Proof of Proposition 1. To establish the convergence to $\sigma_{k,\ell}$ , we divide (13) by n and compare the resulting terms with the definition of $\sigma_{k,\ell}$ . Then $| n^{-1}\mathsf{Cov}(\mathcal C_{n, k}, \mathcal C_{n, \ell}) - \sigma_{k,\ell} |$ can be bounded by the sum of the single-integral and double-integral terms. For the single-integral term (i.e. the first two terms in (13)), the difference from the limiting expectation $\int_0^1 \mathbb E[\mathcal C_k(u)\mathcal C_\ell(u)] \textrm{d} u$ is controlled by Lemma 3 (which handles the product of expectations) and Lemma 4 (which handles the covariance term). Consequently, these lemmas ensure that the error is of order $O(n^{-\zeta(\gamma)})$ . For the double-integral term (i.e. the third term in (13)), the difference arises from the truncation of the spatial domain to $\overline{\mathbb{T}}_n$ and the approximation of the clique counts. Lemma 5(i) bounds the error due to truncating the domain (i.e. restricting from $\bar{\mathbb{R}}$ to $\overline{\mathbb T}_n$ ), while Lemma 5(ii) bounds the error due to approximating $\mathcal C_{k,\ell}$ by $\mathcal C_{n,k,\ell}$ . Both errors are shown to be $O(n^{-(1-\gamma)})$ . Since $\zeta(\gamma) < 1-\gamma$ , the total error is dominated by $O(n^{-\zeta(\gamma)})$ , which completes the proof.

Proof of Lemma 3. To prove part (i), we consider the highest marked point $Q=(Z, W)$ of the k-clique. Apart from a point (0, u), the remaining $k - 2$ points of the k-clique lie in $N^{\mathsf{do}}(Q)$ . We recall from Lemma 2 that the number of down-neighbors of Q is Poisson distributed with a parameter bounded above by a value independent of W and n. Therefore, the number of k-cliques having (0, u) as the lowest marked vertex is bounded by the number of $ (k-2)$ -tuples of points in the down-neighborhood of Q. It follows from Lemma 2 that the desired upper bound is given by

(14) \begin{align} {\mathbb E \Bigg[\sum_{Q\in N^{\mathsf{up}}(u)\cap \mathcal P}\hspace{-.3cm}\mathcal P(N_{}^{\mathsf{do}}(Q)^{k-2})\Bigg]=\int_{N^\mathsf{up}(u)}\hspace{-.3cm}\mathbb{E}[\mathcal P(N_{}^{\mathsf{do}}(q))^{k-2}]\,\textrm{d} q \leqslant\int_{N^{\mathsf{up}}(u)}\hspace{-.4cm}\mathbb E\big[\mathsf{Poi}(c_-)^{k-2}\big]\,\textrm{d} q \in O(u^{-\gamma})}.\end{align}

Next we continue with part (ii). As stated in the paragraph preceding the proof of Lemma 3, it suffices to give an upper bound on the expected number of cliques in $\mathbb C_{n,k}(u)$ . Then we distinguish two (non-exclusive) cases. In the first case, the highest marked point (Z, W) is outside $[\!-n_1, n_1]\times [0, 1]$ . In the other case, at least one of the down-neighbors of (Z, W) has at least distance $n_1$ from it.

Note that if $q = (z, w)\in \overline{\mathbb T}_n$ is such that $q \to (0, u)$ , with $|z| > n_1$ , then $ | z | u^{ \gamma }w^{ 1-\gamma }<\beta$ , and so $ u = u^{ \gamma + 1-\gamma }\leqslant u^{ \gamma }w^{ 1-\gamma }< n_2^{-1}$ . Hence, in the first case, using the same technique as in (14) and writing $B(0,n_1)= \{ z \colon |z|\leqslant n_1 \}$ , we obtain

(15) \begin{align} & \int_0^{n_2^{-1}} u^{-\gamma} \mathbb E\big[\mathcal P(N^\mathsf{up}(u)\setminus B(0, n_1))\big] \,\textrm{d} u \leqslant\int_{0}^{n_2^{-1}}u^{-\gamma }\int_{B(0,n_1)^c\times (u,1)}\unicode{x1D7D9}\big\{q\to (0,u)\big\}\,\textrm{d} q \nonumber\\ & \quad \leqslant \int_{0}^{n_2^{-1}}u^{-\gamma }\int_{B(0,n_1)^c\times (u,1)}\unicode{x1D7D9}\{ | z | <\beta u^{-\gamma }w^{\gamma -1}\}\,\textrm{d} z \,\textrm{d} w \nonumber\\ & \quad \leqslant c\int_0^{n_2^{-1}}\int_u^1 u^{-\gamma} (\beta u^{-\gamma}w^{\gamma - 1} - n_1)_+ \,\textrm{d} w\, \textrm{d} u \nonumber\\ & \quad \leqslant C\int_0^{n_2^{-1}} u^{-2\gamma}(1\wedge ( u^\gamma n_2 )^{-1/({1 - \gamma})})^\gamma \,\textrm{d} u.\end{align}

Hence, there are two cases, namely $u \leqslant n_2^{-1/\gamma}$ and $u > n_2^{-1/\gamma}$ . In the first case, we obtain an upper bound of order $cu^{-2\gamma}\unicode{x1D7D9}\{u \leqslant n_2^{-1/\gamma}\}$ . In this case, integrating over u, we obtain an upper bound of order $O(n^{-(1-2\gamma)/\gamma})$ . In the second case, we also deduce from $u\leqslant w$ that $u\leqslant n_1^{-1}$ , and hence we obtain the upper bound $c\unicode{x1D7D9}\{ n_2^{-1/\gamma} \leqslant u \leqslant n_2^{-1}\}n^{-\gamma/(1-\gamma)}u^{-2\gamma - \gamma^2/(1 - \gamma)}.$ By integrating over u, we obtain an upper bound of order $O(n^{-\zeta(\gamma)} )$ .

Finally, consider the case where at least one down-neighbor of (Z, W) has distance at least $n_1$ . The computation is similar, but we still need to bound the integral of the relevant expression $u^{-2\gamma}w^{\gamma -1}\mathbb E\big[\mathcal P(N^\mathsf{do}((z,w))\setminus B(z,n_1))\big]$ over $u \leqslant w \in [0, 1]$ . Note that $ \mathbb E\big[\mathcal P\big(N^\mathsf{do}((z,w))\setminus B(z,n_1)\big)\big] \leqslant cw^{\gamma -1}(w \wedge (w^{ 1- \gamma}n_2)^{-1/\gamma})^{1 - \gamma}.$ If $w \leqslant n_2^{-1}$ then integrating over $u\leqslant w \leqslant n_1^{-1}$ gives an expression of order $O(n^{-(1 - \gamma)})$ . Second, in the case where $w \geqslant u \geqslant n_1^{-1}$ , we obtain an expression of order $\int_{n_2^{-1}}^1u^{-1}u^{-(1 - \gamma)^2/\gamma} n_2^{-(1 - \gamma)/\gamma} \textrm{d} u \in O(n^{-(1-\gamma)}).$ Finally, consider the case where $w \geqslant n_2^{-1} \geqslant u$ . Then we obtain $n_2^{-\gamma}\int_0^{n_2^{-1}} u^{-2\gamma}\, \textrm{d} u$ , which is again of order $O(n^{-(1- \gamma)})$ , thereby concluding the proof.

Proof. First, if $w \leqslant n_2^{-1}$ then we obtain an upper bound of order $O(n^{a + \gamma - 2})$ . This follows from $a < 2- \gamma$ . If $w \geqslant n_2^{-1}$ , we again obtain a bound of order $n^{-(1 - \gamma)/\gamma} \int_{n_2^{-1}}^1 w^{-(1 - \gamma)^2/\gamma - a} \textrm{d} w \in O( n^{ a +\gamma - 2}),$ which follows from $a > 3- \gamma - 1/\gamma$ .

Case 1. We assume without loss of generality that $(Z', W') \in N^\mathsf{do}((Z, W))$ , since the other case $(Z, W) \in N^\mathsf{do}((Z', W'))$ is analogous. Note that any clique point is in the down-neighborhood of (Z, W) or (Z ′, W ′). Since the number of points in the down-neighborhood is Poisson distributed with parameter of constant order, this provides a constant contribution. Hence, we need to bound the expected number of configurations

\begin{align*}(0, U) \leftarrow (Z, W) \to (Z', W') \to q\,=\!:\,(Y, V)\end{align*}

such that at least one of these edges has a length of at least $n_1$ .

Case 2. The two highest points (Z, W) and (Z ′, W ′) are different and none of them belongs to the other clique. However, by the covariance property, we know that there exists at least one common point, denoted (Z ^″, W ^″), between the two cliques. Hence, we need to bound the expected number of configurations

(16) \begin{align} (0, U) \leftarrow (Z, W) \to (Z'', W'') \leftarrow (Z', W') \to (Y, V)\end{align}

such that at least one of these edges has a length of at least $n_1$ .

Proof of Lemma 5(ii). As in the proof of part (i), to avoid redundancies, we do not treat explicitly the cases $(Z, W) = q$ or $(Z', W') = (0, u)$ . We first consider again the configuration of the form

\begin{align*}(0, U) \leftarrow (Z, W) \to (Z'', W'') \leftarrow (Z', W') \to (Y, V)\end{align*}

as in (16). If one of the edges at (0, U) or (Y, V) is longer than $n_1$ , then we conclude as in part (i).

Hence, it remains to consider the case where one of the edges $(Z, W) \to (Z'', W'')$ or $(Z', W') \to (Z'', W'')$ is at least of length $n_1$ . We deal only with the former case, since the argument for the latter is similar. If this edge goes to (Z ′, W ′) or (Z ^″, W ^″) then we can again argue as for part (i). Therefore, it remains to deal with the case where the long edge goes to a different vertex.

First, the expected number of edges of the form $(z,w)\to(\bar Z, \bar W)$ of length at least $n_1$ is of order

(17) \begin{align} Cw^{\gamma - 1}(w\wedge (w^{1 - \gamma}n_1)^{-1/\gamma})^{1 - \gamma}.\end{align}

Second, the expected number of configurations $(z, w) \to (Z'', W'')\leftarrow (Z', W')$ is of order

(18) \begin{align} C w^{\gamma - 1}\int_0^w (w'')^{-2\gamma} \,\textrm{d} w \in O\big(w^{-\gamma}\big).\end{align}

Hence, combining (17) and (18) yields the upper bound $C\int_0^1 w^{-1}(w\wedge (w^{1 - \gamma}n_2)^{-1/\gamma})^{1 - \gamma}\, \textrm{d} w$ . By Lemma 6 with $a = 1$ , this expression is of order $O(n^{-(1 - \gamma)})$ , thereby concluding the proof.

