# Asymptotic Betti numbers of random subcomplexes

Nermin Salepci, Jean-Yves Welschinger

► To cite this version:

Nermin Salepci, Jean-Yves Welschinger. Asymptotic Betti numbers of random subcomplexes. Topology and its Applications, 2025, 373, pp.109476. 10.1016/j.topol.2025.109476. hal-05347398

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Distributed under a Creative Commons Attribution 4.0 International License
ASYMPTOTIC BETTI NUMBERS OF RANDOM
SUBCOMPLEXES

NERMİN SALEPCİ AND JEAN-YVES WELSCHINGER

ABSTRACT. We prove that the normalized expected Betti numbers of a random subcomplex in the $d$-th barycentric subdivision of a finite simplicial complex converge to universal limits as $d$ grows to $+\infty$.

Keywords : Simplicial complex, Barycentric subdivision, Betti numbers.

MATHEMATICS SUBJECT CLASSIFICATION 2010: 55U10, 60C05

# 1. INTRODUCTION

Let $K$ be a finite $n$-dimensional simplicial complex, $n > 0$, and $\mathrm{Sd}(K)$ be its first barycentric subdivision. Inspired by real toric geometry [21, 22], we introduced [16] a construction of codimension $k$ subcomplexes $V_\epsilon$ of $\mathrm{Sd}(K)$ parametrized by the $(k-1)$-dimensional simplicial cochains $\epsilon \in C^{k-1}(K; \mathbb{Z}/2\mathbb{Z})$ for every $k \in \{1, \dots, n\}$, see Definition 2.1. The finite set $C^{k-1}(K; \mathbb{Z}/2\mathbb{Z})$ being canonically a probability space, or rather equipped with a family of probability measures $(\mu_\nu)_{\nu \in [0,1]}$, where $\mu_\nu$ is the product Bernoulli measure for which the probability that the cochain $\epsilon$ takes the value 0 on a given $(k-1)$-simplex of $K$ is $\nu$, this construction produces a random variable $\epsilon \in C^{k-1}(K; \mathbb{Z}/2\mathbb{Z}) \mapsto V_\epsilon \subset \mathrm{Sd}(K)$ which raises the following questions of random topology: What is the expected topology of $V_\epsilon$, e.g. its expected Betti numbers? How does this expected topology behave under large iterated barycentric subdivisions? For every $p \in \{0, \dots, n-k\}$, we denote the $p$-th Betti number of $V_\epsilon$ by $b_p(V_\epsilon) = \dim H_p(V_\epsilon; \mathbb{Z}/2\mathbb{Z})$ and its mathematical expectation after $d$ barycentric subdivisions by $\mathbb{E}_{\nu,d}(b_p) = \int_{C^{k-1}(\mathrm{Sd}^d(K); \mathbb{Z}/2\mathbb{Z})} b_p(V_\epsilon)d\mu_\nu(\epsilon)$, $d \ge 0$. We proved [16] that there exist universal constants $c_p^\pm(n, k)$ such that

$$c_p^-(n, k) \le \liminf_{d \to +\infty} \frac{\mathbb{E}_{\nu,d}(b_p)}{(n+1)!^d f_n(K)} \le \limsup_{d \to +\infty} \frac{\mathbb{E}_{\nu,d}(b_p)}{(n+1)!^d f_n(K)} \le c_p^+(n, k),$$

where $f_n(K)$ denotes the number of $n$-dimensional simplices of $K$, so that its $d$-th barycentric subdivision $\mathrm{Sd}^d(K)$ contains $(n+1)!^d f_n(K)$ many such simplices. These results produced counterparts in this discrete framework to the ones obtained in [6] and [8, 9] on the expected Betti numbers of random real algebraic submanifolds of real projective
manifolds or random nodal sets in smooth manifolds respectively, see
also [11, 13, 14]. This model inspired by real toric geometry [21, 22]
turns out to be more computable and our main result is the following:

**Theorem 1.1.** Let $n > 0$, $k \in \{1, \dots, n\}$ and $p \in \{0, \dots, n-k\}$. Then, for every finite $n$-dimensional simplicial complex $K$ and every $\nu \in [0, 1]$, the sequence $\left( \frac{\mathbb{E}_{\nu,d}(b_p)}{f_n(K)(n+1)!^d} \right)_{d \ge 0}$ converges and its limit does not depend on $K$.

We denote by $e_{p,\nu}(n, k)$ the limit given by Theorem 1.1. Recall that with the exception of the case $p = 0$ [14], the counterpart of Theorem 1.1 in the theory of random polynomials [10, 5, 6] or random nodal sets [13, 14, 9, 8, 11, 19] remains open. It may be thus somewhat surprising that we are able to prove the convergence of the sequence $\left( \frac{\mathbb{E}_{\nu,d}(b_p)}{f_n(K)(n+1)!^d} \right)_{d \ge 0}$. The fact that the limit is then independent of $K$ can be understood as follows. The contribution of the $(n-1)$-skeleton of $K$ to the total number of faces becomes asymptotically negligible after $d \gg 0$ barycentric subdivisions, in comparison to the number of faces arising from the subdivided $n$-simplices. Consequently, the asymptotic behavior of $\mathbb{E}_{\nu,d}(b_p)$ is governed by the contributions from the $n$-simplices of $K$ and so the total asymptotic is just the same as the one observed in $f_n(K)$ copies of a single $n$-simplex.

Recall [3, 4] that the asymptotic behavior of the face numbers of a simplicial complex $K$ under successive barycentric subdivision is itself universal. Furthermore, in each dimension, these faces asymptotically equidistribute over the underlying topological space $|K|$ with respect to its Lebesgue measure and the asymptotic behaviors of their links and dual blocks are universal as well, see [15].

When $k = 1$, we have been able to obtain the following upper estimates for $e_{p,\nu}(n, k)$ in [17, 18].

