which starts with the full
$1$-dimensional skeleton of the complete simplicial complex and adds each
$2$-dimensional face, independently with probability $p$. This complex was
extended to its $d$-dimensional version by Meshulam and Wallach ([14]). In
this paper we study the _multiparameter random simplicial complex_ introduced
by Costa and Farber ([3, 4]) as a generalization of all the models described
above. This is an abstract simplicial complex on $[n]$, which is parameterized
by a sequence of connectivity probabilities ${\bf
p}=(p_{1},p_{2},\dots,p_{n-1})\in[0,1]^{n-1}$. One can construct this complex
recursively. First we take $[n]$ as the $0$-dimensional skeleton and add edges
between pairs of vertices independently with probability $p_{1}$, just like in
the Erdős-Rényi random graph. For each $i=2,\dots,n-1$, an $i$-dimensional
face (henceforth we simply say “$i$-face”) is added with probability $p_{i}$,
independently of all other potential faces, conditioned on the presence of all
the $(i-1)$-faces that constitute a boundary of that $i$-face. The complex
constructed this way is denoted as $X([n],{\bf p})$.

A number of studies of the multiparameter random simplicial complexes have
appeared recently, many of which focus on the asymptotic, as $n\to\infty$,
behavior of topological invariants of the complex, such as the _Betti numbers_
and the _Euler characteristic_. The model typically contains a particular
dimension in which the Betti number dominates the Betti numbers in other
dimensions. This dimension is called the _critical dimension_ , and much of
the effort has been devoted to the analysis of Betti numbers at the critical
dimension [10, 21, 5, 16, 11]. There are also results on the Betti numbers in
dimensions lower than the critical dimension. For example, by employing the
cohomology vanishing theorem [1] and the spectral gap theorem [7], Fowler
proved that the cohomology groups in dimensions lower than the critical
dimension vanish with high probability ([5]). However, very little has been
done about the asymptotics, as $n\to\infty$, of Betti numbers in dimension
higher than the critical dimension, except for certain special cases in the
context of random clique complexes; see [10, 21].

In this paper we derive various limit theorems for the Betti numbers of
dimension greater than the critical dimension. We start by establishing the
strong law of large numbers (SLLN) and the central limit theorem (CLT) for
these high-dimensional Betti numbers. The obtained limit theorems characterize
the “average” behavior and the “likely deviations” from the average behavior
of these Betti numbers. We also derive a lower bound for the lower tail large
deviation probabilities for the Betti numbers of dimension greater than the
critical dimension.

This paper is structured as follows. Section 2 provides a precise setup of our
multiparameter random simplicial complex. In this section we also present the
SLLN, the CLT, and large deviations results on Betti numbers in dimensions
greater than the critical dimension. Some of our results indicate presence of
phase transitions in terms of the scaling constants in the CLT and the
exponentially decaying rate of convergence for the lower tail large deviation
probabilities. The proofs are in Section 3, with some auxiliary results
deferred to the Appendix.

We conclude this Introduction with several comments on our setup and our
results. We treat only the “static” version of the multiparameter simplicial
complex, where the complex is sampled once and for all. The questions we are
pursuing in this paper make sense also for the dynamic multiparameter
simplicial complexes considered in our previous work [16] and in [21, 20, 6].
In particular, the models in [21, 16] involve on/off processes to describe the
state of various ingredients, so that the complex, though distributionally
stationary, may change with time. Since even the “static” case is technically
involved enough, we have decided not to pursuit this direction in the present
work. Further, in our large deviation results, we present only “lower bound”
for the lower tail probabilities for the Betti numbers. We conjecture that the
logarithmic order of magnitude of our lower bounds in Theorem 2.5 is the
tightest possible, but we do not yet have corresponding upper bounds to
justify that conjecture. There are, clearly, a number of other follow-up
questions on large deviations. These questions are postponed to future work.

Throughout the paper, we will use the following notation. The cardinality of a
set $A$ is denoted by $|A|$. Given a sequence $(a_{i})_{i=1}^{\infty}$ of real
numbers, we denote $\bigvee_{i=1}^{\infty}a_{i}:=\sup_{i\geq 1}a_{i}$ and
$\bigwedge_{i=1}^{\infty}a_{i}:=\inf_{i\geq 1}a_{i}$. Moreover, $\Rightarrow$
denotes weak convergence in ${\mathbb{R}}$, $\mathcal{N}(a,b^{2})$ is a normal
distribution with mean $a$ and variance $b^{2}$, and
${\mathbbm{1}}\\{\cdot\\}$ represents an indicator function. Finally, we will
use the capital letter $C$ and its versions (such as $C_{j},\,C^{*}$, etc) to
denote finite positive constants whose exact values are not important and can
change from one appearance to the next.

##  2\. Setup and main results

We start with a precise setup for the multiparameter random simplicial complex
$X([n],{\bf p})$ described in the previous section. As in the previous studies
[3, 4, 5, 16], we choose the connectivity probabilities ${\bf
p}=(p_{1},p_{2},\dots,p_{n-1})$ to be the following functions of $n$:

$p_{i}=n^{-\alpha_{i}},\ \ i=1,2,\dots,n-1,$

where each $\alpha_{i}\in[0,\infty]$ is a parameter. Given a sequence of
parameters $(\alpha_{1},\alpha_{2},\dots,\alpha_{n-1})$, let

(2.1) $\psi_{j}:=\sum_{i=1}^{j}\binom{j}{i}\alpha_{i},\ \ j\geq 1.$

Define $q:=\min\\{i\geq 1:\alpha_{i}>0\\}$, the smallest dimension $i$ such
that $p_{i}$ goes to $0$ as $n\to\infty$. Notice that (2.1) is increasing in
the sense of $\psi_{i}<\psi_{j}$ for all $q\leq i<j$. We assume, as is common
in the literature ([3, 4, 5, 16]), that the system has a _critical_ dimension
$k$, which is a dimension satisfying

$0<\psi_{q}<\dots<\psi_{k}<1<\psi_{k+1}<\dots,\ \ \text{with }\ k\geq q.$

Next, define

$\tau_{j}:=j+1-\sum_{i=1}^{j}\psi_{i}=j+1-\sum_{i=1}^{j}\binom{j+1}{i+1}\alpha_{i},\
\ 0\leq j\leq n-1$

(by convention we set $\sum_{i=1}^{0}\equiv 0$). Then, the monotonicity of
$(\psi_{j})$ ensures that

(2.2) $q=\tau_{q-1}<\tau_{q}<\cdots<\tau_{k}>\tau_{k+1}>\cdots,$

which also implies that $(\tau_{j})$ is maximized at the critical dimension
$k$. Furthermore, $\tau_{j}$ can be negative for large $j$.

The main theme of this paper is the asymptotic behavior of the Betti numbers
of dimension higher than the critical dimension, i.e.,
$\beta_{m,n}:=\beta_{m}\big{(}X([n],{\bf p})\big{)}$ for $m\geq k+1$. We
obtain the SLLN and the CLT for $(\beta_{m,n})_{n\geq 1}$. Moreover, a lower
bound for the lower tail large deviations for $(\beta_{m,n})_{n\geq 1}$ will
be derived as well. To state our main results, fix $m\geq k+1$. We prove the
the SLLN and the CLT under the assumption

(2.3) ${\tau_{m+1}+\alpha_{m+1}}>0,\ \ \text{ and }\ \
\alpha_{m+1}\in(0,\infty).$

Our first result is the SLLN for $(\beta_{m,n})_{n\geq 1}$.

######  Theorem 2.1.

Under the assumption (2.3), we have, as $n\to\infty$,

$\frac{\beta_{m,n}}{n^{\tau_{m+1}+\alpha_{m+1}}}\to\frac{1}{(m+2)!},\ \
\text{a.s.}$

######  Remark 2.2.

For comparison, the SLLN for the Betti numbers at the critical dimension $k$
was established in [16]: as $n\to\infty$,

(2.4) $\frac{\beta_{k,n}}{n^{\tau_{k}}}\to\frac{1}{(k+1)!},\ \ \text{a.s.}$

The proof of (2.4) heavily depends on the Morse inequality,

$f_{k,n}-f_{k+1,n}-f_{k-1,n}\leq\beta_{k,n}\leq f_{k,n},$

where $f_{j,n}$ denotes the simplex count of dimension $j$ in $X([n],{\bf
p})$. It was shown in [16] that $f_{k,n}$ dominates $f_{k\pm 1,n}$ as
$n\to\infty$, at least in the sense of expectations, and the SLLN for
$(\beta_{k,n})_{n\geq 1}$ was obtained as a direct result of the SLLN for
$(f_{k,n})_{n\geq 1}$. A much more delicate argument is required in the
current study, and it is based on an explicit representation of the higher-
dimensional Betti numbers.

Next, we state the CLT for $(\beta_{m,n})_{n\geq 1}$ with $m\geq k+1$. It
requires a condition in addition to (2.3). For $n\geq m+2$ and
$A_{1},A_{2},A_{3}\subset[n]$ with $|A_{i}|=m+2$, set

(2.5) $\ell_{ij}=|A_{i}\cap A_{j}|,\ \ 1\leq i<j\leq 3,\ \
\ell_{123}=|A_{1}\cap A_{2}\cap A_{3}|$

and denote

$\mathcal{A}_{m,k}:=\big{\\{}(A_{1},A_{2},A_{3})\subset[n]^{3}:|A_{i}|=m+2,\,i=1,2,3,\
\ell_{ij}\in\\{k+2,\dots,m+1\\},\,1\leq i<j\leq 3\big{\\}}.$

We assume that

(2.6)
$\frac{3}{2}\big{(}\tau_{q}\wedge(\tau_{m+1}+\alpha_{m+1})\big{)}+\bigvee_{(A_{1},A_{2},A_{3})\in\mathcal{A}_{m,k}}\Big{\\{}\tau_{\ell_{123}-1}-\tau_{\ell_{12}-1}-\tau_{\ell_{13}-1}-\tau_{\ell_{23}-1}\Big{\\}}<0.$

Note that it is possible to avoid including a selection of sets from $[n]$ as
a part of the assumption. It is not difficult to check that numbers
$\ell_{12},\ell_{13},\ell_{23},\ell_{123}$ in (2.5) are feasible in (2.6) if
and only if the following conditions hold:

(2.7) $\displaystyle\ell_{ij}\in\\{k+2,\dots,m+1\\},\,1\leq i<j\leq 3,$
$\displaystyle\max\bigl{\\{}\ell_{12}+\ell_{13},\ell_{12}+\ell_{23},\ell_{13}+\ell_{23}\bigr{\\}}-(m+2)\leq\ell_{123}\leq\min\bigl{\\{}\ell_{12},\ell_{13},\ell_{23}\bigr{\\}}.$

Instead of referring to the restrictions (2.7), we keep (2.6) as a “more
operational” formulation in the proof.

######  Theorem 2.3.

Assume conditions (2.3) and (2.6).
$(i)$ If ${\tau_{m+1}+\alpha_{m+1}}>\tau_{q}$, then,

$\frac{\beta_{m,n}-\mathbb{E}[\beta_{m,n}]}{n^{{\tau_{m+1}+\alpha_{m+1}}-\tau_{q}/2}}\Rightarrow\mathcal{N}\big{(}0,(q+1)!((m-q+1)!)^{2}\big{)},\
\ \text{as }n\to\infty.$

$(ii)$ If ${\tau_{m+1}+\alpha_{m+1}}<\tau_{q}$, then,

$\frac{\beta_{m,n}-\mathbb{E}[\beta_{m,n}]}{n^{\frac{1}{2}({\tau_{m+1}+\alpha_{m+1}})}}\Rightarrow\mathcal{N}(0,(m+2)!),\
\ \text{as }n\to\infty.$

$(iii)$ If ${\tau_{m+1}+\alpha_{m+1}}=\tau_{q}$, then,

$\frac{\beta_{m,n}-\mathbb{E}[\beta_{m,n}]}{n^{\frac{1}{2}({\tau_{m+1}+\alpha_{m+1}})}}\Rightarrow\mathcal{N}\left(0,\Big{(}\frac{1}{(q+1)!((m-q+1)!)^{2}}+\frac{1}{(m+2)!}\Big{)}^{-1}\right),\
\ \text{as }n\to\infty.$

######  Remark 2.4.

It is not difficult to verify that in some cases the conditions (2.3) and
(2.6) hold at the same time. For example, if $m=k+1$, then (2.6) always holds!
Indeed, we know by (2.7) that, in this case,
$\ell_{12}=\ell_{13}=\ell_{23}=k+2$ and $\ell_{123}\in\\{k+1,k+2\\}$. Since
$\tau_{k+1}<\tau_{k}$, one can upper bound (2.6) by

$\displaystyle\frac{3}{2}(\tau_{k+2}+\alpha_{k+2})+\tau_{k}-3\tau_{k+1}=\frac{3}{2}(\tau_{k}+2-\psi_{k+1}-\psi_{k+2}+\alpha_{k+2})+\tau_{k}-3(\tau_{k}+1-\psi_{k+1})$
$\displaystyle=$
$\displaystyle-\frac{1}{2}\tau_{k}-\frac{3}{2}\left[\psi_{k+2}-\alpha_{k+2}-\psi_{k+1}\right]=-\frac{1}{2}\tau_{k}-\frac{3}{2}\sum_{i=1}^{k+1}\left({k+2\choose
i}-{k+1\choose i}\right)\alpha_{i}<0.$

On the other hand, the assumption (2.6) is not vacuous. A simple example of a
situation where (2.6) does not hold can be constructed by taking
$\ell_{12}=\ell_{13}=\ell_{23}=m+1$, $\ell_{123}=m$ for a large even $m$, and
choosing a large $\alpha_{m/2}$.

It is shown in [16] that the Betti number in the critical dimension satisfies

$\frac{\beta_{k,n}-\mathbb{E}[\beta_{k,n}]}{n^{\tau_{k}-\tau_{q}/2}}\Rightarrow\mathcal{N}\big{(}0,(q+1)!((k-q)!)^{2}\big{)},\
\ \text{as }n\to\infty,$

so the power of $n$ in the scaling depends directly on the value of
$\tau_{k}$. In contrast, in Theorem 2.3 the power of $n$ in the scaling
depends on $\tau_{m+1}+\alpha_{m+1}$. Furthermore, Theorem 2.3 describes a
phase transition not present in the critical dimension.

Our final result takes a step towards understanding the lower tail large
deviations for $(\beta_{m,n})_{n\geq 1}$ with $m\geq k+1$. In contrast to
other limit theorems, such as the SLLN and CLT ([11, 16, 21, 10]), very little
is known about large deviations for Betti numbers in the multiparameter
simplicial complex or versions of that model. The only exception is [19],
which investigates the upper tail large deviations for the Betti number in the
critical dimension. In the result below, we are able to derive a lower bound
for lower tail large deviation probabilities for $(\beta_{m,n})_{n\geq 1}$.

######  Theorem 2.5.

Assume condition (2.3). Let $\varepsilon\in(0,1)$.
$(i)$ If ${\tau_{m+1}+\alpha_{m+1}}\geq\tau_{q}$,

$\liminf_{n\to\infty}\frac{1}{n^{\tau_{q}}}\log\mathbb{P}\Big{(}\beta_{m,n}\leq(1-\varepsilon)\,\frac{n^{\tau_{m+1}+\alpha_{m+1}}}{(m+2)!}\Big{)}\geq-
C_{m,q}\varepsilon^{2},$

where

(2.8) $C_{m,q}=\frac{5^{4}\cdot 2^{2(m+6)-q}\vee 2^{2(m+5)}}{(q+1)!}.$

$(ii)$ If ${\tau_{m+1}+\alpha_{m+1}}<\tau_{q}$,

$\liminf_{n\to\infty}\frac{1}{n^{{\tau_{m+1}+\alpha_{m+1}}}}\log\mathbb{P}\Big{(}\beta_{m,n}\leq(1-\varepsilon)\,\frac{n^{\tau_{m+1}+\alpha_{m+1}}}{(m+2)!}\Big{)}\geq-\frac{5000}{(m+2)!}\,\varepsilon^{2}.$

##  3\. Proofs

We prove Theorem 2.5 first, as its proof requires developing certain new
techniques that are best seen first. The proofs of Theorems 2.1 and 2.3 are
postponed to later sections.

###  3.1. Preliminaries

Our proof relies on an explicit representation of Betti numbers. Recall that
an $\ell$-dimensional complex $X$ is said to be _pure_ if every face of $X$ is
contained in an $\ell$-face.

######  Definition 3.1.

Given a positive integer $\ell$, a simplicial complex $K$ is called _strongly
connected_ of order $\ell$ (we abbreviate it as “$\ell$-strongly connected”)
if

$(i)$ every pair of $\ell$-faces $\sigma,\tau\in K$ can be connected by a
sequence of $\ell$-faces

$\sigma=\sigma_{0},\sigma_{1},\dots,\sigma_{j-1},\sigma_{j}=\tau,$

such that dim$(\sigma_{i}\cap\sigma_{i+1})=\ell-1$ for each $0\leq i\leq j-1$.
$(ii)$ the $\ell$-skeleton of $K$ is pure.

Note that the dimension of $K$ itself can be greater than $\ell$. Given a
simplicial complex $X$, let $K$ be an $\ell$-strongly connected subcomplex of
$X$. We say that $K$ is _maximal_ if there is no $\ell$-strongly connected
subcomplex $K^{\prime}\subset X$ containing $K$ as a strict subset.

