2. Approximation of the beta and product beta variables

Let $B_i$ and $S_{n,i}$ be as in (1.8) and (1.9), where we treat $\textbf{x}$ in $B_i$ and $S_{n,i}$ as a realisation of the fitness sequence $\textbf{X}\,:\!=\,(X_i,i\geqslant 1)$ . In this section, we state the approximation results for $B_i$ and $S_{n,i}$ , assuming that $X_1\,:\!=\,x_1>-1$ is fixed and $(X_i,i\geqslant 2)$ are i.i.d. positive variables with $\mu\,:\!=\,\mathbb{E} X_2<\infty$ and $\mathbb{E}\big[X^p_2\big]<\infty$ for some $p>2$ . These results are later used to derive the limiting degree distribution of the uniformly chosen vertex of the PA tree and the corresponding convergence rate, where the fitness is not necessarily bounded (Theorem 2). The proofs of the upcoming lemmas are deferred to Section 9. For the approximations, we require that w.h.p., $\textbf{X}$ is such that $\sum^i_{h=2} X_h$ is close enough to its mean $(i-1)\mu$ for all i sufficiently large. So given $0<\alpha<1$ and n, we define

(2.1) \begin{equation}A_{\alpha,n} = \Bigg\{ \bigcap^\infty_{i=\lceil{\phi(n)\rceil}} \Bigg\{\Bigg|\sum^i_{h=2} X_h-(i-1)\mu\Bigg|\leqslant i^{\alpha} \Bigg\} \Bigg\},\end{equation}

where $\phi(n)=\Omega(n^{\chi})$ , with $\chi$ as in (1.2). The first lemma is due to an application of standard moment inequalities.

In the remainder of this article, we mostly work with a realisation $\textbf{x}$ of $\textbf{X}$ such that $A_{\alpha,n}$ holds, which we denote (abusively) by $\textbf{x}\in A_{\alpha,n}$ . Below we write $\mathbb{P}_{\textbf{x}}$ and $\mathbb{E}_{\textbf{x}}$ to indicate the conditioning on a specific realisation of the fitness sequence $\textbf{x}$ . The next lemma, which extends [Reference Berger, Borgs, Chayes and Saberi4, Lemma 3.1], states that $S_{n,k}\approx(k/n)^\chi$ for large enough k and n when $\textbf{x}\in A_{\alpha,n}$ . The proof relies on the fact that we can replace $\mathbb{E}_{\textbf{x}}[S_{n,k}]$ with $(k/n)^\chi$ when $\textbf{x}\in A_{\alpha,n}$ , and approximate $S_{n,k}$ with $\mathbb{E}_{\textbf{x}}[S_{n,k}]$ using a martingale argument. As $A_{\alpha,n}$ occurs w.h.p., the result allows us to substitute these $S_{n,k}$ with $(k/n)^\chi$ when we construct a coupling for the $(\textbf{x},n)$ -Pólya urn tree (Definition 4).

Lemma 2. Given a positive integer n and $1/2<\alpha<1$ , assume that $\textbf{x}\in A_{\alpha,n}$ . Then there are positive constants $C\,:\!=\,C(x_1,\mu,\alpha)$ and $c\,:\!=\,c(x_1,\mu,\alpha)$ such that

(2.2) \begin{equation}\mathbb{P}_{\textbf{x}}\left[\max\limits_{\lceil{\phi(n)}\rceil \leqslant k \leqslant n}\left|S_{n,k}-\left(\frac{k}{n} \right)^{\chi}\right|\leqslant \delta_n \right] \geqslant 1-\varepsilon_n,\end{equation}

where $\phi(n)=\Omega(n^\chi)$ , $\delta_n\,:\!=\,Cn^{-\chi(1-\alpha)/4}$ , and $\varepsilon_n\,:\!=\,cn^{-\chi(1-\alpha)/2}$ .

The last lemma is an extension of [Reference Berger, Borgs, Chayes and Saberi4, Lemma 3.2]. It says that on the event $A_{\alpha,n}$ , we can construct the urn tree by generating suitable gamma variables in place of $B_j$ in (1.8). These gamma variables are comparable to $Z_{\bar v}$ in the construction of the $\pi$ -Pólya point tree (Definition 3), after assigning Ulam–Harris labels to the vertices of the urn tree. To state the result, we recall a distributional identity. Independently of $B_j$ , let $\big(\big(\mathcal{Z}_j,\mathcal{\widetilde Z}_{j-1}\big), 2\leqslant j\leqslant n\big)$ be conditionally independent variables such that $\mathcal{Z}_j\sim \textrm{Gamma}(x_j,1)$ and $\mathcal{\widetilde Z}_j \sim \textrm{Gamma}(T_j+j,1)$ , where $T_j\,:\!=\,\sum^j_{i=1}x_i$ . Then by the beta–gamma algebra (see e.g. [Reference Lukacs26]),

\begin{equation*}\Big(B_j, \mathcal{\widetilde Z}_{j-1}+\mathcal{Z}_{j}\Big)=_d \left(\frac{\mathcal{Z}_{j}}{\mathcal{Z}_{j}+\mathcal{\widetilde Z}_{j-1}}, \mathcal{\widetilde Z}_{j-1}+\mathcal{Z}_{j}\right) \quad \text{for $2\leqslant j\leqslant n$,}\end{equation*}

where the two random variables on the right-hand side are independent. Using the law of large numbers, we prove the following.

Lemma 3. Given a positive integer n and $1/2<\alpha<3/4$ , let $\mathcal{Z}_{j}$ and $\mathcal{\widetilde Z}_j$ be as above. Define the event

(2.3) \begin{equation}E_{\varepsilon,j}\,:\!=\, \left\{\left|\frac{\mathcal{Z}_{j}}{\mathcal{Z}_{j}+\mathcal{\widetilde Z}_{j-1}}-\frac{\mathcal{Z}_{j}}{(\mu+1)j}\right|\leqslant \frac{\mathcal{Z}_{j}}{(\mu+1)j}\varepsilon\right\} \quad for\ 2\leqslant j\leqslant n.\end{equation}

When $\textbf{x}\in A_{\alpha,n}$ , there is a positive constant $C\,:\!=\,C(x_1,\alpha,\mu)$ such that

(2.4) \begin{equation}\mathbb{P}_{\textbf{x}}\Bigg[\bigcap^n_{j= \lceil{\phi(n)}\rceil}E_{\varepsilon,j} \Bigg]\geqslant 1-C(1+\varepsilon)^4\varepsilon^{-4}n^{\chi(4\alpha-3)},\end{equation}

where $\phi(n)=\Omega(n^\chi)$ . In addition,

(2.5) \begin{equation}\mathbb{P}_{\textbf{x}}\Bigg[\bigcap^n_{j= \lceil{\phi(n)}\rceil} \big\{\mathcal{Z}_{j}\leqslant j^{1/2}\big\}\Bigg] \geqslant 1-\sum^n_{j= \lceil{\phi(n)}\rceil} j^{-2} \prod^{3}_{\ell=0} (x_j+\ell),\end{equation}

and if $x_i \in (0,\kappa]$ for all $i\geqslant 2$ , then there is a positive constant C such that

(2.6) \begin{equation}\mathbb{P}_{\textbf{x}}\Bigg[\bigcap^n_{j= \lceil{\phi(n)}\rceil} \big\{\mathcal{Z}_{j}\leqslant j^{1/2}\big\}\Bigg]\geqslant 1- C \kappa^4 n^{-\chi}.\end{equation}

3. Offspring distributions of the type-L and type-R parents in the urn tree

Recall that a vertex in the local neighbourhood of the uniformly chosen vertex $k_0$ in the PA tree is a type-L parent if it lies on the path from the uniformly chosen vertex to the initial vertex; otherwise it is a type-R parent. In this section, we state two main lemmas for the $(\textbf{x},n)$ -Pólya urn tree (Definition 4). In Lemma 5, for the root $k_0$ or any type-L or type-R parent in the subsequent generations of the local neighbourhood, we encode its type-R children in a suitable Bernoulli sequence that we introduce later. In Lemma 6, we construct the distribution of the type-L children of the uniformly chosen vertex and any type-L parent in the subsequent generations. For vertex $k_0$ , these results follow immediately from the urn representation in Definition 4 and Theorem 3. For the non-root vertices, these lemmas cannot be deduced from Definition 4. Instead, we need an urn representation that accounts for the conditioning effects of the edges uncovered in the neighbourhood exploration. For more detail on this variant, we refer to the supplementary article [Reference Lo24, Section 10].

3.1. Breadth-first search

To construct the offspring distributions, we need to keep track of the vertices that we discover in the local neighbourhood. For this purpose, we have to precisely define the exploration process. We start with a definition.

Definition 5. (Breadth-first order.) Write $\bar w <_{UH} \bar y$ if the Ulam–Harris label $\bar w$ is smaller than $\bar y$ in the breadth-first order. This means that either $|\bar w|< |\bar y|$ , or when $\bar w=(0,w_1,\ldots,w_q)$ and $\bar y=(0,v_1,\ldots,v_q)$ , $w_j<v_j$ for $j=\min\{l\,:\, v_l\not= w_l\}$ . If $\bar w$ is either smaller than or equal to $\bar y$ in the breadth-first order, then we write $\bar w \leqslant_{UH} \bar y$ .

As examples, we have $(0,2,3)<_{UH} (0,1,1,1)$ and $(0,3,1,5)<_{UH} (0,3,4,2)$ . We run a breadth-first search on the $(\textbf{x},n)$ -Pólya urn tree $G_n$ as follows.

Definition 6. (Breadth-first search (BFS).) A BFS of $G_n$ splits the vertex set $V(G_n)=[n]$ into the random subsets $(\mathcal{A}_t, \mathcal{P}_t, \mathcal{N}_t)_{t\geqslant 0}$ as follows, where the letters respectively stand for active, probed, and neutral. We initialise the search with

\begin{equation*}(\mathcal{A}_0, \mathcal{P}_0, \mathcal{N}_0)=(\{k_0\}, \varnothing, V(G_n)\setminus\{k_0\}),\end{equation*}

where $k_0$ is uniform in [n]. Given $(\mathcal{A}_{t-1}, \mathcal{P}_{t-1}, \mathcal{N}_{t-1})$ , where each vertex in $\mathcal{A}_{t-1}\cup\mathcal{P}_{t-1}$ already receives an additional Ulam–Harris label of the form $\bar w$ , $(\mathcal{A}_t, \mathcal{P}_t, \mathcal{N}_t)$ is generated as follows. Let $v[1]=k_0$ , and let $v[t]\in \mathbb{N}$ be the vertex in $\mathcal{A}_{t-1}$ that is the smallest in the breadth-first order:

(3.1) \begin{equation}v[t]=k_{\bar y}\in \mathcal{A}_{t-1} \quad\text{if $\bar y<_{UH} \bar w$ for all $k_{\bar w}\in \mathcal{A}_{t-1}\setminus\{k_{\bar y}\}$.}\end{equation}

Note that the Ulam–Harris labels appear as subscripts. Denote by $\mathcal{D}_t$ the set of vertices in $\mathcal{N}_{t-1}$ that either receive an incoming edge from v[t] or send an outgoing edge to v[t]:

\begin{equation*}\mathcal{D}_t \,:\!=\,\{j\in \mathcal{N}_{t-1}\,:\,\{j,v[t]\}\text{ or }\{v[t],j\}\in E(G_n)\},\end{equation*}

where $\{i,j\}$ is the edge directed from vertex j to vertex $i<j$ . Then in the tth exploration step, we probe vertex v[t] by marking the neutral vertices attached to v[t] as active. That is,

(3.2) \begin{equation}(\mathcal{A}_t, \mathcal{P}_t, \mathcal{N}_t)=(\mathcal{A}_{t-1}\setminus\{v[t]\}\cup \mathcal{D}_t, \mathcal{P}_{t-1}\cup\{v[t]\}, \mathcal{N}_{t-1}\setminus \mathcal{D}_t),\end{equation}

and if $v[t]=k_{\bar y}$ , we use the Ulam–Harris scheme to label the newly active vertices as $k_{\bar y,j}\,:\!=\,k_{(\bar y,j)}$ , $j\in \mathbb{N}$ , in increasing order of their vertex arrival times, so that $k_{\bar y,i}<k_{\bar y,j}$ for any $i<j$ . If $\mathcal{A}_{t-1}=\varnothing$ , we set $(\mathcal{A}_t, \mathcal{P}_t, \mathcal{N}_t)=(\mathcal{A}_{t-1}, \mathcal{P}_{t-1}, \mathcal{N}_{t-1})$ .