**Theorem 1.2.** [18, Theorem 1.4] For every $n > 0$, $\nu \in [0, 1]$ and $p \in \{1, \dots, n-1\}$,

$$e_{p,\nu}(n, 1) \le \nu(1-\nu)(\nu^p + (1-\nu)^p) \sum_{l=p+1}^{n} (-1)^{l+1-p} q_{l,n} - \lambda_{p,\nu}(n).$$

In Theorem 1.2, both sides of the inequality are continuous functions of $\nu \in [0, 1]$ which vanish for $\nu \in \{0, 1\}$. We do not know the actual value of $\lambda_{p,\nu}(n)$ and refer to [18] for its definition, where it is estimated from below [18, Theorem 3.5] and where explicit upper estimates for $e_{p,\nu}(n, 1)$ are given for $n \le 4$ [18, Corollary 3.6]. As for the quantity $q_{l,n}$, it denotes the asymptotic face number $\lim_{d \to +\infty} \frac{f_l(\mathrm{Sd}^d(\Delta_n))}{(n+1)!^d}$ of the standard $n$-simplex $\Delta_n$. It is a rational number [3] which has been explicitly computed for $n \le 9$ in [4]. Moreover, by Theorem 1.3 of [15],
for every $0 \le l < n$,

$$q_{l,n} = \sum_{(l_1, \dots, l_j) \in \mathcal{P}_{l,n}} \frac{\gamma_{n+1,l_j} \gamma_{l_j, l_{j-1}} \dots \gamma_{l_2,l_1}}{(\gamma_{n+1,n+1} - \gamma_{l_j,l_j}) \dots (\gamma_{n+1,n+1} - \gamma_{l_1,l_1})},$$

where $\mathcal{P}_{l,n} = \{(l_1, \dots, l_j) \in \mathbb{N}^j | j \ge 1 \text{ and } l+1 = l_1 < \dots < l_j < n+1\}$ and where $\gamma_{i,j}$ denotes the number of interior $(j-1)$-faces of the subdivided standard $(i-1)$-simplex $\mathrm{Sd}(\Delta_{i-1})$. The latter agrees with $j!S(i, j)$, where $S(i, j)$ is the Stirling number of the second kind [20, 2], that is the number of partitions of a set of $i$ elements into $j$ disjoint nonempty subsets, see [15, Lemma 2.6] or [3, Lemma 1].

The first values of its generating function $q_n(X) = \sum_{l=0}^{n} q_{l,n}X^l$ are as follows.

$$q_0(X) = 1,$$

$$q_1(X) = 1 + X,$$

$$q_2(X) = \frac{1}{2} + \frac{3}{2}X + X^2,$$

$$q_3(X) = \frac{2}{11} + \frac{13}{11}X + 2X^2 + X^3,$$

$$q_4(X) = \frac{1}{19} + \frac{25}{38}X + \frac{40}{19}X^2 + \frac{5}{2}X^3 + X^4,$$

$$q_5(X) = \frac{132}{10411} + \frac{3004}{10411}X + \frac{45}{29}X^2 + \frac{95}{29}X^3 + 3X^4 + X^5.$$

Finally, for every $n > 0$, $q_{n,n} = 1$ and $q_{n-1,n} = \frac{n+1}{2}$, see [18]. It would be interesting to investigate the asymptotic behavior of $q_{l,n}$ or of $\sum_{l=p+1}^{n} (-1)^{l+1-p}q_{l,n}$ as $n$ grows to $+\infty$, compare [7, Theorem 1.6].

In the case of the standard $n$-simplex, we likewise set $\tilde{b}_p(V_\epsilon) = \sum b_p(\Sigma_\epsilon)$, where the sum is taken over all connected components $\Sigma_\epsilon$ of $V_\epsilon$ which do not intersect the boundary of $\Delta_n$. We then set $\mathbb{E}_{\nu,d}(\tilde{b}_p) = \int_{C^{k-1}(\mathrm{Sd}^d(\Delta_n);\mathbb{Z}/2\mathbb{Z})} \tilde{b}_p(V_\epsilon)d\mu_\nu(\epsilon)$ and get:

**Theorem 1.3.** Let $n > 0$, $k \in \{1, \dots, n\}$ and $p \in \{0, \dots, n-k\}$. Then, for every $\nu \in [0, 1]$, the sequence $\left(\frac{\mathbb{E}_{\nu,d}(\tilde{b}_p)}{(n+1)!^d}\right)_{d \ge 0}$ is increasing and bounded from above.

We denote by $\tilde{e}_{p,\nu}(n, k)$ the limit of the sequence given by Theorem 1.3. We do not know whether $e_{p,\nu}(n, k) = \tilde{e}_{p,\nu}(n, k)$ or not, except when $p = 0$. It is related to a problem of percolation which we introduce and discuss in § 2.3, see Theorem 2.7 (or [1] for another related problem of percolation). Nevertheless we get:

**Theorem 1.4.** Under the hypothesis of Theorem 1.3, $e_{p,\nu}(n, k) \ge \tilde{e}_{p,\nu}(n, k) > 0$. Moreover, $e_{0,\nu}(n, k) = \tilde{e}_{0,\nu}(n, k)$.
## 2. ASYMPTOTIC BEHAVIOR OF THE EXPECTED BETTI NUMBERS

In the sequel, we denote by $K^{[i]}$ the set of $i$-dimensional simplices of a simplicial complex $K$ and by $K^{(i)}$ its $i$-skeleton, that is the set of its simplices of dimension at most $i$. We first prove Theorems 1.3 and 1.4 and then Theorem 1.1. In § 2.3, we study a problem of percolation related to these results.

2.1. **Proofs of Theorems 1.3 and 1.4.** Given a simplicial complex $K$, a $q$-simplex of $Sd(K)$ reads $[\hat{\sigma}_0, \dots, \hat{\sigma}_q]$, where $\hat{\sigma}_i$ denotes the barycenter of the simplex $\sigma_i$ of $K$, $i \in \{0, \dots, q\}$, and where $\sigma_i$ is a proper face of $\sigma_{i+1}$, $i \in \{0, \dots, q-1\}$. The block $D(\sigma)$ dual to a simplex $\sigma \in K$ is the union of all open simplices $[\hat{\sigma}_0, \dots, \hat{\sigma}_q]$ of $Sd(K)$ such that $\sigma_0 = \sigma$. The union of closed such simplices is denoted by $\bar{D}(\sigma)$, see [12] and Figure 1. We recall the following construction from [16].

<figure><img src="image_2.png" /><figcaption>FIGURE 1. Dual block of the vertex $\sigma_0$ of the two-simplex $\sigma_2$.</figcaption></figure>

**Definition 2.1.** Let $K$ be an $n$-dimensional simplicial complex, $n > 0$, and let $k \in \{1, \dots, n\}$. For every $\epsilon \in C^{k-1}(K; \mathbb{Z}/2\mathbb{Z})$, we denote by $V_\epsilon$ the subcomplex of $Sd(K)$ dual to the cocycle $d\epsilon$, where $d: C^{k-1}(K; \mathbb{Z}/2\mathbb{Z}) \to C^k(K; \mathbb{Z}/2\mathbb{Z})$ denotes the coboundary operator. Hence, $V_\epsilon$ is the union over all the $k$-simplices $\sigma$ of $K$ for which $\langle d\epsilon, \sigma \rangle \neq 0$ of their dual blocks $\bar{D}(\sigma)$.