######  Proposition 3.2 (Proposition A.6 in [16]).

For every $m\geq 1$,

(3.1) $\beta_{m,n}=\beta_{m}\big{(}X([n],{\bf
p})\big{)}=\sum_{j=m+2}^{n}\sum_{r\geq
1}\sum_{\sigma\subset[n],\,|\sigma|=j}r\eta_{\sigma}^{(j,r,m)},$

where $\eta_{\sigma}^{(j,r,m)}$ is the indicator function of the event that
$X(\sigma,{\bf p})$, representing the restriction of $X([n],{\bf p})$ to
$\sigma$, forms a maximal $m$-strongly connected subcomplex in $X([n],{\bf
p})$ such that $\beta_{m}\big{(}X(\sigma,{\bf p})\big{)}=r$.

We next provide an estimate of the probability that the restriction of
$X([n],{\bf p})$ to a finite number of points in $[n]$ contains an
$m$-strongly connected complex, which has at least one trivial or non-trivial
$m$-cycle. Since the proof is almost identical to Lemma 7.1 of [16], we omit
it.

######  Lemma 3.3.

Let $K$ be an $m$-strongly connected simplicial complex on $j\geq m+2$ points,
which contains at least one trivial or non-trivial $m$-cycle. Then $K$
contains at least

$\binom{m+2}{i+1}+(j-m-2)\binom{m}{i}$

faces of dimension $i=q,\ldots,m$ and, hence, for $\sigma\subset[n]$ with
$|\sigma|=j$,

$\mathbb{P}\big{(}X(\sigma,{\bf p})\ \text{contains a subcomplex isomorphic
to}\ K\big{)}\leq
j!\,\prod_{i=q}^{m}p_{i}^{\binom{m+2}{i+1}}\Big{(}\prod_{i=q}^{m}p_{i}^{\binom{m}{i}}\Big{)}^{j-m-2}.$

Here, the isomorphism is that of simplicial complexes.

###  3.2. Concepts and representations

The main purpose of this section is to introduce a number of concepts together
with their short-hand notations, that are used throughout the proof of main
theorems. First, recalling the indicator function in (3.1), define, for $j\geq
m+2$ and $r\geq 1$,

(3.2) $T_{j,r}:=\sum_{\sigma\subset[n],\,|\sigma|=j}\eta_{\sigma}^{(j,r,m)}.$

If $j=m+2$, then $\eta_{\sigma}^{(m+2,r,m)}$ is identically zero for all
$r\geq 2$. In other words, $\eta_{\sigma}^{(m+2,1,m)}$ is the only indicator
which can be non-zero. Because of that we will drop the subscript “$1$” from
$T_{m+2,1}$ and simply write $T_{m+2}:=T_{m+2,1}$. In particular, a non-zero
value of $\eta_{\sigma}^{(m+2,1,m)}$ requires that $X(\sigma,{\bf p})$ forms a
maximal non-trivial $m$-cycle, such that $\beta_{m}\big{(}X(\sigma,{\bf
p})\big{)}=1$. By Proposition 3.2, $\beta_{m,n}$ can be represented as

(3.3) $\beta_{m,n}=\sum_{j=m+2}^{n}\sum_{r\geq
1}rT_{j,r}=T_{m+2}+\sum_{j=m+3}^{n}\sum_{r\geq 1}rT_{j,r}.$

Next, let us define for $j\geq m+2$,

(3.4) $S_{j}:=\sum_{\sigma\subset[n],\,|\sigma|=j}\xi_{\sigma}^{(j,m)},$

where $\xi_{\sigma}^{(j,m)}$ is the indicator function of the event on which
$X(\sigma,{\bf p})$ contains an $m$-strongly connected subcomplex with at
least one trivial or non-trivial $m$-cycle. If $j=m+2$, a non-zero value of
the indicator $\xi_{\sigma}^{(m+2,m)}$ requires that $X(\sigma,{\bf p})$ forms
a trivial or non-trivial $m$-cycle. When $X(\sigma,{\bf p})$ forms a non-
trivial $m$-cycle, we have that $\beta_{m}\big{(}X(\sigma,{\bf p})\big{)}=1$.
Note also that a “trivial $m$-cycle” means a simplicial complex of a single
$(m+1)$-simplex. Similarly, for $j\geq m+2$, let

(3.5)
$R_{j}:=\sum_{\sigma\subset[n],\,|\sigma|=j}\widetilde{\xi}_{\sigma}^{(j,m)},$

where $\widetilde{\xi}_{\sigma}^{(j,m)}$ is the indicator function of the
event on which $X(\sigma,{\bf p})$ contains an $m$-strongly connected
subcomplex. Clearly, for every $j\geq m+2$ we have

(3.6) $\sum_{r\geq 1}T_{j,r}\leq S_{j}\leq R_{j}.$

For the proof of Theorem 2.5, one needs to reformulate the probability space
on which (3.3), (3.4), and (3.5) are defined, in the following way. Let
$K_{n}$ be the $(m+1)$-skeleton of a complete simplicial complex on $[n]$. In
other words, $K_{n}$ is the set of all simplices in $[n]$ with dimension $m+1$
or less. Given connectivity probabilities in ${\bf p}$, let $\Gamma_{\bf p}$
be a random subset of $K_{n}$ such that for every $i=1,\dots,m+1$, a word of
length $i+1$ is included in $\Gamma_{\bf p}$ with probability $p_{i}$. More
precisely, $\Gamma_{\bf p}=\Gamma_{\bf p}(\omega)$ is a random subset of
$K_{n}$ defined on the product space
$\\{0,1\\}^{\sum_{\ell=2}^{m+2}\binom{n}{\ell}}$ of Bernoulli random variables
with parameters

$(\underbrace{p_{1},\dots,p_{1}}_{\binom{n}{2}},\underbrace{p_{2},\dots,p_{2}}_{\binom{n}{3}},\dots,\underbrace{p_{m+1},\dots,p_{m+1}}_{\binom{n}{m+2}})\in[0,1]^{\sum_{\ell=2}^{m+2}\binom{n}{\ell}}.$

Throughout the paper we may and will assume that the $(m+1)$-skeleton of
$X([n],{\bf p})$ is defined on this probability space.

###  3.3. Proof of Theorem 2.5

We begin with the following proposition.

######  Proposition 3.4.

Under the assumption (2.3), for every
$\eta\in\big{(}0,(1-\varepsilon)/3\big{)}$, we have

(3.7)
$\displaystyle\begin{split}&\liminf_{n\to\infty}\frac{1}{n^{\tau_{q}\wedge({\tau_{m+1}+\alpha_{m+1}})}}\log\mathbb{P}\Big{(}\beta_{m,n}\leq(1-\varepsilon)\,\frac{n^{\tau_{m+1}+\alpha_{m+1}}}{(m+2)!}\Big{)}\\\
&\geq\liminf_{n\to\infty}\frac{1}{n^{\tau_{q}\wedge({\tau_{m+1}+\alpha_{m+1}})}}\log\mathbb{P}\Big{(}S_{m+2}\leq(1-\varepsilon-2\eta)\,\frac{n^{\tau_{m+1}+\alpha_{m+1}}}{(m+2)!}\Big{)}.\end{split}$

###### Proof.

Fix a positive integer $D$ such that

(3.8) $D>\frac{m+2+\tau_{m}}{\psi_{m}-1};$

note that the fraction in the right hand side of (3.8) is positive due to the
first assumption in (2.3). We decompose the representation (3.3) of
$\beta_{m,n}$ as

(3.9) $\beta_{m,n}=T_{m+2}+\sum_{j=m+3}^{D+m}\sum_{r\geq
1}rT_{j,r}+\sum_{j=D+m+1}^{n}\sum_{r\geq 1}rT_{j,r}.$

We derive an upper bound for each of the terms in (3.9). First, it is clear
that $T_{m+2}\leq S_{m+2}$. Recall next that the $m$-dimensional Betti number
of any simplicial complex on $j$ vertexes is upper bounded by the
corresponding $m$-face counts, which is further upper bounded by
$\binom{j}{m+1}$. This, together with (3.6), implies that

$\sum_{j=m+3}^{D+m}\sum_{r\geq
1}rT_{j,r}\leq\sum_{j=m+3}^{D+m}\binom{j}{m+1}S_{j}.$

For the remaining terms in (3.9), we use the bound

$\displaystyle\sum_{j=D+m+1}^{n}\sum_{r\geq 1}rT_{j,r}$ $\displaystyle\leq
n^{m+1}\sum_{j=D+m+1}^{n}\sum_{\sigma\subset[n],\,|\sigma|=j}{\mathbbm{1}}\big{\\{}X(\sigma,{\bf
p})\text{ is a maximal }$ $\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad
m\text{-strongly connected subcomplex of}\ X([n],{\bf p})\big{\\}}$
$\displaystyle\leq n^{m+1}R_{D+m+1},$

where $R_{D+m+1}$ is given in (3.5). The last inequality above follows from
the fact that a maximal $m$-strongly connected subcomplex of size $j\geq
D+m+1$ always contains a further $m$-strongly connected subcomplex on $D+m+1$
vertices, and the maximality prevents such an $m$-strongly connected
subcomplex on $D+m+1$ vertices from being counted more than once.

Let $\eta\in\big{(}0,(1-\varepsilon)/3\big{)}$. Denoting
$a_{m}:=n^{\tau_{m+1}+\alpha_{m+1}}/(m+2)!$ we use (3.9) and the above bounds
to write

(3.10)
$\displaystyle\mathbb{P}\big{(}\beta_{m,n}\leq(1-\varepsilon)a_{m}\big{)}\geq\mathbb{P}\Big{(}S_{m+2}\leq(1-\varepsilon-2\eta)a_{m},$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\sum_{j=m+3}^{D+m}\binom{j}{m+1}S_{j}\leq\eta
a_{m},\,n^{m+1}R_{D+m+1}\leq\eta a_{m}\Big{)}.$

Since the indicators in (3.4) and (3.5) are all increasing functions of
$\Gamma_{\bf p}$, we can use the FKG inequality (see, e.g., [13, Section 2.3])
to see that (3.10) is bounded below by

$\mathbb{P}\big{(}S_{m+2}\leq(1-\varepsilon-2\eta)a_{m}\big{)}\,\mathbb{P}\Big{(}\sum_{j=m+3}^{D+m}\binom{j}{m+1}S_{j}\leq\eta
a_{m}\Big{)}\,\mathbb{P}(n^{m+1}R_{D+m+1}\leq\eta a_{m}).$

We claim that for every $j\geq m+3$,

$\mathbb{E}[S_{j}]\leq
C_{j}n^{\tau_{m+1}+\alpha_{m+1}}(n^{1-\psi_{m}})^{j-m-2}$

for a finite $C_{j}$ independent of $n$. Indeed, by (3.4),

$\mathbb{E}[S_{j}]=\sum_{K:|K|=j}\binom{n}{j}\mathbb{P}\Big{(}\text{$X(\sigma_{j},{\bf
p})$ contains a subcomplex isomorphic to $K$}\Big{)},$

where the sum above is taken over all isomorphism classes of $m$-strongly
connected complexes on $j$ vertices, with at least one trivial or non-trivial
$m$-cycles. Here, and in similar situations in the sequel, we use a notation
of the type $\sigma_{j}$ to denote any fixed word of length $j$ in $[n]$; this
is legitimate since the indicator functions in (3.4) all have the same
expectations. By Lemma 3.3 and the fact that the number of terms in the above
sum depends only on $j$,

$\mathbb{E}[S_{j}]\leq
C_{j}n^{j}\prod_{i=q}^{m}p_{i}^{\binom{m+2}{i+1}}\Big{(}\prod_{i=q}^{m}p_{i}^{\binom{m}{i}}\Big{)}^{j-m-2}=C_{j}n^{\tau_{m+1}+\alpha_{m+1}}(n^{1-\psi_{m}})^{j-m-2}.$

By Markov’s inequality,

(3.11)
$\displaystyle\begin{split}\mathbb{P}\Big{(}\sum_{j=m+3}^{D+m}\binom{j}{m+1}S_{j}\leq\eta
a_{m}\Big{)}&\geq
1-\eta^{-1}a_{m}^{-1}\sum_{j=m+3}^{D+m}C_{j}n^{\tau_{m+1}+\alpha_{m+1}}(n^{1-\psi_{m}})^{j-m-2}\\\
&\geq 1-C^{*}n^{1-\psi_{m}}\to 1,\ \ \text{ as }n\to\infty.\end{split}$

Next, we claim that

(3.12) $\mathbb{E}[R_{D+m+1}]\leq C^{*}n^{\tau_{m}-D(\psi_{m}-1)}.$

To see this, note that by (3.5),

(3.13) $\displaystyle\mathbb{E}[R_{D+m+1}]=$
$\displaystyle\sideset{}{{}^{\prime}}{\sum}_{K:|K|=D+m+1}\binom{n}{D+m+1}$
$\displaystyle\hskip
57.81621pt\times\mathbb{P}\Big{(}\text{$X(\sigma_{D+m+1},{\bf p})$ contains a
subcomplex isomorphic to $K$}\Big{)},$

where the above sum is taken over all isomorphism classes of $m$-strongly
connected complexes on $D+m+1$ vertices. Since any simplicial complex $K$ in
the above sum contains at least $\binom{m+1}{i+1}+D\binom{m}{i}$ faces of
dimension $i$ for each $1\leq i\leq m$ (the first step of the argument in [5,
Lemma 8.1] is a detailed derivation of this fact), we have for every $K$,

(3.14)
$\displaystyle\binom{n}{D+m+1}\mathbb{P}\Big{(}\text{$X(\sigma_{D+m+1},{\bf
p})$ contains a subcomplex isomorphic to $K$}\Big{)}$ $\displaystyle\hskip
144.54pt\leq
n^{D+m+1}\prod_{i=q}^{m}p_{i}^{\binom{m+1}{i+1}+D\binom{m}{i}}=n^{\tau_{m}-D(\psi_{m}-1)},$

and substituting (3.14) into (3.13) proves (3.12). Now Markov’s inequality and
(3.12) show that for large $n$,

(3.15) $\displaystyle\begin{split}\mathbb{P}(n^{m+1}R_{D+m+1}\leq\eta
a_{m})&\geq 1-n^{m+1}\mathbb{E}[R_{D+m+1}]\\\ &\geq
1-C^{*}n^{m+1+\tau_{m}-D(\psi_{m}-1)}\to 1,\ \ \ n\to\infty,\end{split}$

where the last convergence is due to the choice of $D$. The proof is completed
by combining (3.11) and (3.15). ∎

Proposition 3.5 below establishes a lower bound for the expression in the
right hand side of (LABEL:e:first.leading.term). Theorem 2.5 will then follow
from Propositions 3.4 and 3.5 by letting $\eta\downarrow 0$. We keep the above
notation $a_{m}=n^{\tau_{m+1}+\alpha_{m+1}}/(m+2)!$.

For the proof of Proposition 3.5, we will exploit Theorem 2 in [8] and
Proposition 4.2 of the Appendix; for this purpose, we need to reformulate
$S_{m+2}$ in the following way. For $\alpha\subset[n]$ with $|\alpha|=m+2$,
denote by

$\Big{(}\alpha_{1}^{(i)},\alpha_{2}^{(i)},\dots,\alpha_{\binom{m+2}{i+1}}^{(i)}\Big{)},\
\ i=q,\dots,m,$

the collection of words of length $i+1$, that are contained in $\alpha$.
Define

(3.16)
$Q_{m+2}(\alpha):=\Big{\\{}\alpha_{p_{2}}^{(p_{1})}:p_{1}=q,\dots,m,\,p_{2}=1,\dots,\binom{m+2}{p_{1}+1}\Big{\\}},$

and
$I_{\alpha}^{(m+2)}:={\mathbbm{1}}\big{\\{}Q_{m+2}(\alpha)\subset\Gamma_{\bf
p}\big{\\}}$; then, we have

(3.17) $S_{m+2}=\sum_{\alpha\subset[n],\,|\alpha|=m+2}I_{\alpha}^{(m+2)}.$

######  Proposition 3.5.

Assume that (2.3) holds. Let $\varepsilon\in(0,1)$ and
$\eta\in\big{(}0,(1-\varepsilon)/3\big{)}$.
$(i)$ If ${\tau_{m+1}+\alpha_{m+1}}\geq\tau_{q}$, then

$\liminf_{n\to\infty}\frac{1}{n^{\tau_{q}}}\,\log\mathbb{P}\big{(}S_{m+2}\leq(1-\varepsilon-2\eta)a_{m}\big{)}\geq-
C_{m,q}(\varepsilon+3\eta)^{2},$

where $C_{m,q}$ is given in (2.8).
$(ii)$ If ${\tau_{m+1}+\alpha_{m+1}}<\tau_{q}$,

$\liminf_{n\to\infty}\frac{1}{n^{\tau_{m+1}+\alpha_{m+1}}}\,\log\mathbb{P}\big{(}S_{m+2}\leq(1-\varepsilon-2\eta)a_{m}\big{)}\geq-\frac{5000}{(m+2)!}\,(\varepsilon+3\eta)^{2}.$

###### Proof.

Set $X_{m}$ to be the right hand side in (3.17) and let $K$ be either a
trivial or non-trivial $m$-cycle on $m+2$ points. Clearly $K$ contains
$\binom{m+2}{i+1}$ faces of dimension $i$ for each $1\leq i\leq m$, and hence,

(3.18)
$\mathbb{E}[X_{m}]=\binom{n}{m+2}\prod_{i=q}^{m}p_{i}^{\binom{m+2}{i+1}}\sim\frac{n^{\tau_{m+1}+\alpha_{m+1}}}{(m+2)!}=a_{m},\
\ \ n\to\infty.$

Therefore, for large $n$ enough,

$(1-\varepsilon-3\eta)\mathbb{E}[X_{m}]\leq(1-\varepsilon-2\eta)a_{m}.$

It thus suffices to lower bound the probability

(3.19)
$\mathbb{P}\big{(}X_{m}\leq(1-\varepsilon-3\eta)\mathbb{E}[X_{m}]\big{)}.$

Set, for $\alpha,\beta\subset[n]$, $|\alpha|=|\beta|=m+2$,

(3.20) $\alpha\sim\beta\ \text{ if and only if }\ Q_{m+2}(\alpha)\cap
Q_{m+2}(\beta)\neq\emptyset,\ \alpha\neq\beta.$

In order to apply Theorem 2 in [8] and Proposition 4.2 of the Appendix, we
have to deal with the following quantities:

(3.21) $\displaystyle\begin{split}\mu(X_{m})&:=\mathbb{E}[X_{m}],\\\
\Lambda(X_{m})&:=\mu(X_{m})+\sum_{\begin{subarray}{c}\alpha,\beta\subset[n],|\alpha|=|\beta|=m+2,\\\
\alpha\sim\beta\end{subarray}}\mathbb{E}[I_{\alpha}^{(m+2)}I_{\beta}^{(m+2)}],\\\
\delta(X_{m})&:=\Lambda(X_{m})/\mu(X_{m})-1,\\\
\Pi(X_{m})&:=\max_{\alpha\subset[n],\,|\alpha|=m+2}\mathbb{E}[I_{\alpha}^{(m+2)}]\
\text{(the biggest among identical values).}\end{split}$

The behaviour of $\mu(X_{m})$ is given in (3.18). Furthermore,
$\Pi(X_{m})=n^{-\sum_{i=q}^{m}\binom{m+2}{i+1}\alpha_{i}}\to 0$ as
$n\to\infty$. The asymptotics of $\Lambda(X_{m})$ and $\delta(X_{m})$ are more
involved and postponed to Lemma 4.1 in the Appendix.