Proof of Proposition 2; $\Gamma_3'$ -term in (8). Let $\eta \in (1,2)$ with $(\eta + 1)\gamma < 1$ . Then, by Proposition 3, $\sup_{n \geqslant 1}\mathbb E[(D_u\mathcal C_{n, k})^{\eta+ 1}] \leqslant u^{-(\eta + 1)\gamma + o(1)}$ . Hence, as asserted, $ \sup_{n \geqslant 1}\int_0^1 \mathbb E[(D_u\mathcal C_{n, k})^{\eta + 1}]\,\textrm{d} u < \infty.$

Proof. Since $|[x- u^{-\gamma}, x+ u^{-\gamma}]| =2u^{-\gamma}$ , we may assume that $| y| \geqslant \beta u^{-\gamma}$ . Also,

\begin{align*}s_\wedge(u, |y|)^{a\gamma} \leqslant u^{-a\gamma^2/(1 - \gamma)} (\beta ^{-1}|y|)^{-a\gamma/(1 -\gamma)}.\end{align*}

We first assume $a\gamma/(1 - \gamma) > 1$ so that $\int_1^{\infty} r^{-a\gamma/(1- \gamma)} \textrm{d} r < \infty.$ Hence, using the change of variables $r\,:\!=\, \beta^{-1}u^\gamma | y| $ , we have $\int_{\mathbb R}s_\wedge(u,|y|)^{a\gamma}\,\textrm{d} y\in O(u^{-\gamma}).$ Second, we consider the case $a\gamma/(1 - \gamma) < 1$ . Then, by (19), we have $| y| \leqslant2\beta u^{-1}$ . Moreover, $u^{-a\gamma^2/(1-\gamma)}\int_0^{2\beta u^{- 1}} r^{-a\gamma/(1- \gamma)} \textrm{d} r \in O(u^{-(1 - a\gamma)}),$ as asserted. The case $a\gamma=(1 - \gamma)$ is similar except for the appearance of a new $\log (1/u)$ term.

Proof of Proposition 2; $\Gamma'_2$ -term in (7). We need to control the expression

(20) \begin{align} \int_{0}^1 \left(\int_{\overline{\mathbb T}_n} \max_{k \leqslant k_0} \mathbb E\big[(D_{u,q} \mathcal C_{n, k})^{2\eta} \big]^{1/\eta} \, \textrm{d} q \right)^\eta \,\textrm{d} u,\end{align}

where $q=(y,v)$ . We first assume that $u \leqslant v$ . Then, by Proposition 4 with $a=2\eta$ ,

(21) \begin{align} \max_{k\leqslant k_0}\mathbb E\left[( D_{ u, q}\mathcal C_{n, k})^{2\eta}\right]^{1/\eta} \leqslant C\big(\unicode{x1D7D9} \{q \in N^{\mathsf{up}}(u)\} + v^{-2\gamma} s_\wedge(u, |y|)^{2\gamma}\big).\end{align}

For the inner integral, note that $|N^{\mathsf{up}}(u)|\in O( u^{-\gamma})$ . Also, we can apply Lemma 7, which gives $\int_{\mathbb R} s_\wedge(u,|y|)^{2\gamma} \,\textrm{d} y \in O\big(u^{-\tau(\gamma)}\log\!(1/u)\big)$ . Hence, we bound (20) as $\int_0^1u^{-\tau(\gamma)\eta} (\log u^{-1})^\eta\, \textrm{d} u <\infty$ .

Next we assume $v < u$ . Then (21) holds with the roles of u and v reversed. Note that $|N^{\mathsf{do}}(u)|\leqslant C$ . Moreover, we apply Lemma 7 to compute the integral over y, resulting in

\begin{align*}\int_0^1\int_{\mathbb R}s_\wedge(v, |y|)^{2\gamma}\, \textrm{d} y\,\textrm{d} v \leqslant c\int_0^1 v^{-\tau(\gamma)}\log\!(1/v) \,\textrm{d} v< \infty.\end{align*}

Since $\int_0^1u^{-2\gamma\eta} \textrm{d} u<\infty$ , we can conclude the proof.

Proof of Proposition 2; $\Gamma'_1$ -term in (6). First, note that $\mathbb E\left[(D_q\mathcal C_{n, k})^{2\eta}\right]^{1/(2\eta)} \in O(v^{-\gamma})$ by Proposition 3. We now first consider the case, where $u \leqslant v$ . We note that integrating $u^{-\gamma}v^{-1}$ for $v \geqslant u$ gives an expression of order $O(u^{-\gamma}\log\!(1/u))$ . This quantity is integrable, even when raised to the power $\eta$ . Hence, by Proposition 4 with $a=2\eta$ it remains to deal with the quantity

(22) \begin{align} Cv^{-2\gamma} s_\wedge(u, |y|)^\gamma.\end{align}

We integrate this expression over y, applying Lemma 7. This yields a term of order $O\big(u^{-(1-\gamma)}\log\!(1/u)v^{-2\gamma}\big)$ . Hence, integration over u and v shows the asserted finiteness, even after raising to the power $\eta$ .

Next we deal with the case $v<u$ . Here we need to first consider the integral of $v^{-2\gamma}u^{\gamma - 1}$ over $v \leqslant u$ , which is of order $O(u^{-\gamma})$ . Hence, this quantity is integrable even after raising to the power $\eta$ . Now, analogous to (22), we need to deal with $Cu^{-\gamma}v^{-\gamma} s_\wedge(v, |y|)^\gamma$ . If $|y| \leqslant v^{-\gamma}$ , then we obtain an expression of order $Cu^{-\gamma}v^{-2\gamma}$ . Hence, we can from now on assume that $|y|>v^{-\gamma}$ . Again, we can integrate out y using Lemma 7 and are reduced to the quantity $Cu^{-\gamma}v^{-1}$ . Now, integrating v in the domain $n^{-1/\gamma} \leqslant v \leqslant u$ , we obtain an expression of order $C u^{-\gamma} \log\!(n^{1/\gamma}u)$ . Raising to the power $\eta$ and integrating, we obtain a quantity of order $(\log n)^\eta$ .

Proof of Lemma 1. Proof of (i): For ease of description, we will establish an upper bound for $\mathsf{Cov}(T_1^{(n)}, \, T_{k+2}^{(n)})$ . First, the Mecke formula yields

(29) \begin{align}\mathsf{Cov}(T_1^{(n)},T_{k+2}^{(n)}) &= \int_{[0, 1]^2} \int_{[k+1,k+2] \times [0, 1]}\mathsf{Cov} \big( D_\mathsf{in}(x,u), D_\mathsf{in}(y,v) \big)\,\textrm{d} y \textrm{d} v \textrm{d} x \textrm{d} u + R_k^{(n)} \nonumber\\&= \int_{[0, 1]^2} \int_{[k+1,k+2] \times [0, 1]} \nonumber\\ & \quad\times \mathbb E \Bigg[ \sum_{(P_1, \dots, P_{m-1})\in (\mathcal P_n)_{\neq}^{m-1}}\sum_{\substack{(Q_1,\dots,Q_{m-1})\in (\mathcal P_n)_{\neq}^{m-1}, \nonumber\\ |(P_1,\dots,P_{m-1})\cap (Q_1,\dots,Q_{m-1})|\geqslant 1}} \hspace{-10pt}h\big( (x,u),P_1,\dots,P_{m-1} \big)\,\nonumber\\ &\quad\times h\big( (y,v),Q_1,\dots,Q_{m-1} \big)\Bigg] \, \textrm{d} y \,\textrm{d} v \,\textrm{d} x \,\textrm{d} u + R_k^{(n)}, \end{align}

where h is the indicator function for the event that there exists a graph homomorphism from $\mathsf T$ to an induced graph defined on $\big\{ (0,u), P_1,\dots, P_{m-1} \big\}$ , with the root $\mathsf{r}$ mapped to (0, u). See also (47) below. Moreover, $R_k^{(n)}$ is a negligible remainder term; see [Reference Hirsch and Juhasz8, Lemma 10] for its detailed calculations.

As argued in [Reference Hirsch and Owada9] (see the proof of Proposition 2.2 therein), it suffices to consider the case $\big|(P_1,\dots,P_{m-1})\cap (Q_1,\dots,Q_{m-1})\big|= 1$ . Let $\mathsf T_1$ and $\mathsf T_2$ be two identical trees isomorphic to $\mathsf T$ , and select a node from each of $\mathsf T_1$ and $\mathsf T_2$ , and identify these nodes together. For the calculation of (29), it is essential to consider the configuration of nodes and edges, defined by

(30) \begin{align}\begin{split} &(z_0,w_0) \to (z_1,w_1) \to \cdots \to (z_d,w_d) \to (x,u), \\&(z_0,w_0) \to (z_1',w_1') \to \cdots \to (z_e',w_e') \to (y,v),\end{split}\end{align}

where $d, e \geqslant 0$ , and $(z_0,w_0)$ corresponds to the common node between $(P_1,\dots, P_{m-1})$ and $(Q_1, \dots, Q_{m-1})$ . If $d=0$ (respectively $e=0$ ) then $(z_0,w_0)$ directly connects to (x, u) (respectively (y, v)) without any intermediate nodes. To clarify, we assume that the nodes in the first line of (30) are taken from $\mathsf T_1$ , while all the nodes in the second line belong to $\mathsf T_2$ .

We introduce a non-negative integer $\ell_0$ as the number of leaves that have no further children and are connected to $(z_0,w_0)$ by paths consisting of nodes whose marks are all higher than $w_0$ . In addition, for every $i=1,\dots,d$ (respectively $j=1,\dots,e$ ), there are $\ell_i$ (respectively $\ell_j'$ ) leaves that have no further children and are connected to $(z_i,w_i)$ (respectively $(z_j',w_j')$ ) by the paths not including $(z_{i-1}, w_{i-1})$ (respectively $(z_{j-1}', w_{j-1}')$ ), but consisting of nodes whose marks are all higher than $w_i$ (respectively $w_j'$ ). Moreover, let p (respectively p′) denote the number of leaves that have no further children and are connected to (x, u) (respectively (y, v)) by the paths not including $(z_d,w_d)$ (respectively $(z_e',w_e')$ ), but consisting of nodes whose marks are higher than u (respectively v).