The characterisation of the BFS above is standard, and more details can be found in works such as [Reference Van der Hofstad19, Chapter 4] and [Reference Karp23]. The vertex labelling v[t] is very useful for the construction here, as the offspring distribution of v[t] depends on the vertex partition $(\mathcal{A}_{t-1}, \mathcal{P}_{t-1}, \mathcal{N}_{t-1})$ , whereas the vertex arrival times are helpful for identifying a type-L vertex in the BFS (see Lemma 4 below) and constructing the offspring distributions. On the other hand, we shall use the Ulam–Harris labels to match the vertices when we couple $G_n$ and in the $\pi$ -Pólya point tree (Definition 3). Hereafter we ignore the possibility that $v[t]=1$ , because when t (or equivalently the number of discovered vertices) is not too large, the probability that $v[t]=o(n)$ tends to zero as $n\to\infty$ . For $t\geqslant 1$ , let $v^{(op)}[t]$ (resp. $v^{(oa)}[t]$ ) be the vertex in $\mathcal{P}_{t-1}$ (resp. $\mathcal{A}_{t-1}$ ) that has the earliest arrival time, where op and oa stand for oldest probed and oldest active. That is,

(3.3) \begin{equation}v^{(op)}[t]\,:\!=\,\min\{ j\,:\, j\in \mathcal{P}_{t-1}\}\quad\text{and}\quad v^{(oa)}[t]\,:\!=\,\min\{ j\,:\, j\in \mathcal{A}_{t-1}\}.\end{equation}

3.2. Construction of the offspring distributions

To distinguish the urn tree $G_n$ from the other models, we apply Ulam–Harris labels of the form $k^{(\textrm{u})}_{\bar y}$ to its vertices; the superscript $^{(\textrm{u})}$ stands for ‘urn’. An example is given in Figure 4. Define

(3.4)

noting that $R^{(\textrm{u})}_0$ is the in-degree of the uniformly chosen vertex $k^{(\textrm{u})}_0$ in $G_n$ . Before we proceed further, we prove a simple lemma to help us identify when v[t] in (3.1) is a type-L vertex, which will be useful for constructing the offspring distributions later. The result can be understood as a consequence of the facts that in the tree setting, we uncover a new active type-L vertex immediately after we probe an active type-L vertex, and that the oldest probed vertex cannot be rediscovered as a child of another type-R vertex in the subsequent explorations.

Figure 4.

Each level corresponds to each time step of the BFS $(\textbf{x},n)$ -Pólya urn tree $G_n$ . The vertices are arranged from left to right in increasing order of their arrival times. The small and large dots correspond to the probed and active vertices, respectively. The densely dotted path joins the uniformly chosen vertex and the discovered type-L vertices. Here, $\mathcal{P}_{3}=\big\{k^{(\textrm{u})}_{0},k^{(\textrm{u})}_{0,1},k^{(\textrm{u})}_{0,2}\big\}$ , $\mathcal{A}_{3}=\big\{k^{(\textrm{u})}_{0,3},k^{(\textrm{u})}_{0,1,1},k^{(\textrm{u})}_{0,1,2},k^{(\textrm{u})}_{0,1,3},k^{(\textrm{u})}_{0,2,1}\big\}$ , $v[4]=k^{(\textrm{u})}_{0,3}$ , $v^{(op)}[4]=k^{(\textrm{u})}_{0,1}$ , and $v^{(oa)}[4]=k^{(\textrm{u})}_{0,1,1}$ .

Lemma 4. Assume that $\mathcal{A}_{t-1}\cup \mathcal{P}_{t-1}$ does not contain vertex 1. If $t=2$ , then $v^{(op)}[2]=k^{(\textrm{u})}_0$ and $v^{(oa)}[2]=k^{(\textrm{u})}_{0,1}$ , while if $2<i\leqslant t$ , then $v^{(op)}[i]$ and $v^{(oa)}[i]$ are type-L children, where $v^{(oa)}[i]$ is the only type-L child in $\mathcal{A}_{i-1}$ , and it receives an incoming edge from $v^{(op)}[i]$ .

Proof. We prove the lemma by induction on $2\leqslant i\leqslant t$ . The base case is clear, since $\mathcal{P}_{1}=\big\{k^{(\textrm{u})}_0\big\}$ and $\mathcal{A}_{1}$ consists of its type-L and type-R children. Assume that the lemma holds for some $2\leqslant i<t$ . If we probe a type-L child at time i, then $v[i]=v^{(oa)}[i]$ , and there is a vertex $u\in \mathcal{N}_{i-1}$ that receives the incoming edge emanating from v[i]. Hence, vertex u belongs to type L and $v^{(oa)}[i+1]=u$ . Furthermore, $v^{(op)}[i+1]=v[i]$ , as $v^{(op)}[i]$ sends an outgoing edge to v[i] by assumption, implying $v[i]<v^{(op)}[i]$ . If we probe a type-R vertex at time i, then $v[i]>v^{(oa)}[i]$ and we uncover vertices in $\mathcal{N}_{i-1}$ that have later arrival times than v[i]. Setting $\mathcal{P}_{i}=\mathcal{P}_{i-1}\cup \{v[i]\}$ , we have $v^{(op)}[i+1]=v^{(op)}[i]$ and $v^{(oa)}[i+1]=v^{(oa)}[i]$ , which are of type L.

To construct the offspring distributions for each parent in the local neighbourhood, we also need to define some notation and variables. Let $\mathcal{E}_0 =\varnothing$ , and for $t\geqslant 1$ , let $\mathcal{E}_t$ be the set of edges connecting the vertices in $\mathcal{A}_t \cup \mathcal{P}_t$ . Given a positive integer m, denote the set of vertices and edges in $\mathcal{P}_t$ and $\mathcal{E}_{t}$ whose arrival time in $G_n$ is earlier than that of vertex m by

(3.5) \begin{align}\mathcal{P}_{t, m}= \mathcal{P}_t\cap[m-1]\quad\text{and}\quad\mathcal{E}_{t, m}=\{\{h,i\}\in \mathcal{E}_t\,:\,i<m\},\end{align}

noting that $\{h,i\}$ is the edge directed from i to $h<i$ , and $(\mathcal{A}_0, \mathcal{P}_0, \mathcal{N}_0)=(\{k_0\}, \varnothing, V(G_n)\setminus\{k_0\})$ and $\mathcal{E}_0=\varnothing$ . Below we use [t] in the notation to indicate the exploration step, and omit $\textbf{x}$ for simplicity. Let $\big(\big(\mathcal{Z}_{i}[t], \mathcal{\widetilde Z}_{i}[t]\big), 2\leqslant i\leqslant n, i\not \in \mathcal{P}_{t-1}\big)$ be independent variables, where

(3.6) \begin{equation}\mathcal{Z}_i[1]\sim \textrm{Gamma}(x_i,1)\quad\text{and}\quad \mathcal{\widetilde Z}_i[1]\sim \textrm{Gamma}(T_{i-1}+i-1,1),\end{equation}

with $T_i\,:\!=\,\sum^i_{j=1}x_j$ . Now, suppose that $t\geqslant 2$ . Because of the size-bias effect of the edge $\{v^{(op)}[t], v^{(oa)}[t]\}\in \mathcal{E}_{t-1}$ , we define

(3.7) \begin{align}\mathcal{Z}_{i}[t]\sim \textrm{Gamma}\big(x_{i}+\mathbb{1}\big[i=v^{(oa)}[t]\big],1\big), \quad i \in \mathcal{A}_{t-1}\cup \mathcal{N}_{t-1},\end{align}

so that the initial attractiveness of the type-L child $v^{(oa)}[t]$ is $x_{v^{(oa)}[t]}+1$ . The shape parameter of $\mathcal{\widetilde Z}_i[t]$ defined below is the total weight of the vertices in $\mathcal{A}_{t-1}\cup \mathcal{N}_{t-1}$ whose arrival time is earlier than the ith time step. This is because when a new vertex is added to the $(\textbf{x},n)$ -sequential model conditional on having the edges $\mathcal{E}_{t-1}$ , the recipient of its outgoing edge cannot be a vertex in $\mathcal{P}_{t-1}$ , and it is chosen with probability proportional to the current weights of the vertices in $\mathcal{A}_{t-1}\cup \mathcal{N}_{t-1}$ that arrive before the new vertex. Hence, define

(3.8) \begin{align}\mathcal{\widetilde Z}_{i}[t]\sim\begin{cases}\textrm{Gamma}(T_{i-1}+i-1,1),&\text{if $2\leqslant i\leqslant v^{(oa)}[t]$;}\\\textrm{Gamma}(T_{i-1}+i,1),& \text{if $v^{(oa)}[t]<i<v^{(op)}[t]$;}\\\textrm{Gamma}\Big(T_{i-1}+i-\sum_{h\in \mathcal{P}_{t-1,i}}x_h - |\mathcal{E}_{t-1,i}|,1\Big),& \text{if $v^{(op)}[t]<i\leqslant n$.} \end{cases}\end{align}

Let $B_1[t]\,:\!=\,1$ and $B_i[t]\,:\!=\,0$ for $i \in \mathcal{P}_{t-1}$ , as the edges attached to $\mathcal{P}_{t-1}$ are already determined. Furthermore, define

(3.9) \begin{equation}B_i[t]\,:\!=\,\frac{\mathcal{Z}_i[t]}{\mathcal{Z}_i[t]+\mathcal{\widetilde Z}_i[t]}\quad \text{for $i \in \mathcal{A}_{t-1}\cup \mathcal{N}_{t-1}$}.\end{equation}