In the case $k=1$, the condition $\langle d\epsilon, \sigma \rangle \neq 0$ means that the restriction of $\epsilon$ to $\sigma$ is not constant. In each simplex of $K$, $V_\epsilon$ then separates the vertices where $\epsilon$ vanishes from the vertices where $\epsilon$ equals one, see Figure 2.

<figure><img src="image_3.png" /><figcaption>FIGURE 2. Subcomplexes $V_\epsilon$ in standard simplices.</figcaption></figure>
Proof of Theorem 1.3. For every $m \in \{0, \dots, d\}$, we observe that

$$
\begin{align*}
\mathbb{E}_{\nu,d}(\tilde{b}_p) &= \int_{C^{k-1}(\mathrm{Sd}^d(\Delta_n);\mathbb{Z}/2\mathbb{Z})} \tilde{b}_p(V_\epsilon)d\mu_\nu(\epsilon) \\
&\geq \int_{C^{k-1}(\mathrm{Sd}^d(\Delta_n);\mathbb{Z}/2\mathbb{Z})} \left(\sum_{\sigma \in \mathrm{Sd}^{d-m}(\Delta_n)[n]} \tilde{b}_p(V_\epsilon \cap \sigma)\right) d\mu(\epsilon) \\
&= \sum_{\sigma \in \mathrm{Sd}^{d-m}(\Delta_n)[n]} \int_{C^{k-1}(\mathrm{Sd}^m(\sigma);\mathbb{Z}/2\mathbb{Z})} \tilde{b}_p(V_\epsilon)d\mu(\epsilon) \\
&= (n+1)!^{d-m} \mathbb{E}_{\nu,m}(\tilde{b}_p),
\end{align*}
$$

where the third line follows from the fact that $\mu_\nu$ is a product measure and the last one from the equality $f_n(\mathrm{Sd}^{d-m}(\Delta_n)) = (n+1)!^{d-m}$.

The sequence $\left( \frac{\mathbb{E}_{\nu,d}(\tilde{b}_p)}{(n+1)!^d} \right)_{d \in \mathbb{N}}$ is thus increasing. Moreover, it is bounded from above by the converging sequence $\left( \frac{f_p(\mathrm{Sd}^{d+1}(\Delta_n))}{(n+1)!^d} \right)_{d \in \mathbb{N}}$, since $V_\epsilon$ is a subcomplex of $\mathrm{Sd}^{d+1}(\Delta_n)$, see [3, 4] or [15]. Hence the result. $\square$

**Remark 2.2.** For every $m \ge 1$, let $\mathbf{C}(m)$ be the finite set of homeomorphism classes of pairs $(\mathbb{R}^n, \Sigma)$, where $\Sigma$ denotes a closed manifold of dimension $n-k$ embedded in $\mathbb{R}^n$ by an embedding of complexity $m$, see Definition 5.2 of [16]. For every $\epsilon \in \mathbf{C}^{k-1}(\mathrm{Sd}^d(K))$ and every $\Sigma$ of complexity $m$, let $N_\Sigma(\epsilon)$ be the maximum number of disjoint open simplices $(\mathring{\sigma}_j)_{j \in J}$ which can be packed in $|K|$ in such a way that the pair $(\mathring{\sigma}_j, V_\epsilon \cap \mathring{\sigma}_j)$ is homeomorphic to $(\mathbb{R}^n, \Sigma)$, where for every $j \in J$, $\sigma_j$ is a simplex of $\mathrm{Sd}^{d-m_j}(K)$ for some $m_j \in \{m, \dots, d\}$. We deduce the following lower estimate for every $d \ge 0$.

$$
\begin{align*}
\tilde{e}_{p,\nu}(n,k) &\ge \frac{1}{(n+1)!^d} \mathbb{E}_{\nu,d}(\tilde{b}_p) \text{ by Theorem 1.3} \\
&= \sum_{m=1}^d \sum_{\Sigma \in \mathbf{C}(m)} \frac{1}{(n+1)!^d} b_p(\Sigma) \mathbb{E}_{\nu,d}(N_\Sigma) \\
&\ge \sum_{m=1}^d \sum_{\Sigma \in \mathbf{C}(m)} b_p(\Sigma) c_\Sigma,
\end{align*}
$$

where $c_\Sigma > 0$ is related to the probability of appearance of $\Sigma$, see Theorem 5.11 of [16]. As $d$ grows to $+\infty$, we deduce in the limit that $\tilde{e}_{p,\nu}(n,k) \ge c_p^-(n,k)$, by Definition 5.12 of [16].

**Proof of Theorem 1.4.** By definition, for every $\epsilon \in \mathbf{C}^{k-1}(\mathrm{Sd}^d(\Delta_n);\mathbb{Z}/2\mathbb{Z})$ and $p \in \{0, \dots, n-k\}$, $\tilde{b}_p(V_\epsilon) \le b_p(V_\epsilon)$. After integration over $\mathbf{C}^{k-1}(\mathrm{Sd}^d(\Delta_n);\mathbb{Z}/2\mathbb{Z})$, we thus get $\mathbb{E}_{\nu,d}(\tilde{b}_p) \le \mathbb{E}_{\nu,d}(b_p)$. The inequality $\tilde{e}_{p,\nu}(n,k) \le e_{p,\nu}(n,k)$ is deduced from Theorems 1.1 and 1.3 by passing to the limits as $d$ tends to $+\infty$, after dividing by $(n+1)!^d$.

When $p=0$, we observe that by definition, for every $\epsilon \in \mathbf{C}^{k-1}(\mathrm{Sd}^d(\Delta_n);\mathbb{Z}/2\mathbb{Z})$, $b_0(V_\epsilon) \le \tilde{b}_0(V_\epsilon) + b_0(V_\epsilon \cap \partial\Delta_n)$, since the number of connected components of $V_\epsilon$ which meet $\partial\Delta_n$ is bounded from above by $b_0(V_\epsilon \cap \partial\Delta_n)$. We deduce the inequality

$$
(1) \quad \frac{\mathbb{E}_{\nu,d}(b_0)}{(n+1)!^d} \le \frac{\mathbb{E}_{\nu,d}(\tilde{b}_0)}{(n+1)!^d} + \frac{\mathbb{E}_{\nu,d}(b_0(V_\epsilon \cap \partial \Delta_n))}{(n+1)!^d}.
$$
If $k = n$, $\mathbb{E}_{\nu,d}(b_0(V_\epsilon \cap \partial\Delta_n)) = 0$ and the result follows. If $k < n$,

$$
\begin{align*}
\mathbb{E}_{\nu,d}(b_0(V_\epsilon \cap \partial \Delta_n)) &= \int_{C^{k-1}(\mathrm{Sd}^d(\Delta_n);\mathbb{Z}/2\mathbb{Z})} b_0(V_\epsilon \cap \partial \Delta_n) d\mu_\nu(\epsilon) \\
&= \int_{C^{k-1}(\mathrm{Sd}^d(\partial \Delta_n);\mathbb{Z}/2\mathbb{Z})} b_0(V_\epsilon) d\mu_\nu(\epsilon) \\
&= O(n!^d) \text{ by Theorem 1.2 of [16],}
\end{align*}
$$

where the second equality holds true since $\mu_\nu$ is a product measure.
By letting $d$ tend to $+\infty$, we thus deduce from (1) that $e_{0,\nu}(n,k) \le \tilde{e}_{0,\nu}(n,k)$. Hence the result.
□