In this setting, if the common node $(z_0,w_0)$ is not taken from any leaves of either $\mathsf T_1$ or $\mathsf T_2$ , then

(31) \begin{equation} 2\ell = p + p' + \ell_0 + \sum_{i=1}^d \ell_i + \sum_{j=1}^e \ell_j'.\end{equation}

If $(z_0,w_0)$ corresponds to a leaf of either $\mathsf T_1$ or $\mathsf T_2$ but not the other, then we have

(32) \begin{equation} 2\ell = p + p' + \ell_0 + \sum_{i=1}^d \ell_i + \sum_{j=1}^e \ell_j' + 1.\end{equation}

Finally, if $(z_0,w_0)$ is taken from the leaves of both $\mathsf T_1$ and $\mathsf T_2$ , then $\ell_0=0$ , so

(33) \begin{equation} \ell = p + \sum_{i=1}^d \ell_i + 1 = p' + \sum_{j=1}^e \ell_j' + 1.\end{equation}

Notice that if $\ell_0 \geqslant 2$ , either (31) or (32) applies, whereas (32) always holds when $\ell_0 = 1$ . Moreover, (33) is valid only when $\ell_0 = 0$ .

In the sequel, we focus only on the case $\ell_0 \geqslant 1$ , as $\ell_0 = 0$ is an easier case that can be treated analogously. We assume further that either d or e is positive, because the case $d = e = 0$ is significantly simpler to study. Without loss of generality, we take $d \geqslant 1$ . In the setup and notation above, we note

\begin{align*}\max_{0\leqslant i \leqslant d-1} \mathsf{dist}_n(z_i, z_{i+1})& \vee \mathsf{dist}_n(z_d, x) \vee \mathsf{dist}_n(z_0, z_1') \vee \max_{1\leqslant j \leqslant e-1} \mathsf{dist}_n(z_j', z_{j+1}') \\ &\vee \mathsf{dist}_n(z_e', y) \geqslant \frac{k}{d+e+2},\end{align*}

where $\mathsf{dist}_n(x,y) \,:\!=\, \min_{z\in \mathbb Z} |x-y+nz|$ denotes the toroidal metric on $\mathbb T_n$ . By the symmetry between $(z_i)$ and $(z_j')$ , along with the assumption $d \geqslant 1$ , it is sufficient to consider the following two cases.

(34) \begin{align}\begin{split} &\text{Case (i)}: \ \mathsf{dist}_n(z_q, z_{q+1}) \geqslant\frac{k}{d+e+2} \text{ for some } q=0,\dots,d-1. \\&\text{Case (ii)}: \ \mathsf{dist}_n(z_d, x) \geqslant \frac{k}{d+e+2}.\end{split}\end{align}

The required arguments in both cases are similar, so we discuss case (i) only. In case (i), we have

\begin{align*}\frac{k}{d+e+2} \leqslant \mathsf{dist}_n(z_q, z_{q+1}) \leqslant \beta w_{q+1}^{-\gamma} w_q^{\gamma-1} \leqslant \beta u^{-\gamma} w_q^{\gamma-1} \leqslant \beta u^{-1},\end{align*}

and thus

(35) \begin{equation} w_q\leqslant \Big( \frac{(d+e+2)\beta}{ku^\gamma} \Big)^{1/(1-\gamma)}\, \unicode{x1D7D9} \Big\{ u \leqslant \frac{(d+e+2)\beta}{k} \Big\} \,=\!:\, s(u,k).\end{equation}

Define also $s_\wedge (u,k) \,:\!=\, s(u,k)\wedge 1$ . For simplicity, we may set, without loss of generality, $(d+e+2)\beta \equiv 1$ and then recover the definition from Section 5.

Here we need to prove that the following integral is of order $O(k^{-(2 - 2\ell \gamma)})$ uniformly over n:

(36) \begin{align} &\int_{u=0}^{1} \int_{v=0}^1 \int_{w_e'=v}^1 \int_{w_{e-1}'=w_e'}^1 \cdots \int_{w_1'=w_2'}^1 \int_{w_0=w_1'}^1 \nonumber\\&\quad \times \int_{w_d=u}^1 \int_{w_{d-1}=w_d}^1 \cdots \int_{w_1=w_2}^1 w_0^{-\ell_0\gamma} \Bigg( \prod_{i=1}^d w_i^{-\ell_i\gamma} \Bigg) \Bigg( \prod_{j=1}^e (w_j')^{-\ell_j'\gamma} \Bigg)u^{-p\gamma} v^{-p'\gamma} \unicode{x1D7D9} \{ w_q\leqslant s_\wedge(u,k) \} \nonumber\\&\quad \times \int_{x=0}^1 \int_{y=k+1}^{k+2} \prod_{j=1}^e \int_{z_j'\in [0,n]} \int_{z_0\in [0,n]} \prod_{i=1}^d \int_{z_i\in [0,n]} \nonumber\\&\quad \times \unicode{x1D7D9} \big\{ (z_0,w_0) \to (z_1,w_1) \to \cdots \to (z_d,w_d) \to (x,u) \big\} \nonumber\\&\quad \times \unicode{x1D7D9} \big\{ (z_0,w_0) \to (z_1',w_1') \to \cdots \to (z_e',w_e') \to (y,v) \big\}.\end{align}

Here the integrand in the second line is obtained by repeatedly integrating out both spatial and time coordinates of all the vertices, except for those in (30). As detailed after (54) below, additional logarithmic factors may need to be multiplied into the above integrands. However, in such cases, it suffices to bound these logarithmic factors by terms such as $w_0^{-\eta \gamma}$ . For simplicity of presentation, we will ignore the appearance of logarithmic factors throughout the proof.

We start with the case $q=0$ . Dropping the condition $(z_0,w_0) \to (z_1,w_1)$ from (36) (with $q=0$ ), we calculate integrals over part of the variables in (36):

(37) \begin{align}A_{u,k}: &= \int_{v=0}^{s_\wedge(u,k)} \int_{w_e'=v}^{s_\wedge(u,k)} \int_{w_{e-1}'=w_e'}^{s_\wedge(u,k)} \cdots \int_{w_1'=w_2'}^{s_\wedge(u,k)} \int_{w_0=w_1'}^{s_\wedge(u,k)} w_0^{-\ell_0\gamma} \prod_{j=1}^e (w_j')^{-\ell_j'\gamma} v^{-p'\gamma} \\&\quad \times \int_{y=k+1}^{k+2} \prod_{j=1}^e \int_{z_j'\in [0,n]} \int_{z_0\in [0,n]} \unicode{x1D7D9} \big\{ (z_0,w_0) \to (z_1',w_1') \to \cdots \to (z_e',w_e') \to (y,v) \big\}. \notag\end{align}

Here we have suppressed the dependence on n from $A_{u,k}$ . By integrating out all spatial coordinates in (37), as well as using a bound of the type (53) below, we obtain

\begin{align*}A_{u,k} &\leqslant \int_{v=0}^{s_\wedge(u,k)} v^{-(p'+1)\gamma} \int_{w_e'=v}^{s_\wedge(u,k)} (w_e')^{-1-\ell_e'\gamma} \int_{w_{e-1}'=w_e'}^{s_\wedge(u,k)} (w_{e-1}')^{-1-\ell_{e-1}'\gamma}\times \cdots \\&\quad \times \int_{w_1'=w_2'}^{s_\wedge(u,k)} (w_1')^{-1-\ell_1'\gamma} \int_{w_0=w_1'}^{s_\wedge(u,k)} w_0^{-1-(\ell_0-1)\gamma}.\end{align*}

Since we have assumed $\ell_0\geqslant1$ , it holds that $ \int_{w_0=w_1'}^{s_\wedge(u,k)} w_0^{-1-(\ell_0-1)\gamma}\leqslant C(w_1')^{-(\ell_0-1)\gamma}.$ By carrying out all the remaining integrals, we obtain

\begin{align*}A_{u,k} \leqslant C\int_{v=0}^{s_\wedge(u,k)} v^{-(p'+\ell_0+\sum_{j=1}^e \ell_j')\gamma} = C s_\wedge(u,k)^{1-(p'+\ell_0+\sum_{j=1}^e \ell_j')\gamma},\end{align*}

where the above equality is assured by the constraint $\gamma<1/(2\ell)$ .

We continue by calculating integrals over the remaining coordinates, i.e. $(z_1,w_1), \dots, (z_d,w_d)$ , and (x, u) in (36), to estimate the expression

\begin{align*}B_k &\,:\!=\, \int_{u=0}^{1}\int_{w_d=u}^{s_\wedge(u,k)} \int_{w_{d-1}=w_d}^{s_\wedge(u,k)} \cdots \int_{w_1=w_2}^{s_\wedge(u,k)} u^{-p\gamma} s_\wedge(u,k)^{1-(p'+\ell_0+\sum_{j=1}^e \ell_j')\gamma} \prod_{i=1}^d w_i^{-\ell_i\gamma} \\&\quad \times \int_{x=0}^1 \prod_{i=1}^d \int_{z_i\in [0,n]} \unicode{x1D7D9} \big\{ (z_1,w_1) \to \cdots \to (z_d,w_d) \to (x,u) \big\} \\&\leqslant C \int_{u=0}^{1} u^{-(1+p)\gamma} s_\wedge(u,k)^{1-(p'+\ell_0+\sum_{j=1}^e \ell_j')\gamma} \int_{w_d=u}^{s_\wedge(u,k)} w_d^{-1-\ell_d\gamma} \int_{w_{d-1}=w_d}^{s_\wedge(u,k)} w_{d-1}^{-1-\ell_{d-1}\gamma} \times \cdots \\&\quad \times \int_{w_2=w_3}^{s_\wedge(u,k)} w_2^{-1-\ell_2\gamma}\int_{w_1=w_2}^{s_\wedge(u,k)} w_1^{-1-(\ell_1-1)\gamma}.\end{align*}

Observe that

(38) \begin{equation} \int_{w_1=w_2}^{s_\wedge(u,k)} w_1^{-1-(\ell_1-1)\gamma} \leqslant \begin{cases}Cw_2^{-(\ell_1-1)\gamma} & \text{if } \ell_1\geqslant1, \\Cs_\wedge(u,k)^\gamma & \text{if } \ell_1=0.\end{cases}\end{equation}

If $\ell_1\geqslant1$ , it follows from the first bound in (38), as well as (35), that

\begin{align*}B_k &\leqslant C\int_{0}^{1} u^{-(p+\sum_{i=1}^d \ell_i)\gamma} s(u,k)^{1-(p'+\ell_0+\sum_{j=1}^e \ell_j')\gamma} \,\textrm{d} u \\&= C\int_0^{1/k} u^{-(p+\sum_{i=1}^d \ell_i)\gamma} \Big( \frac{1}{ku^\gamma} \Big)^{\frac{1-(p'+\ell_0+\sum_{j=1}^e \ell_j')\gamma}{1-\gamma}} \,\textrm{d} u \\&=Ck^{-(2-(p+p'+\ell_0+\sum_{i=1}^d\ell_i +\sum_{j=1}^e \ell_j')\gamma)} \leqslant Ck^{-(2-2\ell \gamma)}.\end{align*}