Let $S_{n,0}[t]\,:\!=\,0$ , $S_{n,n}[t]\,:\!=\,1$ , and

(3.10) \begin{equation}S_{n,i}[t]\,:\!=\,\prod^n_{j=i+1}\big(1-B_j[t]\big)= \prod^n_{j=i+1;j\not \in \mathcal{P}_{t-1}}\big(1-B_j[t]\big)\quad\text{for $1\leqslant i\leqslant n-1$,}\end{equation}

where the second equality is true because $B_i[t]=0$ for $i\in \mathcal{P}_{t-1}$ . Observe that by the beta–gamma algebra, $(B_i[1], 1\leqslant i\leqslant n)=_d (B_i,1\leqslant i\leqslant n)$ and $(S_{n,i}[1],1\leqslant i\leqslant n)=_d (S_{n,i},1\leqslant i\leqslant n)$ , where $B_i$ and $S_{n,i}$ are as in (1.8) and (1.9). For $t\geqslant 2$ , $(B_i[t], 1\leqslant i\leqslant n)$ are the beta variables in the urn representation of the $(\textbf{x},n)$ -sequential model, conditional on having the edges $\mathcal{E}_{t-1}$ ; see [Reference Lo24, Section 10] for details. We construct a Bernoulli sequence that encodes the type-R children of vertex v[t] as follows. For $j\in \mathcal{N}_{t-1}$ and $v[t]+1\leqslant j\leqslant n$ , let $\mathbb{1}_{\textrm{R}}[j\to v[t]]$ be an indicator variable that takes value one if and only if vertex j sends an outgoing edge to v[t]; for $j\not \in \mathcal{N}_{t-1}$ , let $\mathbb{1}_{\textrm{R}}[j\to v[t]]=0$ with probability one, since the recipient of the incoming edge from vertex j is already in $\mathcal{P}_{t-1}$ . Note that if $v[t]=k^{(\textrm{u})}_{\bar y}$ , then $R^{(\textrm{u})}_{\bar y}$ in (3.4) is equal to $\sum^n_{j=v[t]+1}\mathbb{1}_{\textrm{R}}[j\to v[t]]$ . We also assume $\mathcal{N}_{t-1}\not =\varnothing$ , because for large n, w.h.p. the local neighbourhood of vertex $k_0$ does not contain all the vertices of $G_n$ . To state the distribution of $(\mathbb{1}_{\textrm{R}}[j\to v[t]],v[t]+1\leqslant j\leqslant n)$ , we use (3.9) and (3.10) to define

(3.11) \begin{align}P_{j\to v[t]} \,:\!=\,\begin{cases}\frac{S_{n,v[t]}[t]}{ S_{n,j-1}[t]} B_{v[t]}[t], &\text{if $j \in \{v[t]+1,\ldots,n\}\cap\mathcal{N}_{t-1}$;}\\[6pt]0, &\text{if $j \in \{v[t]+1,\ldots,n\}\setminus \mathcal{N}_{t-1}$.}\end{cases}\end{align}

Definition 7. Given $(\mathcal{A}_{t-1},\mathcal{P}_{t-1},\mathcal{N}_{t-1})$ and $(B_j[t],v[t]\leqslant j\leqslant n)$ , let $Y_{j\to v[t]}$ , $v[t]+1\leqslant j\leqslant n$ be conditionally independent Bernoulli variables, each with parameter $P_{j\to v[t]}$ . Define this Bernoulli sequence by the random vector

\begin{equation*}\textbf{Y}^{(v[t],n)}_{\textrm{Be}}\,:\!=\,\left(Y_{(v[t]+1)\to v[t]},Y_{(v[t]+2)\to v[t]},\ldots,Y_{n\to v[t]}\right).\end{equation*}

With the preparations above, we are ready to state the main results of this section. For $t\geqslant 2$ , the following lemmas are immediate consequences of the urn representation of the $(\textbf{x},n)$ -sequential model conditional on the discovered edges; see [Reference Lo24, Section 10]. For $t=1$ , they follow directly from Theorem 3. The first lemma states that we can encode the type-R children of the uniformly chosen vertex (the root) or a non-root parent in the local neighbourhood in a Bernoulli sequence; see Figure 3 in the case of the root.

Lemma 5. Assume that $\mathcal{N}_{t-1} \not= \varnothing$ and $\mathcal{A}_{t-1}\cup\mathcal{P}_{t-1}$ does not contain vertex 1. Then given $\big(\mathcal{A}_{t-1},\mathcal{P}_{t-1},\mathcal{N}_{t-1}\big)$ , the random vector $\big(\mathbb{1}_{\textrm{R}}[j\to v[t]],v[t]+1\leqslant j\leqslant n\big)$ is distributed as $\textbf{Y}^{(v[t],n)}_{\textrm{Be}}$ .

Remark 3. When $|\mathcal{E}_{t-1}|=o(n)$ for some $t\geqslant 2$ , $P_{j\to v[t]}$ in (3.11) is approximately distributed as $\big(S_{n,v[t]}/S_{n,j-1}\big)B_{v[t]}$ for n sufficiently large, with $B_j$ and $S_{n,j}$ as in (1.8) and (1.9). As we shall see later in the proof, $|\mathcal{E}_{t-1}|=o(n)$ indeed occurs w.h.p.

The next lemma states that when v[t] is the uniformly chosen vertex or a type-L parent, we can use the beta variables in (3.9) to obtain the distribution of the type-L child of v[t]. Observe that $(B_j[t],2\leqslant j\leqslant v[t]-1)$ does not appear in Definition 7, but is required for this purpose.

Lemma 6. Assume that $\mathcal{N}_{t-1} \not= \varnothing$ , $\mathcal{A}_{t-1}\cup\mathcal{P}_{t-1}$ does not contain vertex 1, and v[t] is either the uniformly chosen vertex or of type L. Given $(\mathcal{A}_{t-1},\mathcal{P}_{t-1}, \mathcal{N}_{t-1})$ , let $S_{n,j}[t]$ be as in (3.10), and $U\sim \textrm{U}[0,S_{n,v[t]-1}{[t]}]$ . For $1\leqslant j\leqslant v[t]-1$ , the probability that vertex j receives the only incoming edge from v[t] is given by the probability that $S_{n,j-1}[t] \leqslant U < S_{n,j}[t]$ .

4. An intermediate tree for graph couplings

As the $\pi$ -Pólya point tree $(\mathcal{T},0)$ in Definition 3 does not have vertex labels that are comparable to the vertex arrival times of the $(\textbf{x},n)$ -Pólya urn tree $G_n$ , we define a suitable conditional analogue of the $\pi$ -Pólya point tree. We call this analogue the intermediate Pólya point tree $(\mathcal{T}_{\textbf{x},n},0)$ , where vertex 0 is its root, and the subscripts are the parameters corresponding to $\textbf{x}$ and n of $G_n$ . All the variables of $(\mathcal{T}_{\textbf{x},n},0)$ have the superscript $^{(\textrm{i})}$ (for ‘intermediate’). Each vertex of $(\mathcal{T}_{\textbf{x},n},0)$ has an Ulam–Harris label $\bar v$ , an age $a^{(\textrm{i})}_{\bar v}$ , and a type (except for the root), which are defined similarly as in Section 1.2 for $(\mathcal{T},0)$ . Moreover, vertex $\bar v$ has an additional PA label $k^{(\textrm{i})}_{\bar v}$ , which determines its initial attractiveness by taking $x_{k^{(\textrm{i})}_{\bar v}}$ . The distributions of the PA labels are constructed using gamma variables similar to (3.6), (3.7), and (3.8), and as we remark after the definition, the $k^{(\textrm{i})}_{\bar v}$ are approximately distributed as the vertices (or equivalently the arrival times) $k^{(\textrm{u})}_{\bar v}$ in the local neighbourhood in $G_n$ . Define

(4.1) \begin{equation}R^{(\textrm{i})}_{\bar v}=\text{the number of type-R children of vertex $\bar v$ in $(\mathcal{T}_{\textbf{x},n},0)$,}\end{equation}

which is analogous to $R_{\bar v}$ in (1.3) and $R^{(\textrm{u})}_{\bar v}$ in (3.4). Finally, recall $\chi$ in (1.2) and $\bar w\leqslant_{UH} \bar v$ whenever the Ulam–Harris label $\bar w$ is smaller than $\bar v$ in the breadth-first order (Definition 5).

We now discuss the distributions of the PA labels above. Clearly, $k^{(\textrm{i})}_0$ is uniform in [n]. Observe that t above is essentially the number of breadth-first exploration steps in $(\mathcal{T}_{\textbf{x},n},0)$ , starting from the root 0. Suppose that we have completed $t-1$ exploration steps in $G_n$ , starting from its uniformly chosen vertex $k^{(\textrm{u})}_0$ , and the resulting neighbourhood is coupled to that of vertex $0\in V((\mathcal{T}_{\textbf{x},n},0))$ so that they are isomorphic and $k^{(\textrm{i})}_{\bar w}=k^{(\textrm{u})}_{\bar w}$ for all ${\bar w}$ in the neighbourhoods. If $v[t]=k^{(\textrm{u})}_{\bar v}$ , a moment’s thought shows that the total weight of the vertices $\big\{k^{(\textrm{i})}_{\bar w}<j\,:\,\bar w<_{UH} \bar v\big\}$ and the vertex set $\mathcal{P}_{t-1,j}$ satisfy

\begin{equation*}W_{\bar v,j}+|\bar v|-1=\sum_{h\in \mathcal{P}_{t-1,j}}x_h+|\mathcal{E}_{t-1,j}|,\end{equation*}

where $\mathcal{P}_{t-1,j}$ and $\mathcal{E}_{t-1,j}$ are as in (3.5). Hence, recalling the gammas in (3.7) and (3.8), it follows that $\mathcal{Z}_j[t]=_d\mathcal{Z}^{(\textrm{i})}_j[t]$ and $\mathcal{\widetilde Z}_j[t]=_d\mathcal{\widetilde Z}^{(\textrm{i})}_j[t]$ for any j. So by Lemma 6, $k^{(\textrm{i})}_{\bar v}=_d k^{(\textrm{u})}_{\bar v}$ if $\bar v$ is a type-L child. However, $k^{(\textrm{i})}_{\bar v}$ is only approximately distributed as $k^{(\textrm{u})}_{\bar v}$ if $\bar v$ is of type R. The discretised Poisson point process that generates these PA labels will be coupled to the Bernoulli sequence in Lemma 5 that encodes the type-R children of $k^{(\textrm{u})}_{\bar v}$ . When $k^{(\textrm{i})}_{\bar v}=1$ , it must be the case that $\bar v=0$ or $\bar v=(0,1,\ldots,1)$ . We stop the construction in this case to avoid an ill-defined $\mathcal{Z}^{(\textrm{i})}_1[t]$ when $-1<x_1\leqslant 0$ and $k^{(\textrm{i})}_0=1$ , and also because vertex 1 cannot have a type-L neighbour, so Steps 1 and 2 are unnecessary in this case. Nevertheless, for $r<\infty$ and any vertex $\bar v$ in the r-neighbourhood of vertex $0\in V((\mathcal{T}_{\textbf{x},n},0))$ , the probability that $k^{(\textrm{i})}_{\bar v}=1$ tends to zero as $n\to\infty$ .