Denote by $\mathcal{H}_{q+1}$ the collection of words of length $q+1$ in
$[n]$. For $\alpha\in\mathcal{H}_{q+1}$ let

(3.22) $J_{\alpha}^{(q+1)}={\mathbbm{1}}\\{\alpha\subset\Gamma_{\bf p}\\}.$

We set

(3.23) $\mathcal{U}=\big{[}\lfloor n/2\rfloor\big{]}=\\{1,2,\dots,\lfloor
n/2\rfloor\\},$

and

$H_{q+1}(\mathcal{U})=\\{\alpha\in\mathcal{H}_{q+1}:V(\alpha)\subset\mathcal{U}\\},$

where $V(\alpha)$ denotes a vertex set of $\alpha$. Then

(3.24)
$Y_{q}:=\sum_{\alpha\in\mathcal{H}_{q+1}(\mathcal{U})}J_{\alpha}^{(q+1)}$

represents the $q$-face counts in $X(\mathcal{U},{\bf p})$. Analogously to
(3.21), we also define

$\displaystyle\begin{split}\mu(Y_{q})&:=\mathbb{E}[Y_{q}],\\\
\Pi(Y_{q})&:=\max_{\alpha\in\mathcal{H}_{q+1}(\mathcal{U})}\mathbb{E}[J_{\alpha}^{(q+1)}]\
\text{(again, the biggest among identical values).}\end{split}$

Since (3.24) is the sum of i.i.d. random variables, one can set
$\Lambda(Y_{q})\equiv\mu(Y_{q})$ and $\delta(Y_{q})\equiv 0$. Clearly,

(3.25) $\mu(Y_{q})=\binom{\lfloor
n/2\rfloor}{q+1}p_{q}\sim\frac{n^{\tau_{q}}}{2^{q+1}(q+1)!},\ \text{ and }\
\Pi(Y_{q})=n^{-\alpha_{q}}\to 0.$

Case I: ${\tau_{m+1}+\alpha_{m+1}}\geq\tau_{q}$.
Suppose first that $\varepsilon+3\eta\in[2^{-(m+5)},1]$. Then,

$\displaystyle\mathbb{P}\big{(}X_{m}\leq(1-\varepsilon-3\eta)\mathbb{E}[X_{m}]\big{)}$
$\displaystyle\geq\mathbb{P}(X_{m}=0)\geq\mathbb{P}\Big{(}\bigcap_{\alpha\in\mathcal{H}_{q+1}}\\{J_{\alpha}^{(q+1)}=0\\}\Big{)}.$

Since $\\{J_{\alpha}^{(q+1)}=0\\}$ is a decreasing event for each
$\alpha\in\mathcal{H}_{q+1}$ in terms of the Bernoulli random variables
defining $\Gamma_{\bf p}$, the FKG inequality ensures that the last expression
is at least

$\displaystyle\prod_{\alpha\in\mathcal{H}_{q+1}}\mathbb{P}(J_{\alpha}^{(q+1)}=0)$
$\displaystyle=(1-n^{-\alpha_{q}})^{\binom{n}{q+1}}\sim\exp\Big{\\{}-\frac{n^{\tau_{q}}}{(q+1)!}\Big{\\}}\geq\exp\Big{\\{}-\frac{2^{2(m+5)}(\varepsilon+3\eta)^{2}n^{\tau_{q}}}{(q+1)!}\Big{\\}}.$

It thus follows that

$\liminf_{n\to\infty}\frac{1}{n^{\tau_{q}}}\log\mathbb{P}\big{(}X_{m}\leq(1-\varepsilon-3\eta)\mathbb{E}[X_{m}]\big{)}\geq-\frac{2^{2(m+5)}(\varepsilon+3\eta)^{2}}{(q+1)!}.$

Suppose next that $\varepsilon+3\eta\in(0,2^{-(m+5)})$. By Proposition 4.2
with $i=q$,

(3.26)
$\mathbb{P}\big{(}X_{m}\leq(1-\varepsilon-3\eta)\mathbb{E}[X_{m}]\big{)}\geq\frac{1}{6}\mathbb{P}\big{(}Y_{q}\leq(1-2^{m+5}(\varepsilon+3\eta))\mathbb{E}[Y_{q}]\big{)}.$

Notice that $\Pi(Y_{q})<1$ and for large $n$, by (3.25),

$(\varepsilon+3\eta)^{2}\mu(Y_{q})\geq{\mathbbm{1}}\\{\varepsilon+3\eta<1/50\\}\,\big{(}1+\delta(Y_{q})\big{)}^{-1/2}={\mathbbm{1}}\\{\varepsilon+3\eta<1/50\\}.$

Therefore, Theorem 2 in [8] is applicable, yielding that for large $n$,

$\displaystyle\frac{1}{6}\mathbb{P}\big{(}Y_{q}\leq(1-2^{m+5}(\varepsilon+3\eta))\mathbb{E}[Y_{q}]\big{)}$
$\displaystyle\geq\frac{1}{6}\exp\Big{\\{}-\frac{5000}{\big{(}1-\Pi(Y_{q})\big{)}^{5}}\,\big{(}2^{m+5}(\varepsilon+3\eta)\big{)}^{2}\mu(Y_{q})\big{(}1+\delta(Y_{q})\big{)}\Big{\\}}$
$\displaystyle=\exp\Big{\\{}-\log 6-\frac{5000\cdot
2^{2(m+5)}(\varepsilon+3\eta)^{2}}{(1-n^{-\alpha_{q}})^{5}}\,\mu(Y_{q})\Big{\\}}.$

We conclude that

$\liminf_{n\to\infty}\frac{1}{n^{\tau_{q}}}\log\mathbb{P}\big{(}X_{m}\leq(1-\varepsilon-3\eta)\mathbb{E}[X_{m}]\big{)}\geq-\frac{5^{4}\cdot
2^{2(m+6)-q}(\varepsilon+3\eta)^{2}}{(q+1)!}.$

Case II: ${\tau_{m+1}+\alpha_{m+1}}<\tau_{q}$.
Unlike Case I, we use [8, Theorem 2] only. By Lemma 4.1 and (3.18), it holds
that $\Pi(X_{m})<1$ and for large $n$,

$(\varepsilon+3\eta)^{2}\mu(X_{m})\geq{\mathbbm{1}}\\{\varepsilon+3\eta<1/50\\}\,\big{(}1+\delta(X_{m})\big{)}^{-1/2}.$

Hence, Theorem 2 in [8] shows that

$\displaystyle\liminf_{n\to\infty}\frac{1}{n^{{\tau_{m+1}+\alpha_{m+1}}}}\log\mathbb{P}\big{(}X_{m}\leq(1-\varepsilon-3\eta)\mathbb{E}[X_{m}]\big{)}$
$\displaystyle\geq\liminf_{n\to\infty}\frac{1}{n^{{\tau_{m+1}+\alpha_{m+1}}}}\bigg{\\{}-\frac{5000}{\big{(}1-\Pi(X_{m})\big{)}^{5}}\,(\varepsilon+3\eta)^{2}\mu(X_{m})\big{(}1+\delta(X_{m})\big{)}\bigg{\\}}$
$\displaystyle=-\frac{5000}{(m+2)!}\,(\varepsilon+3\eta)^{2}.$

∎

###  3.4. Proof of Theorem 2.1

We use the representation in (3.3). We begin with a proposition showing that
$T_{m+2}$, the first term in (3.3), itself satisfies a SLLN with the same
scaling, and the same limit, as stated in Theorem 2.1. After that,
Propositions 3.7 and 3.8 verify that the rest of the terms in (3.3) are all
negligible under the same scaling.

######  Proposition 3.6.

Under the assumption (2.3) we have as $n\to\infty$,

$\frac{T_{m+2}}{n^{\tau_{m+1}+\alpha_{m+1}}}\to\frac{1}{(m+2)!}\ \
\text{a.s.}$

The main difficulty here is that the power of $n$ in the scaling constant of
$T_{m+2}$ above may not be large enough for a standard use of the Borel-
Cantelli lemma. We overcome this issue using a known technique of
concentrating first on subsequences increasing faster than linearly fast. In
our context this requires properly bounding $T_{m+2}$ from above and below
first. This approach is a higher-dimensional analog of a technique in Chapter
3 of the monograph [18], which can be used, for example, to show the SLLN for
subgraph counts in random geometric graphs. There are several related works
[15, 17] that used this method to show SLLNs for topological invariants of a
random geometric complex.

The proof of Proposition 3.6 requires some setup first. Let $v_{N}=N^{\gamma}$
for $\gamma=1,2,\dots$ such that

(3.27) $\gamma({\tau_{m+1}+\alpha_{m+1}})>1.$

Then, for every $n\geq 1$, there exists a unique $N=N(n)\in{\mathbb{N}}$ such
that $v_{N}\leq n<v_{N+1}$. For each $i\geq 1$, let

$p_{i}^{\uparrow}=v_{N}^{-\alpha_{i}},\ \ \text{ and }\ \
p_{i}^{\downarrow}=v_{N+1}^{-\alpha_{i}}.$

Now, we are ready to set up the desired bounds for $T_{m+2}$. We define

(3.28)
$S_{m+2}^{\uparrow}=\sum_{\sigma\subset[v_{N+1}],\,|\sigma|=m+2}\xi_{\sigma}^{(m+2,m,\uparrow)},$

where $\xi_{\sigma}^{(m+2,m,\uparrow)}$ is the indicator of the event on which
the complex
$X\big{(}\sigma,p_{1}^{\uparrow},\dots,p_{m}^{\uparrow},p_{m+1}^{\uparrow}\big{)}$
forms a trivial or non-trivial $m$-cycle (since (3.28) counts the number of
$m$-cycles, there is no need to specify connectivity probabilities of
dimension greater than $m+1$). Similarly, we define

(3.29)
$T_{m+2}^{\downarrow}=\sum_{\sigma\subset[v_{N}],\,|\sigma|=m+2}\eta_{\sigma}^{(m+2,1,m,\downarrow)}.$

Here, $\eta_{\sigma}^{(m+2,1,m,\downarrow)}$ is the indicator function of the
event on which the complex
$X\big{(}\sigma,p_{1}^{\downarrow},\dots,p_{m}^{\downarrow},p_{m+1}^{\uparrow}\big{)}$
forms a non-trivial $m$-cycle and no vertex $s\in[v_{N+1}]\setminus\sigma$
forms an $m$-face with an $(m-1)$-face in $\sigma$, under the connectivity
probabilities $(p_{1}^{\uparrow},\dots,p_{m}^{\uparrow})$.

Given random variables $X$ and $Y$, denote by $X\stackrel{{\scriptstyle
st}}{{\leq}}Y$ the stochastic domination meaning that $\mathbb{P}(X\leq
x)\geq\mathbb{P}(Y\leq x)$ for all $x\in{\mathbb{R}}$. Recalling (3.2) and
(3.4), it is clear that

(3.30) $S_{m+2}\stackrel{{\scriptstyle st}}{{\leq}}S_{m+2}^{\uparrow},\ \
\text{ and }\ \ T_{m+2}^{\downarrow}\stackrel{{\scriptstyle
st}}{{\leq}}T_{m+2}.$

###### Proof of Proposition 3.6.

According to the stochastic domination in (3.30),

$\frac{T_{m+2}^{\downarrow}}{v_{N+1}^{\tau_{m+1}+\alpha_{m+1}}}\stackrel{{\scriptstyle
st}}{{\leq}}\frac{T_{m+2}}{n^{\tau_{m+1}+\alpha_{m+1}}}\stackrel{{\scriptstyle
st}}{{\leq}}\frac{S_{m+2}^{\uparrow}}{v_{N}^{\tau_{m+1}+\alpha_{m+1}}},$

for every $n\geq 1$. By the first Borel-Cantelli lemma, the claim of the
proposition will follow once we show that for every $\varepsilon>0$,

(3.31)
$\displaystyle\sum_{N=1}^{\infty}\mathbb{P}\Big{(}S_{m+2}^{\uparrow}\geq\frac{(1+\varepsilon)^{2}}{(m+2)!}\,v_{N}^{\tau_{m+1}+\alpha_{m+1}}\Big{)}<\infty,$
(3.32)
$\displaystyle\sum_{N=1}^{\infty}\mathbb{P}\Big{(}T_{m+2}^{\downarrow}\leq\frac{(1+\varepsilon)^{-2}}{(m+2)!}\,v_{N+1}^{\tau_{m+1}+\alpha_{m+1}}\Big{)}<\infty.$

For the proof of (3.31), we have

$\mathbb{E}[S_{m+2}^{\uparrow}]=\binom{v_{N+1}}{m+2}\prod_{i=q}^{m}(p_{i}^{\uparrow})^{\binom{m+2}{i+1}}=\binom{v_{N+1}}{m+2}v_{N}^{-\sum_{i=q}^{m}\binom{m+2}{i+1}\alpha_{i}}.$

As $v_{N+1}/v_{N}\to 1$ as $N\to\infty$, we have

$\mathbb{E}[S_{m+2}^{\uparrow}]\sim\frac{v_{N}^{\tau_{m+1}+\alpha_{m+1}}}{(m+2)!},\
\ \ N\to\infty,$

hence, there exists $N_{0}\in{\mathbb{N}}$ such that for all $N\geq N_{0}$,

$\mathbb{E}[S_{m+2}^{\uparrow}]\leq\frac{1+\varepsilon}{(m+2)!}\,v_{N}^{\tau_{m+1}+\alpha_{m+1}}.$

By Chebyshev’s inequality and (4.16) in Lemma 4.4 of the Appendix,

$\displaystyle\sum_{N=1}^{\infty}\mathbb{P}\Big{(}S_{m+2}^{\uparrow}\geq\frac{(1+\varepsilon)^{2}}{(m+2)!}\,v_{N}^{\tau_{m+1}+\alpha_{m+1}}\Big{)}$
$\displaystyle\leq\sum_{N=1}^{\infty}\mathbb{P}\Big{(}S_{m+2}^{\uparrow}-\mathbb{E}[S_{m+2}^{\uparrow}]\geq\frac{\varepsilon(1+\varepsilon)}{(m+2)!}\,v_{N}^{\tau_{m+1}+\alpha_{m+1}}\Big{)}$
$\displaystyle\leq
C^{*}\sum_{N=1}^{\infty}\frac{\text{Var}(S_{m+2}^{\uparrow})}{v_{N}^{2({\tau_{m+1}+\alpha_{m+1}})}}$
$\displaystyle\leq
C^{*}\sum_{N=1}^{\infty}\frac{v_{N+1}^{2({\tau_{m+1}+\alpha_{m+1}})-\tau_{q}}\vee
v_{N+1}^{\tau_{m+1}+\alpha_{m+1}}}{v_{N}^{2({\tau_{m+1}+\alpha_{m+1}})}}$
$\displaystyle\leq C^{*}\sum_{N=1}^{\infty}\big{(}N^{-\gamma\tau_{q}}\vee
N^{-\gamma({\tau_{m+1}+\alpha_{m+1}})}\big{)}.$

By (3.27) and the fact that $\tau_{q}>q\geq 1$, the last sum is finite.