**Remark 2.3.** The equality $e_{0,\nu}(n,k) = \tilde{e}_{0,\nu}(n,k)$ is similar to the one established in [14]. Moreover, Theorem 1.4 combined with Remark 2.2 provides the lower bound $e_{p,\nu}(n,k) \ge c_p^-(n,k)$ given by Corollary 5.13 of [16].

Theorem 1.4 raises the following question: Are the limits $e_{p,\nu}(n,k)$ and $\tilde{e}_{p,\nu}(n,k)$ equal in general or not? This question is related to a problem of percolation that we discuss in § 2.3.

2.2. **Proof of Theorem 1.1.** In order to prove Theorem 1.1, we first need several preliminary results. For every $m \in \{0, \dots, d\}$, we consider

$$
A_{d,m} = \bigcup_{\sigma \in \mathrm{Sd}^{d-m}(K)^{[n]}} \mathring{\sigma} \quad \text{and} \quad B_{d,m} = \bigcup_{\tau \in \mathrm{Sd}^m(\mathrm{Sd}^{d-m}(K)^{(n-1)})} D(\tau),
$$

so that $A_{d,m}$, $B_{d,m}$ are two open sets covering the underlying topological
space $|K|$. For every $q$-simplex $\theta = [\hat{\theta}_0, \dots, \hat{\theta}_q] \in \mathrm{Sd}^{d+1}(K)$, let $l_m(\theta) \in
\{-1, \dots, q\}$ be the greatest integer $i$ such that $\theta_i \in \mathrm{Sd}^m(\mathrm{Sd}^{d-m}(K)^{(n-1)})$,
where $l_m(\theta) = -1$ if and only if there is no such integer $i$. When
$l_m(\theta) \ne -1$, we moreover set

$$
r_{\theta} : \begin{array}{l} (\theta \setminus [\hat{\theta}_{l_m(\theta)+1}, \dots, \hat{\theta}_q]) \times [0, 1] \rightarrow \theta \setminus [\hat{\theta}_{l_m(\theta)+1}, \dots, \hat{\theta}_q] \\ ((1-\alpha)y + \alpha z, t) \rightarrow (1-t\alpha)y + t\alpha z, \end{array}
$$

where $\alpha \in [0, 1[, y \in [\hat{\theta}_0, \dots, \hat{\theta}_{l_m(\theta)}]$ and $z \in [\hat{\theta}_{l_m(\theta)+1}, \dots, \hat{\theta}_q]$.

**Proposition 2.4.** Let $K$ be a finite simplicial complex of positive dimension $n$ and let $\theta = [\hat{\theta}_0, \dots, \hat{\theta}_q]$ be a $q$-simplex of $\mathrm{Sd}^{d+1}(K)$, $q \in \{0, \dots, n\}$, $d \ge 0$. Then, for every $m \in \{0, \dots, d\}$,

(1) $B_{d,m} \cap \theta = \theta \setminus [\hat{\theta}_{l_m(\theta)+1}, \dots, \hat{\theta}_q]$, where $[\hat{\theta}_{l_m(\theta)+1}, \dots, \hat{\theta}_q] = \emptyset$ if $l_m(\theta) = q$.

(2) If $l_m(\theta) \ne -1$, $r_\theta$ retracts $\theta \setminus [\hat{\theta}_{l_m(\theta)+1}, \dots, \hat{\theta}_q]$ by deformation onto $[\hat{\theta}_0, \dots, \hat{\theta}_{l_m(\theta)}]$.

(3) For every face $\sigma \in \theta$ such that $l_m(\sigma) \ne -1$, the restriction of $r_\theta$ to $\sigma$ is $r_\sigma$.

*Proof.* By definition, $B_{d,m}$ is the union of the open simplices $\sigma$ of $\mathrm{Sd}^{d+1}(K)$ such that $l_m(\sigma) \ne -1$. An open face of $\theta$ is thus included in $B_{d,m}$ if and only if it contains a vertex $\hat{\theta}_i$ with $i \le l_m(\theta)$. Hence the first part. The second and third parts follow from the definition of $r_\theta$. $\square$
**Corollary 2.5.** *Under the hypothesis of Proposition 2.4, for every $m \in \{0, \dots, d\}$,*

(1) $B_{d,m}$ is an open subset of $|\mathrm{Sd}^{d+1}(K)| = |K|$ and $\mathrm{Sd}^{d+1}(K) \setminus B_{d,m}$ is a subcomplex of $\mathrm{Sd}^{d+1}(K)$.

(2) For every subcomplex $L$ of $\mathrm{Sd}^{d+1}(K)$, the retractions $(r_\theta)_{\theta \in L}$ glue together to define $r_L: L \cap B_{d,m} \to L \cap B_{d,m}$. Moreover, $r_L$ retracts $L \cap B_{d,m}$ by deformation onto $L \cap \mathrm{Sd}^{m+1}(\mathrm{Sd}^{d-m}(K)^{(n-1)})$.

(3) For every subcomplexes $M < L < \mathrm{Sd}^{d+1}(K)$, the restriction of $r_L$ to $M$ is $r_M$.