5. Proof of the base case

Recall that, as in Section 3.2, $k^{(\textrm{u})}_{\bar v}$ are the vertices in the local neighbourhood of the uniformly chosen vertex $k^{\textrm{(u)}}_0$ of the $(\textbf{x},n)$ -Pólya urn tree $G_n$ in Definition 4. Let $\chi$ be as in (1.2), and let $(\mathcal{T}_{\textbf{x},n},0)$ be the intermediate Pólya point tree in Definition 8, with $k^{(\textrm{i})}_{\bar v}$ and $a^{(\textrm{i})}_{\bar v}$ being the PA label and the age of vertex $\bar v$ in the tree. Here, we couple $\big(G_n,k^{(\textrm{u})}_0\big)$ and $(\mathcal{T}_{\textbf{x},n},0)$ so that with probability tending to one, $(B_1\big(G_n,k^{(\textrm{u})}_0\big),k^{(\textrm{u})}_0)\cong (B_1(\mathcal{T}_{\textbf{x},n},0),0)$ , $\big(k^{(\textrm{u})}_{0,j}/n\big)^\chi \approx a^{(\textrm{i})}_{0,j}$ , and $k^{(\textrm{u})}_{0,j}=k^{(\textrm{i})}_{0,j}$ . For convenience, we also refer to the rescaled arrival time $(k/n)^\chi$ of vertex k in $G_n$ simply as its age. Let $U_0\sim \textrm{U}[0,1]$ , $a^{(\textrm{i})}_0=U^{\chi}_0$ , and $k^{(\textrm{u})}_0=\lceil{nU_0}\rceil$ , where $a^{(\textrm{i})}_0$ is the age of vertex 0. By a direct comparison to Definition 8, $a^{(\textrm{i})}_0\approx \big(k^{(\textrm{u})}_0/n\big)^\chi$ and the initial attractiveness of vertex 0 is $x_{k^{(\textrm{u})}_0}$ . Under this coupling we define the event

(5.1) \begin{equation}\mathcal{H}_{1,0}=\big\{a^{(\textrm{i})}_0>(\log \log n)^{-\chi} \big\},\end{equation}

which guarantees that we choose a vertex of low degree in $G_n$ . To prepare for the coupling, let $\big(\big(\mathcal{Z}_j[1],\mathcal{\widetilde Z}_{j}[1]\big),2\leqslant j\leqslant n\big)$ and $\big(S_{n,j}[1],1\leqslant j\leqslant n\big)$ be as in (3.6) and (3.10). We use $\big(S_{n,j}[1],1\leqslant j\leqslant n\big)$ to construct the distribution of the type-R children of $k^{(\textrm{u})}_0$ , and for sampling the initial attractiveness of vertex $(0,1)\in V((\mathcal{T}_{\textbf{x},n},0))$ . Furthermore, let the ages $\Big(a^{(\textrm{i})}_{0,j},1\leqslant j\leqslant 1+R^{(\textrm{i})}_0\Big)$ and $R^{(\textrm{u})}_0$ be as in Definition 8 and (3.4). Below we define a coupling of $\big(G_n,k^{(\textrm{u})}_0\big)$ and $(\mathcal{T}_{\textbf{x},n},0)$ , and for this coupling we define the events that the children of $k^{(\textrm{u})}_0$ have low degrees, the 1-neighbourhoods are isomorphic and of size at most $(\log n)^{1/r}$ , and the ages are close enough as

(5.2) \begin{align}&\mathcal{H}_{1,1} = \Big\{a^{(\textrm{i})}_{0,1}>(\log \log n)^{-2\chi}\Big\},\nonumber\\&\mathcal{H}_{1,2}= \bigg\{ R^{(\textrm{u})}_0= R^{(\textrm{i})}_0,\text{ for $\bar v\in V(B_1(\mathcal{T}_{\textbf{x},n},0))$, $k^{(\textrm{u})}_{\bar v}=k^{(\textrm{i})}_{\bar v}$,}\bigg|a^{(\textrm{i})}_{\bar v}-\bigg(\frac{k^{(\textrm{u})}_{\bar v}}{n}\bigg)^\chi \bigg|\leqslant C_1 b(n)\bigg\},\nonumber\\&\mathcal{H}_{1,3}=\Big\{R^{(\textrm{i})}_0<(\log n)^{1/r}\Big\},\end{align}

where $R^{(\textrm{i})}_{0}$ and $R^{(\textrm{u})}_{0}$ are as in (3.4) and (4.1), $b(n)\,:\!=\,n^{-\frac{\chi}{12}}(\log\log n)^{\chi}$ , and $C_1\,:\!=\,C_1(x_1,\mu)$ will be chosen in the proof of Lemma 8 below. Because the initial attractivenesses of the vertices (0,j) and $k^{(\textrm{u})}_{0,j}$ match and their ages are close enough on the event $\mathcal{H}_{1,2}$ , we can couple the Bernoulli and the mixed Poisson sequences that encode their type-R children, and hence the 2-neighbourhoods. The event $\mathcal{H}_{1,1}$ ensures that the local neighbourhood of $k^{(\textrm{u})}_0$ grows slowly, and on the event $\mathcal{H}_{1,3}$ , the number of vertex pairs that we need to couple for the 2-neighbourhoods is not too large. So by a union bound argument, the probability that any of the subsequent couplings fail tends to zero as $n\to\infty$ . The aim of this section is to show that when $\sum^j_{i=2}x_i\approx (j-1)\mu$ for all j sufficiently large, we can couple the two graphs so that $\bigcap^3_{i=0}\mathcal{H}_{1,i}$ occurs w.h.p. Thus, let $A_n\,:\!=\,A_{2/3,n}$ be as in (2.1) with $\alpha=2/3$ ; this $\alpha$ is chosen to simplify the calculation, and can be chosen under the assumption of bounded fitness. Recall that $\mathbb{P}_{\textbf{x}}$ indicates the conditioning on a specific realisation of the fitness sequence $\textbf{x}$ . The main result is the lemma below; it serves as the base case when we inductively prove an analogous result for the general radius $r<\infty$ . Because $B_1(\mathcal{T}_{\textbf{x},n},0)$ approximately distributes as the 1-neighbourhood of the $\pi$ -Pólya point tree after randomisation of $\textbf{x}$ , and Lemma 1 says that $\mathbb{P}[A^c_n]=O\big(n^{-b}\big)$ for some $b>0$ , it follows from the lemma below and (1.4) that (1.5) of Theorem 1 holds for $r=1$ .

Lemma 7. Assume that $x_j\in (0,\kappa]$ for $j\geqslant 2$ and $\textbf{x}\in A_n$ . Let $\mathcal{H}_{1,j}$ , $j=0,\ldots,3$ , be as in (5.1) and (5.2). Then there is a coupling of $\big(G_n,k^{(\textrm{u})}_0\big)$ and $(\mathcal{T}_{\textbf{x},n},0)$ , with a positive constant $C\,:\!=\,C(x_1,\mu,\kappa)$ such that

(5.3) \begin{align}\mathbb{P}_{\textbf{x}}\Bigg[\Bigg(\bigcap^3_{j=0}\mathcal{H}_{1,j}\Bigg)^c\Bigg]\leqslant C (\log \log n)^{-\chi}\quad for\ all\ n\geqslant 3.\end{align}

The proof of Lemma 7 consists of several lemmas, which we now develop. From the definitions of $a^{(\textrm{i})}_0$ and $a^{(\textrm{i})}_{0,1}$ , it is obvious that the probabilities of the events $\mathcal{H}_{1,0}$ and $\mathcal{H}_{1,1}$ tend to one as $n\to\infty$ , and by Chebyshev’s inequality, we can show that this is also true for the event $\mathcal{H}_{1,3}$ . We take care of $\mathcal{H}_{1,2}$ in the lemma below, whose proof is the core of this section.

Lemma 8. Retaining the assumptions and the notation of Lemma 7, there is a coupling of $\big(G_n,k^{(\textrm{u})}_0\big)$ and $(\mathcal{T}_{\textbf{x},n},0)$ , with a positive constant $C\,:\!=\,C(x_1,\mu,\kappa)$ such that

\begin{align*}\mathbb{P}_{\textbf{x}}\left[\mathcal{H}_{1,0}\cap \mathcal{H}_{1,1} \cap \mathcal{H}^c_{1,2}\right]&\leqslant C n^{-\beta}(\log \log n)^{1-\chi}\quad for\ all\ n\geqslant 3, \end{align*}

where $0<\gamma<\chi/12$ and $\beta=\min\{\chi/3-4\gamma,\gamma\}$ .

Given that the vertices $k^{(\textrm{u})}_0\in V\big(\big(G_n,k^{(\textrm{u})}_0\big)\big)$ and $0\in V((\mathcal{T}_{\textbf{x},n},0))$ are coupled, we prove Lemma 8 as follows. Note that $v[1]=k^{(\textrm{u})}_0$ , where v[t] is as in (3.1). We first couple $\textbf{Y}^{(v[1],n)}_{\textrm{Be}}$ in Definition 7 and a discretisation of the mixed Poisson process that encodes the ages and the PA labels that are of type R. Then we couple the type-L vertices $k^{(\textrm{u})}_{0,1}$ and $(0,1)\in V((\mathcal{T}_{\textbf{x},n},0))$ so that on the event $\mathcal{H}_{1,0}\cap \mathcal{H}_{1,1}$ , we have $k^{(\textrm{i})}_{0,1}=k^{(\textrm{u})}_{0,1}$ and $\big(k^{(\textrm{u})}_{0,1}/n\big)^\chi\approx a^{(\textrm{i})}_{0,1}$ w.h.p. For the means of the discretised Poisson process, we define

(5.4) \begin{align}\lambda^{[1]}_{v[1]+1}\,:\!=\,\int^{\left(\frac{v[1]+1}{n}\right)^\chi}_{a^{(\textrm{i})}_0} \frac{\mathcal{Z}_{v[1]}[1]}{\big(a^{(\textrm{i})}_0\big)^{1/\mu}\mu} y^{1/\mu-1} dy\quad \text{and} \quad\lambda^{[1]}_{j}\,:\!=\,\int^{\left(\frac{j}{n}\right)^\chi}_{\left(\frac{j-1}{n}\right)^\chi} \frac{\mathcal{Z}_{v[1]}[1]}{\big(a^{(\textrm{i})}_0\big)^{1/\mu}\mu} y^{1/\mu-1} dy\nonumber\\\end{align}

for $v[1]+2\leqslant j \leqslant n$ , where $\mathcal{Z}_{v[1]}[1]$ is the gamma variable in (3.6).