We now prove (3.32). We claim that one can write

(3.33)
$\mathbb{E}[T_{m+2}^{\downarrow}]=\binom{v_{N}}{m+2}\prod_{i=q}^{m}(p_{i}^{\downarrow})^{\binom{m+2}{i+1}}(1-p_{m+1}^{\uparrow})\Big{(}1-\binom{m+2}{m}v_{N}^{-\psi_{m}}+u_{N}\Big{)}^{v_{N+1}-(m+2)},$

where $u_{N}=o(v_{N}^{-\psi_{m}})$ as $N\to\infty$. Indeed, for every
$\sigma\subset[v_{N}]$ with $|\sigma|=m+2$, the probability that
$X\big{(}\sigma,p_{1}^{\downarrow},\dots,p_{m}^{\downarrow},p_{m+1}^{\uparrow}\big{)}$
forms a non-trivial $m$-cycle is given by
$\prod_{i=q}^{m}(p_{i}^{\downarrow})^{\binom{m+2}{i+1}}(1-p_{m+1}^{\uparrow})$.
Further, $\sigma$ contains $\binom{m+2}{m}$ faces of dimension $m-1$. Consider
a vertex $w\in[v_{N+1}]\setminus\sigma$ and an $(m-1)$-face $f$ in $\sigma$.
Then, $w$ and $f$ form an $m$-face with probability
$\prod_{i=q}^{m}(p_{i}^{\uparrow})^{\binom{m}{i}}=v_{N}^{-\psi_{m}}$. By the
inclusion-exclusion formula, $w$ forms an $m$-face with at least one of the
$(m-1)$-faces in $\sigma$, with probability
$\binom{m+2}{m}v_{N}^{-\psi_{m}}-u_{N}$, with $u_{N}$ accounting for the
probability that multiple $m$-faces are formed between $w$ and the
$(m-1)$-faces in $\sigma$. Since there are at least $v_{N+1}-(m+2)$ vertices
in $[v_{N+1}]\setminus\sigma$, the probability that no vertex
$s\in[v_{N+1}]\setminus\sigma$ forms an $m$-face with an $(m-1)$-face in
$\sigma$ can be written as

(3.34)
$\Big{(}1-\binom{m+2}{m}v_{N}^{-\psi_{m}}+u_{N}\Big{)}^{v_{N+1}-(m+2)},$

and (3.33) follows.

As $\psi_{m}>1$ and $\alpha_{m+1}>0$, we see that (3.34) tends to $1$ as
$N\to\infty$, and $1-p_{m+1}^{\uparrow}\to 1$, $N\to\infty$. Therefore,

$\mathbb{E}[T_{m+2}^{\downarrow}]\sim\frac{v_{N}^{m+2}}{(m+2)!}\,v_{N+1}^{-\sum_{i=q}^{m}\binom{m+2}{i+1}\alpha_{i}}\sim\frac{v_{N+1}^{\tau_{m+1}+\alpha_{m+1}}}{(m+2)!},\
\ \ N\to\infty,$

and so, there exists $N_{1}\in{\mathbb{N}}$ such that for all $N\geq N_{1}$,

$\mathbb{E}[T_{m+2}^{\downarrow}]\geq\frac{(1+\varepsilon)^{-1}}{(m+2)!}\,v_{N+1}^{\tau_{m+1}+\alpha_{m+1}}.$

Once again, by Chebyshev’s inequality and (4.17) of Lemma 4.4 conclude that

$\displaystyle\sum_{N=1}^{\infty}\mathbb{P}\Big{(}T_{m+2}^{\downarrow}\leq\frac{(1+\varepsilon)^{-2}}{(m+2)!}\,v_{N+1}^{\tau_{m+1}+\alpha_{m+1}}\Big{)}$
$\displaystyle\leq\sum_{N=1}^{\infty}\mathbb{P}\Big{(}T_{m+2}^{\downarrow}-\mathbb{E}[T_{m+2}^{\downarrow}]\leq-\frac{\varepsilon(1+\varepsilon)^{-2}}{(m+2)!}\,v_{N+1}^{\tau_{m+1}+\alpha_{m+1}}\Big{)}$
$\displaystyle\leq C^{*}\sum_{N=1}^{\infty}\big{(}N^{-\gamma\tau_{q}}\vee
N^{-\gamma({\tau_{m+1}+\alpha_{m+1}})}\big{)}<\infty,$

as required. ∎

In the next proposition we start addressing the rest of the terms in the
representation (3.3) of the Betti numbers.

######  Proposition 3.7.

Assume (2.3). Then for every $j\geq m+3$ and $r\geq 1$,

$\frac{T_{j,r}}{n^{\tau_{m+1}+\alpha_{m+1}}}\to 0\ \ \text{a.s. as }\
n\to\infty.$

###### Proof.

Analogously to (3.28) we define

(3.35)
$S_{j}^{\uparrow}:=\sum_{\sigma\subset[v_{N+1}],\,|\sigma|=j}\xi_{\sigma}^{(j,m,\uparrow)},\
\ \ j\geq m+3,$

where, $\xi_{\sigma}^{(j,m,\uparrow)}$ is the indicator of the event on which
$X\big{(}\sigma,p_{1}^{\uparrow},\dots,p_{m}^{\uparrow},p_{m+1}^{\uparrow}\big{)}$
contains an $m$-strongly connected subcomplex with at least one trivial or
non-trivial $m$-cycle. Clearly, $S_{j}\stackrel{{\scriptstyle
st}}{{\leq}}S_{j}^{\uparrow}$ for all $j\geq m+3$, so that

$\frac{T_{j,r}}{n^{\tau_{m+1}+\alpha_{m+1}}}\leq\frac{S_{j}}{n^{\tau_{m+1}+\alpha_{m+1}}}\stackrel{{\scriptstyle
st}}{{\leq}}\frac{S_{j}^{\uparrow}}{v_{N}^{\tau_{m+1}+\alpha_{m+1}}}$

for every $n\geq 1$ and $r\geq 1$, and by the first Borel-Cantelli lemma, it
is suffices to show that for every $\varepsilon>0$,

(3.36) $\sum_{N=1}^{\infty}\mathbb{P}(S_{j}^{\uparrow}\geq\varepsilon
v_{N}^{{\tau_{m+1}+\alpha_{m+1}}})<\infty.$

Note that

$\mathbb{E}[S_{j}^{\uparrow}]=\sum_{K:|K|=j}\binom{v_{N+1}}{j}\mathbb{P}\big{(}X\big{(}\sigma,p_{1}^{\uparrow},\dots,p_{m}^{\uparrow},p_{m+1}^{\uparrow}\big{)}\
\text{contains a subcomplex isomorphic to}\ K\big{)},$

where the sum is taken over all isomorphism classes of $m$-strongly connected
complexes on $j$ vertices, with at least one trivial or non-trivial $m$-cycle.
It follows from Lemma 3.3 and $v_{N+1}/v_{N}\to 1$ as $N\to\infty$, that for a
$j$-dependent constant $C_{j}$,

$\displaystyle\mathbb{E}[S_{j}^{\uparrow}]$ $\displaystyle\leq
C_{j}v_{N+1}^{j}\prod_{i=q}^{m}(p_{i}^{\uparrow})^{\binom{m+2}{i+1}}\Big{\\{}\prod_{i=q}^{m}(p_{i}^{\uparrow})^{\binom{m}{i}}\Big{\\}}^{j-m-2}$
$\displaystyle\leq
C_{j}v_{N}^{\tau_{m+1}+\alpha_{m+1}}\big{(}v_{N}^{1-\psi_{m}}\big{)}^{j-m-2}=o\big{(}v_{N}^{\tau_{m+1}+\alpha_{m+1}}\big{)},\
\ \ N\to\infty,$

where the last equality is due to $\psi_{m}>1$ (and $j\geq m+3$). Hence, for
sufficiently large $N$,

(3.37) $\displaystyle\begin{split}\mathbb{P}(S_{j}^{\uparrow}\geq\varepsilon
v_{N}^{{\tau_{m+1}+\alpha_{m+1}}})&\leq\mathbb{P}\Big{(}S_{j}^{\uparrow}-E[S_{j}^{\uparrow}]\geq\frac{\varepsilon}{2}v_{N}^{\tau_{m+1}+\alpha_{m+1}}\Big{)}\\\
&\leq\frac{4}{\varepsilon^{2}}\cdot\frac{\text{Var}(S_{j}^{\uparrow})}{v_{N}^{2({\tau_{m+1}+\alpha_{m+1}})}}\leq
C_{j}\big{(}N^{-\gamma\tau_{q}}\vee
N^{-\gamma({\tau_{m+1}+\alpha_{m+1}})}\big{)},\end{split}$

where the last inequality follows from (4.16) in Lemma 4.4 of the Appendix.
The last term in (3.37) is summable, so (3.36) follows. ∎

It follows from Propositions 3.6 and 3.7 that

$\frac{1}{n^{\tau_{m+1}+\alpha_{m+1}}}\Big{(}T_{m+2}+\sum_{j=m+3}^{D+m}\sum_{r\geq
1}rT_{j,r}\Big{)}\to\frac{1}{(m+2)!}\ \ \text{a.s. as }\ n\to\infty,$

where $D$ is a positive integer satisfying (3.8). The proof of Theorem 2.1 is
now completed by Proposition 3.8 below and (3.3).

######  Proposition 3.8.

Assume (2.3). We have, as $n\to\infty$,

$\frac{1}{n^{\tau_{m+1}+\alpha_{m+1}}}\sum_{j=D+m+1}^{n}\sum_{r\geq
1}rT_{j,r}\to 0,\ \ \ \text{a.s.}$

###### Proof.

Repeating the same argument as in (3.13) and (3.14), along with Markov’s
inequality, we have,

$\displaystyle\sum_{n=1}^{\infty}\mathbb{P}\Big{(}\sum_{j=D+m+1}^{n}\sum_{r\geq
1}rT_{j,r}\geq 1\Big{)}\leq\sum_{n=1}^{\infty}\sum_{j=D+m+1}^{n}\sum_{r\geq
1}r\mathbb{E}[T_{j,r}]$ $\displaystyle\leq
C^{*}\sum_{n=1}^{\infty}n^{m+1+\tau_{m}-D(\psi_{m}-1)}<\infty$

by the choice of $D$. The first Borel-Cantelli lemma completes the proof of
Proposition 3.8. ∎

###  3.5. Proof of Theorem 2.3

Theorem 2.3 will be established by combining a series of propositions. We
start with a CLT for $S_{m+2}$, a special case of (3.4).

######  Proposition 3.9.

Assume conditions (2.3) and (2.6). As $n\to\infty$,

$\frac{S_{m+2}-\mathbb{E}[S_{m+2}]}{\sqrt{\text{Var}(S_{m+2})}}\Rightarrow\mathcal{N}(0,1).$

###### Proof.

Our proof takes an approach similar to that of Proposition 3.6 in [16]. Using
(3.17) we write

$W=:\frac{S_{m+2}-\mathbb{E}[S_{m+2}]}{\sqrt{\text{Var}(S_{m+2})}}=\sum_{\sigma\subset[n],\,|\sigma|=m+2}Z_{\sigma},$

where

$Z_{\sigma}=\frac{I_{\sigma}^{(m+2)}-\mathbb{E}[I_{\sigma}^{(m+2)}]}{\sqrt{\text{Var}(S_{m+2})}}.$

Evidently we have $\mathbb{E}[W]=0$, $\text{Var}(W)=1$, and further,
$(Z_{\sigma})$ constitute a _dissociated set_ of random variables in the sense
of Barbour et al. [2]. Namely, as in (3.16), we identify each
$\sigma\subset[n],\,|\sigma|=m+2$ with the collection $Q_{q}(\sigma)$ of its
subsets of size $q+1$. Then for every two collections $K,L$ of subsets of
$[n]$ of length $m+2$, if

$\bigcup_{\sigma\in K}Q_{q}(\sigma)\cap\bigcup_{\sigma\in
L}Q_{q}(\sigma)=\emptyset,$

then $(Z_{\sigma},\,\sigma\in K)$ and $(Z_{\sigma},\,\sigma\in L)$ are
independent. The weak convergence claimed in the proposition will follow once
we show that the $L_{1}$-Wasserstein distance between the laws of $W$ and the
standard normal random variable $Y$,

(3.38)
$d_{1}(W,Y):=\sup_{\phi:{\mathbb{R}}\to{\mathbb{R}}}\Big{|}\,\mathbb{E}\big{[}\phi(W)\big{]}-\mathbb{E}\big{[}\phi(Y)\big{]}\,\Big{|}\to
0,\ \ \ n\to\infty,$

where the supremum is taken over all $1$-Lipschitz functions
$\phi:{\mathbb{R}}\to{\mathbb{R}}$.

For $\alpha\subset[n],\,|\alpha|=m+2$, let
$\mathcal{L}_{\alpha}:=\\{\beta\subset[n],\,|\beta|=m+2:\,|\beta\cap\alpha|\geq
q+1\\}$ be the dependency neighborhood of $\alpha$. By Theorem 1 in Barbour et
al. [2] (see also the bound (2.7) therein), the distance (3.38) is bounded by
a constant multiple of

(3.39)
$\displaystyle\begin{split}&\sum_{\alpha\subset[n],\,|\alpha|=m+2}\sum_{\beta\in\mathcal{L}_{\alpha}}\sum_{\gamma\in\mathcal{L}_{\alpha}}\Big{\\{}\mathbb{E}\big{[}|Z_{\alpha}Z_{\beta}Z_{\gamma}|\big{]}+\mathbb{E}\big{[}|Z_{\alpha}Z_{\beta}|\big{]}\,\mathbb{E}\big{[}|Z_{\gamma}|\big{]}\Big{\\}}\\\
&\leq\frac{C^{*}}{\big{(}\text{Var}(S_{m+2})\big{)}^{3/2}}\sum_{\alpha\subset[n],\,|\alpha|=m+2}\sum_{\beta\in\mathcal{L}_{\alpha}}\sum_{\gamma\in\mathcal{L}_{\alpha}}\Big{\\{}\mathbb{E}\big{[}(I_{\alpha}^{(m+2)}+\mathbb{E}[I_{\alpha}^{(m+2)}])(I_{\beta}^{(m+2)}+\mathbb{E}[I_{\beta}^{(m+2)}])\\\
&\qquad\qquad\times(I_{\gamma}^{(m+2)}+\mathbb{E}[I_{\gamma}^{(m+2)}])\big{]}+2\mathbb{E}\big{[}(I_{\alpha}^{(m+2)}+\mathbb{E}[I_{\alpha}^{(m+2)}])(I_{\beta}^{(m+2)}+\mathbb{E}[I_{\beta}^{(m+2)}])\big{]}\,\mathbb{E}[I_{\gamma}^{(m+2)}]\Big{\\}}.\end{split}$

Given $\alpha\subset[n],\,|\alpha|=m+2$, $\beta\in\mathcal{L}_{\alpha}$ and
$\gamma\in\mathcal{L}_{\alpha}$, denote

$\ell_{12}=|\alpha\cap\beta|,\ \ \ell_{13}=|\alpha\cap\gamma|,\ \
\ell_{23}=|\beta\cap\gamma|,\ \ \ell_{123}=|\alpha\cap\beta\cap\gamma|.$

Since $\beta\in\mathcal{L}_{\alpha}$ and $\gamma\in\mathcal{L}_{\alpha}$, it
must be that $\ell_{12}\geq q+1$, $\ell_{13}\geq q+1$, whereas $\ell_{23}$,
$\ell_{123}$ can be less than $q+1$.