*Proof.* From the first part of Proposition 2.4, the complement of $B_{d,m}$ in $\mathrm{Sd}^{d+1}(K)$ is the union of the simplices $\theta$ for which $l_m(\theta) = -1$. It is a subcomplex of $\mathrm{Sd}^{d+1}(K)$, which is closed. Hence the first part. The third part of Proposition 2.4 guarantees that the retractions $(r_\theta)_{\theta \in L}$ glue together to define $r_L: L \cap B_{d,m} \to L \cap B_{d,m}$ and the second part of Proposition 2.4 guarantees that the latter is a retraction of $L \cap B_{d,m}$ onto $L \cap \mathrm{Sd}^{m+1}(\mathrm{Sd}^{d-m}(K)^{(n-1)})$. Finally, the last part of Corollary 2.5 follows from the last part of Proposition 2.4. $\square$

**Proposition 2.6.** *Under the hypothesis of Theorem 1.1, there exists a universal constant $c(n)$ such that for every $m \in \{0, \dots, d\}$,*

$$|\mathbb{E}_{\nu,d}(b_p) - \mathbb{E}_{\nu,d}(b_p(V_{\epsilon} \cap A_{d,m}))| \le (n+1)!^d f_n(K) \frac{c(n)}{(n+1)^m} + O(n!^d).$$

*Proof.* It follows from Corollary 2.5 that for every $\epsilon \in C^{k-1}(\mathrm{Sd}^d(K); \mathbb{Z}/2\mathbb{Z})$, $(A_{d,m} \cap V_{\epsilon}) \cup (B_{d,m} \cap V_{\epsilon})$ is an open cover of $V_{\epsilon}$. The long exact sequence of Mayer-Vietoris [12] associated to this open cover reads

$$\dots \xrightarrow{i_p} H_p(A_{d,m} \cap V_{\epsilon}) \oplus H_p(B_{d,m} \cap V_{\epsilon}) \xrightarrow{j_p} H_p(V_{\epsilon}) \xrightarrow{\partial_p} H_{p-1}(A_{d,m} \cap B_{d,m} \cap V_{\epsilon}) \to \dots$$

where the coefficients are in $\mathbb{Z}/2\mathbb{Z}$. We thus deduce that

$$\begin{aligned} b_p(V_{\epsilon}) &= \dim \mathrm{Im}(\partial_p) + \dim \mathrm{Im}(j_p) \\ &= \dim \mathrm{Im}(\partial_p) + b_p(A_{d,m} \cap V_{\epsilon}) + b_p(B_{d,m} \cap V_{\epsilon}) - \dim \mathrm{Im}(i_p). \end{aligned}$$

After integration over $C^{k-1}(\mathrm{Sd}^d(K); \mathbb{Z}/2\mathbb{Z})$, we get

$$|\mathbb{E}_{\nu,d}(b_p) - \mathbb{E}_{\nu,d}(A_{d,m} \cap V_{\epsilon})| \le \mathbb{E}_{\nu,d}(b_p(B_{d,m} \cap V_{\epsilon})) + \mathbb{E}_{\nu,d}(b_p(A_{d,m} \cap B_{d,m} \cap V_{\epsilon})) \\ (2) \qquad \qquad \qquad \qquad \qquad \qquad + \mathbb{E}_{\nu,d}(b_{p-1}(A_{d,m} \cap B_{d,m} \cap V_{\epsilon})).$$

Let us now bound each term of the right hand side of (2). We know from Corollary 2.5 that for every $\epsilon \in C^{k-1}(\mathrm{Sd}^d(K); \mathbb{Z}/2\mathbb{Z})$, $r_{V_{\epsilon}}$ retracts $V_{\epsilon} \cap B_{d,m}$ by deformation onto $V_{\epsilon} \cap \mathrm{Sd}^{m+1}(\mathrm{Sd}^{d-m}(K)^{(n-1)})$. Thus,

$$\begin{aligned} \mathbb{E}_{\nu,d}(b_p(V_{\epsilon} \cap B_{d,m})) &= \mathbb{E}_{\nu,d}(b_p(V_{\epsilon} \cap \mathrm{Sd}^{m+1}(\mathrm{Sd}^{d-m}(K)^{(n-1)}))) \\ &\le f_p(\mathrm{Sd}^{m+1}(\mathrm{Sd}^{d-m}(K)^{(n-1)})) \\ &\le \sum_{i=p}^{n-1} \sum_{\sigma \in \mathrm{Sd}^{d-m}(K)^{[i]}} f_p(\mathrm{Sd}^{m+1}(\sigma)) \\ &\le f_*(\mathrm{Sd}^{d-m}(K)) f_p(\mathrm{Sd}^{m+1}(\Delta_{n-1})), \end{aligned}$$
where $f_*(\mathrm{Sd}^{d-m}(K))$ denotes the total face number of $\mathrm{Sd}^{d-m}(K)$.

Let $\tilde{K}$ be the subcomplex of $K$ consisting of the faces of the $n$-simplices of $K$. Then,

$$ 
\begin{align*}
f_*(\mathrm{Sd}^{d-m}(K)) &= f_*(\mathrm{Sd}^{d-m}(\tilde{K})) + f_*(\mathrm{Sd}^{d-m}(K \setminus \tilde{K})) \\
&\leq f_n(K)f_*(\mathrm{Sd}^{d-m}(\Delta_n)) + f_*(\mathrm{Sd}^{d-m}(K \setminus \tilde{K})) \\
&\leq f_n(K)f_*(\mathrm{Sd}^{d-m}(\Delta_n)) + O(n!^{d-m}),
\end{align*}
$$

from [3], since the simplices of $K \setminus \tilde{K}$ are of dimension at most $(n-1)$, see also [4, 15]. Likewise, the sequences $\left(\frac{f_*(\mathrm{Sd}^{d-m}(\Delta_n))}{(n+1)!^{d-m}}\right)_{d \in \mathbb{N}}$ and $\left(\frac{f_p(\mathrm{Sd}^{m+1}(\Delta_{n-1}))}{n!^{m+1}}\right)_{m \in \mathbb{N}}$ are universal and convergent, see [3, 4]. They are thus bounded by universal constants. We deduce the existence of a universal constant $c_1(n)$ such that $\mathbb{E}_{\nu,d}(b_p(B_{d,m} \cap V_\epsilon)) \leq f_n(K)(n+1)!^{d} \frac{c_1(n)}{(n+1)^m} + O(n!^d)$.