Definition 9. Given $v[1]=k^{(\textrm{u})}_0$ , $a^{(\textrm{i})}_0$ , and $\mathcal{Z}_{v[1]}[1]$ , let $V_{j\to v[1]}$ , $v[1]+1\leqslant j\leqslant n$ , be conditionally independent Poisson random variables, each with parameter $\lambda^{[1]}_{j}$ as in (5.4). Define this Poisson sequence by the random vector

\begin{equation*}\textbf{V}^{(v[1],n)}_{\textrm{Po}}\,:\!=\,\left(V_{(v[1]+1)\to v[1]},V_{(v[1]+2)\to v[1]},\ldots,V_{n\to v[1]}\right).\end{equation*}

Next, we define the events that ensure that $P_{j\to v[1]}$ is close enough to $\lambda^{[1]}_j$ . Let $\phi(n)=\Omega(n^\chi)$ , and let $C^\star\,:\!=\,C^\star(x_1,\mu)$ be a positive constant such that (2.2) of Lemma 2 holds with $\delta_n=C^\star n^{-\frac{\chi}{12}}$ . Let $\mathcal{Z}_{j}[1]$ , $B_j[1]$ , and $S_j[1]$ be as in (3.6), (3.9), and (3.10). Given $0<\gamma<\chi/12$ , define the events

(5.5) \begin{equation} \begin{split}&F_{1,1}=\left\{\max_{\lceil{\phi(n)}\rceil\leqslant j \leqslant n}\left|S_{n,j}[1] - \left(\frac{j}{n}\right)^\chi\right|\leqslant C^\star n^{-\frac{\chi}{12}}\right\},\\& F_{1,2} = \bigcap^n_{j=\lceil{\phi(n)}\rceil}\left\{\left| B_{j}[1]-\frac{\mathcal{Z}_{j}[1]}{(\mu+1)j}\right|\leqslant \frac{\mathcal{Z}_{j}[1]n^{-\gamma}}{(\mu+1)j}\right\}, \\& F_{1,3} = \bigcap^n_{j=\lceil{\phi(n)}\rceil}\big\{\mathcal{Z}_{j}[1]\leqslant j^{1/2}\big\}.\end{split}\end{equation}

The next lemma is the major step towards proving Lemma 8, as it implies that we can couple the ages and the initial attractivenesses of the type-R children of the uniformly chosen vertex $k^{(\textrm{u})}_0\,:\!=\,v[1]\in V\big(G_n,k^{(\textrm{u})}_0\big)$ and those of the root $0\in V(\mathcal{T}_{\textbf{x},n},0)$ .

Lemma 9. Retaining the assumptions and the notation in Lemma 8, let $\textbf{Y}^{(v[1],n)}_{\textrm{Be}}$ and $\textbf{V}^{(v[1],n)}_{\textrm{Po}}$ be as in Definitions 7 and 9, and let $F_{1,i}$ , $i=1,2,3$ , be as in (5.5). Then we can couple the random vectors so that there is a positive constant $C\,:\!=\,C(x_1,\mu,\kappa)$ such that

\begin{align*}\mathbb{P}_{\textbf{x}}\bigg[\Big\{\textbf{Y}^{(v[1],n)}_{\textrm{Be}}\not=\textbf{V}^{(v[1],n)}_{\textrm{Po}}\Big\}\cap \bigcap^3_{i=1} F_{1,i} \cap \mathcal{H}_{1,0}\bigg]\leqslant C n^{-\gamma} (\log \log n)^{1-\chi}\quad for\ all\ n\geqslant 3.\end{align*}

Sketch of proof. We summarise the proof here and defer the details to [Reference Lo24]. On the event $F_{1,1}\cap F_{1,2}\cap \mathcal{H}_{1,0}$ , a little calculation shows that the Bernoulli success probability $P_{j\to v[1]}$ in (3.11) is close enough to

(5.6) \begin{equation}\widehat P_{j\to v[1]}\,:\!=\,\left(\frac{v[1]}{j}\right)^\chi \frac{\mathcal{Z}_{v[1]}[1]}{(\mu+1)v[1]},\end{equation}

while the event $F_{1,3}$ ensures that $\widehat P_{j\to v[1]}\leqslant 1$ . The Poisson mean $\lambda^{[1]}_j\approx \widehat P_{j\to v[1]}$ as $a^{(\textrm{i})}_0\approx (v[1]/n)^\chi$ . Hence, we use $\widehat P_{j\to v[1]}$ in (5.6) as means of constructing two intermediate Bernoulli and Poisson random vectors, and then explicitly couple the four processes. It is enough to consider the coupling under the event $\bigcap^3_{j=1} F_{1,j}\cap \mathcal{H}_{1,0}$ , because when $\textbf{x}\in A_{n}$ , Lemmas 2 and 3 imply that $F_{1,1}$ , $F_{1,2}$ , and $F_{1,3}$ occur with probability tending to one as $n\to\infty$ .

Below we use Lemmas 2, 3, and 9 to prove Lemma 8.

Proof of Lemma 8. We bound the right-hand side of

(5.7) \begin{align}&\mathbb{P}_{\textbf{x}}\big[\mathcal{H}^c_{1,2}\cap \mathcal{H}_{1,1}\cap\mathcal{H}_{1,0}\big] \nonumber\\&\leqslant \mathbb{P}_{\textbf{x}}\Bigg[\bigcap^3_{j=1} F_{1,j} \cap \bigcap^1_{j=0}\mathcal{H}_{1,j}\cap \mathcal{H}^c_{1,2} \Bigg] + \mathbb{P}_{\textbf{x}}\Bigg[\Bigg(\bigcap^3_{j=1} F_{1,j}\Bigg)^c\Bigg]\nonumber\\&\leqslant \mathbb{P}_{\textbf{x}}\Bigg[\bigcap^3_{j=1} F_{1,j} \cap \bigcap^1_{j=0}\mathcal{H}_{1,j}\cap \mathcal{H}^c_{1,2} \Bigg]+\mathbb{P}_{\textbf{x}}\big[ F^c_{1,3}\big]+ \mathbb{P}_{\textbf{x}}\big[F^c_{1,2} \big]+\mathbb{P}_{\textbf{x}}\big[ F^c_{1,1} \big],\end{align}

under a suitable coupling of $\big(G_n,k^{(\textrm{u})}_0\big)$ and $(\mathcal{T}_{\textbf{x},n},0)$ . We start by handling the last three terms. First, apply (2.6) of Lemma 3 to obtain

(5.8) \begin{align}\mathbb{P}_{\textbf{x}}\big[ F^c_{1,3} \big]=\mathbb{P}_{\textbf{x}}\Bigg[\bigcup^n_{j= \phi(n)}\big\{\mathcal{Z}_j[1]\geqslant j^{1/2}\big\}\Bigg]\leqslant C \kappa^4 n^{-\chi},\end{align}

where C is a positive constant. Since $\textbf{x}\in A_n$ , we can apply (2.4) of Lemma 3 (with $\varepsilon=n^{-\gamma}$ , $0<\gamma<\chi/12$ , and $\alpha=2/3$ ) and Lemma 2 to deduce that

(5.9) \begin{align}\mathbb{P}_{\textbf{x}}\big[F^c_{1,2}\big]=\mathbb{P}_{\textbf{x}}\Bigg[\bigcup^n_{j=\lceil{\phi(n)}\rceil}\left\{\left| B_j[1]-\frac{\mathcal{Z}_j[1]}{(\mu+1)j}\right|\geqslant\frac{\mathcal{Z}_j[1] n^{-\gamma}}{(\mu+1)j}\right\}\Bigg]\leqslant Cn^{4\gamma-\frac{\chi}{3}},\nonumber\\\mathbb{P}_{\textbf{x}}\big[ F^c_{1,1} \big]\leqslant \mathbb{P}_{\textbf{x}}\left[\max_{\lceil{\phi(n)}\rceil\leqslant k\leqslant n} \left|S_{n,k}[1]-\left(\frac{k}{n}\right)^\chi\right|\geqslant C^\star n^{-\frac{\chi}{12}}\right]\leqslant c n^{-\frac{\chi}{6}}, \end{align}

where $C\,:\!=\,C(x_1,\gamma,\mu)$ and $c\,:\!=\,c(x_1,\mu)$ . Note that $ \mathbb{P}_{\textbf{x}}\big[F^c_{1,2}\big]\to 0$ as $n\to\infty$ thanks to our choice of $\gamma$ . Next, we give the appropriate coupling to bound the first probability of (5.7). Let the vertices $k^{(\textrm{u})}_0\in V\big(G_n,k^{(\textrm{u})}_0\big)$ and $0\in V(\mathcal{T}_{\textbf{x},n},0)$ be coupled as in the beginning of this section. Assume that they are such that the event $\mathcal{H}_{1,0}$ occurs, and the variables $\big(\big(\mathcal{Z}_j[1],\mathcal{\widetilde Z}_{j}[1]\big),2\leqslant j\leqslant n\big)$ are such that $\bigcap^3_{j=1}F_{1,j}$ holds. We first argue that under the event $\bigcap^3_{j=1}F_{1,j}\cap \mathcal{H}_{1,0}$ , their type-R children can be coupled so that w.h.p., $R^{(\textrm{u})}_0=R^{(\textrm{i})}_0$ and $k^{(\textrm{u})}_{0,j}=k^{(\textrm{i})}_{0,j}$ for $j=2,\ldots,1+R^{(\textrm{i})}_0$ . In view of Definitions 4 and 8, this follows readily from Lemma 9. It remains to prove that under this coupling, $\big|(k^{(\textrm{u})}_0/n)^\chi-a^{(\textrm{i})}_0\big|$ and $\big|a^{(\textrm{i})}_{0,j}-(k^{(\textrm{u})}_{0,j}/n)^\chi\big|$ are sufficiently small. Since $k^{(\textrm{u})}_0=\Big\lceil{n \Big(a^{(\textrm{i})}_0\Big)^{1/\chi}}\Big\rceil$ , and for $j=2,\ldots,1+R^{(\textrm{i})}_0$ , $k^{(\textrm{u})}_{0,j}$ satisfies $k^{(\textrm{u})}_{0,j}>k^{(\textrm{u})}_0$ and $\Big(\Big(k^{(\textrm{u})}_{0,j}-1\Big)/n\Big)^\chi<a^{(\textrm{i})}_{0,j}\leqslant \Big(k^{(\textrm{u})}_{0,j}/n\Big)^\chi$ , it is enough to bound $(\ell/n)^\chi-((\ell-1)/n)^\chi$ for $k^{(\textrm{u})}_0+1\leqslant \ell\leqslant n$ . For such $\ell$ , we use the fact that $k^{(\textrm{u})}_0>n(\log\log n)^{-1}$ on the event $\mathcal{H}_{1,0}$ and the mean value theorem to obtain

(5.10) \begin{equation}\mathbb{1}[\mathcal{H}_{1,0}] \left[\left(\frac{\ell}{n}\right)^\chi- \left(\frac{\ell-1}{n}\right)^\chi\right] \leqslant \mathbb{1}[\mathcal{H}_{1,0}] \frac{\chi}{n} \left(\frac{n}{\ell-1}\right)^{1-\chi} \leqslant \frac{\chi(\log\log n)^{1-\chi}}{n}.\end{equation}