Part I: We first deal with the case when $(\alpha,\beta,\gamma)$ satisfies
$\ell_{12}\leq m+1$, $\ell_{13}\leq m+1$, $\ell_{23}\leq m+1$. Then

(3.40)
$\mathbb{E}[I_{\alpha}^{(m+2)}I_{\beta}^{(m+2)}I_{\gamma}^{(m+2)}]=\prod_{i=q}^{m}p_{i}^{3\binom{m+2}{i+1}-\binom{\ell_{12}}{i+1}-\binom{\ell_{13}}{i+1}-\binom{\ell_{23}}{i+1}+\binom{\ell_{123}}{i+1}},$

and all other expectation terms in (LABEL:e:Wasserstein.bound) are bounded by
the right hand side of (3.40). Appealing to these bounds and using the fact
that the number of ways to choose $(\alpha,\beta,\gamma)$ corresponding to
given $\ell_{12},\ell_{13},\ell_{23},\ell_{123}$ is bounded by
$n^{3(m+2)-\ell_{12}-\ell_{13}-\ell_{23}+\ell_{123}}$, one can bound the
corresponding terms in (LABEL:e:Wasserstein.bound), up to a constant factor,
by

$\frac{1}{\big{(}\text{Var}(S_{m+2})\big{)}^{3/2}}\sum_{\ell_{12}=q+1}^{m+1}\sum_{\ell_{13}=q+1}^{m+1}\sum_{\ell_{23}=0}^{m+1}\sum_{\ell_{123}=0}^{\ell_{12}\wedge\ell_{13}\wedge\ell_{23}}n^{3({\tau_{m+1}+\alpha_{m+1}})-\tau_{\ell_{12}-1}-\tau_{\ell_{13}-1}-\tau_{\ell_{23}-1}+\tau_{\ell_{123}-1}}.$

If, say, $\ell_{23}\leq k+1$; then,
$\tau_{\ell_{23}-1}-\tau_{\ell_{123}-1}\geq 0$ (see (2.2)). Therefore,

$\displaystyle\sum_{\ell_{12}=q+1}^{m+1}\sum_{\ell_{13}=q+1}^{m+1}\sum_{\ell_{23}=0}^{k+1}\sum_{\ell_{123}=0}^{\ell_{12}\wedge\ell_{13}\wedge\ell_{23}}n^{3({\tau_{m+1}+\alpha_{m+1}})-\tau_{\ell_{12}-1}-\tau_{\ell_{13}-1}-\tau_{\ell_{23}-1}+\tau_{\ell_{123}-1}}$
$\displaystyle\leq
C^{*}\sum_{\ell_{12}=q+1}^{m+1}\sum_{\ell_{13}=q+1}^{m+1}n^{3({\tau_{m+1}+\alpha_{m+1}})-\tau_{\ell_{12}-1}-\tau_{\ell_{13}-1}}$
$\displaystyle\leq
C^{*}\big{\\{}n^{3({\tau_{m+1}+\alpha_{m+1}})-2\tau_{q}}\vee
n^{3({\tau_{m+1}+\alpha_{m+1}})-2\tau_{m}}\big{\\}},$

and similar bounds apply when $\ell_{12}\leq k+1$ or $\ell_{13}\leq k+1$.
Therefore, it is sufficient to bound

(3.41)
$\displaystyle\begin{split}&\frac{1}{\big{(}\text{Var}(S_{m+2})\big{)}^{3/2}}\Big{\\{}n^{3({\tau_{m+1}+\alpha_{m+1}})-2\tau_{q}}\vee
n^{3({\tau_{m+1}+\alpha_{m+1}})-2\tau_{m}}\\\
&\qquad+\sum_{\ell_{12}=k+2}^{m+1}\sum_{\ell_{13}=k+2}^{m+1}\sum_{\ell_{23}=k+2}^{m+1}\sum_{\ell_{123}=0}^{\ell_{12}\wedge\ell_{13}\wedge\ell_{23}}n^{3({\tau_{m+1}+\alpha_{m+1}})-\tau_{\ell_{12}-1}-\tau_{\ell_{13}-1}-\tau_{\ell_{23}-1}+\tau_{\ell_{123}-1}}\Big{\\}}.\end{split}$

By (4.4) and Lemma 4.3 in the Appendix, the contribution of the first term in
(LABEL:e:part1.bound) is bounded by $C^{*}\big{(}n^{-\tau_{q}/2}\vee
n^{-\frac{1}{2}({\tau_{m+1}+\alpha_{m+1}})}\big{)}$, which clearly converges
to $0$ as $n\to\infty$. Moreover, the second line of (LABEL:e:part1.bound) is
bounded by

$C^{*}\sum_{\ell_{12}=k+2}^{m+1}\sum_{\ell_{13}=k+2}^{m+1}\sum_{\ell_{23}=k+2}^{m+1}\sum_{\ell_{123}=0}^{\ell_{12}\wedge\ell_{13}\wedge\ell_{23}}n^{\frac{3}{2}(\tau_{q}\wedge({\tau_{m+1}+\alpha_{m+1}}))-\tau_{\ell_{12}-1}-\tau_{\ell_{13}-1}-\tau_{\ell_{23}-1}+\tau_{\ell_{123}-1}},$

which also tends to $0$ as $n\to\infty$ by the assumption (2.6).

Part II: Next we deal with the case when $(\alpha,\beta,\gamma)$ satisfies one
of the following conditions.

  * •

$\ell_{12}=m+2$, $\ell_{13}\vee\ell_{23}\leq m+1$,

  * •

$\ell_{13}=m+2$, $\ell_{12}\vee\ell_{23}\leq m+1$,

  * •

$\ell_{23}=m+2$, $\ell_{12}\vee\ell_{13}\leq m+1$.

We treat in detail only the first case, as the other two cases are similar.
The conditions imply that $\alpha=\beta$, and it follows that
$\ell_{13}=\ell_{23}=\ell_{123}$. Then the contribution of the corresponding
terms in (LABEL:e:Wasserstein.bound) is, up to a constant multiplier, bounded
by

$\displaystyle\frac{1}{\big{(}\text{Var}(S_{m+2})\big{)}^{3/2}}\sum_{\ell_{13}=q+1}^{m+1}\sum_{\alpha\subset[n],\,|\alpha|=m+2}\sum_{\gamma\in\mathcal{L}_{\alpha}:\,|\alpha\cap\gamma|=\ell_{13}}\mathbb{E}[I_{\alpha}^{(m+2)}I_{\gamma}^{(m+2)}]$
$\displaystyle=\frac{1}{\big{(}\text{Var}(S_{m+2})\big{)}^{3/2}}\sum_{\ell_{13}=q+1}^{m+1}\sum_{\alpha\subset[n],\,|\alpha|=m+2}\sum_{\gamma\in\mathcal{L}_{\alpha}:\,|\alpha\cap\gamma|=\ell_{13}}\prod_{i=q}^{m}p_{i}^{2\binom{m+2}{i+1}-\binom{\ell_{13}}{i+1}}$
$\displaystyle\leq\frac{1}{\big{(}\text{Var}(S_{m+2})\big{)}^{3/2}}\sum_{\ell_{13}=q+1}^{m+1}n^{2({\tau_{m+1}+\alpha_{m+1}})-\tau_{\ell_{13}-1}}$
$\displaystyle\leq\frac{C^{*}}{\big{(}\text{Var}(S_{m+2})\big{)}^{3/2}}\big{\\{}n^{2({\tau_{m+1}+\alpha_{m+1}})-\tau_{q}}\vee
n^{2({\tau_{m+1}+\alpha_{m+1}})-\tau_{m}}\big{\\}}$ $\displaystyle\leq
C^{*}\big{\\{}n^{-\tau_{q}/2}\vee
n^{-\frac{1}{2}({\tau_{m+1}+\alpha_{m+1}})}\big{\\}}\to 0,\ \ \text{as
}n\to\infty.$

Part III: It remains to consider the case when $(\alpha,\beta,\gamma)$
satisfies $\ell_{12}=\ell_{13}=\ell_{23}=m+2$. Then $\alpha=\beta=\gamma$ and
$\ell_{123}=m+2$, and the contribution of the corresponding terms in
(LABEL:e:Wasserstein.bound) is bounded, up to a constant multiplier, by

$\displaystyle\frac{1}{\big{(}\text{Var}(S_{m+2})\big{)}^{3/2}}\sum_{\alpha\subset[n],\,|\alpha|=m+2}\mathbb{E}[I_{\alpha}^{(m+2)}]$
$\displaystyle\leq\frac{C^{*}n^{\tau_{m+1}+\alpha_{m+1}}}{\big{\\{}n^{2({\tau_{m+1}+\alpha_{m+1}})-\tau_{q}}\vee
n^{\tau_{m+1}+\alpha_{m+1}}\big{\\}}^{3/2}}$ $\displaystyle\leq
C^{*}n^{-\frac{1}{2}({\tau_{m+1}+\alpha_{m+1}})}\to 0,\ \ \text{as
}n\to\infty.$

∎

In the next proposition we show that the CLT for $S_{m+2}$ can be translated
into a CLT for $T_{m+2}$, a special case of (3.2).

######  Proposition 3.10.

Assume conditions (2.3) and (2.6). As $n\to\infty$,

$\frac{T_{m+2}-\mathbb{E}[T_{m+2}]}{\sqrt{\text{Var}(S_{m+2})}}\Rightarrow\mathcal{N}(0,1).$

###### Proof.

Define

(3.42)
$V_{m+2}:=\sum_{\sigma\subset[n],\,|\sigma|=m+2}\lambda_{\sigma}^{(m+2,m)},$

where $\lambda_{\sigma}^{(m+2,m)}$ is the indicator function of the event that
$X(\sigma,{\bf p})$ forms a non-trivial $m$-cycle. By virtue of the last
proposition, it is enough to verify that, as $n\to\infty$,

$\frac{S_{m+2}-T_{m+2}-\mathbb{E}[S_{m+2}-T_{m+2}]}{\sqrt{\text{Var}(S_{m+2})}}\stackrel{{\scriptstyle
p}}{{\to}}0.$

For every $\varepsilon>0$, Chebyshev’s inequality gives that

(3.43)
$\displaystyle\begin{split}&\mathbb{P}\big{(}\,|S_{m+2}-T_{m+2}-\mathbb{E}[S_{m+2}-T_{m+2}]|\geq\varepsilon\sqrt{\text{Var}(S_{m+2})}\big{)}\\\
&\leq\mathbb{P}\Big{(}\,|S_{m+2}-V_{m+2}-\mathbb{E}[S_{m+2}-V_{m+2}]|\geq\frac{\varepsilon}{2}\sqrt{\text{Var}(S_{m+2})}\Big{)}\\\
&\qquad+\mathbb{P}\Big{(}\,|V_{m+2}-T_{m+2}-\mathbb{E}[V_{m+2}-T_{m+2}]|\geq\frac{\varepsilon}{2}\sqrt{\text{Var}(S_{m+2})}\Big{)}\\\
&\leq\frac{4}{\varepsilon^{2}}\,\Big{\\{}\frac{\text{Var}(S_{m+2}-V_{m+2})}{\text{Var}(S_{m+2})}+\frac{\text{Var}(V_{m+2}-T_{m+2})}{\text{Var}(S_{m+2})}\Big{\\}}.\end{split}$

By definition, $S_{m+2}-V_{m+2}$ equals the number of $(m+1)$-faces in
$X([n],{\bf p})$, so by Proposition 3.1 in [16],

(3.44) $\text{Var}(S_{m+2}-V_{m+2})\leq C^{*}(n^{2\tau_{m+1}-\tau_{q}}\vee
n^{\tau_{m+1}}).$

It now follows from (4.11) and (4.15) of Lemma 4.3, as well as (3.44), that
the final expression in (LABEL:e:after.Chebyshev) is at most

$\displaystyle\frac{C^{*}}{n^{2({\tau_{m+1}+\alpha_{m+1}})-\tau_{q}}\vee
n^{{\tau_{m+1}+\alpha_{m+1}}}}\,\Big{\\{}n^{2\tau_{m+1}-\tau_{q}}\vee
n^{\tau_{m+1}}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad+\big{(}n^{2({\tau_{m+1}+\alpha_{m+1}})-\tau_{q}}\vee
n^{\tau_{m+1}+\alpha_{m+1}}\big{)}n^{(1\vee\psi_{m-1})-\psi_{m}}\Big{\\}}$
$\displaystyle\leq C^{*}(n^{-\alpha_{m+1}}+n^{(1\vee\psi_{m-1})-\psi_{m}})\to
0,\ \ \text{as }n\to\infty.$

∎

Finally, we show that the remaining terms in the expression (3.3) of the Betti
numbers are negligible in the context of the required CLT.

######  Proposition 3.11.

Assume (2.3) and (2.6). As $n\to\infty$,

$\frac{\sum_{j=m+3}^{n}\sum_{r\geq
1}rT_{j,r}-\mathbb{E}\big{[}\sum_{j=m+3}^{n}\sum_{r\geq
1}rT_{j,r}\big{]}}{\sqrt{\text{Var}(S_{m+2})}}\stackrel{{\scriptstyle
p}}{{\to}}0.$

###### Proof.

We first check that for every $j\geq m+3$ and $r\geq 1$,

$\frac{T_{j,r}-\mathbb{E}[T_{j,r}]}{\sqrt{\text{Var}(S_{m+2})}}\stackrel{{\scriptstyle
p}}{{\to}}0,\ \ \ n\to\infty.$

Indeed, for every $\varepsilon>0$, Chebyshev’s inequality, along with (4.11)
and (4.14) of Lemma 4.3, demonstrate that as $n\to\infty$,

$\displaystyle\mathbb{P}\big{(}\,|T_{j,r}-\mathbb{E}[T_{j,r}]|\geq\varepsilon\sqrt{\text{Var}(S_{m+2})}\big{)}\leq\frac{1}{\varepsilon^{2}}\cdot\frac{\text{Var}(T_{j,r})}{\text{Var}(S_{m+2})}\leq
C_{j}n^{1-\psi_{m}}\to 0.$

It remains to show that

$\frac{\sum_{j=D+m+1}^{n}\sum_{r\geq
1}rT_{j,r}-\mathbb{E}\big{[}\sum_{j=D+m+1}^{n}\sum_{r\geq
1}rT_{j,r}\big{]}}{\sqrt{\text{Var}(S_{m+2})}}\stackrel{{\scriptstyle
p}}{{\to}}0,$

where $D$ is a positive constant satisfying (3.8). However, the denominator
diverges to infinity, while proceeding the same way as in (3.13) and (3.14),
one sees that the mean of the absolute value of the numerator is bounded by a
constant multiple of $n^{m+1+\tau_{m}-D(\psi_{m}-1)}$, which goes to $0$ as
$n\to\infty$. This proves the claim. ∎

###### Proof of Theorem 2.3.

By Propositions 3.10 and 3.11, along with the representation (3.3),

$\frac{\beta_{m,n}-\mathbb{E}[\beta_{m,n}]}{\sqrt{\text{Var}(S_{m+2})}}\Rightarrow\mathcal{N}(0,1).$

Now, (4.11) in Lemma 4.3 completes the proof. ∎

##  4\. Appendix

We start with the asymptotic behaviour of $\Lambda(X_{m})$ and $\delta(X_{m})$
in (3.21).

######  Lemma 4.1.

$(i)$ As $n\to\infty$,

(4.1)
$\Lambda(X_{m})\sim\begin{cases}\frac{n^{2({\tau_{m+1}+\alpha_{m+1}})-\tau_{q}}}{(q+1)!((m-q+1)!)^{2}}&\text{
if }{\tau_{m+1}+\alpha_{m+1}}>\tau_{q},\\\\[7.0pt]
\frac{n^{\tau_{m+1}+\alpha_{m+1}}}{(m+2)!}&\text{ if
}{\tau_{m+1}+\alpha_{m+1}}<\tau_{q},\\\\[7.0pt]
\Big{\\{}\frac{1}{(q+1)!((m-q+1)!)^{2}}+\frac{1}{(m+2)!}\Big{\\}}n^{\tau_{m+1}+\alpha_{m+1}}&\text{
if }{\tau_{m+1}+\alpha_{m+1}}=\tau_{q}.\end{cases}$

$(ii)$ As $n\to\infty$,

(4.2)
$\delta(X_{m})\sim\begin{cases}\frac{(m+2)!n^{{\tau_{m+1}+\alpha_{m+1}}-\tau_{q}}}{(q+1)!((m-q+1)!)^{2}}&\text{
if }{\tau_{m+1}+\alpha_{m+1}}\geq\tau_{q},\\\\[3.0pt] 0&\text{ if
}{\tau_{m+1}+\alpha_{m+1}}<\tau_{q}.\end{cases}$

###### Proof.

For $\sigma\subset[n],|\sigma|=m+2$, we use the notation
$\xi_{\sigma}:=\xi_{\sigma}^{(m+2,m)}$ in (3.4). By the definition of the
relation (3.20) we have

$\displaystyle\sum_{\begin{subarray}{c}\alpha,\beta\subset[n],|\alpha|=|\beta|=m+2,\\\
\alpha\sim\beta\end{subarray}}\mathbb{E}\big{[}I_{\alpha}^{(m+2)}I_{\beta}^{(m+2)}\big{]}$
$\displaystyle=\sum_{\ell=q+1}^{m+1}\sum_{\begin{subarray}{c}\sigma\subset[n],\\\
|\sigma|=m+2\end{subarray}}\sum_{\begin{subarray}{c}\tau\subset[n],\\\
|\tau|=m+2\end{subarray}}\mathbb{E}\big{[}\xi_{\sigma}\xi_{\tau}\big{]}\,{\mathbbm{1}}\big{\\{}|\sigma\cap\tau|=\ell\big{\\}}$
$\displaystyle=\sum_{\ell=q+1}^{m+1}\binom{n}{m+2}\binom{m+2}{\ell}\binom{n-m-2}{m+2-\ell}\mathbb{E}\big{[}\xi_{\sigma_{m+2}}\xi_{\tau_{m+2}}\big{]}\,{\mathbbm{1}}\big{\\{}|\sigma_{m+2}\cap\tau_{m+2}|=\ell\big{\\}}$
$\displaystyle\sim\sum_{\ell=q+1}^{m+1}\frac{n^{2(m+2)-\ell}}{\ell!\big{(}(m+2-\ell)!\big{)}^{2}}\,\mathbb{E}\big{[}\xi_{\sigma_{m+2}}\xi_{\tau_{m+2}}\big{]}\,{\mathbbm{1}}\big{\\{}|\sigma_{m+2}\cap\tau_{m+2}|=\ell\big{\\}}.$

For
$\sigma,\tau\subset[n],\,|\sigma|=|\tau|=m+2,\,\ell=|\sigma\cap\tau|\in\\{q+1,\dots,m+1\\}$,
we have

$\mathbb{E}\big{[}\xi_{\sigma}\xi_{\tau}\big{]}=\prod_{i=q}^{m}p_{i}^{2\binom{m+2}{i+1}-\binom{\ell}{i+1}}=n^{-2\sum_{i=q}^{m}\binom{m+2}{i+1}\alpha_{i}+\sum_{i=q}^{\ell-1}\binom{\ell}{i+1}\alpha_{i}},$

where $\prod_{i=q}^{m}p_{i}^{-\binom{\ell}{i+1}}$ accounts for the double
count of the $(\ell-1)$-face, which is common to $\sigma$ and $\tau$.
Therefore, as $n\to\infty$,