The last step above holds as an equality if (31) is satisfied. On the other hand, under the condition (32), the last step is a strict inequality. If $\ell_1 = 0$ , using the second bound in (38), we observe that

\begin{align*}B_k &\leqslant C \int_{0}^{1} u^{-(1+p+\sum_{i=1}^d \ell_i)\gamma} s(u,k)^{1-(p'-1+\ell_0+\sum_{j=1}^e \ell_j')\gamma} \,\textrm{d} u\\&= C\int_0^{1/k} u^{-(1+p+\sum_{i=1}^d \ell_i)\gamma} \Big( \frac{1}{ku^\gamma} \Big)^{\frac{1-(p'-1+\ell_0+\sum_{j=1}^e \ell_j')\gamma}{1-\gamma}}\, \textrm{d} u \\&=Ck^{-(2-(p+p'+\ell_0+\sum_{i=1}^d\ell_i +\sum_{j=1}^e \ell_j')\gamma)} \leqslant Ck^{-(2-2\ell \gamma)}.\end{align*}

Now let us return to case (i) in (34) and examine the remaining case $q \in \{1, \dots, d-1\}$ . Once again, our goal is to show that the integral in (36) is bounded by the term of order $O(k^{-(2-2\ell\gamma)})$ uniformly over n. We will consider the following distinct cases, based on the size of the time coordinates. Once these cases have been treated, the proof of the assertion will be complete:

1. (I) $v > 1/k$ ;

2. (II) $w_e' > 1/k$ and $v \leqslant 1/k$ ;

3. (III) $w_{s-1}' > 1/k$ and $w_s' \leqslant 1/k$ for some $s = 2, \dots, e$ ;

4. (IV) $w_0 > 1/k$ and $w_1' \leqslant 1/k$ ;

5. (V) $w_0 \leqslant 1/k$ .

We first discuss cases (I)–(IV), which always indicates that $w_0>1/k$ . Removing the condition $(z_q,w_q)\to (z_{q+1}, w_{q+1})$ from (36), we first need to bound the integral:

\begin{align*}C_{u,k} &\,:\!=\, \int_{v=0}^1 \int_{w_e'=v}^1 \int_{w_{e-1}'=w_e'}^1 \cdots \int_{w_1'=w_2'}^1 \int_{w_0=w_1'\vee \frac{1}{k}}^1 \int_{w_q=0}^{s_\wedge(u,k)} \int_{w_{q-1} = w_q}^1 \cdots \int_{w_1=w_2}^1 \\&\qquad \qquad w_0^{-\ell_0\gamma} \Bigg( \prod_{j=1}^e (w_j')^{-\ell_j'\gamma}\Bigg) \Bigg( \prod_{i=1}^q w_i^{-\ell_i\gamma} \Bigg) v^{-p'\gamma} \\&\quad \times \int_{y=k+1}^{k+2} \prod_{j=1}^e \int_{z_j'\in [0,n]} \int_{z_0\in [0,n]} \unicode{x1D7D9} \big\{ (z_0,w_0) \to (z_1',w_1') \to \cdots \to (z_e',w_e') \to (y,v) \big\} \\&\quad \times \prod_{i=1}^q \int_{z_i\in [0,n]} \unicode{x1D7D9} \big\{ (z_0,w_0) \to (z_1,w_1) \to \cdots \to (z_q,w_q) \big\}.\end{align*}

By integrating the above expression over all spatial coordinates, we obtain

\begin{align*}&\int_{y=k+1}^{k+2} \prod_{j=1}^e \int_{z_j'\in [0,n]} \int_{z_0\in [0,n]} \unicode{x1D7D9} \big\{ (z_0,w_0) \to (z_1',w_1') \to \cdots \to (z_e',w_e') \to (y,v) \big\} \\&\quad \times \prod_{i=1}^q \int_{z_i\in [0,n]} \unicode{x1D7D9} \big\{ (z_0,w_0) \to (z_1,w_1) \to \cdots \to (z_q,w_q) \big\} \\&\leqslant C w_0^{2\gamma-2} \Bigg( \prod_{i=1}^{q-1} w_i^{-1} \Bigg) w_q^{-\gamma} \prod_{j=1}^e (w_j')^{-1} v^{-\gamma},\end{align*}

and hence

\begin{align*}C_{u,k} &\leqslant C \int_{v=0}^1 v^{-(1+p')\gamma} \int_{w_e'=v}^1 (w_e')^{-1-\ell_e'\gamma} \int_{w_{e-1}'=w_e'}^1 (w_{e-1}')^{-1-\ell_{e-1}'\gamma} \times \cdots \\&\quad \times \int_{w_1'=w_2'}^1 (w_1')^{-1-\ell_1'\gamma} \int_{w_0=w_1'\vee \frac{1}{k}}^1 w_0^{2\gamma-2-\ell_0\gamma} \\&\quad \times \int_{w_q=0}^{s_\wedge(u,k)} w_q^{-(1+\ell_q)\gamma} \int_{w_{q-1}=w_q}^1 w_{q-1}^{-1-\ell_{q-1}\gamma} \times \cdots\times \int_{w_1=w_2}^1 w_1^{-1-\ell_1\gamma}.\end{align*}

Since $\gamma<1/(2\ell)$ , it follows that

\begin{align*}\int_{w_q=0}^{s_\wedge(u,k)} w_q^{-(1+\ell_q)\gamma} \int_{w_{q-1}=w_q}^1 w_{q-1}^{-1-\ell_{q-1}\gamma} \times\cdots\times \int_{w_1=w_2}^1 w_1^{-1-\ell_1\gamma} \leqslant Cs(u,k)^{1-(1+\sum_{i=1}^q\ell_i)\gamma}.\end{align*}

Moreover, since $\ell_0\geqslant1$ with $2\gamma-1-\ell_0\gamma \leqslant \gamma-1< \frac{1}{2\ell}-1 <0$ , we deduce that $\int_{w_0=w_1'\vee \frac{1}{k}}^1 w_0^{2\gamma-2-\ell_0\gamma} \leqslant C \Big( w_1' \vee \frac{1}{k} \Big)^{2\gamma-1-\ell_0\gamma},$ so

(39) \begin{align}C_{u,k} &\leqslant C s(u,k)^{1-(1+\sum_{i=1}^q\ell_i)\gamma} \int_{v=0}^1 v^{-(1+p')\gamma} \int_{w_e'=v}^1 (w_e')^{-1-\ell_e'\gamma} \int_{w_{e-1}'=w_e'}^1 (w_{e-1}')^{-1-\ell_{e-1}'\gamma} \times \cdots \nonumber\\&\quad \times \int_{w_2'=w_3'}^1 (w_2')^{-1-\ell_2'\gamma} \int_{w_1'=w_2'}^1 (w_1')^{-1-\ell_1'\gamma} \Big( w_1' \vee \frac{1}{k} \Big)^{2\gamma-1-\ell_0\gamma}. \end{align}

We now claim that

(40) \begin{equation} C_{u,k} \leqslant Cs(u,k)^{1-(1+\sum_{i=1}^q\ell_i)\gamma}\cdot k^{-(1-p'-\ell_0-\sum_{j=1}^e \ell_j')\gamma}.\end{equation}

To prove this, we first assume the condition in case (I); that is, $v > 1/k$ . Then the corresponding terms on the right-hand side of (39) are given by

(41) \begin{align}&Cs(u,k)^{1-(1+\sum_{i=1}^q\ell_i)\gamma} \int_{v=1/k}^1 v^{-(1+p')\gamma} \int_{w_e'=v}^1 (w_e')^{-1-\ell_e'\gamma} \int_{w_{e-1}'=w_e'}^1 (w_{e-1}')^{-1-\ell_{e-1}'\gamma} \nonumber\\&\quad \times \cdots \times \int_{w_2'=w_3'}^1 (w_2')^{-1-\ell_2'\gamma} \int_{w_1'=w_2'}^1 (w_1')^{2\gamma-2-(\ell_0+\ell_1')\gamma}. \end{align}

Since $\gamma<1/(2\ell)$ and $\ell_0\geqslant1$ , we have $2\gamma-1-(\ell_0+\ell_1')\gamma<0$ , and thus

\begin{align*}\int_{w_1'=w_2'}^1 (w_1')^{2\gamma-2-(\ell_0+\ell_1')\gamma} \leqslant C (w_2')^{2\gamma-1-(\ell_0+\ell_1')\gamma}.\end{align*}

Substituting this bound and integrating the resulting expression over $(w_2', \dots, w_d')$ , we find that (41) is upper-bounded by the desired quantity

\begin{align*}&Cs(u,k)^{1-(1+\sum_{i=1}^q\ell_i)\gamma} \int_{1/k}^1 v^{-1+(1-p'-\ell_0-\sum_{j=1}^e \ell_j')\gamma} \,\textrm{d} v \\ &\quad \leqslant Cs(u,k)^{1-(1+\sum_{i=1}^q\ell_i)\gamma} \cdot k^{-(1-p'-\ell_0-\sum_{j=1}^e \ell_j')\gamma}.\end{align*}

We next turn to case (III) and assume that $w_{s-1}' >1/k$ and $w_s'\leqslant 1/k$ for some $s=2,\dots,e$ . Then the corresponding terms on the right-hand side of (39) can be bounded as follows:

\begin{align*}&C s(u,k)^{1-(1+\sum_{i=1}^q\ell_i)\gamma} \int_{v=0}^{1/k} v^{-(1+p')\gamma} \int_{w_e'=v}^{1/k} (w_e')^{-1-\ell_e'\gamma} \int_{w_{e-1}'=w_e'}^{1/k} (w_{e-1}')^{-1-\ell_{e-1}'\gamma}\\&\qquad \times \cdots\times \int_{w_{s}'=w_{s+1}'}^{1/k} (w_{s}')^{-1-\ell_{s}'\gamma} \int_{w_{s-1}'=1/k}^1 (w_{s-1}')^{-1-\ell_{s-1}'\gamma} \int_{w_{s-2}'=w_{s-1}'}^1 (w_{s-2}')^{-1-\ell_{s-2}'\gamma}\\&\qquad \times \cdots \times \int_{w_{2}'= w_{3}'}^1 (w_{2}')^{-1-\ell_{2}'\gamma} \int_{w_1'=w_2'}^1 (w_1')^{2\gamma-2-(\ell_0+\ell_1')\gamma} \\&\leqslant C s(u,k)^{1-(1+\sum_{i=1}^q\ell_i)\gamma} \cdot k^{-(2\gamma-1-\ell_0\gamma-\sum_{j=1}^{s-1} \ell_j'\gamma)} \\&\qquad \times \int_{v=0}^{1/k} v^{-(1+p')\gamma}\! \int_{w_e'=v}^{1/k} (w_e')^{-1-\ell_e'\gamma} \int_{w_{e-1}'=w_e'}^{1/k} (w_{e-1}')^{-1-\ell_{e-1}'\gamma}\times \cdots\times \! \int_{w_{s}'=w_{s+1}'}^{1/k} \!(w_{s}')^{-1-\ell_{s}'\gamma} \\&\leqslant C s(u,k)^{1-(1+\sum_{i=1}^q\ell_i)\gamma} \cdot k^{-(2\gamma-1-\ell_0\gamma-\sum_{j=1}^{s-1} \ell_j'\gamma)} \cdot k^{-1+(1+p'+\sum_{j=s}^e \ell_j')\gamma} \\&=C s(u,k)^{1-(1+\sum_{i=1}^q\ell_i)\gamma} \cdot k^{-(1-p'-\ell_0-\sum_{j=1}^e \ell_j')\gamma}.\end{align*}

For cases (II) and (IV), the required computations are almost identical to those above, and we omit detailed arguments. Now (40) has been established.