Next, we couple the type-L child of $k^{(\textrm{u})}_0\in V\big(G_n,k^{(\textrm{u})}_0\big)$ and $0\in V(\mathcal{T}_{\textbf{x},n},0)$ so that $k^{(\textrm{u})}_{0,1}=k^{(\textrm{i})}_{0,1}$ and $\Big(k^{(\textrm{u})}_{0,1}/n\Big)^\chi\approx a^{(\textrm{i})}_{0,1}$ . Independently from $a^{(\textrm{i})}_0$ , let $U_{0,1}\sim \textrm{U}[0,1]$ and $a^{(\textrm{i})}_{0,1}=U_{0,1}a^{(\textrm{i})}_0$ . Recalling $v[2]=k^{(\textrm{u})}_{0,1}$ in the BFS, it follows from Definitions 4 and 8 that we can define $k^{(\textrm{u})}_{0,1}$ to satisfy

\begin{equation*}S_{n,v[2]-1}[1]\leqslant U_{0,1}S_{n,v[1]-1}[1]< S_{n,v[2]}[1],\end{equation*}

or equivalently,

(5.11) \begin{equation}S_{n,v[2]-1}[1]\leqslant \frac{a^{(\textrm{i})}_{0,1}}{a^{(\textrm{i})}_0}S_{n,v[1]-1}[1]< S_{n,v[2]}[1],\end{equation}

and take $k^{(\textrm{u})}_{0,1}=k^{(\textrm{i})}_{0,1}$ . Now we show that under this coupling, $a^{(\textrm{i})}_{0,1}\approx \big(k^{(\textrm{u})}_{0,1}/n\big)^\chi$ on the ‘good’ event $\mathcal{H}_{1,0}\cap \mathcal{H}_{1,1}\cap F_{1,1}$ . Observe that $S_{n,v[2]}[1]=\Omega\big((\log \log n)^{-3\chi}\big)$ on the good event, because for n large enough,

\begin{align*}S_{n,v[2]}[1]&\geqslant U_{0,1}S_{n,v[1]-1}[1]\geqslant (\log \log n)^{-2\chi}\left[\left(\frac{v[1]-1}{n}\right)^\chi-C^\star n^{-\frac{\chi}{12}}\right]\\&\geqslant (\log \log n)^{-2\chi}\big[(\log \log n)^{-\chi} -2C^\star n^{-\frac{\chi}{12}}\big].\end{align*}

Since $S_{n,j}[1]$ increases with j, the last calculation implies that $|S_{n,j}[1]-(j/n)^\chi|\leqslant C^\star n^{-\frac{\chi}{12}}$ for $j=k^{(\textrm{u})}_{0,1},k^{(\textrm{u})}_{0,1}-1$ . Furthermore, a little calculation shows that on the event $\mathcal{H}_{1,0}\cap F_{1,1}$ , there is a constant $c\,:\!=\,c(x_1,\mu)$ such that

\begin{align*}\Big|\Big(S_{n,v[1]-1}[1]/a^{(\textrm{i})}_0\Big)-1\Big| = c n^{-\frac{\chi}{12}}(\log \log n)^{\chi}.\end{align*}

Swapping $\Big(S_{n,v[1]-1}[1]/a^{(\textrm{i})}_0\Big)$ , $S_{n,v[2]}[1]$ , and $S_{n,v[2]-1}[1]$ in (5.11) for one, $(v[2]/n)^\chi$ , and $((v[2]-1)/n)^\chi$ at the costs above, we conclude that there exists $\widehat C\,:\!=\,\widehat C(x_1,\mu)$ such that on the good event, $\big|a^{(\textrm{i})}_{0,1}-\big(k^{(\textrm{u})}_{0,1}/n\big)^\chi\big|\leqslant \widehat Cn^{-\frac{\chi}{12}}(\log \log n)^{\chi}$ . Hence, we can take $C_1\,:\!=\,\widehat C \vee \chi$ for the event $\mathcal{H}_{1,2}$ and apply Lemma 9 to obtain

(5.12) \begin{align}\mathbb{P}_{\textbf{x}}\Bigg[\bigcap^3_{j=1} F_{1,j} \cap \mathcal{H}^c_{1,2} \cap \mathcal{H}_{1,1}\cap\mathcal{H}_{1,0}\Bigg]&= \mathbb{P}_{\textbf{x}}\bigg[\Big\{\textbf{Y}^{(v[1],n)}_{\textrm{Be}}\not=\textbf{V}^{(v[1],n)}_{\textrm{Po}}\Big\}\cap \bigcap^3_{j=1} F_{1,j} \cap \bigcap^1_{j=0}\mathcal{H}_{1,j}\bigg]\nonumber\\&\leqslant \mathbb{P}_{\textbf{x}}\bigg[\Big\{\textbf{Y}^{(v[1],n)}_{\textrm{Be}}\not=\textbf{V}^{(v[1],n)}_{\textrm{Po}}\Big\}\cap \bigcap^3_{j=1} F_{1,j} \cap \mathcal{H}_{1,0}\bigg]\nonumber\\&\leqslant C n^{-\gamma} (\log \log n)^{1-\chi},\end{align}

where $C\,:\!=\,C(x_1,\mu,\kappa)$ is as in Lemma 9. The lemma is proved by applying (5.8), (5.9), and (5.12) to (5.7).

Before proving Lemma 7, we require a final result that says that under the graph coupling, the event that vertex $k^{(\textrm{u})}_0$ has a low degree occurs w.h.p. The next lemma is proved in [Reference Lo24] using Chebyshev’s inequality. Recall that $A_n\,:\!=\,A_{2/3,n}$ is the event in (2.1) with $\alpha=2/3$ .

Lemma 10. Assume that $x_i\in (0,\kappa]$ for $i\geqslant 2$ and $\textbf{x}\in A_n$ . Let $\mathcal{H}_{1,i}$ , $0,\ldots,3$ , be as in (5.2). There is a coupling of $\big(G_n,k^{(\textrm{u})}_0\big)$ and $(\mathcal{T}_{\textbf{x},n},0)$ , with a positive integer p and a positive constant $C\,:\!=\,C(p,\kappa)$ such that

\begin{equation*}\mathbb{P}_{\textbf{x}}\Bigg[\bigcap^2_{i=0} \mathcal{H}_{1,i}\cap \mathcal{H}^c_{1,3}\Bigg]\leqslant C (\log n)^{-\frac{p}{r}}(\log \log n)^{\frac{p}{\mu+1}}\quad for\ all\ n\geqslant 3. \end{equation*}

We now complete the proof of Lemma 7 by applying Lemmas 8 and 10.

Proof of Lemma 7. The lemma follows from bounding the right-hand side of

\begin{align*}\mathbb{P}_{\textbf{x}}\Bigg[\Bigg(\bigcap^3_{i=0} \mathcal{H}_{1,i}\Bigg)^c\Bigg]&=\mathbb{P}_{\textbf{x}}\big[ \mathcal{H}^c_{1,0}\big]+\sum^3_{i=1}\mathbb{P}_{\textbf{x}}\Bigg[\bigcap^{i-1}_{j=0}\mathcal{H}_{1,j}\cap \mathcal{H}^c_{1,i}\Bigg],\end{align*}

under the coupling in the proof of Lemma 8. To bound $\mathbb{P}_{\textbf{x}}\big[ \mathcal{H}^c_{1,0}\big]$ and $\mathbb{P}_{\textbf{x}}\big[\mathcal{H}_{1,0}\cap \mathcal{H}^c_{1,1}\big]$ , recall that $a^{(\textrm{i})}_0=U^\chi_0$ and $a^{(\textrm{i})}_{0,1}=U_{0,1} a^{(\textrm{i})}_{0}$ , where $U_{0}$ and $U_{0,1}$ are independent standard uniform variables. Hence,

(5.13) \begin{align}\mathbb{P}_{\textbf{x}}\big[\mathcal{H}_{1,0}\cap \mathcal{H}^c_{1,1}\big]&=\mathbb{E}\Big[\mathbb{1}\big[ \mathcal{H}_{1,0}\big]\mathbb{P}_{\textbf{x}}\Big[U_{0,1}\leqslant \big(a^{(\textrm{i})}_0\big)^{-1}(\log \log n)^{-2\chi}|U_0\Big]\Big]\nonumber\\&\leqslant \mathbb{P}_{\textbf{x}}\big[U_{0,1}\leqslant (\log \log n)^{-\chi}\big]\nonumber\\&= (\log \log n)^{-\chi}, \end{align}

and $\mathbb{P}_{\textbf{x}}\big[ \mathcal{H}^c_{1,0}\big]=\mathbb{P}_{\textbf{x}}\big[U_0\leqslant(\log \log n)^{-1}\big]=(\log \log n)^{-1}$ . Bounding the remaining probabilities using Lemmas 8 and 10 completes the proof.

6. Proof of the general case

Recall the definitions of $k^{(\textrm{u})}_{\bar w}, a^{(\textrm{i})}_{\bar w}$ , and $k^{(\textrm{i})}_{\bar w}$ in the beginning of Section 3.2 and Definition 8. In this section, we inductively couple the uniformly rooted Pólya urn graph $\big(G_n,k^{(\textrm{u})}_0\big)$ and the intermediate Pólya point tree $(\mathcal{T}_{\textbf{x},n},0)$ in Definition 8 so that w.h.p., $\Big(B_r\big(G_n,k^{(\textrm{u})}_0\big),k^{(\textrm{u})}_0\Big)\cong (B_r(\mathcal{T}_{\textbf{x},n},0),0)$ , and for $k^{(\textrm{u})}_{\bar w}\in V\Big(B_r\big(G_n,k^{(\textrm{u})}_0\big)\Big)$ and $\bar w\in V(B_r(\mathcal{T}_{\textbf{x},n},0)$ , their ages are close enough $\Big(\Big(k^{(\textrm{u})}_{\bar w}/n\Big)^\chi\approx a^{(\textrm{i})}_{\bar w}\Big)$ and their PA labels and arrival times match $\big(k^{(\textrm{i})}_{\bar w}=k^{(\textrm{u})}_{\bar w}\big)$ . Given a positive integer r, let $L[q]\,:\!=\,(0,1,\ldots,1)$ and $|L[q]|=q+1$ for $2\leqslant q\leqslant r$ , so that L[q] and $k^{(\textrm{u})}_{L[q]}$ are the type-L children in $\partial \mathfrak{B}_q\,:\!=\,V(B_q(\mathcal{T}_{\textbf{x},n},0))\setminus V(B_{q-1}(\mathcal{T}_{\textbf{x},n},0))$ and $\partial \mathcal{B}_q\,:\!=\,V\Big(B_q\big(G_n,k^{(\textrm{u})}_0\big)\Big)\setminus V\Big(B_{q-1}\big(G_n,k^{(\textrm{u})}_0\big)\Big)$ . Let $\chi/12\,:\!=\,\beta_1>\beta_2>\cdots>\beta_r>0$ , so that $n^{-\beta_q}>n^{-\beta_{q-1}}(\log \log n)^{q\chi}$ for n large enough. To ensure that we can couple the $(q+1)$ -neighbourhoods for $1\leqslant q\leqslant r-1$ , we define the coupling events analogous to $\mathcal{H}_{1,i}$ , $i=1,2,3$ , in (5.2):