(4.3)
$\displaystyle\begin{split}\sum_{\begin{subarray}{c}\alpha,\beta\subset[n],|\alpha|=|\beta|=m+2,\\\
\alpha\sim\beta\end{subarray}}\mathbb{E}\big{[}I_{\alpha}^{(m+2)}I_{\beta}^{(m+2)}\big{]}&\sim\sum_{\ell=q+1}^{m+1}\frac{n^{2({\tau_{m+1}+\alpha_{m+1}})-\tau_{\ell-1}}}{\ell!\big{(}(m+2-\ell)!\big{)}^{2}}\\\
&\sim\frac{n^{2({\tau_{m+1}+\alpha_{m+1}})-\tau_{q}}}{(q+1)!\big{(}(m-q+1)!\big{)}^{2}}\vee\frac{n^{2({\tau_{m+1}+\alpha_{m+1}})-\tau_{m}}}{(m+1)!}.\end{split}$

The key observation is that

(4.4)
$\tau_{m}-({\tau_{m+1}+\alpha_{m+1}})=-1+\sum_{i=q}^{m}\binom{m+1}{i}\alpha_{i}>-1+\psi_{m}>0.$

Combining (3.18), (4.3), and (4.4) gives us (4.1), and (4.2) follows from
(3.18) and (4.1). ∎

Before stating the next proposition, for $j\geq q+1$ we let $\mathcal{H}_{j}$
be the collection of words of length $j$ in $[n]$. Let us fix an arbitrary
$i$-simplex for $i\in\\{q,\dots,m\\}$, and let $L$ be a simplicial complex (of
dimension $i$) that is induced by this $i$-simplex. Given
$\beta\in\mathcal{H}_{i+1}$, define

$J_{\beta}^{(i+1)}={\mathbbm{1}}\big{\\{}R_{i+1}(\beta)\subset\Gamma_{\bf
p}\big{\\}},$

where $R_{i+1}(\beta)$ denotes the collection of words of length
$q+1,\dots,i+1$, that are contained in $\beta$; it is isomorphic to
$L\setminus L^{(q-1)}$, where $L^{(q-1)}$ is the $(q-1)$-skeleton of $L$. In
particular, $R_{q+1}(\beta)=\beta$ for $\beta\in\mathcal{H}_{q+1}$ (see
(3.22)). For a subset $S\subset[n]$ define

$\mathcal{H}_{i+1}(S)=\\{\beta\in\mathcal{H}_{i+1}:V(\beta)\subset S\\},$

where $V(\beta)$ is the vertex set of $\beta$. We will be particularly
interested in $\mathcal{H}_{i+1}(\mathcal{U})$, where $\mathcal{U}$ is defined
in (3.23). As an extension of (3.24), define
$Y_{i}:=\sum_{\beta\in\mathcal{H}_{i+1}(\mathcal{U})}J_{\beta}^{(i+1)}$, which
represents the number of $i$-faces in $X(\mathcal{U},{\bf p})$.

The following proposition is used for the proof of Proposition 3.5 only in the
special case $i=q$ (see (3.26)). Nevertheless, we will prove it under a more
general setup. The proof uses ideas from Theorem 20 in [8].

######  Proposition 4.2.

Assume that (2.3) holds. Let $i\in\\{q,\ldots,m\\}$. Then for every
$\varepsilon\in(0,2^{-(m+5)})$, there exists $N\in{\mathbb{N}}$ such that for
all $n\geq N$,

$\mathbb{P}\big{(}X_{m}\leq(1-\varepsilon)\mathbb{E}[X_{m}]\big{)}\geq\frac{1}{6}\,\mathbb{P}\big{(}Y_{i}\leq(1-2^{m+5}\varepsilon)\mathbb{E}[Y_{i}]\big{)}.$

###### Proof.

Set

$Z:=\sum_{(\alpha,\beta)\in\mathcal{H}_{m+2}\times\mathcal{H}_{i+1}}{\mathbbm{1}}\\{\beta\subset\alpha\\}I_{\alpha}^{(m+2)}=\tau
X_{m},$

where (independently of $\alpha$),
$\tau:=\sum_{\beta\in\mathcal{H}_{i+1}}{\mathbbm{1}}\\{\beta\subset\alpha\\}=\binom{m+2}{i+1}$.
For $\beta\in\mathcal{H}_{i+1}$ and $S\subset[n]$, define

$\mathcal{K}(S,\beta)=\big{\\{}\alpha\in\mathcal{H}_{m+2}:\beta\subset\alpha,\,V(\alpha)\setminus
V(\beta)\subset[n]\setminus S\big{\\}}.$

Define, for $S\subset[n]$,

$Z_{S}:=\sum_{(\alpha,\beta):\,\beta\in\mathcal{H}_{i+1}(S),\,\alpha\in
K(S,\beta)}I_{\alpha}^{(m+2)},$

and $R_{\mathcal{U}}:=Z-Z_{\mathcal{U}}$. Additionally, let
$z=(1-2^{m+4}\varepsilon)\mathbb{E}[Z_{\mathcal{U}}]$ and
$r=(1-\varepsilon)\mathbb{E}[Z]-z$. By the FKG inequality, we have

$\displaystyle\mathbb{P}\big{(}X_{m}\leq(1-\varepsilon)\mathbb{E}[X_{m}]\big{)}$
$\displaystyle=\mathbb{P}(R_{\mathcal{U}}+Z_{\mathcal{U}}\leq r+z)$
$\displaystyle\geq\mathbb{P}(R_{\mathcal{U}}\leq r,\,Z_{\mathcal{U}}\leq
z)\geq\mathbb{P}(R_{\mathcal{U}}\leq r)\mathbb{P}(Z_{\mathcal{U}}\leq z).$

To complete the proof, it is sufficient to demonstrate that

(4.5) $\displaystyle\mathbb{P}(R_{\mathcal{U}}\leq r)$ $\displaystyle\geq
1-\frac{\Lambda(X_{m})}{\Lambda(X_{m})+(\varepsilon\mu(X_{m}))^{2}},$ (4.6)
$\displaystyle\mathbb{P}(Z_{\mathcal{U}}\leq z)$
$\displaystyle\geq\frac{1}{2}\,\Big{(}1-\frac{\Lambda(X_{m})}{\Lambda(X_{m})+2(\varepsilon\mu(X_{m}))^{2}}\Big{)}\mathbb{P}\big{(}Y_{i}\leq(1-2^{m+5}\varepsilon)\mathbb{E}[Y_{i}]\big{)},$

Indeed, by (3.18), Lemma 4.1 $(i)$ and (2.3) we have
$(\varepsilon\mu(X_{m}))^{2}\geq\Lambda(X_{m})$ for sufficiently large $n$, so
(4.5) and (4.6) imply the claim of Proposition 4.2.

For the proof of (4.5), let $\mathscr{X}$ be a random subset of $[n]$, whose
elements are chosen uniformly and independently of $\Gamma_{\bf p}$, such that
$|\mathscr{X}|=|\mathcal{U}|=\lfloor n/2\rfloor$. Then,

$\mathbb{E}[Z_{\mathscr{X}}|\Gamma_{\bf
p}]=\sum_{(\alpha,\beta)\in\mathcal{H}_{m+2}\times\mathcal{H}_{i+1}}{\mathbbm{1}}\\{\beta\subset\alpha\\}\,\mathbb{P}\big{(}V(\alpha)\setminus
V(\beta)\subset[n]\setminus\mathscr{X},\,V(\beta)\subset\mathscr{X}\big{)}\,I_{\alpha}^{(m+2)}.$

By an easy calculation,

$\mathbb{P}\big{(}V(\alpha)\setminus
V(\beta)\subset[n]\setminus\mathscr{X},\,V(\beta)\subset\mathscr{X}\big{)}=\binom{n}{\lfloor
n/2\rfloor}^{-1}\binom{n-(m+2)}{\lfloor n/2\rfloor-(i+1)}\to 2^{-(m+2)},\ \ \
n\to\infty,$

and so, for large $n$,

$\mathbb{E}[Z_{\mathscr{X}}]=\mathbb{E}\big{[}\mathbb{E}[Z_{\mathscr{X}}|\Gamma_{\bf
p}]\big{]}\geq 2^{-(m+3)}\mathbb{E}[Z].$

It follows from the independence of $\Gamma_{\bf p}$ and $\mathscr{X}$,
together with the symmetry, that

$\mathbb{E}[Z_{\mathscr{X}}]=\sum_{S\subset[n],\,|S|=|\mathcal{U}|}\mathbb{E}[Z_{\mathscr{X}}|\mathscr{X}=S]\,\mathbb{P}(\mathscr{X}=S)=\mathbb{E}[Z_{\mathcal{U}}],$

and hence, for large $n$,

(4.7) $\mathbb{E}[Z_{\mathcal{U}}]\geq 2^{-(m+3)}\mathbb{E}[Z].$

For $\alpha,\alpha^{\prime}\in\mathcal{H}_{m+2}$, the FKG inequality ensures
that Cov$(I_{\alpha}^{(m+2)},I_{\alpha^{\prime}}^{(m+2)})\geq 0$, which in
turn implies that

$\text{Var}(R_{\mathcal{U}})\leq\text{Var}(Z)=\tau^{2}\text{Var}(X_{m})\leq\tau^{2}\Lambda(X_{m}).$

From (4.7) we see that

$r-\mathbb{E}[R_{\mathcal{U}}]\geq\varepsilon\mathbb{E}[Z]=\tau\varepsilon\mu(X_{m}).$

Finally, the one-sided Chebyshev’s inequality, which is given in [8, Claim
16], with $\nu=\tau^{2}\Lambda(X_{m})$ and $t=\tau\varepsilon\mu(X_{m})$,
completes the proof of (4.5).

We now turn our attention to (4.6). We start by introducing additional
notation: for $\alpha_{1},\alpha_{2}\in\mathcal{H}_{m+2}$ and
$\beta_{1},\beta_{2}\in\mathcal{H}_{i+1}$ denote

$K_{\alpha_{1}\setminus\beta_{1}}^{(m+2,i+1)}:={\mathbbm{1}}\big{\\{}\text{all
strict subwords of $\alpha_{1}$ that are not subwords of $\beta_{1}$ are in}\
\Gamma_{\bf p}\big{\\}},$

and

$\displaystyle
K_{(\alpha_{1}\cup\alpha_{2})\setminus(\beta_{1}\cup\beta_{2})}^{(m+2,i+1)}:={\mathbbm{1}}\Big{\\{}\text{all
strict subwords of $\alpha_{1}$ or $\alpha_{2}$}$ $\displaystyle\hskip
144.54pt\text{ that are not subwords of either $\beta_{1}$ or $\beta_{2}$ are
in}\ \Gamma_{\bf p}\Big{\\}}.$

We define two events, $\mathcal{E}$ and $\mathcal{D}$, as follows. Let
$\mathcal{E}:=\big{\\{}Y_{i}\leq(1-2^{m+5}\varepsilon)\mathbb{E}[Y_{i}]\big{\\}}$.
If we denote

$V_{\mathcal{U}}^{+}:=\sum_{(\beta_{1},\beta_{2})\in\mathcal{H}_{i+1}(\mathcal{U})\times\mathcal{H}_{i+1}(\mathcal{U})}J_{\beta_{1}}^{(i+1)}J_{\beta_{2}}^{(i+1)}\sum_{\begin{subarray}{c}(\alpha_{1},\alpha_{2})\in\mathcal{K}(\mathcal{U},\beta_{1})\times\mathcal{K}(\mathcal{U},\beta_{2}),\\\
|\alpha_{1}\cap\alpha_{2}|\geq
q+1\end{subarray}}\mathbb{E}[K_{(\alpha_{1}\cup\alpha_{2})\setminus(\beta_{1}\cup\beta_{2})}^{(m+2,i+1)}],$

then we set $\mathcal{D}:=\big{\\{}V_{\mathcal{U}}^{+}\leq
2\mathbb{E}[V_{\mathcal{U}}^{+}]\big{\\}}$. Appealing to the FKG inequality,
as well as Markov’s inequality,

$\mathbb{P}(\mathcal{D}\cap\mathcal{E})\geq\mathbb{P}(\mathcal{D})\mathbb{P}(\mathcal{E})\geq\frac{1}{2}\,\mathbb{P}\big{(}Y_{i}\leq(1-2^{m+5}\varepsilon)\mathbb{E}[Y_{i}]\big{)}.$

Let $\Gamma_{\bf p}^{(i+1)}(\mathcal{U})$ be the subset of $\Gamma_{\bf p}$
consisting of words on $\mathcal{U}$ with length at most $i+1$. Since
$\mathcal{D}$ and $\mathcal{E}$ are both measurable with respect to
$\Gamma_{\bf p}^{(i+1)}(\mathcal{U})$,

(4.8) $\mathbb{P}(Z_{\mathcal{U}}\leq
z)\geq\mathbb{E}\Big{[}\mathbb{P}\big{(}Z_{\mathcal{U}}\leq z\,|\,\Gamma_{\bf
p}^{(i+1)}(\mathcal{U})\big{)}{\mathbbm{1}}_{\mathcal{D}\cap\mathcal{E}}\Big{]}.$

Consider the conditional probability

$\mathbb{P}^{*}(\cdot):=\mathbb{P}\big{(}\cdot\,|\,\Gamma_{\bf
p}^{(i+1)}(\mathcal{U})\big{)},$

and the corresponding (conditional) mean and variance, $\mathbb{E}^{*}$ and
$\text{Var}^{*}$. We claim that, on the event $\mathcal{D}\cap\mathcal{E}$,

(4.9) $\displaystyle z-\mathbb{E}^{*}[Z_{\mathcal{U}}]$ $\displaystyle\geq
2\tau\varepsilon\mu(X_{m}),$ (4.10)
$\displaystyle\text{Var}^{*}(Z_{\mathcal{U}})$ $\displaystyle\leq
2\tau^{2}\Lambda(X_{m}).$

Once (4.9) and (4.10) have been established, the one-sided Chebyshev’s
inequality says that, on the event $\mathcal{D}\cap\mathcal{E}$,

$\mathbb{P}^{*}(Z_{\mathcal{U}}\geq
z)\leq\mathbb{P}^{*}\big{(}Z_{\mathcal{U}}-\mathbb{E}^{*}[Z_{\mathcal{U}}]\geq
2\tau\varepsilon\mu(X_{m})\big{)}\leq\frac{\Lambda(X_{m})}{\Lambda(X_{m})+2(\varepsilon\mu(X_{m}))^{2}}.$

Substituting this inequality into (4.8), gives us (4.6) as required.

For the proof of (4.9), note that for $\beta\in\mathcal{H}_{i+1}(\mathcal{U})$
and $\alpha\in\mathcal{K}(\mathcal{U},\beta)$, $J_{\beta}^{(i+1)}$ is
measurable with respect to $\Gamma_{\bf p}^{(i+1)}(\mathcal{U})$, whereas
$K_{\alpha\setminus\beta}^{(m+2,i+1)}$ is independent of $\Gamma_{\bf
p}^{(i+1)}(\mathcal{U})$. Therefore,

$\mathbb{E}^{*}[Z_{\mathcal{U}}]=\sum_{\beta\in\mathcal{H}_{i+1}(\mathcal{U})}J_{\beta}^{(i+1)}\mathbb{E}\Big{[}\sum_{\alpha\in\mathcal{K}(\mathcal{U},\beta)}K_{\alpha\setminus\beta}^{(m+2,i+1)}\Big{]}.$

In the last expression, by symmetry,
$W_{\beta}:=\sum_{\alpha\in\mathcal{K}(\mathcal{U},\beta)}K_{\alpha\setminus\beta}^{(m+2,i+1)}$
has expectation independent of the choice of
$\beta\in\mathcal{H}_{i+1}(\mathcal{U})$. It follows that for any fixed
$\tilde{\beta}\in\mathcal{H}_{i+1}(\mathcal{U})$, we have
$\mathbb{E}^{*}[Z_{\mathcal{U}}]=Y_{i}\mathbb{E}[W_{\tilde{\beta}}]$, and,
hence,
$\mathbb{E}[Z_{\mathcal{U}}]=\mathbb{E}[Y_{i}]\mathbb{E}[W_{\tilde{\beta}}]$.
Now on $\mathcal{E}$ we have
$\mathbb{E}^{*}[Z_{\mathcal{U}}]\leq(1-2^{m+5}\varepsilon)\mathbb{E}[Z_{\mathcal{U}}]$
and by (4.7),

$z-\mathbb{E}^{*}[Z_{\mathcal{U}}]\geq
z-(1-2^{m+5}\varepsilon)\mathbb{E}[Z_{\mathcal{U}}]\geq
2\tau\varepsilon\mu(X_{m}),$

proving that (4.9) holds on $\mathcal{E}\supset\mathcal{D}\cap\mathcal{E}$.
Furthermore, arguments similar to those used for Equ. (67) and (68) in [8],
show that (4.10) holds on $\mathcal{D}\supset\mathcal{D}\cap\mathcal{E}$. ∎

Lemma 4.3 below calculates variance asymptotics for $S_{j}$, $T_{j,r}$, and
$V_{m+2}$, which are respectively defined at (3.4), (3.2), and (3.42).
Similarly, Lemma 4.4 gives variance asymptotics for $S_{m+2}^{\uparrow}$,
$S_{j}^{\uparrow}$, and $T_{m+2}^{\downarrow}$ (see (3.28), (3.35), and
(3.29), respectively). Since Lemmas 4.3 and 4.4 can be proven in the same way,
we give only a proof of Lemma 4.3.