It remains to integrate the right-hand side of (40) with respect to the remaining coordinates in (36).

\begin{align*}D_k &\,:\!=\, \int_{u=0}^{1} \int_{w_d=u}^{s_\wedge(u,k)}\int_{w_{d-1}=w_d}^{s_\wedge(u,k)} \dots \int_{w_{q+1}=w_{q+2}}^{s_\wedge(u,k)} s(u,k)^{1-(1+\sum_{i=1}^q\ell_i)\gamma} \cdot k^{-(1-p'-\ell_0-\sum_{j=1}^e \ell_j')\gamma} \\&\quad \times \Bigg( \prod_{i=q+1}^d w_i^{-\ell_i\gamma} \Bigg)u^{-p\gamma} \int_{x=0}^1 \prod_{i=q+1}^d \int_{z_i\in [0,n]} \unicode{x1D7D9} \big\{ (z_{q+1}, w_{q+1}) \to \cdots \to (z_d,w_d)\to (x,u) \big\} \\&\leqslant C k^{-(1-p'-\ell_0-\sum_{j=1}^e \ell_j')\gamma} \int_{u=0}^{1} s(u,k)^{1-(1+\sum_{i=1}^q\ell_i)\gamma} u^{-(p+1)\gamma} \int_{w_d=u}^{s_\wedge(u,k)} w_d^{-1-\ell_d\gamma} \\&\quad \times \int_{w_{d-1}=w_d}^{s_\wedge(u,k)} w_{d-1}^{-1-\ell_{d-1}\gamma} \times \cdots\times \int_{w_{q+2}=w_{q+3}}^{s_\wedge(u,k)} w_{q+2}^{-1-\ell_{q+2}\gamma} \int_{w_{q+1}=w_{q+2}}^{s_\wedge(u,k)} w_{q+1}^{-1-(\ell_{q+1}-1)\gamma}.\end{align*}

The innermost integral is bounded as

(42) \begin{equation} \int_{w_{q+1}=w_{q+2}}^{s_\wedge(u,k)} w_{q+1}^{-1-(\ell_{q+1}-1)\gamma} \leqslant \begin{cases}Cw_{q+2}^{-(\ell_{q+1}-1)\gamma} & \text{if } \ell_{q+1}\geqslant1, \\[5pt]Cs(u,k)^\gamma & \text{if } \ell_{q+1}=0.\end{cases}\end{equation}

If $\ell_{q+1}\geqslant1$ , it follows from the first bound in (42) that

\begin{align*}D_k &\leqslant C k^{-(1-p'-\ell_0-\sum_{j=1}^e \ell_j')\gamma} \int_{0}^{1} s(u,k)^{1-(1+\sum_{i=1}^q\ell_i)\gamma} u^{-(p+1)\gamma} \cdot u^{-(\sum_{i=q+1}^d\ell_i-1)\gamma} \,\textrm{d} u \\&= C k^{-(1-p'-\ell_0-\sum_{j=1}^e \ell_j')\gamma} \times k^{-\frac{1-(1+\sum_{i=1}^q \ell_i)\gamma}{1-\gamma}} \int_0^{1/k} u^{-(p+\sum_{i=q+1}^d \ell_i)\gamma-\gamma \,\frac{1-(1+\sum_{i=1}^q \ell_i)\gamma}{1-\gamma}} \,\textrm{d} u \\&= C k^{-(1-p'-\ell_0-\sum_{j=1}^e \ell_j')\gamma} \times k^{-(2-(p+1+\sum_{i=1}^d \ell_i)\gamma)} \\&= Ck^{-(2-(p+p'+\ell_0+\sum_{i=1}^d \ell_i + \sum_{j=1}^e \ell_j')\gamma)} \leqslant C k^{-(2-2\ell\gamma)},\end{align*}

where the last inequality follows from (31) and (32). On the other hand, even when $\ell_{q+1}=0$ , by using the second bound in (42) and following the same calculations as above, we can obtain the same bound $D_k\leqslant Ck^{-(2-2\ell\gamma)}$ .

Since we have completed all the necessary arguments for cases (I)–(IV), we are left to deal with case (V): $w_0\leqslant 1/k$ . Removing the condition $(z_0, w_0) \to (z_1,w_1)$ from (36), we begin with the integral concerning some of the coordinates in (36).

\begin{align*}E_{u,k} &\,:\!=\, \int_{v=0}^{1/k} \int_{w_e'=v}^{1/k} \int_{w_{e-1}'=w_e'}^{1/k} \cdots \int_{w_1'=w_2'}^{1/k} \int_{w_0=w_1'}^{1/k} v^{-p'\gamma} \Bigg(\prod_{j=1}^e (w_j')^{-\ell_j'\gamma} \Bigg) w_0^{-\ell_0\gamma} \\&\quad \times \int_{y=k+1}^{k+2} \prod_{j=1}^e \int_{z_j'\in [0,n]} \int_{z_0\in [0,n]} \unicode{x1D7D9} \big\{ (z_0,w_0)\to (z_1', w_1') \to \cdots \to (z_e', w_e') \to (y,v)\big\} \\&\leqslant C \int_{v=0}^{1/k} v^{-(1+p')\gamma} \int_{w_e'=v}^{1/k} (w_e')^{-1-\ell_e'\gamma} \int_{w_{e-1}'=w_e'}^{1/k} (w_{e-1}')^{-1-\ell_{e-1}'\gamma} \times \cdots \\&\quad \times \int_{w_1'=w_2'}^{1/k} (w_1')^{-1-\ell_1'\gamma} \int_{w_0=w_1'}^{1/k} w_0^{-1-(\ell_0-1)\gamma}.\end{align*}

Since $\ell_0\geqslant1$ and $\int_{w_1'}^{1/k}w_0^{-1-(\ell_0-1)\gamma} \,\textrm{d} w_0 \leqslant C (w_1')^{-(\ell_0-1)\gamma}$ , we have

\begin{align*}E_{u,k}\leqslant C \int_{0}^{1/k} v^{-(p'+\ell_0+\sum_{j=1}^e \ell_j')\gamma}\,\textrm{d} v = C k^{-1+(p'+\ell_0+\sum_{j=1}^e \ell_j')\gamma}.\end{align*}

Substituting this bound, we calculate the integral over all the remaining coordinates,

\begin{align*}F_k &\,:\!=\, \int_{u=0}^{1/k} \int_{w_d=u}^{1/k} \int_{w_{d-1}=w_d}^{1/k} \cdots \int_{w_1=w_2}^{1/k} k^{-1+(p'+\ell_0+\sum_{j=1}^e \ell_j')\gamma} u^{-p\gamma} \prod_{i=1}^d w_i^{-\ell_i\gamma} \\&\quad \times \int_{x=0}^1 \prod_{i=1}^d \int_{z_i\in [0,n]} \unicode{x1D7D9} \big\{ (z_1,w_1)\to \cdots \to (z_d,w_d)\to (x,u) \big\} \\&\leqslant C k^{-1+(p'+\ell_0+\sum_{j=1}^e \ell_j')\gamma} \int_{u=0}^{1/k} u^{-(p+1)\gamma} \int_{w_d=u}^{1/k} w_d^{-1-\ell_d\gamma} \int_{w_{d-1}=w_d}^{1/k} w_{d-1}^{-1-\ell_{d-1}\gamma} \times \cdots \\&\quad \times \int_{w_2=w_3}^{1/k} w_2^{-1-\ell_2\gamma} \int_{w_1=w_2}^{1/k} w_1^{-1-(\ell_1-1)\gamma}.\end{align*}

The innermost integral satisfies

(43) \begin{align} \int_{w_1=w_2}^{1/k} w_1^{-1-(\ell_1-1)\gamma} \leqslant \begin{cases}Cw_2^{-(\ell_1-1)\gamma} & \text{if } \ell_1\geqslant1, \\[5pt]Ck^{-\gamma} & \text{if } \ell_1=0.\end{cases}\end{align}

If $\ell_1\geqslant1$ then by (43)

\begin{align*}F_k &\leqslant C k^{-1+(p'+\ell_0+\sum_{j=1}^e \ell_j')\gamma} \int_{0}^{1/k} u^{-(p+\sum_{i=1}^d \ell_i)\gamma} \,\textrm{d} u= Ck^{-(2-(p+p'+\ell_0+\sum_{i=1}^d \ell_i + \sum_{j=1}^e \ell_j')\gamma)} \\ &\leqslant C k^{-(2-2\ell\gamma)}.\end{align*}

If $\ell_1=0$ then by (43) again, we have

\begin{align*}F_k &\leqslant C k^{-1+(p'-1+\ell_0+\sum_{j=1}^e \ell_j')\gamma} \int_{0}^{1/k} u^{-(p+1+\sum_{i=1}^d \ell_i)\gamma}\, \textrm{d} u \\&= Ck^{-(2-(p+p'+\ell_0+\sum_{i=1}^d \ell_i + \sum_{j=1}^e \ell_j')\gamma)} \leqslant C k^{-(2-2\ell\gamma)}.\end{align*}