(6.1) \begin{align}\mathcal{H}_{q,1}&=\Big\{a^{(\textrm{i})}_{L[q]}>(\log \log n)^{-\chi(q+1)}\Big\},\nonumber\\\mathcal{H}_{q,2}&= \bigg\{ \Big(B_{q}\big(G_n,k^{(\textrm{u})}_0\big),k^{(\textrm{u})}_0\Big)\cong (B_{q}(\mathcal{T}_{\textbf{x},n},0),0),\text{ with }k^{(\textrm{u})}_{\bar v}=k^{(\textrm{i})}_{\bar v}\nonumber\\&\qquad \text{ and }\bigg|a^{(\textrm{i})}_{\bar v}-\bigg(\frac{k^{(\textrm{u})}_{\bar v}}{n}\bigg)^\chi\bigg|\leqslant C_q n^{-\beta_q} \text{ for $\bar v\in V(B_{q}(\mathcal{T}_{\textbf{x},n},0))$} \bigg\},\nonumber\\\mathcal{H}_{q,3}&=\Big\{R^{(\textrm{i})}_{\bar v}<(\log n)^{1/r} \text{ for $\bar v\in V(B_{q-1}(\mathcal{T}_{\textbf{x},n},0))$}\Big\},\end{align}

where $R^{(\textrm{i})}_{\bar v}$ is as in (4.1) and $C_q\,:\!=\,C_q(x_1,\mu)$ will be chosen in the proof of Lemma 12 below. Let $A_n\,:\!=\,A_{2/3,n}$ be the event in (2.1), and let $\mathbb{P}_{\textbf{x}}$ indicate the conditioning on a specific realisation of the fitness sequence $\textbf{x}$ . The next lemma is the key result of this section; it essentially states that if we can couple $\big(G_n,k^{(\textrm{u})}_0\big)$ and $(\mathcal{T}_{\textbf{x},n},0)$ so that $\Big(B_{q}\big(G_n,k^{(\textrm{u})}_0\big),k^{(\textrm{u})}_0\Big)\cong (B_{q}(\mathcal{T}_{\textbf{x},n},0),0)$ w.h.p., then we can achieve this for the $(q+1)$ -neighbourhoods too.

Lemma 11. Let $\mathcal{H}_{q,i}$ , $1\leqslant q\leqslant r-1$ , $i=1,2,3$ , be as in (5.2) and (6.1). Assume that $x_i\in (0,\kappa]$ for $i\geqslant 2$ and $\textbf{x}\in A_n$ . Given $r<\infty$ and $1\leqslant q\leqslant r-1$ , if there is a coupling of $\big(G_n,k^{(\textrm{u})}_0\big)$ and $(\mathcal{T}_{\textbf{x},n},0)$ , with a positive constant $C\,:\!=\,C(x_1,\mu,\kappa,q)$ such that

\begin{align*}\mathbb{P}_{\textbf{x}}\Bigg[\Bigg(\bigcap^{3}_{i=1}\mathcal{H}_{q,i}\Bigg)^c\Bigg]\leqslant C (\log \log n)^{-\chi} \quad for\ all\ n\geqslant 3,\end{align*}

then there is a coupling of $\big(G_n,k^{(\textrm{u})}_0\big)$ and $(\mathcal{T}_{\textbf{x},n},0)$ , with a positive constant $C^{\prime}\,:\!=\,C^{\prime}(x_1,\mu,\kappa,q)$ such that

\begin{align*}\mathbb{P}_{\textbf{x}}\Bigg[\Bigg(\bigcap^{3}_{i=1}\mathcal{H}_{q+1,i}\Bigg)^c\Bigg]\leqslant C^{\prime} (\log \log n)^{-\chi} \quad for\ all\ n\geqslant 3.\end{align*}

Since Lemma 7 implies that such a graph coupling exists for $r=1$ , combining Lemmas 7 and 11 yields the following corollary, which is instrumental in proving Theorem 1.

Corollary 2. Retaining the assumptions and the notation in Lemma 11, there is a coupling of $\big(G_n,k^{(\textrm{u})}_0\big)$ and $(\mathcal{T}_{\textbf{x},n},0)$ , with a positive constant $C\,:\!=\,C(x_1,\mu,\kappa,r)$ such that

\begin{align*}\mathbb{P}_{\textbf{x}}\Bigg[\Bigg(\bigcap^{3}_{i=1}\mathcal{H}_{r,i}\Bigg)^c\Bigg]\leqslant C (\log \log n)^{-\chi} \quad for\ all\ n\geqslant 3. \end{align*}

The proof of Lemma 11 is in the same vein as that of Lemma 7. Fixing $1\leqslant q\leqslant r-1$ , we may assume that the two graphs are already coupled so that the event $\bigcap^3_{i=1}\mathcal{H}_{q,i}$ has occurred. On the event $\mathcal{H}_{q,1}$ , it is clear from the definitions of $a^{(\textrm{i})}_{L[q+1]}$ and $\mathcal{H}_{q+1,1}$ that $\mathcal{H}_{q+1,1}$ occurs w.h.p., while on the event $\mathcal{H}_{q,1}\cap \mathcal{H}_{q,3}$ , we can easily show that $\mathcal{H}_{q+1,3}$ occurs w.h.p. So for most of the proof, we handle the event $\mathcal{H}_{q+1,2}$ . To couple the graphs so that $\mathcal{H}_{q+1,2}$ occurs w.h.p., we consider the vertices of $\partial \mathcal{B}_q$ and $\partial \mathfrak{B}_q$ in the breadth-first order, and couple $k^{(\textrm{u})}_{\bar w}\in \partial \mathcal{B}_q$ and $\bar w\in \partial \mathfrak{B}_q$ so that w.h.p., the numbers of children they have are the same and not too large, and the children’s ages are close enough. Recall the definition of v[t] in (3.1). If $v[t]=k^{(\textrm{u})}_{\bar w}$ , the associated events are as follows:

(6.2) \begin{align}\mathcal{K}_{t,1}&= \bigg\{R^{(\textrm{u})}_{\bar w}=R^{(\textrm{i})}_{\bar w},\text{ and for $1\leqslant j\leqslant 1+R^{(\textrm{i})}_{\bar w}$,}\nonumber\\&\hspace{2cm}k^{(\textrm{i})}_{\bar w,j}=k^{(\textrm{u})}_{\bar w,j}\text{ and }\bigg|a^{(\textrm{i})}_{\bar w,j}-\bigg(\frac{k^{(\textrm{u})}_{\bar w,j}}{n}\bigg)^\chi\bigg| \leqslant C_{q+1} n^{-\beta_{q+1}}\bigg\}, \\\mathcal{K}_{t,2}&=\Big\{R^{(\textrm{i})}_{\bar w}<(\log n)^{1/r}\Big\},\nonumber\end{align}

where $R^{(\textrm{u})}_{\bar w}$ and $R^{(\textrm{i})}_{\bar w}$ are as in (3.4) and (4.1), and $C_{q+1}$ and $\beta_{q+1}$ are as in the event $\mathcal{H}_{q+1,2}$ . Below we only consider the coupling of the type-L children in detail, because the type-R case can be proved similarly. Observe that for n large enough, there must be a type-L child in $\partial \mathcal{B}_q$ on the event $\bigcap^3_{j=1}\mathcal{H}_{q,j}$ . Define $v[\tau[q]]=k^{(\textrm{u})}_{L[q]}$ , so that

\begin{equation*}\tau[1]=2\quad\text{and}\quad\tau[q]=\Big|V\Big(B_{q-1}\big(G_n,k^{(\textrm{u})}_0\big)\Big)\Big|+1\quad\text{for $2\leqslant q \leqslant r-1$}\end{equation*}

are the exploration times of the type-L children. Noting that

\begin{align*}\big(\mathcal{A}_{\tau[q]-1},\mathcal{P}_{\tau[q]-1}, \mathcal{N}_{\tau[q]-1}\big)=\Big(\partial \mathcal{B}_q,V\Big(B_{q-1}\big(G_n,k^{(\textrm{u})}_0\big)\Big),V(G_n)\setminus V\Big(B_{q}\big(G_n,k^{(\textrm{u})}_0\big)\Big)\Big),\end{align*}

we let

\begin{align*}\Big(\Big(\mathcal{Z}_j[\tau[q]], \mathcal{\widetilde Z}_{j}[\tau[q]]\Big),j\in \mathcal{A}_{\tau[q]-1}\cup \mathcal{N}_{\tau[q]-1}\Big)\quad \text{and} \quad \left(S_{n,j}[\tau[q]],1\leqslant j\leqslant n\right)\end{align*}

be as in (3.7), (3.8), and (3.10). For convenience, we also define

(6.3) \begin{equation}\zeta_q\,:\!=\,\mathcal{Z}_{v[\tau[q]]}{[\tau[q]]},\quad \text{so that }\zeta_q\sim\textrm{Gamma}(x_{v[\tau[q]]}+1,1).\end{equation}

We use $(S_{n,j}[\tau[q]],1\leqslant j\leqslant n)$ to generate the children of vertex $v[\tau[q]]$ , and let $\Big(a^{(\textrm{i})}_{L[q],j},2\leqslant j\leqslant 1+R^{(\textrm{i})}_{L[q]}\Big)$ be the points of the mixed Poisson process on $\Big(a^{(\textrm{i})}_{L[q]},1\Big]$ with intensity

(6.4) \begin{equation}\frac{\zeta_q}{\mu \Big(a^{(\textrm{i})}_{L[q]}\Big)^{1/\mu}} y^{1/\mu-1} dy,\end{equation}

and $a^{(\textrm{i})}_{L[q],1} \sim \textrm{U}\Big[0,a^{(\textrm{i})}_{L[q]}\Big]$ . In the sequel, we develop lemmas analogous to Lemmas 8 and 10 to show that on the event $\bigcap^3_{j=1} \mathcal{H}_{q,j}$ , we can couple the vertices $v[\tau[q]]\in \partial\mathcal{B}_q$ and $\tau[q]\in \partial \mathfrak{B}_q$ so that the events $\mathcal{K}_{\tau[q],1}$ and $\mathcal{K}_{\tau[q],2}$ occur w.h.p. As in the 1-neighbourhood case, the difficult part is proving the claim for $\mathcal{K}_{\tau[q],1}$ . From now on we write

(6.5) \begin{equation}\overline{\mathcal{H}}_{q,l}\,:\!=\,\bigcap^3_{j=1} \mathcal{H}_{q,j}\cap \big\{|\partial\mathcal{B}_q|=l\big\} \quad for\ all\ l\geqslant 1. \end{equation}

Lemma 12. Retaining the assumptions and the notation in Lemma 11, let $\beta_q$ , $\mathcal{K}_{\tau[q],1}$ , and $\overline{\mathcal{H}}_{q,l}$ be as in (6.1), (6.2), and (6.5). Then there is a coupling of $\big(G_n,k^{(\textrm{u})}_0\big)$ and $(\mathcal{T}_{\textbf{x},n},0)$ , with a positive constant $C\,:\!=\,C(x_1,\mu,\kappa,q)$ such that

\begin{align*}\mathbb{P}_{\textbf{x}}\Big(\overline{\mathcal{H}}_{q,l} \cap \mathcal{H}_{q+1,1}\cap \mathcal{K}^c_{\tau[q],1} \Big)\leqslant C n^{-d}(\log\log n)^{q+1}(\log n)^{q/r}\quad for\ all\ n\geqslant 3\ \text{and} l\geqslant 1, \end{align*}

where $0<\gamma<\chi/12$ and $d\,:\!=\,\min\big\{\chi/3-4\gamma,\gamma, 1-\chi,\beta_q\big\}$ .