######  Lemma 4.3.

As $n\to\infty$,

(4.11)
$\displaystyle\text{Var}(S_{m+2})\sim\text{Var}(V_{m+2})\sim\begin{cases}\frac{n^{2({\tau_{m+1}+\alpha_{m+1}})-\tau_{q}}}{(q+1)!((m-q+1)!)^{2}}&\text{
if }{\tau_{m+1}+\alpha_{m+1}}>\tau_{q},\\\\[7.0pt]
\frac{n^{\tau_{m+1}+\alpha_{m+1}}}{(m+2)!}&\text{ if
}{\tau_{m+1}+\alpha_{m+1}}<\tau_{q},\\\\[7.0pt]
\Big{\\{}\frac{1}{(q+1)!((m-q+1)!)^{2}}+\frac{1}{(m+2)!}\Big{\\}}n^{\tau_{m+1}+\alpha_{m+1}}&\text{
if }{\tau_{m+1}+\alpha_{m+1}}=\tau_{q}.\end{cases}$

Further,

(4.12) $\text{Var}(T_{m+2})\leq
C^{*}\big{(}n^{2({\tau_{m+1}+\alpha_{m+1}})-\tau_{q}}\vee
n^{\tau_{m+1}+\alpha_{m+1}}\big{)}.$

For $j\geq m+3$,

(4.13) $\displaystyle\text{Var}(S_{j})\leq
C_{j}\big{(}n^{2({\tau_{m+1}+\alpha_{m+1}})-\tau_{q}}\vee
n^{\tau_{m+1}+\alpha_{m+1}}\big{)}$

for a $j$-dependent $C_{j}>0$. For $j\geq m+3$ and $r\geq 1$,

(4.14) $\text{Var}(T_{j,r})\leq
C_{j}\big{(}n^{2({\tau_{m+1}+\alpha_{m+1}})-\tau_{q}}\vee
n^{\tau_{m+1}+\alpha_{m+1}}\big{)}\,n^{1-\psi_{m}}.$

Finally,

(4.15) $\text{Var}(V_{m+2}-T_{m+2})\leq
C^{*}\big{(}n^{2({\tau_{m+1}+\alpha_{m+1}})-\tau_{q}}\vee
n^{\tau_{m+1}+\alpha_{m+1}}\big{)}\,n^{(1\vee\psi_{m-1})-\psi_{m}}.$

######  Lemma 4.4.

For $j\geq m+2$,

(4.16) $\displaystyle\text{Var}(S_{j}^{\uparrow})$ $\displaystyle\leq
C_{j}\big{(}v_{N+1}^{2({\tau_{m+1}+\alpha_{m+1}})-\tau_{q}}\vee
v_{N+1}^{\tau_{m+1}+\alpha_{m+1}}\big{)},$

and

(4.17) $\displaystyle\text{Var}(T_{m+2}^{\downarrow})$ $\displaystyle\leq
C^{*}\big{(}v_{N}^{2({\tau_{m+1}+\alpha_{m+1}})-\tau_{q}}\vee
v_{N}^{\tau_{m+1}+\alpha_{m+1}}\big{)}.$

###### Proof of Lemma 4.3.

We begin with the proof of (4.11).
Proof of (4.11): For ease of notation, we drop the superscript from
$\xi_{\sigma}^{(m+2,m)}$ and write that

(4.18)
$\displaystyle\begin{split}\text{Var}(S_{m+2})&=\sum_{\begin{subarray}{c}\sigma\subset[n],\\\
|\sigma|=m+2\end{subarray}}\sum_{\begin{subarray}{c}\tau\subset[n],\\\
|\tau|=m+2\end{subarray}}\big{\\{}\mathbb{E}[\xi_{\sigma}\xi_{\tau}]-\mathbb{E}[\xi_{\sigma}]\mathbb{E}[\xi_{\tau}]\big{\\}}\\\
&=\sum_{\ell=0}^{m+2}\sum_{\begin{subarray}{c}\sigma\subset[n],\\\
|\sigma|=m+2\end{subarray}}\sum_{\begin{subarray}{c}\tau\subset[n],\\\
|\tau|=m+2\end{subarray}}\big{\\{}\mathbb{E}[\xi_{\sigma}\xi_{\tau}]-\mathbb{E}[\xi_{\sigma}]\mathbb{E}[\xi_{\tau}]\big{\\}}\,{\mathbbm{1}}\\{|\sigma\cap\tau|=\ell\\}\\\
&=\sum_{\ell=0}^{m+2}\binom{n}{m+2}\binom{m+2}{\ell}\binom{n-m-2}{m+2-\ell}\\\
&\hskip
72.26999pt\big{\\{}\mathbb{E}[\xi_{\sigma_{m+2}}\xi_{\tau_{m+2}}]-\mathbb{E}[\xi_{\sigma_{m+2}}]\mathbb{E}[\xi_{\tau_{m+2}}]\big{\\}}\,{\mathbbm{1}}\\{|\sigma_{m+2}\cap\tau_{m+2}|=\ell\\}.\end{split}$

For every $\sigma\subset[n]$ with $|\sigma|=m+2$,

(4.19)
$\mathbb{E}[\xi_{\sigma}]=\prod_{i=q}^{m}p_{i}^{\binom{m+2}{i+1}}=n^{-\sum_{i=q}^{m}\binom{m+2}{i+1}\alpha_{i}}.$

Let $\sigma,\tau\subset[n]$ with $|\sigma|=|\tau|=m+2$. If
$\ell=|\sigma\cap\tau|\in\\{0,\dots,q\\}$, then $\xi_{\sigma}$ and
$\xi_{\tau}$ are independent, because all the $(\ell-1)$-faces exist in
$X([n],{\bf p})$ with probability $1$. Therefore,
$\mathbb{E}[\xi_{\sigma}\xi_{\tau}]=\mathbb{E}[\xi_{\sigma}]\mathbb{E}[\xi_{\tau}]$.
Next, if $\ell\in\\{q+1,\dots,m+1\\}$, then $\sigma\cap\tau$ contains
$\binom{\ell}{i+1}$ faces of dimension $i$ for each $q\leq i\leq\ell-1$;
hence,

(4.20)
$\mathbb{E}[\xi_{\sigma}\xi_{\tau}]=\prod_{i=q}^{m}p_{i}^{\binom{m+2}{i+1}}\times\prod_{i=q}^{m}p_{i}^{\binom{m+2}{i+1}-\binom{\ell}{i+1}}=n^{-2\sum_{i=q}^{m}\binom{m+2}{i+1}\alpha_{i}+\sum_{i=q}^{\ell-1}\binom{\ell}{i+1}\alpha_{i}}.$

In this case, by (4.19) and (4.20), as $n\to\infty$,

$\displaystyle\binom{n}{m+2}\binom{m+2}{\ell}\binom{n-m-2}{m+2-\ell}\big{\\{}\mathbb{E}[\xi_{\sigma}\xi_{\tau}]-\mathbb{E}[\xi_{\sigma}]\mathbb{E}[\xi_{\tau}]\big{\\}}$
$\displaystyle\sim\frac{n^{2(m+2)-\ell}}{\ell!\big{(}(m+2-\ell)!\big{)}^{2}}\,\Big{(}n^{-2\sum_{i=q}^{m}\binom{m+2}{i+1}\alpha_{i}+\sum_{i=q}^{\ell-1}\binom{\ell}{i+1}\alpha_{i}}-n^{-2\sum_{i=q}^{m}\binom{m+2}{i+1}\alpha_{i}}\Big{)}$
$\displaystyle\sim\frac{n^{2({\tau_{m+1}+\alpha_{m+1}})-\tau_{\ell-1}}}{\ell!\big{(}(m+2-\ell)!\big{)}^{2}}.$

Finally, when $\ell=m+2$, we have as $n\to\infty$,

$\displaystyle\binom{n}{m+2}\binom{m+2}{m+2}\binom{n-m-2}{0}\big{\\{}\mathbb{E}[\xi_{\sigma}\xi_{\tau}]-\mathbb{E}[\xi_{\sigma}]\mathbb{E}[\xi_{\tau}]\big{\\}}$
$\displaystyle\sim\frac{n^{m+2}}{(m+2)!}\,\big{\\{}\mathbb{E}[\xi_{\sigma}]-\big{(}\mathbb{E}[\xi_{\sigma}]\big{)}^{2}\big{\\}}\sim\frac{n^{\tau_{m+1}+\alpha_{m+1}}}{(m+2)!}.$

Substituting these asymptotic results back into (4.18),

$\displaystyle\text{Var}(S_{m+2})$
$\displaystyle\sim\sum_{\ell=q+1}^{m+1}\frac{n^{2({\tau_{m+1}+\alpha_{m+1}})-\tau_{\ell-1}}}{\ell!\big{(}(m+2-\ell)!\big{)}^{2}}+\frac{n^{\tau_{m+1}+\alpha_{m+1}}}{(m+2)!}$
$\displaystyle\sim\left(\frac{n^{2({\tau_{m+1}+\alpha_{m+1}})-\tau_{q}}}{(q+1)!\big{(}(m-q+1)!\big{)}^{2}}\vee\frac{n^{2({\tau_{m+1}+\alpha_{m+1}})-\tau_{m}}}{(m+1)!}\right)+\frac{n^{\tau_{m+1}+\alpha_{m+1}}}{(m+2)!},\
\ \ n\to\infty.$

Since $\tau_{m}>{\tau_{m+1}+\alpha_{m+1}}$ (see (4.4)), this is the
asymptotics claimed for Var$(S_{m+2})$ in (4.11). One can derive variance
asymptotics for $V_{m+2}$ in a nearly identical manner.

Proof of (4.13): Removing again the superscript from $\xi_{\sigma}^{(j,m)}$,
we have

$\text{Var}(S_{j})=\sum_{\ell=0}^{j}\binom{n}{j}\binom{j}{\ell}\binom{n-j}{j-\ell}\big{\\{}\mathbb{E}[\xi_{\sigma_{j}}\xi_{\tau_{j}}]-\mathbb{E}[\xi_{\sigma_{j}}]\mathbb{E}[\xi_{\tau_{j}}]\big{\\}}\,{\mathbbm{1}}\\{|\sigma_{j}\cap\tau_{j}|=\ell\\}.$

Let $\sigma,\tau\subset[n]$ with $|\sigma|=|\tau|=j$. As before, if
$\ell=|\sigma\cap\tau|\in\\{0,\dots,q\\}$, then
$\mathbb{E}[\xi_{\sigma}\xi_{\tau}]=\mathbb{E}[\xi_{\sigma}]\mathbb{E}[\xi_{\tau}]$.
Next, if $\ell\in\\{q+1,\dots,m+1\\}$, then

(4.21)
$\displaystyle\begin{split}\binom{n}{j}\binom{j}{\ell}\binom{n-j}{j-\ell}\big{\\{}\mathbb{E}[\xi_{\sigma}\xi_{\tau}]-\mathbb{E}[\xi_{\sigma}]\mathbb{E}[\xi_{\tau}]\big{\\}}&\leq
n^{2j-\ell}\mathbb{E}[\xi_{\sigma}\xi_{\tau}]\\\
&=n^{2j-\ell}\sum_{K:|K|=j}\sum_{K^{\prime}:|K^{\prime}|=j}\mathbb{P}\big{(}A_{\ell}(K,K^{\prime})\big{)},\end{split}$

where the sum is taken over all $m$-strongly connected complexes on $j$
vertices, with at least one trivial or non-trivial $m$-cycle, up to an
isomorphism class, and

(4.22) $\displaystyle A_{\ell}(K,K^{\prime}):=\big{\\{}$ $\displaystyle
X(\sigma,{\bf p})\ \text{contains a subcomplex isomorphic to $K$,}$
$\displaystyle X(\tau,{\bf p})\ \text{contains a subcomplex isomorphic to
$K^{\prime}$}\big{\\}}.$

In order to bound the sum in (4.21), observe that
$\mathbb{P}\big{(}A_{\ell}(K,K^{\prime})\big{)}$ is maximized when all of the
$\ell$ common points of $\sigma$ and $\tau$ are taken from a single
$(\ell-1)$-face. Then, according to Lemma 3.3,

$\displaystyle\mathbb{P}\big{(}A_{\ell}(K,K^{\prime})\big{)}$
$\displaystyle\leq
C_{j}\Big{\\{}\prod_{i=q}^{m}p_{i}^{\binom{m+2}{i+1}}\Big{(}\prod_{i=q}^{m}p_{i}^{\binom{m}{i}}\Big{)}^{j-m-2}\Big{\\}}^{2}\prod_{i=q}^{\ell-1}p_{i}^{-\binom{\ell}{i+1}}$
$\displaystyle=C_{j}n^{-2\sum_{i=q}^{m}\binom{m+2}{i+1}\alpha_{i}+\sum_{i=q}^{\ell-1}\binom{\ell}{i+1}\alpha_{i}}(n^{-\psi_{m}})^{2(j-m-2)}.$

Here, the factor $\prod_{i=q}^{\ell-1}p_{i}^{-\binom{\ell}{i+1}}$ is added due
to the double count of the common $(\ell-1)$-face. Substituting this bound in
(4.21),

$n^{2j-\ell}\sum_{K:|K|=j}\sum_{K^{\prime}:|K^{\prime}|=j}\mathbb{P}\big{(}A_{\ell}(K,K^{\prime})\big{)}\leq
C_{j}n^{2({\tau_{m+1}+\alpha_{m+1}})-\tau_{\ell-1}}(n^{1-\psi_{m}})^{2(j-m-2)}.$

We next consider the case $\ell\in\\{m+2,\dots,j-1\\}$. Once again, we bound
the final expression in (4.21). In this case, the maximum of
$\mathbb{P}\big{(}A_{\ell}(K,K^{\prime})\big{)}$ is attained when $\sigma$ and
$\tau$ have $\ell-m$ faces of dimension $m$ in common, such that these
$m$-faces are strongly connected, in a way that each of the $m$-faces shares
$m$ points with at least one of the remaining $m$-faces. Taking this into
consideration, we conclude that

(4.23)
$\displaystyle\begin{split}\mathbb{P}\big{(}A_{\ell}(K,K^{\prime})\big{)}&\leq
C_{j}\Big{\\{}\prod_{i=q}^{m}p_{i}^{\binom{m+2}{i+1}}\Big{(}\prod_{i=q}^{m}p_{i}^{\binom{m}{i}}\Big{)}^{j-m-2}\Big{\\}}^{2}\prod_{i=q}^{m}p_{i}^{-\binom{m+1}{i+1}}\Big{(}\prod_{i=q}^{m}p_{i}^{-\binom{m}{i}}\Big{)}^{\ell-m-1}\\\
&=C_{j}n^{-2\sum_{i=q}^{m}\binom{m+2}{i+1}\alpha_{i}+\sum_{i=q}^{m}\binom{m+1}{i+1}\alpha_{i}}(n^{-\psi_{m}})^{2j-m-\ell-3}.\end{split}$

We refer the reader to the proof of [16, Theorem 4.2] for a more detailed
derivation of (4.23). Using this estimate, we see that

$n^{2j-\ell}\sum_{K:|K|=j}\sum_{K^{\prime}:|K^{\prime}|=j}\mathbb{P}\big{(}A_{\ell}(K,K^{\prime})\big{)}\leq
C_{j}n^{2({\tau_{m+1}+\alpha_{m+1}})-\tau_{m}}(n^{1-\psi_{m}})^{2j-m-\ell-3}.$

Finally, for $\ell=j$, a direct application of Lemma 3.3 gives us

$\displaystyle\binom{n}{j}\binom{j}{j}\binom{n-j}{0}\big{\\{}\mathbb{E}[\xi_{\sigma}]-\big{(}\mathbb{E}[\xi_{\sigma}]\big{)}^{2}\big{\\}}$
$\displaystyle\leq$ $\displaystyle
n^{j}\sum_{K:|K|=j}\mathbb{P}\big{(}X(\sigma,{\bf p})\ \text{contains a
subcomplex isomorphic to}\ K\big{)}$ $\displaystyle\leq$ $\displaystyle
C_{j}n^{j}\prod_{i=q}^{m}p_{i}^{\binom{m+2}{i+1}}\Big{(}\prod_{i=q}^{m}p_{i}^{\binom{m}{i}}\Big{)}^{j-m-2}=C_{j}n^{\tau_{m+1}+\alpha_{m+1}}(n^{1-\psi_{m}})^{j-m-2}.$

Putting all of these results together, we obtain

$\displaystyle\text{Var}(S_{j})$ $\displaystyle\leq
C_{j}\Big{\\{}\bigvee_{\ell=q+1}^{m+1}n^{2({\tau_{m+1}+\alpha_{m+1}})-\tau_{\ell-1}}(n^{1-\psi_{m}})^{2(j-m-2)}$
$\displaystyle\qquad\qquad\vee\bigvee_{\ell=m+2}^{j-1}n^{2({\tau_{m+1}+\alpha_{m+1}})-\tau_{m}}(n^{1-\psi_{m}})^{2j-m-\ell-3}\vee
n^{{\tau_{m+1}+\alpha_{m+1}}}(n^{1-\psi_{m}})^{j-m-2}\Big{\\}}$
$\displaystyle\leq C_{j}\big{\\{}n^{2({\tau_{m+1}+\alpha_{m+1}})-\tau_{q}}\vee
n^{\tau_{m+1}+\alpha_{m+1}}\big{\\}}.$

The last inequality follows from (4.4) and $\psi_{m}>1$.