Proof of (ii): Using the Mecke formula for Poisson point processes, we can write

\begin{equation*} \mathsf{Var}(T_1^{(n)}) = \int_0^1 \mathbb E \big[ D_\mathsf{in}(u)^2 \big] \textrm{d} u + \int_{[0, 1]^4} \mathsf{Cov} \big( D_\mathsf{in}(x,u), D_\mathsf{in}(y,v) \big) \,\textrm{d} x \,\textrm{d} u \,\textrm{d} y \,\textrm{d} v + C_1^{(n)},\end{equation*}

where $C_1^{(n)}$ is a positive remainder term corresponding to the integral of the expectation of the count of subtrees rooted at (x, u) (respectively (y, v)) that contain a node (y, v) (respectively (x, u)). A more detailed expression for $C_1^{(n)}$ , along with its analysis, can be found in [Reference Hirsch and Juhasz8, Lemma 10]. By the Harris-Fortuin—Kasteleyn—Ginibre inequality, the covariance term above is non-negative, and hence

\begin{align*}\mathsf{Var}(T_1^{(n)}) \geqslant \int_0^1 \mathbb E \big[ D_\mathsf{in}(u)^2 \big] \,\textrm{d} u\geqslant \int_0^1 \mu_{n,\mathsf T} (u)^2 \,\textrm{d} u.\end{align*}

By appealing to Fatou’s lemma, $\liminf_{n\to\infty} \mathsf{Var}(T_1^{(n)}) \geqslant \int_0^1 \mu_{\mathsf T}(u)^2 \,\textrm{d} u>0.$ By an additional application of the Harris-Fortuin—Kasteleyn—Ginibre inequality, as well as the fact that $T_1 \stackrel{d}{=} \cdots \stackrel{d}{=} T_n^{(n)}$ , we have $\mathsf{Var}(\mathcal T_{n, T}) \geqslant n \mathsf{Var}(T_1^{(n)})$ , and now the proof of (ii) has been completed.

Proof of Theorem 2. Our proof adopts an approach similar to that used for Theorems 4.1 and 4.8 in [Reference Oliveira21]. We first regroup $\bar T_i^{(n)} = T_i^{(n)} - \mathbb E \big[T_i^{(n)} \big]$ , $i=1,\dots, n$ , into multiple blocks of size $\ell_n$ , where $\ell_n\to\infty$ and $\ell_n=o(n)$ as $n\to\infty$ . Then $m_n\,:\!=\,\lfloor n/\ell_n \rfloor$ represents, asymptotically, the number of blocks (here $\lfloor z \rfloor$ is the largest integer that does not exceed z). We define $s_n\,:\!=\,\sqrt {\mathsf{Var}(\mathcal T_{n, T})}$ and

\begin{align*}Y_{j,\ell_n}^{(n)} \,:\!=\, \sum_{i=(\,j-1)\ell_n+1}^{j\ell_n} \bar T_i^{(n)}, \ \ j=1,\dots,m_n, \ \ \ Y_{m_n+1,\ell_n}^{(n)} \,:\!=\, \sum_{i=m_n\ell_n+1}^{n} \bar T_i^{(n)}.\end{align*}

We first claim that as $n\to\infty$ ,

\begin{align*}\mathbb E \Big[\! \exp \Big\{i\frac{u}{s_n}\big(\mathcal T_{n, T}-\mathbb E[\mathcal T_{n, T}]\big)\Big\} \Big] \sim \prod_{j=1}^{m_n} \mathbb E \Big[ \exp\Big\{ i\frac{u}{s_n} Y_{j,\ell_n}^{(n)} \Big\} \Big], \ \ \ u \in \mathbb R.\end{align*}

For the proof, it is sufficient to show that for any $u\in \mathbb R$ ,

(44) \begin{align}&\bigg| \mathbb E \Big[ \exp \Big\{i\frac{u}{s_n}\big(\mathcal T_{n, T}-\mathbb E[\mathcal T_{n, T}]\big)\Big\} \Big] - \mathbb E \Big[ \exp \Big\{i\frac{u}{s_n}\sum_{j=1}^{m_n}Y_{j,\ell_n}^{(n)}\Big\} \Big] \bigg| \to 0, \ \ \ n\to\infty, \end{align}

(45) \begin{align}&\bigg| \mathbb E \Big[ \exp \Big\{i\frac{u}{s_n}\sum_{j=1}^{m_n}Y_{j,\ell_n}^{(n)}\Big\} \Big] - \prod_{j=1}^{m_n} \mathbb E \Big[ \exp\Big\{ i\frac{u}{s_n} Y_{j,\ell_n}^{(n)} \Big\} \Big] \bigg| \to 0, \ \ \ n\to\infty. \end{align}

Since $|e^{is}-e^{it}| \leqslant |s-t|$ for $s,t\in \mathbb R$ , the Cauchy–Schwarz inequality and Lemma 1(ii) give

\begin{align*}\bigg| \mathbb E \Big[ \exp \Big\{i\frac{u}{s_n}\big(\mathcal T_{n, T}-\mathbb E[\mathcal T_{n, T}]\big)\Big\} \Big] - \mathbb E \Big[ \exp \Big\{i\frac{u}{s_n}\sum_{j=1}^{m_n}Y_{j,\ell_n}^{(n)}\Big\} \Big] \bigg| \leqslant C|u| \Big\{ n^{-1} \mathbb E \big[ (Y_{m_n+1, \ell_n}^{(n)})^2 \big] \Big\}^{1/2}.\end{align*}

By Lemma 1(i), we see that $\mathbb E \big[ (Y_{m_n+1, \ell_n}^{(n)})^2 \big] \leqslant C \ell_n(1+ u_n(1)) \leqslant C\ell_n$ , and thus (44) converges to 0 as $n\to\infty$ .

For the proof of (45), we use Newman’s inequality (see Theorem 3) to ensure that (45) can be upper-bounded by

\begin{align*}\frac{u^2}{s_n^2}\, \sum_{j=1}^{m_n-1}\sum_{k=j+1}^{m_n} \mathsf{Cov} (Y_{j,\ell_n}^{(n)}, \, Y_{k,\ell_n}^{(n)}) &\leqslant \frac{Cu^2 m_n}{s_n^2}\, \sum_{p=1}^{\ell_n} \sum_{q=\ell_n+1}^n \mathsf{Cov} (T_p^{(n)}, T_q^{(n)}) \\&\leqslant \frac{Cu^2m_n\ell_n}{s_n^2}\cdot \frac{1}{\ell_n} \!\sum_{p=1}^{\ell_n} u_n(p) \leqslant \frac{C}{\ell_n} \!\sum_{p=1}^{\ell_n} p^{-(1-2\ell\gamma)}\! \to 0, \text{as } n\to\infty.\end{align*}

In the last inequality, we used Lemma 1(ii) and (12).

The proof of Theorem 2 will be completed if we can demonstrate that

\begin{align*}\prod_{j=1}^{m_n} \mathbb E \Big[\exp\Big\{ i\frac{u}{s_n}Y_{j,\ell_n}^{(n)} \Big\} \Big] \to e^{-u^2/2} \ \ \text{for all } u \in \mathbb R;\end{align*}

equivalently, $s_n^{-1} \sum_{j=1}^{m_n} \widetilde Y_{j,\ell_n}^{(n)} \stackrel{d}{\to} \mathcal N(0,1)$ , where $\big(\widetilde Y_{j,\ell_n}^{(n)}\big)_{j=1}^{m_n}$ is an i.i.d. copy of $\big(Y_{j,\ell_n}^{(n)}\big)_{j=1}^{m_n}$ . For this purpose, we note that $s_n^2 \sim \mathsf{Var}\Big( \sum_{j=1}^{m_n} \widetilde Y_{j,\ell_n}^{(n)} \Big) = m_n \mathsf{Var}(Y_{1,\ell_n}^{(n)}) \ \ \text{as } n\to\infty.$ Thus, by Slutsky’s theorem, it suffices to verify that as $n\to\infty$ ,

\begin{equation*} \frac{1}{\sqrt{m_n \mathsf{Var}(Y_{1,\ell_n}^{(n)})}} \sum_{j=1}^{m_n} \widetilde Y_{j,\ell_n}^{(n)} \stackrel{d}{\to} \mathcal N(0,1).\end{equation*}

According to the Lindeberg–Feller CLT (see Theorem 2 on p. 334 in [Reference Shiryaev26]), it is sufficient to show that for every $\epsilon>0$ ,

(46) \begin{equation} \frac{1}{m_n\mathsf{Var}(Y_{1,\ell_n}^{(n)})}\, \sum_{j=1}^{m_n} \mathbb E \bigg[ (Y_{j,\ell_n}^{(n)})^2 \, \unicode{x1D7D9} \Big\{ |Y_{j,\ell_n}^{(n)}| \geqslant \epsilon \sqrt{m_n\text{Var}(Y_{1,\ell_n}^{(n)})} \Big\}\bigg] \to 0, \ \ \ n\to\infty;\end{equation}

equivalently, $\frac{1}{s_n^2}\, \sum_{j=1}^{m_n} \mathbb E \bigg[ (Y_{j,\ell_n}^{(n)})^2 \, \unicode{x1D7D9} \Big\{ |Y_{j,\ell_n}^{(n)}| \geqslant \epsilon s_n \Big\}\bigg] \to 0$ as $n\to\infty. $ We observe the bound $(Y_{j,\ell_n}^{(n)})^2 \leqslant \ell_n \sum_{i=(j-1)\ell_n+1}^{j\ell_n}(\bar T_i^{(n)})^2$ . Thus, together with the union bound, (46) is upper-bounded by

\begin{align*}&\frac{1}{s_n^2} \sum_{j=1}^{m_n}\ell_n \sum_{i=(j-1)\ell_n+1}^{j\ell_n} \sum_{k=(j-1)\ell_n+1}^{j\ell_n} \mathbb E \Big[ (\bar T_i^{(n)})^2 \unicode{x1D7D9} \big\{ |\bar T_i^{(n)}| \geqslant \epsilon s_n/\ell_n \big\} \Big] \\ & \quad=\frac{m_n\ell_n^3}{s_n^2} \mathbb E \Big[ (\bar T_1^{(n)})^2 \unicode{x1D7D9} \big\{ |\bar T_1^{(n)}| \geqslant \epsilon s_n/\ell_n \big\} \Big].\end{align*}