Similarly, from Corollary 2.5 we know that for every $\epsilon \in C^{k-1}(\mathrm{Sd}^d(K); \mathbb{Z}/2\mathbb{Z})$ and every $\sigma \in \mathrm{Sd}^{d-m}(K)^{[n]}$, $r_{V_\epsilon \cap \sigma}$ is a deformation retract of $B_{d,m} \cap \mathring{\sigma} \cap V_\epsilon$ onto $\partial\sigma \cap V_\epsilon$. So we have

$$ 
\begin{align*}
\mathbb{E}_{\nu,d}(b_p(A_{d,m} \cap B_{d,m} \cap V_\epsilon)) &= \sum_{\sigma \in \mathrm{Sd}^{d-m}(K)^{[n]}} \mathbb{E}_{\nu,d}(b_p(B_{d,m} \cap \mathring{\sigma} \cap V_\epsilon)) \\
&= \sum_{\sigma \in \mathrm{Sd}^{d-m}(K)^{[n]}} \mathbb{E}_{\nu,d}(b_p(V_\epsilon \cap \partial\sigma)) \\
&\leq \sum_{\sigma \in \mathrm{Sd}^{d-m}(K)^{[n]}} f_p(\mathrm{Sd}^{m+1}(\partial\sigma)) \\
&\leq f_n(K) \frac{(n+1)!^d}{(n+1)^m} \frac{f_p(\mathrm{Sd}^{m+1}(\partial\Delta_n))}{n!^m}.
\end{align*}
$$

Once again the sequence $\left(\frac{f_p(\mathrm{Sd}^{m+1}(\partial\Delta_n))}{n!^m}\right)_{m \in \mathbb{N}}$ is universal and convergent and thus gets bounded by a universal constant. The third term in (2) is analogous to the second one, and gets bounded by a universal constant as well. Hence the result. $\square$

**Proof of Theorem 1.1.** We first prove the result for $K = \Delta_n$ and then deduce it for a general simplicial complex $K$. For every $d \geq 0$, we set

$$ u_d = \frac{1}{(n+1)!^d} \int_{C^{k-1}(\mathrm{Sd}^d(\Delta_n); \mathbb{Z}/2\mathbb{Z})} b_p(V_\epsilon) d\mu_\nu(\epsilon). $$

By definition of $A_{d,m}$, for every $\epsilon \in C^{k-1}(\mathrm{Sd}^d(\Delta_n); \mathbb{Z}/2\mathbb{Z})$ and every $m \in \{0, \dots, d\}$,

$$ b_p(V_\epsilon \cap A_{d,m}) = \sum_{\sigma \in \mathrm{Sd}^{d-m}(\Delta_n)^{[n]}} b_p(V_\epsilon \cap \mathring{\sigma}). $$

After integration over $C^{k-1}(\mathrm{Sd}^d(\Delta_n); \mathbb{Z}/2\mathbb{Z})$, we get $\mathbb{E}_{\nu,d}(b_p(V_\epsilon \cap A_{d,m})) = (n+1)!^{d-m} \mathbb{E}_{\nu,m}(b_p)$ since $\mu_\nu$ is a product measure, $\#\mathrm{Sd}^{d-m}(\Delta_n)^{[n]} = (n+1)!^{d-m}$ by [3] and $b_p(V_\epsilon \cap \mathring{\sigma}) = b_p(V_\epsilon \cap \sigma)$ by Corollary 2.5. Dividing
by $(n+1)!^d$, we deduce from Proposition 2.6 the upper bound

$$|u_d - u_m| \le \frac{c(n)}{(n+1)^m} + O\left(\frac{1}{(n+1)^d}\right).$$

The sequence $(u_d)_{d \in \mathbb{N}}$ is thus a Cauchy sequence and hence a converging one. The result follows for $K = \Delta_n$. Let us now suppose that $K$ is any finite simplicial complex of dimension $n$ and set, for every $d \ge 0$,

$$v_d = \frac{1}{f_n(K)(n+1)!^d} \int_{C^{k-1}(\mathrm{Sd}^d(K);\mathbb{Z}/2\mathbb{Z})} b_p(V_\epsilon)d\mu_\nu(\epsilon).$$

We deduce from Proposition 2.6, after dividing by $f_n(K)(n+1)!^d$, that $|v_d - u_m| \le \frac{c(n)}{(n+1)^m} + O\left(\frac{1}{(n+1)^d}\right)$. Since $(u_m)_{m \in \mathbb{N}}$ converges, this implies that $(v_d)_{d \in \mathbb{N}}$ converges as well and that $\lim_{d \to +\infty} v_d = \lim_{m \to +\infty} u_m$. Hence the result. □

**2.3. Percolation.** Let $n > 0$ and $k \in \{1, \dots, n\}$. For every $0 \le m \le d$ and every $\sigma \in \mathrm{Sd}^{d-m}(\Delta_n)^{[n]}$, we denote by $P_{d,m}(\sigma)$ the set of $\epsilon \in C^{k-1}(\mathrm{Sd}^m(\mathrm{Sd}^{d-m}(\Delta_n)\setminus\hat{\sigma}))$ for which there exists a path in $V_\epsilon$ connecting $\partial\Delta_n$ to $\partial\sigma$. In other words, $\epsilon$ belongs to $P_{d,m}(\sigma)$ if and only if $V_\epsilon\setminus\hat{\sigma}$ percolates between the boundaries of $\Delta_n$ and $\sigma$, see Figure 3.

<figure><img src="image_4.png" /><figcaption>FIGURE 3. Examples and counterexamples of percolation in the third barycentric subdivision of a two-simplex.</figcaption></figure>

**Theorem 2.7.** Let $\nu \in [0, 1]$. If there exist $n > 0$, $k \in \{1, \dots, n\}$ and $p \in \{0, \dots, n-k\}$ such that $e_{p,\nu}(n, k) > \tilde{e}_{p,\nu}(n, k)$, then $n \ge 3$, $p \ge 1$ and $\lim_{m \to +\infty} \mu_\nu(P_{m+m',m}(\sigma)) = 1$ for every $\sigma \in \mathrm{Sd}^{m'}(\Delta_n)^{[n]}$ and $m' \in \mathbb{N}$.
*Proof.* If $p=0$, Theorem 1.4 implies that $e_{p,\nu}(n,k) = \tilde{e}_{p,\nu}(n,k)$ for every $k \in \{1, \dots, n\}$ and $n > 0$. If $n=2$, we need to prove the equality for $k=p=1$. In this case, for every $\epsilon \in C^0(\mathrm{Sd}^d(\Delta_2); \mathbb{Z}/2\mathbb{Z})$, $V_\epsilon$ is a homological manifold of dimension one by Theorem 1.1 of [16]. The connected components of $V_\epsilon$ which meet $\partial\Delta_2$ are thus homeomorphic to $\Delta_1$, so that $b_1(V_\epsilon) = \tilde{b}_1(V_\epsilon)$. The equality $e_1(2,1) = \tilde{e}_1(2,1)$ is then obtained by integrating over $C^0(\mathrm{Sd}^d(\Delta_2); \mathbb{Z}/2\mathbb{Z})$.