Proof of (4.12) and (4.14): For $j\geq m+2$ and $r\geq 1$, dropping the
superscript from $\eta_{\sigma}^{(j,r,m)}$ as before, we have that

$\text{Var}(T_{j,r})=\sum_{\ell=0}^{j}\binom{n}{j}\binom{j}{\ell}\binom{n-j}{j-\ell}\big{\\{}\mathbb{E}[\eta_{\sigma_{j}}\eta_{\tau_{j}}]-\mathbb{E}[\eta_{\sigma_{j}}]\mathbb{E}[\eta_{\tau_{j}}]\big{\\}}\,{\mathbbm{1}}\\{|\sigma_{j}\cap\tau_{j}|=\ell\\}=:\sum_{\ell=0}^{j}\mathcal{T}_{\ell,j}.$

The following estimates were established in the proof of [16, Theorem 4.2]:

(4.24) $\mathcal{T}_{\ell,j}=\begin{cases}0&\text{ if
}\ell\in\\{0,\dots,q-2\\},\\\\[3.0pt]
\mathcal{O}\big{(}n^{2({\tau_{m+1}+\alpha_{m+1}})-\tau_{q}}(n^{1-\psi_{m}})^{2(j-m)-2}\big{)}&\text{
if }\ell=q-1,\\\\[3.0pt]
\mathcal{O}\big{(}n^{2({\tau_{m+1}+\alpha_{m+1}})-\tau_{q}}(n^{1-\psi_{m}})^{2(j-m)-3}\big{)}&\text{
if }\ell=q,\\\\[3.0pt]
\mathcal{O}\big{(}n^{2({\tau_{m+1}+\alpha_{m+1}})-\tau_{\ell-1}}(n^{1-\psi_{m}})^{2(j-m)-4}\big{)}&\text{
if }\ell\in\\{q+1,\dots,m+1\\},\\\\[3.0pt]
\mathcal{O}\big{(}n^{2({\tau_{m+1}+\alpha_{m+1}})-\tau_{m}}(n^{1-\psi_{m}})^{2j-m-\ell-3}\big{)}&\text{
if }\ell\in\\{m+2,\dots,j-1\\},\\\\[3.0pt]
\mathcal{O}\big{(}n^{{\tau_{m+1}+\alpha_{m+1}}}(n^{1-\psi_{m}})^{j-m-2}\big{)}&\text{
if }\ell=j,\\\\[3.0pt] \end{cases}$

where $\beta_{n}=\mathcal{O}(\gamma_{n})$ means that there exists a
$j$-dependent $C_{j}>0$ such that $\beta_{n}/\gamma_{n}\leq C_{j}$ for all
$n\geq 1$. Recalling (4.4), it follows from (4.24) that
$\text{Var}(T_{j,r})\leq C_{j}(a_{n}\vee b_{n})$, where

$\displaystyle a_{n}$
$\displaystyle:=\bigvee_{\ell=q+1}^{m+1}n^{2({\tau_{m+1}+\alpha_{m+1}})-\tau_{\ell-1}}(n^{1-\psi_{m}})^{2(j-m)-4},$
$\displaystyle b_{n}$
$\displaystyle:=n^{\tau_{m+1}+\alpha_{m+1}}(n^{1-\psi_{m}})^{j-m-2}.$

If $j=m+2$, then by (4.4), $a_{n}\vee
b_{n}=n^{2({\tau_{m+1}+\alpha_{m+1}})-\tau_{q}}\vee
n^{\tau_{m+1}+\alpha_{m+1}}$, proving (4.12). If $j\geq m+3$ and $r\geq 1$,
then

$\displaystyle a_{n}$
$\displaystyle\leq\big{\\{}n^{2({\tau_{m+1}+\alpha_{m+1}})-\tau_{q}}\vee
n^{2({\tau_{m+1}+\alpha_{m+1}})-\tau_{m}}\big{\\}}\,n^{1-\psi_{m}},$
$\displaystyle b_{n}$ $\displaystyle\leq
n^{{\tau_{m+1}+\alpha_{m+1}}+1-\psi_{m}},$

so that

$a_{n}\vee b_{n}\leq\big{\\{}n^{2({\tau_{m+1}+\alpha_{m+1}})-\tau_{q}}\vee
n^{\tau_{m+1}+\alpha_{m+1}}\big{\\}}\,n^{1-\psi_{m}},$

and (4.14) follows.

Proof of (4.15): We use the same style of notation as in (4.18). We have

$\displaystyle\text{Var}(V_{m+2}-T_{m+2})$
$\displaystyle=\sum_{\ell=0}^{m+2}\binom{n}{m+2}\binom{m+2}{\ell}\binom{n-m-2}{m+2-\ell}$
$\displaystyle\times\Big{\\{}\mathbb{E}\big{[}(\lambda_{\sigma_{m+2}}-\eta_{\sigma_{m+2}})(\lambda_{\tau_{m+2}}-\eta_{\tau_{m+2}})\big{]}-\mathbb{E}[\lambda_{\sigma_{m+2}}-\eta_{\sigma_{m+2}}]\mathbb{E}[\lambda_{\tau_{m+2}}-\eta_{\tau_{m+2}}]\Big{\\}}$
$\displaystyle\hskip
252.94499pt\times{\mathbbm{1}}\\{|\sigma_{m+2}\cap\tau_{m+2}|=\ell\\}$
$\displaystyle=\sum_{\ell=0}^{m+2}\binom{n}{m+2}\binom{m+2}{\ell}\binom{n-m-2}{m+2-\ell}(A_{n}-2B_{n}+C_{n}){\mathbbm{1}}\\{|\sigma_{m+2}\cap\tau_{m+2}|=\ell\\}$
$\displaystyle:=\sum_{\ell=0}^{m+2}\mathcal{U}_{\ell},$

where

$\displaystyle A_{n}$
$\displaystyle:=\mathbb{E}[\lambda_{\sigma_{m+2}}\lambda_{\tau_{m+2}}]-\mathbb{E}[\lambda_{\sigma_{m+2}}]\mathbb{E}[\lambda_{\tau_{m+2}}],\
\ \
B_{n}:=\mathbb{E}[\lambda_{\sigma_{m+2}}\eta_{\tau_{m+2}}]-\mathbb{E}[\lambda_{\sigma_{m+2}}]\mathbb{E}[\eta_{\tau_{m+2}}],$
$\displaystyle C_{n}$
$\displaystyle:=\mathbb{E}[\eta_{\sigma_{m+2}}\eta_{\tau_{m+2}}]-\mathbb{E}[\eta_{\sigma_{m+2}}]\mathbb{E}[\eta_{\tau_{m+2}}].$

Let $K$ be the non-trivial $m$-cycle on $m+2$ points. For $\sigma\subset[n]$
with $|\sigma|=m+2$, define

$A(K):=\big{\\{}X(\sigma,{\bf p})\text{ is isomorphic to }K\big{\\}},$

so that

(4.25)
$q_{K}:=\mathbb{P}(A(K))=\prod_{i=q}^{m}p_{i}^{\binom{m+2}{i+1}}(1-p_{m+1}).$

Let $\sigma,\tau\subset[n]$ with $|\sigma|=|\tau|=m+2$. We break our
discussion into four different cases, depending on the value of
$\ell=|\sigma\cap\tau|$. We continue using the notation (4.22).

Case I: $\ell\in\\{0,\dots,q-1\\}$.
Since all faces of dimension smaller than $q$ exist in $X([n],{\bf p})$ with
probability $1$, we have
$\mathbb{E}[\lambda_{\sigma}\lambda_{\tau}]=q_{K}^{2}=\mathbb{E}[\lambda_{\sigma}]\mathbb{E}[\lambda_{\tau}]$,
so that $A_{n}=0$. Moreover,

$\mathbb{E}[\lambda_{\sigma}\eta_{\tau}]=\mathbb{P}\big{(}A_{\ell}(K,K)\big{)}\mathbb{P}\big{(}\tau\text{
is maximal }|\,A_{\ell}(K,K)\big{)}.$

Here, it is clear that $\mathbb{P}\big{(}A_{\ell}(K,K)\big{)}=q_{K}^{2}$.
Furthermore, the event $\\{\tau\text{ is maximal}\\}$ is, clearly, independent
of the restriction of the complex to $\sigma$. It thus follows that $B_{n}=0$.
We conclude by (4.24) with $j=m+2$ that

(4.26) $\mathcal{U}_{\ell}=\mathcal{T}_{\ell,m+2}=\begin{cases}0&\text{ if
}\ell\in\\{0,\dots,q-2\\},\\\\[3.0pt]
\mathcal{O}\big{(}n^{2({\tau_{m+1}+\alpha_{m+1}})-\tau_{q}+2(1-\psi_{m})}\big{)}&\text{
if }\ell=q-1.\end{cases}$

Case II: $\ell=q$.
Since all the $(q-1)$-faces exist with probability $1$, we still have
$\mathbb{E}[\lambda_{\sigma}\lambda_{\tau}]=\mathbb{E}[\lambda_{\sigma}]\mathbb{E}[\lambda_{\tau}]$,
and, hence, $A_{n}=0$. Next,

$\mathbb{E}[\lambda_{\sigma}\eta_{\tau}]=q_{K}^{2}\mathbb{P}\big{(}\tau\text{
is maximal }|\,A_{q}(K,K)\big{)}\leq
q_{K}^{2}\mathbb{P}\Big{(}\bigcap_{v\in(\sigma\cap\tau)^{c}}B_{v}^{c}\,\Big{|}\,A_{q}(K,K)\Big{)},$

where $B_{v}$ is the event that $v$ forms an $m$-face with one of the
$(m-1)$-faces in $\tau$. Using independence and the argument in (3.33), we
have

$\displaystyle\mathbb{P}\Big{(}\bigcap_{v\in(\sigma\cap\tau)^{c}}B_{v}^{c}\,\Big{|}\,A_{q}(K,K)\Big{)}$
$\displaystyle=\prod_{v\in(\sigma\cap\tau)^{c}}\Big{(}1-\mathbb{P}\big{(}B_{v}\,|\,A_{q}(K,K)\big{)}\Big{)}$
$\displaystyle=\Big{(}1-\binom{m+2}{m}n^{-\psi_{m}}+u_{n}\Big{)}^{n-2(m+2)+q},$

where $u_{n}=o(n^{-\psi_{m}})$. Therefore,

$\displaystyle B_{n}$ $\displaystyle\leq
q_{K}^{2}\Big{(}1-\binom{m+2}{m}n^{-\psi_{m}}+u_{n}\Big{)}^{n-2(m+2)+q}-q_{K}^{2}\Big{(}1-\binom{m+2}{m}n^{-\psi_{m}}+u_{n}\Big{)}^{n-m-2}\leq
C^{*}q_{K}^{2}n^{-\psi_{m}},$

and, by (4.25),

(4.27)
$\displaystyle\begin{split}\binom{n}{m+2}\binom{m+2}{q}\binom{n-m-2}{m+2-q}B_{n}&\leq
C^{*}n^{2(m+2)-q}q_{K}^{2}n^{-\psi_{m}}\leq
C^{*}n^{2({\tau_{m+1}+\alpha_{m+1}})-\tau_{q}+1-\psi_{m}}.\end{split}$

Finally, by (4.24),

(4.28)
$\binom{n}{m+2}\binom{m+2}{q}\binom{n-m-2}{m+2-q}C_{n}=\mathcal{T}_{q,m+2}=\mathcal{O}\big{(}n^{2({\tau_{m+1}+\alpha_{m+1}})-\tau_{q}+1-\psi_{m}}\big{)}.$

Combining (4.27) and (4.28) gives us

(4.29) $\mathcal{U}_{q}\leq
C^{*}n^{2({\tau_{m+1}+\alpha_{m+1}})-\tau_{q}+1-\psi_{m}}.$

Case III: $\ell\in\\{q+1,\dots,m+1\\}$.
It follows from $\lambda_{\sigma}\geq\eta_{\sigma}$ and
$\lambda_{\tau}\geq\eta_{\tau}$ that

$\displaystyle\mathcal{U}_{\ell}$ $\displaystyle\leq
n^{2(m+2)-\ell}\mathbb{E}\big{[}(\lambda_{\tau}-\eta_{\tau})\lambda_{\sigma}\big{]}$
$\displaystyle=n^{2(m+2)-\ell}\mathbb{P}\big{(}A_{\ell}(K,K)\big{)}\mathbb{P}\big{(}\tau\text{
is not maximal }\big{|}\,A_{\ell}(K,K)\big{)}.$

Since there are $\ell$ common points between $\sigma$ and $\tau$,

$\displaystyle\mathbb{P}\big{(}A_{\ell}(K,K)\big{)}$
$\displaystyle=\prod_{i=q}^{m}p_{i}^{\binom{m+2}{i+1}}(1-p_{m+1})\times\prod_{i=q}^{m}p_{i}^{\binom{m+2}{i+1}-\binom{\ell}{i+1}}(1-p_{m+1})\leq\prod_{i=q}^{m}p_{i}^{2\binom{m+2}{i+1}-\binom{\ell}{i+1}}.$

Suppose first that $\ell\in\\{q+1,\dots,m-1\\}$. By the union bound, the
probability that $\tau$ is not maximal, conditioned on $A_{\ell}(K,K)$, is
bounded by $\binom{m+2}{m}$ times the conditional probability that there
exists at least one $m$-face between a fixed $(m-1)$-face in $\tau$ and any
points outside of $\tau$. The conditional probability of forming an $m$-face
from an $(m-1)$-face in $\tau$ and a point in $[n]\setminus(\tau\cup\sigma)$
is given by $\prod_{i=q}^{m}p_{i}^{\binom{m}{i}}=n^{-\psi_{m}}$, and there are
$n-2(m+2)+\ell$ such points. On the other hand, if the candidate point is in
$\sigma\setminus\tau$, it suffices to consider the $(m-1)$-face in $\tau$
containing all $\ell$ common points between $\sigma$ and $\tau$. Then the
above conditional probability of forming an $m$-face is

$\prod_{i=q}^{m}p_{i}^{\binom{m}{i}}\left(\prod_{i=q}^{\ell}p_{i}^{\binom{\ell}{i}}\right)^{-1}=n^{\psi_{\ell}-\psi_{m}},$

which is further bounded by $n^{(1\vee\psi_{m-1})-\psi_{m}}$.

On the other hand, if $\ell\in\\{m,m+1\\}$, then $\mathbb{P}\big{(}\tau\text{
is not maximal }\big{|}\,A_{\ell}(K,K)\big{)}=1$. We thus conclude that

(4.30) $\mathcal{U}_{\ell}\leq
C^{*}n^{2(m+2)-\ell}\prod_{i=q}^{m}p_{i}^{2\binom{m+2}{i+1}-\binom{\ell}{i+1}}n^{(1\vee\psi_{m-1})-\psi_{m}}=C^{*}n^{2({\tau_{m+1}+\alpha_{m+1}})-\tau_{\ell-1}+(1\vee\psi_{m-1})-\psi_{m}}$

for $\ell\in\\{q+1,\ldots,m-1\\}$ and

(4.31) $\mathcal{U}_{\ell}\leq
C^{*}n^{2({\tau_{m+1}+\alpha_{m+1}})-\tau_{\ell-1}}\leq
C^{*}n^{2({\tau_{m+1}+\alpha_{m+1}})-\tau_{m}}$

for $\ell\in\\{m,m+1\\}$.

Case IV: $\ell=m+2$.
In this case $\sigma=\tau$ and, as above,

(4.32) $\displaystyle\begin{split}\mathcal{U}_{m+2}&\leq
n^{m+2}\mathbb{E}[\lambda_{\tau}-\eta_{\tau}]=n^{m+2}q_{K}\mathbb{P}(\tau\text{
is not maximal }|\,A(K))\\\ &\leq
C^{*}n^{m+2}\prod_{i=q}^{m}p_{i}^{\binom{m+2}{i+1}}n^{1-\psi_{m}}=C^{*}n^{{\tau_{m+1}+\alpha_{m+1}}+1-\psi_{m}}.\end{split}$

Combining all the results in (4.26), (4.29), (4.30), (4.31) and (4.32), we
conclude that

$\displaystyle\text{Var}(V_{m+2}-T_{m+2})\leq
C^{*}\Big{(}n^{2({\tau_{m+1}+\alpha_{m+1}})-\tau_{q}+2(1-\psi_{m})}\vee
n^{2({\tau_{m+1}+\alpha_{m+1}})-\tau_{q}+1-\psi_{m}}$ $\displaystyle\hskip
72.26999pt\vee\bigvee_{\ell=q+1}^{m-1}n^{2({\tau_{m+1}+\alpha_{m+1}})-\tau_{\ell-1}+(1\vee\psi_{m-1})-\psi_{m}}\vee
n^{2({\tau_{m+1}+\alpha_{m+1}})-\tau_{m}}\vee
n^{{\tau_{m+1}+\alpha_{m+1}}+1-\psi_{m}}\Big{)}$ $\displaystyle\hskip
72.26999pt\leq C^{*}\big{(}n^{2({\tau_{m+1}+\alpha_{m+1}})-\tau_{q}}\vee
n^{\tau_{m+1}+\alpha_{m+1}}\big{)}\,n^{(1\vee\psi_{m-1})-\psi_{m}}.$

∎

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