Here we assume temporarily that there exists $\delta\in (0,1)$ such that $\sup_{n} \mathbb E \big[ (\bar T_1^{(n)})^{2+\delta} \big] <\infty$ . Then Hölder’s inequality yields

\begin{align*} \mathbb E \Big[ (\bar T_1^{(n)})^2 \unicode{x1D7D9} \big\{ |\bar T_1^{(n)}| \geqslant \epsilon s_n/\ell_n \big\} \Big] & \leqslant \Big\{ \mathbb E \big[ (\bar T_1^{(n)})^{2+\delta} \big] \Big\}^{\delta/({2+\delta})} \mathbb P \Big( |\bar T_1^{(n)}| \geqslant \frac{\epsilon s_n}{\ell_n} \Big)^{\delta/(2+\delta)} \\ & \leqslant C \mathbb P \Big( |\bar T_1^{(n)}| \geqslant \frac{\epsilon s_n}{\ell_n} \Big)^{\delta/({2+\delta})}.\end{align*}

By Markov’s inequality, $\mathbb P \Big( |\bar T_1^{(n)}| \geqslant \frac{\epsilon s_n}{\ell_n} \Big) \leqslant \frac{2\ell_n}{\epsilon s_n}\int_0^1 \mu^{(n)}(u)\,\textrm{d} u \leqslant \frac{C\ell_n}{s_n}.$ Substituting them, we get

\begin{align*}\frac{1}{s_n^2}\, \sum_{j=1}^{m_n} \mathbb E \Big[ (Y_{j,\ell_n}^{(n)})^2 \, \unicode{x1D7D9} \big\{ |Y_{j,\ell_n}^{(n)}| \geqslant \epsilon s_n \big\}\Big] \leqslant C\, \frac{m_n\ell_n^{3+\delta/({2+\delta})}}{s_n^{2+\delta/({2+\delta})}} \leqslant C \Big( \frac{\ell_n^{4+3\delta}}{n^{\delta/2}} \Big)^{1/(2+\delta)},\end{align*}

where the last inequality follows from Lemma 1(ii). By our taking $\ell_n\to \infty$ to ensure that $\ell_n = o\big( n^{\delta/(2(4+3\delta))} \big)$ , the final term above tends to 0 as $n\to\infty$ .

To conclude the proof, it remains to verify $\sup_{n} \mathbb E \big[ (\bar T_1^{(n)})^{2+\delta} \big] <\infty$ . Here it is enough to show that $\sup_{n} \mathbb E \big[ ( T_1^{(n)})^{2+\delta} \big] <\infty$ . By Hölder’s inequality,

\begin{align*}\mathbb E \big[ (T_1^{(n)})^{2+\delta} \big] &\leqslant \mathbb E \Big[ \mathcal P\big( [0, 1]^2\big)^{2+\delta} \max_{P\in \mathcal P\cap [0, 1]^2} D_\mathsf{in}(P)^{2+\delta} \Big] \\&\leqslant \bigg\{ \mathbb E\Big[\mathcal P\big( [0, 1]^2\big)^{(2+\delta)(1+\epsilon_0)/\epsilon_0} \Big]\bigg\}^{\epsilon_0/(1+\epsilon_0)} \bigg\{ \mathbb E\Big[ \max_{P\in \mathcal P\cap [0, 1]^2} D_\mathsf{in}(P)^{(2+\delta)(1+\epsilon_0)} \Big] \bigg\}^{1/(1+\epsilon_0)},\end{align*}

where $\epsilon_0>0$ satisfies $(2+\delta)(1+\epsilon_0)<3$ . Since the $((2+\delta)(1+\epsilon_0)/\epsilon_0)$ th moment of $\mathcal P( [0, 1]^2 )$ is clearly finite, we demonstrate only that $\sup_{n} \mathbb E [ \max_{P\in \mathcal P\cap [0, 1]^2} D_\mathsf{in}(P)^{(2+\delta)(1+\epsilon_0)} ]<\infty.$ By applying again Hölder’s inequality and Proposition 5(iii), we obtain

\begin{align*} \mathbb E \Bigg[ \max_{P\in \mathcal P\cap [0, 1]^2} D_\mathsf{in}(P)^{(2+\delta)(1+\epsilon_0)} \Bigg] \leqslant \mathbb E\Bigg[ \sum_{P\in \mathcal P\cap [0, 1]^2}\hspace{-.4cm} D_\mathsf{in}(P)^{(2+\delta)(1+\epsilon_0)} \Bigg] = \int_0^1 \mathbb E \big[ D_\mathsf{in}(u)^{(2+\delta)(1+\epsilon_0)} \big] \,\textrm{d} u.\end{align*}

Now $\mathbb E \big[ D_\mathsf{in}(u)^{(2+\delta)(1+\epsilon_0)} \big] \leqslant\Big\{ \mathbb E \big[ D_\mathsf{in}(u)^3 \big] \Big\}^{(2+\delta)(1+\epsilon_0)/3} \leqslant C (u^{-3\ell\gamma-\eta})^{(2+\delta)(1+\epsilon_0)/3}$ . It can be assured from the assumption $0 < \gamma < 1/(2\ell)$ that the integral over the final term becomes finite if we set $\delta, \epsilon_0$ , and $\eta$ sufficiently small.

Proof of Proposition 5. The proof of part (i) is provided in [Reference Hirsch and Owada9, Proposition 2.1]. For the proof of part (ii), given a directed tree $\mathsf T$ on m vertices with root $\mathsf{r}$ , we start by expressing $D_\mathsf{in}(u)$ as follows:

(47) \begin{equation} D_\mathsf{in}(u) = \sum_{(P_1,\dots,P_{m-1})\in (\mathcal P_n)_{\neq}^{m-1}} h\big( (0,u), P_1,\dots, P_{m-1}\big),\end{equation}

where $(\mathcal P_n)_{\neq}^{m-1} =\big\{ (P_1,\dots,P_{m-1})\in \mathcal P_n^{m-1}\,:\, P_i \neq P_j \text{ for } i \neq j \big\}$ and h is an indicator function for the event that there exists a graph homomorphism from $\mathsf T$ to an induced graph defined on $\big\{ (0,u), P_1,\dots, P_{m-1} \big\}$ , with the root $\mathsf{r}$ mapped to (0, u).

By the Mecke formula for Poisson point processes,

(48) \begin{align} \mathsf{Var}\big( D_\mathsf{in}(u)\big) &=\mathbb E \Bigg[ \sum_{(P_1, \dots, P_{m-1})\in (\mathcal P_n)_{\neq}^{m-1}}\sum_{(Q_1,\dots,Q_{m-1})\in (\mathcal P_n)_{\neq}^{m-1}} \hspace{-10pt}h\big( (0,u),P_1,\dots,P_{m-1} \big)\, \nonumber\\&\quad\times h\big( (0,u),Q_1,\dots,Q_{m-1} \big) \Bigg] - \mu_{n,\mathsf T}(u)^2 \nonumber\\&=\mu_{n,\mathsf T}(u) + \mathbb E \Bigg[ \sum_{(P_1, \dots, P_{m-1})\in (\mathcal P_n)_{\neq}^{m-1}} \hspace{-10pt}\sum_{\substack{(Q_1,\dots,Q_{m-1})\in (\mathcal P_n)_{\neq}^{m-1}, \nonumber\\ 1\leqslant |(P_1,\dots,P_{m-1})\cap (Q_1,\dots,Q_{m-1})|\leqslant m-2}} \hspace{-20pt}h\big( (0,u),P_1,\dots,P_{m-1} \big)\, \nonumber\\ &\quad \times h\big( (0,u),Q_1,\dots,Q_{m-1} \big) \Bigg].\end{align}

The asymptotics of $\mu_{n,\mathsf T}(u)$ is already given in part (i), so we focus on the second term above. As discussed in the proof of Proposition 2.2 in [Reference Hirsch and Owada9], it suffices to consider the case where $|(P_1,\dots,P_{m-1})\cap (Q_1, \dots, Q_{m-1})|=1$ . Let $\mathsf T_i$ , $i=1,2$ , be copies of $\mathsf T$ , and select one node from each of $\mathsf T_1$ and $\mathsf T_2$ , identifying the two selected nodes together. For the calculation of the second term in (48), we focus on the configuration of nodes and edges defined by

(49) \begin{align}\begin{split} &(z_0, w_0) \to (z_1,w_1) \to \cdots \to (z_d,w_d) \to (0,u), \\&(z_0, w_0) \to (z_1',w_1') \to \cdots \to (z_e',w_e') \to (0,u),\end{split}\end{align}

for some $d, e \geq 0$ , where $(z_0,w_0)$ represents a common node between $(P_1,\dots,P_{m-1})$ and $(Q_1,\dots, Q_{m-1})$ . In the special case of $d=0$ or $e=0$ , the point $(z_0,w_0)$ directly connects to (0, u). If $d=e=0$ then (49) does not imply a cycle, whereas a cycle occurs only when either d or e is positive. Since the latter case is more fundamental, we assume below that $e\geqslant1$ . To clarify the configuration in (49), we assume, without loss of generality, that the path in the first line of (49) is taken from $\mathsf T_1$ , while the path in the second line is taken from $\mathsf T_2$ . Furthermore, assume there are $\ell_0$ leaves with no further children that are connected to $(z_0,w_0)$ by the paths, all consisting of nodes with marks higher than $(z_0,w_0)$ . Similarly, for each $i=1,\dots,d$ (respectively $j=1,\dots,e$ ), there are $\ell_i$ (respectively $\ell_j'$ ) leaves with no further children that are connected to $(z_i,w_i)$ (respectively $(z_j',w_j')$ ) through paths that do not include $(z_{i-1}, w_{i-1})$ (respectively $(z_{j-1}', w_{j-1}')$ ) and consist entirely of nodes with marks higher than $(z_i,w_i)$ (respectively $(z_j',w_j')$ ). In addition, let p denote the number of leaves with no further children that are connected to (0, u) through paths containing neither $(z_d,w_d)$ nor $(z_e',w_e')$ , where all nodes on these paths have marks higher than (0, u).

Under this setup, there are three possible cases regarding the relation between $\ell$ and the other integers p, $\ell_i$ , and $\ell_j$ . First, if the node $(z_0,w_0)$ is not taken from the leaves of either $\mathsf T_1$ or $\mathsf T_2$ , then

(50) \begin{equation} 2\ell = p + \ell_0 + \sum_{i=1}^d \ell_i + \sum_{j=1}^e \ell_j'.\end{equation}

Second, if $(z_0,w_0)$ corresponds to a leaf in only one of $\mathsf T_1$ or $\mathsf T_2$ but not both, then

(51) \begin{equation} 2\ell = p + \ell_0 + \sum_{i=1}^d \ell_i + \sum_{j=1}^e \ell_j' + 1.\end{equation}

Finally, if $(z_0,w_0)$ corresponds to a leaf in both $\mathsf T_1$ and $\mathsf T_2$ , then $\ell_0=0$ and

(52) \begin{equation} \ell = \frac{p}{2} + \sum_{i=1}^d \ell_i +1 = \frac{p}{2} + \sum_{j=1}^e \ell_j' +1.\end{equation}

It is important to note that if $\ell_0 \geq 2$ then either (50) or (51) applies. If $\ell_0 = 1$ , (51) always applies. Lastly, (52) applies only when $\ell_0 = 0$ .