Let us suppose now that there exists $n > 0$, $k \in \{1, \dots, n\}$ and $p \in \{0, \dots, n-k\}$ such that $e_{p,\nu}(n,k) > \tilde{e}_{p,\nu}(n,k)$. Let $m' \in \mathbb{N}$ and $\sigma \in \mathrm{Sd}^{m'}(\Delta_n)^{[n]}$. For every $m \ge 1$, we denote by $\tilde{P}_{m'+m,m}(\sigma)$ the set of all $\tilde{\epsilon} \in C^{k-1}(\mathrm{Sd}^{m'+m}(\Delta_n) \setminus \mathrm{Sd}^m(\sigma))$ for which there exists $\epsilon \in P_{m'+m,m}(\sigma)$ which agrees with $\tilde{\epsilon}$ on the complement of $\sigma$ —note that $\mathrm{Sd}^{m+m'}(\Delta_n) \setminus \mathrm{Sd}^m(\sigma)$ is not a simplicial complex. It is enough to show that $\lim_{m \to +\infty} \mu_\nu(\tilde{P}_{m+m',m}(\sigma)) = 1$. Indeed, let $\tilde{\sigma} \in \mathrm{Sd}^2(\sigma)$ be a face which does not meet the boundary of $\sigma$. Then, for every $m \ge 1$ and $\tilde{\epsilon} \in \tilde{P}_{m+m'+2,m}(\tilde{\sigma})$, the restriction of $\tilde{\epsilon}$ to $\mathrm{Sd}^{m+2}(\mathrm{Sd}^{m'}(\Delta_n) \setminus \mathring{\sigma})$ lies in $P_{m+m'+2,m+2}(\sigma)$, since a path in $V_\epsilon$ connecting $\partial\Delta_n$ to $\partial\tilde{\sigma}$ has to cross $\partial\sigma$. Let $\tilde{P}_{m+m'+2,m+2}(\tilde{\sigma})$ be the image of $\tilde{P}_{m+m'+2,m+2}(\tilde{\sigma})$ in $P_{m+m'+2,m+2}(\sigma)$ under this restriction map, so that it satisfies $\mu_\nu(\tilde{P}_{m+m'+2,m+2}(\tilde{\sigma})) \le \mu_\nu(P_{m+m'+2,m+2}(\sigma))$. Since $\mu_\nu$ is a product measure, $\mu_\nu(\tilde{P}_{m+m'+2,m+2}(\tilde{\sigma})) \ge \mu_\nu(\tilde{P}_{m+m'+2,m+2}(\tilde{\sigma}))$ and we deduce the inequality $\mu_\nu(\tilde{P}_{m+m'+2,m+2}(\tilde{\sigma})) \le \mu_\nu(P_{m+m'+2,m+2}(\sigma))$. As a result, the fact that $\lim_{m \to +\infty} \mu_\nu(\tilde{P}_{m+m'+2,m+2}(\tilde{\sigma})) = 1$ implies that $\lim_{m \to +\infty} \mu_\nu(P_{m+m'+2,m+2}(\sigma)) = 1$.

Let us now prove that $\lim_{m \to +\infty} \mu_\nu(\tilde{P}_{m+m',m}(\sigma)) = 1$, a result which does not depend on the choices of $m'$ and $\sigma$. For every $\epsilon \in C^{k-1}(\mathrm{Sd}^{m'+m}(\Delta_n))$, let $\Lambda_\epsilon$ be the union of the connected components of $V_\epsilon$ which meet $\partial\Delta_n$, so that $b_p(V_\epsilon) = \tilde{b}_p(V_\epsilon) + b_p(\Lambda_\epsilon)$. Replacing $V_\epsilon$ by $\Lambda_\epsilon$ in Proposition 2.6, we deduce that

$$ \lim_{m \to +\infty} \frac{1}{(n+1)!^{m'+m}} |\mathbb{E}_{m'+m}(b_p(\Lambda_\epsilon)) - \mathbb{E}_{m'+m}(b_p(\Lambda_\epsilon \cap A_{m+m',m}))| = 0. $$

Under our hypothesis, we then deduce that

$$ \lim_{m \to +\infty} \frac{1}{(n+1)!^{m'+m}} \mathbb{E}_{m'+m}(b_p(\Lambda_\epsilon \cap A_{m+m',m})) = e_{p,\nu}(n,k) - \tilde{e}_{p,\nu}(n,k) > 0. $$

However, by definition,

$$
\begin{align*}
& \mathbb{E}_{m+m'}(b_p(\Lambda_\epsilon \cap A_{m+m',m})) \\
&= \sum_{\sigma \in \mathrm{Sd}^{m'}(\Delta_n)^{[n]}} \int_{C^{k-1}(\mathrm{Sd}^{m+m'}(\Delta_n) \setminus \mathrm{Sd}^m(\sigma))} \int_{C^{k-1}(\mathrm{Sd}^m(\sigma))} b_p(\Lambda_\epsilon \cap \mathring{\sigma}) d\mu_\nu(\epsilon) \\
&= \sum_{\sigma \in \mathrm{Sd}^{m'}(\Delta_n)^{[n]}} \int_{\tilde{P}_{m+m',m}(\sigma)} \int_{C^{k-1}(\mathrm{Sd}^m(\sigma))} b_p(\Lambda_\epsilon \cap \mathring{\sigma}) d\mu_\nu(\epsilon) \\
&\le \sum_{\sigma \in \mathrm{Sd}^{m'}(\Delta_n)^{[n]}} \mu_\nu(\tilde{P}_{m+m',m}(\sigma)) \mathbb{E}_{\nu,m}(b_p - \tilde{b}_p),
\end{align*}
$$
so that

$$ \frac{1}{(n+1)!^{m'}} \sum_{\sigma \in \mathrm{Sd}^{m'}(\Delta_n)^{[n]}} \mu_{\nu}(\tilde{P}_{m+m',m}(\sigma)) \ge \frac{\mathbb{E}_{\nu,m'+m}(b_p(\Lambda_{\epsilon} \cap A_{m+m',m}))}{(n+1)!^{m'} \mathbb{E}_{\nu,m}(b_p - \tilde{b}_p)}. $$

From what precedes, the right hand side of (3) converges to 1 as $m$ grows to $+\infty$. We thus deduce that for every $m' \in \mathbb{N}$ and every $\sigma \in \mathrm{Sd}^{m'}(\Delta_n)^{[n]}$, $\lim_{m \to +\infty} \mu_{\nu}(\tilde{P}_{m+m',m}(\sigma)) = 1$. Hence the result. $\square$

**Remark 2.8.** (1) It would be interesting to prove that for every $m' \in \mathbb{N}$, the sequence

$$ \frac{1}{(n+1)!^{m'}} \sum_{\sigma \in \mathrm{Sd}^{m'}(\Delta_n)^{[n]}} \mu_{\nu}(\tilde{P}_{m+m',m}(\sigma)) $$

which appeared in (3) converges to a limit $u_{m'}$ as $m$ grows to $+\infty$, without assuming that $e_{p,\nu}(n,k) > \tilde{e}_{p,\nu}(n,k)$. Then, the sequence $(u_{m'})_{m' \in \mathbb{N}}$ would be submultiplicative.