Before continuing, we describe how to integrate out the indicator for the presence of an edge over spatial coordinates under the toroidal metric: for $(x,u), (y,v) \in \overline{\mathbb T}_n$ with $0 \leqslant u \leqslant v \leqslant 1$ ,

(53) \begin{align}\int_{[0,n]} \unicode{x1D7D9} \big\{ (y,v) \to (x,u) \big\} \,\textrm{d} x &= \int_{[0,n]} \unicode{x1D7D9} \big\{ \mathsf{dist}_n (x,y) \leqslant \beta u^{-\gamma}v^{\gamma-1}\big\} \,\textrm{d} x \leqslant 2\beta u^{-\gamma}v^{\gamma-1}.\end{align}

Clearly, the same bound is obtained even when $\textrm{d} x$ above is replaced with $\textrm{d} y$ .

Returning to the second term in (48), applying the Mecke formula, and integrating out all variables except those in (49), it remains for us to bound the expression

(54) \begin{align} A_u &\,:\!=\, \int_{w_0=u}^1 \int_{w_d=u}^{w_0} \int_{w_{d-1}=w_d}^{w_0} \cdots \int_{w_1=w_2} ^{w_0} \int_{w_e'=u}^{w_0} \int_{w_{e-1}'=w_e'}^{w_0} \cdots \int_{w_1'=w_2'}^{w_0} u^{-p\gamma} \nonumber\\ &\quad \times \Big( \prod_{i=0}^d w_i^{-\ell_i\gamma}\Big)\Big( \prod_{j=1}^e (w_j')^{-\ell_j'\gamma}\Big) \nonumber\\&\quad \times \int_{z_0\in [0,n]} \prod_{i=1}^d\int_{z_i\in [0,n]} \prod_{j=1}^e \int_{z_j'\in [0,n]} \unicode{x1D7D9} \big\{ (z_0, w_0) \to (z_1,w_1) \to \cdots \to (z_d,w_d) \to (0,u) \big\} \nonumber\\&\quad \times \unicode{x1D7D9} \big\{ (z_0, w_0) \to (z_1',w_1') \to \cdots \to (z_e',w_e') \to (0,u) \big\}.\end{align}

More precisely, it may be necessary to include additional logarithmic factors in the integrands in (54). For instance, $u^{-p\gamma} (\log u^{-1})^s$ could appear instead of $u^{-p\gamma}$ for some $s=1,2,\dots$ . In such cases, we bound $(\log u^{-1})^s$ by $Cu^{-\eta}$ for a sufficiently small $\eta > 0$ . To simplify the presentation, however, these logarithmic factors are omitted throughout the remainder of the proof. Note further that $A_u$ depends on n, owing to the integration domain [0, n] (and the related toroidal metric). However, if we use bounds of the form (53), the dependence on n can be immediately removed. Consequently, the dependence on n is suppressed in (54) for simplicity, and this convention is applied throughout the proof.

By removing the condition $(z_0,w_0)\to (z_1',w_1')$ and calculating the integral (54) sequentially over all spatial coordinates, we obtain

(55) \begin{align}&\int_{z_0\in [0,n]} \prod_{i=1}^d\int_{z_i\in [0,n]} \prod_{j=1}^e \int_{z_j'\in [0,n]} \unicode{x1D7D9} \big\{ (z_0, w_0) \to (z_1,w_1) \to \cdots \to (z_d,w_d) \to (0,u) \big\} \nonumber\\&\quad \times \unicode{x1D7D9} \big\{ (z_0, w_0) \to (z_1',w_1') \to \cdots \to (z_e',w_e') \to (0,u) \big\} \nonumber \\&\leqslant C u^{-2\gamma} w_0^{\gamma-1} \Big( \prod_{i=1}^d w_i^{-1} \Big) (w_1')^{\gamma-1} \Big\{ \prod_{j=2}^e (w_j')^{-1} \Big\}. \end{align}

Substituting this bound into $A_u$ , we have

(56) \begin{align} A_u &\leqslant Cu^{-(p+2)\gamma} \int_{w_0=u}^1 w_0^{-1-(\ell_0-1)\gamma} \int_{w_d=u}^{w_0} w_d^{-1-\ell_d\gamma} \int_{w_{d-1}=w_d}^{w_0} w_{d-1}^{-1-\ell_{d-1}\gamma} \nonumber\\&\quad \times \cdots \times \int_{w_1=w_2} ^{w_0} w_1^{-1-\ell_1\gamma} \int_{w_e'=u}^{w_0} (w_e')^{-1-\ell_e'\gamma} \int_{w_{e-1}'=w_e'}^{w_0} (w_{e-1}')^{-1-\ell_{e-1}'\gamma} \nonumber\\ &\quad \times \cdots \times \int_{w_{2}'=w_3'}^{w_0} (w_{2}')^{-1-\ell_{2}'\gamma} \int_{w_1'=w_2'}^{w_0} (w_1')^{-1-(\ell_1'-1)\gamma}.\end{align}

The innermost integral can be bounded as

(57) \begin{equation} \int_{w_1'=w_2'}^{w_0} (w_1')^{-1-(\ell_1'-1)\gamma} \leqslant \begin{cases}C(w_2')^{-(\ell_1'-1)\gamma} & \text{if } \ell_1'\geqslant1, \\[5pt]Cw_0^\gamma & \text{if } \ell_1'=0.\end{cases}\end{equation}

(If $\ell_1'=1$ , the upper bound takes a logarithmic form, but as noted earlier, such logarithmic terms are systematically ignored.) Suppose first that $\ell_1'\geqslant1$ . Then, by applying the first bound in (57) and sequentially integrating the resulting expression, we obtain

(58) \begin{equation} A_u \leqslant C u^{-(p+1+\sum_{i=1}^d \ell_i + \sum_{j=1}^e \ell_j')\gamma} \int_{w_0=u}^1 w_0^{-1-(\ell_0-1)\gamma} \leqslant \begin{cases}C u^{-(p+\ell_0+\sum_{i=1}^d \ell_i + \sum_{j=1}^e \ell_j')\gamma} & \text{if } \ell_0\geqslant1, \\[5pt]Cu^{-(p+1+\sum_{i=1}^d \ell_i + \sum_{j=1}^e \ell_j')\gamma} & \text{if } \ell_0=0.\end{cases}\end{equation}

Recall that if $\ell_0\geqslant1$ , either (50) or (51) holds. In both cases, the first term in (58) is further bounded by $Cu^{-2\ell\gamma}$ . If $\ell_0=0$ , (52) always holds and the second term in (58) is again upper-bounded by $Cu^{-2\ell\gamma}$ . Throughout our discussion, logarithmic terms have been systematically ignored. However, whenever they appear, they can be bounded by $Cu^{-\eta}$ for sufficiently small $\eta > 0$ . From this perspective, $A_u$ is ultimately bounded by $Cu^{-2\ell\gamma-\eta}$ , as required.

Let us return to (56) and suppose next that $\ell_1'=0$ . Then we use the second bound in (57). Similarly to the last case, we conclude that

(59) \begin{align} A_u \leqslant C u^{-(p+2+\sum_{i=1}^d \ell_i + \sum_{j=1}^e \ell_j')\gamma} \int_{w_0=u}^1 w_0^{-1-(\ell_0-2)\gamma} \leqslant \begin{cases}C u^{-(p+\ell_0+\sum_{i=1}^d \ell_i + \sum_{j=1}^e \ell_j')\gamma} & \text{if } \ell_0\geqslant2, \\[5pt]Cu^{-(p+2+\sum_{i=1}^d \ell_i + \sum_{j=1}^e \ell_j')\gamma} & \text{if } \ell_0 \in \{ 0,1 \}.\end{cases}\end{align}

Once again, it can be shown that, for any $\ell_0$ , the final term in (59) is bounded by $Cu^{-2\ell\gamma}$ . More precisely, because of the possible contribution of logarithmic terms, we have $A_u \leq Cu^{-2\ell\gamma - \eta}$ for sufficiently small $\eta > 0$ .

Proof of (iii): It follows from the Mecke formula that

(60) \begin{align} \mathbb E \big[ D_\mathsf{in}(u)^3 \big] &= \mu_{n,\mathsf T}(u)^3 \nonumber\\&\quad + \mathbb E \bigg[ \sum_{(P_1,\dots, P_{m-1}) \in (\mathcal P_n)_{\neq}^{m-1}} \sum_{(Q_1,\dots, Q_{m-1}) \in (\mathcal P_n)_{\neq}^{m-1}} \sum_{(R_1,\dots, R_{m-1}) \in (\mathcal P_n)_{\neq}^{m-1}} \hspace{-10pt} h \big( (0,u), P_1, \dots, P_{m-1} \big) \nonumber\\&\quad \times h\big( (0,u), Q_1, \dots, Q_{m-1} \big)\, h\big( (0,u), R_1, \dots, R_{m-1} \big) \, \nonumber\\&\quad \times \unicode{x1D7D9} \big\{ \text{there is at least one common point between } (P_1,\dots, P_{m-1}), \nonumber\\&\qquad (Q_1,\dots, Q_{m-1}) \text{ and } (R_1,\dots, R_{m-1}) \big\}\bigg].\end{align}

By Proposition 5(i), we observe that

(61) \begin{equation} \mu_{n,\mathsf T}(u)^3 \leqslant (C u^{-\ell \gamma -\eta/3})^3 =C u^{-3\ell\gamma-\eta}. \end{equation}

For the second term in (60), whenever a cyclic structure arises as in (49), we will collapse it by removing an edge formed by the youngest pair of nodes, such as $(z_0, w_0) \to (z_1', w_1')$ in (55). As implied by the proof of Proposition 5(ii), the removal of such an edge will not lead to a growth rate exceeding that in (61). Specifically, the removal of this edge introduces a new leaf, such as $(z_1, w_1)$ in (55), whose mark is, however, lower than that of the common node as $(z_0, w_0)$ in (55). This constraint ensures that the removal of an edge does not increase the growth rate of the second term in (60) beyond that of (61).

Finally, the remaining bounds for $\mu_{\mathsf T}(u)$ , $\text{Var}(D_\mathsf{in}(u))$ , and $\mathbb E \big[ D_\mathsf{in}(u)^3 \big]$ follow directly by our repeating the proofs of (i)–(iii) and applying the monotone convergence theorem.