(2) Whether the sequence $(\mu_{\nu}(P_{m+m',m}(\sigma)))_{m \in \mathbb{N}}$ converges or not for every $m'$ and every $\sigma \in \mathrm{Sd}^{m'}(\Delta_n)^{[n]}$, and whether or not the limit is 1 without assuming that $e_{p,\nu}(n,k) > \tilde{e}_{p,\nu}(n,k)$ would also be interesting to know. Nevertheless, while by Theorem 2.7 this condition is necessary for $e_{p,\nu}(n,k)$ to be greater than $\tilde{e}_{p,\nu}(n,k)$, it is a priori not sufficient. This question of percolation seems already of interest in dimension two.

(3) These results are taken out from the preprint [17] and consist of its first part, while its remaining part appeared in [18].

## ACKNOWLEDGMENT

The second author is partially supported by the ANR project MI-CROLOCAL (ANR-15CE40-0007-01).

## REFERENCES

[1] V. Beffara and D. Gayet. Percolation of random nodal lines. *Publications Mathématiques de l'Institut des Hautes Études Scientifiques*, 126:131–176, 2017. https://doi.org/10.1007/s10240-017-0093-0

[2] K. N. Boyadzhiev. Close encounters with the Stirling numbers of the second kind. *Math. Mag.* 85(4), 252–266, 2012. https://doi.org/10.4169/math.mag.85.4.252

[3] F. Brenti and V. Welker. *f*-vectors of barycentric subdivisions. *Mathematische Zeitschrift*, 259(4):849–865, 2008. https://doi.org/10.1007/s00209-007-0251-z

[4] E. Delucchi, A. Pixton, and L. Sabalka. Face vectors of subdivided simplicial complexes. *Discrete Mathematics*, 312(2):248–257, 2012. https://doi.org/10.1016/j.disc.2011.08.032

[5] A. Edelman and E. Kostlan. How many zeros of a random polynomial are real? *Bulletin of the American Mathematical Society. New Series.*, 32(1):1–37, 1995. https://doi.org/10.1090/S0273-0979-1995-00571-9
[6] D. Gayet and J.-Y. Welschinger. Expected topology of random real algebraic submanifolds. *Journal de l'Institut de Mathématiques de Jussieu*, 14(4):673-702, 2015. https://doi.org/10.1017/S1474748014000115

[7] D. Gayet and J.-Y. Welschinger. Betti numbers of random real hypersurfaces and determinants of random symmetric matrices. *J. Eur. Math. Soc. (JEMS)*, 18(4):733-772, 2016. https://doi.org/10.4171/JEMS/601

[8] D. Gayet and J.-Y. Welschinger. Universal Components of Random Nodal Sets. *Communications in Mathematical Physics*, 347(3):777-797, 2016. https://doi.org/10.1007/s00220-016-2595-x

[9] D. Gayet and J.-Y. Welschinger. Betti numbers of random nodal sets of elliptic pseudo-differential operators. *Asian Journal of Mathematics*, 21(5):811-840, 2017. https://doi.org/10.4310/AJM.2017.v21.n5.a2

[10] M. Kac. On the average number of real roots of a random algebraic equation. *Bulletin of the American Mathematical Society*, 49:314-320, 1943. https://doi.org/10.1090/S0002-9904-1943-07912-8

[11] A. Lerario and E. Lundberg. Statistics on Hilbert's 16th problem. *International Mathematics Research Notices*, (12):4293-4321, 2015. https://doi.org/10.1093/imrn/rnu069

[12] J. R. Munkres. *Elements of algebraic topology*. Addison-Wesley Publishing Company, Menlo Park, CA, 1984.

[13] F. Nazarov and M. Sodin. On the number of nodal domains of random spherical harmonics. *American Journal of Mathematics*, 131(5):1337-1357, 2009. https://doi.org/10.1353/ajm.0.0070

[14] F. Nazarov and M. Sodin. Asymptotic laws for the spatial distribution and the number of connected components of zero sets of Gaussian random functions. *Journal of Mathematical Physics, Analysis, Geometry*, 12(3):205-278, 2016. https://doi.org/10.15407/mag12.03.205

[15] N. Salepci and J.-Y. Welschinger. Asymptotic measures and links in simplicial complexes. *Discrete & Computational Geometry*, 62(1):164-179, 2019. https://doi.org/10.1007/s00454-019-00091-0

[16] N. Salepci and J.-Y. Welschinger. Asymptotic topology of random subcomplexes in a finite simplicial complex. *International Mathematics Research Notices*, (17):5219-5255, 2019. https://doi.org/10.1093/imrn/rnx272

[17] N. Salepci and J.-Y. Welschinger. Tilings, packings and expected Betti numbers in simplicial complexes. *arXiv:1806.05084*, 2018.

[18] N. Salepci and J.-Y. Welschinger. Upper estimates for the expected Betti numbers of random subcomplexes. *Topology Appl.*, 355 (2024), Paper No. 109010. https://doi.org/10.1016/j.topol.2024.109010

[19] P. Sarnak and I. Wigman. Topologies of nodal sets of random band limited functions. In *Advances in the theory of automorphic forms and their L-functions*, volume 664 of *Contemporary Mathematics*, pages 351-365. Amer. Math. Soc., Providence, RI, 2016. https://doi.org/10.1090/conm/664/13040

[20] J. Stirling. *Methodus Differentialis: sive Tractatus de Summatione et Interpolatione Serierum Infinitarum*. Londini, 1730.

[21] O. Viro. Dequantization of real algebraic geometry on logarithmic paper. In *European Congress of Mathematics, Vol. I (Barcelona, 2000)*, volume 201 of *Progress in Mathematics*, pages 135-146. Birkhäuser, Basel, 2001.

[22] O. Y. Viro. Gluing of plane real algebraic curves and constructions of curves of degrees 6 and 7. In *Topology (Leningrad, 1982)*, volume 1060 of *Lecture Notes in Mathematics*, pages 187-200. Springer, Berlin, 1984. https://doi.org/10.1007/BFb0099934
UNIVERSITÉ CLAUDE BERNARD LYON 1, CNRS UMR 5208, INSTITUT CAMILLE JORDAN, 43 BLVD. DU 11 NOVEMBRE 1918, F-69622 VILLEURBANNE CEDEX, FRANCE

Email address: salepci@math.univ-lyon1.fr, welschinger@math.univ-lyon1.fr