Similar to Lemma 8, the key step is to couple the type-R children of vertex $k^{(\textrm{u})}_{L[q]}$ . Let the Bernoulli sequence $\textbf{Y}_\textrm{Be}^{(v[\tau[q]],n)}$ be as in Definition 7, which by Lemma 5 encodes the type-R children of vertex $k^{(\textrm{u})}_{L[q]}$ . Additionally, define

(6.6) \begin{align}M_{L[q]}\,:\!=\,\min\Big\{j\in [n]\,:\, (j/n)^\chi\geqslant a^{(\textrm{i})}_{L[q]}\Big\}.\end{align}

We want to use the bins $\Big(a^{(\textrm{i})}_{L[q]}, ((j/n)^\chi)^n_{j=M_{L[q]}}\Big)$ to construct the discretised mixed Poisson process that encodes the ages and the PA labels, i.e. $\Big(\Big(a^{(\textrm{i})}_{L[q],j},k^{(\textrm{i})}_{L[q],j}\Big),2\leqslant j\leqslant 1+R^{(\textrm{i})}_{L[q]}\Big)$ . However, it is possible that $M_{L[q]}\not =v[\tau[q]]+1$ , in which case the numbers of Bernoulli and Poisson variables do not match and a modification of $\textbf{Y}_\textrm{Be}^{(v[\tau[q]],n)}$ is needed. If $M_{L[q]}\leqslant v[\tau[q]]$ , define $Y_{j\to v[\tau[q]]}$ , $M_{L[q]}\leqslant j \leqslant v[\tau[q]]$ , as Bernoulli variables with means $P_{j\to v[\tau[q]]}\,:\!=\,0$ , and concatenate the vectors $\big(Y_{j\to v[\tau[q]]},M_{L[q]}\leqslant j \leqslant v[\tau[q]]\big)$ and $\textbf{Y}_\textrm{Be}^{(v[\tau[q]],n)}$ . This corresponds to the fact that vertex j cannot send an outgoing edge to vertex $v[\tau[q]]$ in $\big(G_n,k^{(\textrm{u})}_0\big)$ . If $M_{L[q]}\geqslant v[\tau[q]]+1$ , let $\textbf{Y}_\textrm{Be}^{(v[\tau[q]],n)}$ be as in Definition 7. Saving notation, we redefine

(6.7) \begin{align}\textbf{Y}_\textrm{Be}^{(v[\tau[q]],n)}\,:\!=\, \left(Y_{\widetilde M_{L[q]}\to v[\tau[q]]}, Y_{(\widetilde M_{L[q]}+1)\to v[\tau[q]]}, \ldots, Y_{n\to v[\tau[q]]}\right),\end{align}

where

(6.8) \begin{equation}\widetilde M_{L[q]}\,:\!=\,\min\big\{M_{L[q]}, v[\tau[q]]+1\big\}.\end{equation}

We also make the following adjustment so that the upcoming Poisson random vector is a discretisation of the mixed Poisson point process on $\big(a^{(\textrm{i})}_{L[q]},1\big]$ , and is still comparable to the modified $\textbf{Y}_\textrm{Be}^{(v[\tau[q]],n)}$ . When $M_{L[q]}\leqslant v[\tau[q]]+1$ , define the Poisson means

(6.9) \begin{align}\lambda^{[\tau[q]]}_{M_{L[q]}}\,:\!=\,\int^{\left(\frac{M_{L[q]}}{n}\right)^\chi}_{a^{(\textrm{i})}_{L[q]}}\frac{\zeta_q}{\mu \big(a^{(\textrm{i})}_{L[q]}\big)^{1/\mu}} y^{1/\mu-1} dy,\quad\lambda^{[\tau[q]]}_{j}\,:\!=\,\int^{\left(\frac{j}{n}\right)^\chi}_{\left(\frac{j-1}{n}\right)^\chi}\frac{\zeta_q}{\mu \big(a^{(\textrm{i})}_{L[q]}\big)^{1/\mu}} y^{1/\mu-1}dy \end{align}

for $M_{L[q]}+1\leqslant j\leqslant n$ . When $M_{L[q]}\geqslant v[\tau[q]]+2$ , we let

(6.10) \begin{equation}\lambda^{[\tau[q]]}_{j}\,:\!=\,0\quad\text{for $v[\tau[q]]+1\leqslant j< M_{L[q]}$,}\end{equation}

in addition to (6.9), so that there are no Poisson points outside the interval $\big(a^{(\textrm{i})}_{L[q]},1\big]$ .

Definition 10. Given $k^{(\textrm{u})}_{L[q]}$ , $a^{(\textrm{i})}_{L[q]}$ , and $\zeta_q$ defined in (6.3), let $\widetilde M_{L[q]}$ be as in (6.8); for $\widetilde M_{L[q]}+1\leqslant j\leqslant n$ , let $V_{j\to v[\tau[q]]}$ be conditionally independent Poisson random variables, each with parameters $\lambda^{[\tau[q]]}_j$ given in (6.9) and (6.10). Define this (mixed) Poisson sequence by the random vector

\begin{equation*}\textbf{V}_\textrm{Po}^{(v[\tau[q]],n)}\,:\!=\,\Big(V_{\widetilde M_{L[q]}\to v[\tau[q]]},V_{(\widetilde M_{L[q]}+1)\to v[\tau[q]]},\ldots,V_{n\to v[\tau[q]]}\Big).\end{equation*}

We proceed to define the events analogous to $F_{1,j}$ , $j=1,2,3$ , in (5.5). On these events, $P_{j\to v[\tau[q]]}$ in (3.11) and $\lambda^{[\tau[q]]}_j$ are close enough for most j, and so we can couple the point processes as before. Let $\phi(n)=\Omega(n^\chi)$ , $0<\gamma<\chi/12$ , $\mathcal{Z}_{j}{[\tau[q]]}$ , $B_{j}{[\tau[q]]}$ , and $S_{n,j}{[\tau[q]]}$ be as in (3.7), (3.9), and (3.10), and let $C^\star_q\,:\!=\,C^\star_q(x_1,\mu)$ be a positive constant that we specify later. Define

(6.11) \begin{align}& F_{\tau[q],1}=\left\{\max_{\lceil{\phi(n)}\rceil\leqslant j\leqslant n}\left|S_{n,j}{[\tau[q]]}-\left(\frac{j}{n}\right)^\chi\right|\leqslant C^\star_q n^{-\frac{\chi}{12}}\right\},\nonumber\\& F_{\tau[q],2}=\bigcap^{n}_{\substack{j=\lceil{\phi(n)}\rceil;j\not \in\mathcal{P}_{\tau[q]-1}}}\left\{\left|B_{j}{[\tau[q]]}-\frac{\mathcal{Z}_{j}{[\tau[q]]}}{(\mu+1)j}\right|\leqslant \frac{\mathcal{Z}_{j}{[\tau[q]]}n^{-\gamma}}{(\mu+1)j}\right\},\nonumber\\& F_{\tau[q],3} = \bigcap^n_{j=\lceil{\phi(n)}\rceil;j\not \in\mathcal{P}_{\tau[q]-1} }\big\{\mathcal{Z}_{j}{[\tau[q]]}\leqslant j^{1/2}\big\}.\end{align}

The following analogue of Lemma 9 is the main ingredient for proving Lemma 12.

Lemma 13. Retaining the assumptions and the notation in Lemma 12, let $\textbf{Y}_\textrm{Be}^{(v[\tau[q]],n)}$ , $\textbf{V}_\textrm{Po}^{(v[\tau[q]],n)}$ , and $F_{\tau[q],i}$ , $i=1,2,3$ , be as in Definition 7, Definition 10, and (6.11). Then we can couple the random vectors so that there is a positive constant $C=C(x_1,\mu,\kappa,q)$ such that

\begin{align*}\mathbb{P}_{\textbf{x}}\Bigg[\Big\{\textbf{Y}_\textrm{Be}^{(v[\tau[q]],n)}\not = \textbf{V}_\textrm{Po}^{(v[\tau[q]],n)}\Big\}\cap \bigcap^3_{i=1} F_{\tau[q], i}\cap \overline{\mathcal{H}}_{q,l}\Bigg]\leqslant C n^{-d}(\log \log n)^{q+1}(\log n)^{q/r}\end{align*}

for all $n\geqslant 3$ , where $d\,:\!=\,\min\{\beta_q,\gamma,1-\chi \}$ .

Sketch of proof. We only highlight the main steps here, leaving the details to [Reference Lo24]. As in the proof of Lemma 9, we construct a Bernoulli and a Poisson sequence using suitable means $\widehat P_{j\to v[\tau[q]]}$ , $\widetilde M_{L[q]}\leqslant j\leqslant n$ , with $\widetilde M_{L[q]}$ as in (6.8). Then we couple the four processes. However, this time we need to handle the cases where we couple a Bernoulli variable with mean zero and a Poisson variable with positive mean, or vice versa. We take care of these cases by choosing the appropriate $\widehat P_{j\to v[\tau[q]]}$ . First we observe that on the event $\Big(\bigcap^3_{j=1} F_{\tau[q], j}\Big)\cap\Big(\bigcap^3_{j=1} \mathcal{H}_{q,j}\Big)$ , $P_{j\to v[\tau[q]]}$ is close enough to

(6.12) \begin{equation}\widehat P_{j\to v[\tau[q]]}\,:\!=\,\left(\frac{v[\tau[q]]}{j}\right)^\chi \frac{\zeta_q}{(\mu+1)v[\tau[q]]}\quad \leqslant 1\end{equation}

for $j\in \{v[\tau[q]]+1,\ldots, n\}\setminus V\Big(B_q\big(G_n,k^{(\textrm{u})}_0\big)\Big)$ , whereas for $\max\{M_{L[q]},v[\tau[q]]+1\}\leqslant j\leqslant n$ , we have

\begin{equation*}\lambda^{[\tau[q]]}_j\approx \widehat P_{j\to v[\tau[q]]}\end{equation*}

because $a^{(\textrm{i})}_{L[q]}\approx (v[\tau[q]]/n)^\chi$ on the event $\mathcal{H}_{q,2}$ . Hence, we can couple $Y_{j\to v[\tau[q]]}$ and $V_{j\to v[\tau[q]]}$ for $j\in\{\max\{M_{L[q]},v[\tau[q]]+1\},\ldots, n\}\setminus V\Big(B_q\big(G_n,k^{(\textrm{u})}_0\big)\Big)$ as in Lemma 9. When $j \in V\Big(B_q\big(G_n,k^{(\textrm{u})}_0\big)\Big)$ , the Bernoulli variable $Y_{j\to v[\tau[q]]}$ has mean zero, and we couple it to a Bernoulli variable with mean (6.12) as in Lemma 9. By a little computation,

(6.13) \begin{equation}|\partial \mathcal{B}_{j}|<1+j(\log n)^{j/r},\quad\Big|V\Big(B_j\big(G_n,k^{(\textrm{u})}_0\big)\Big)\Big|< 1+j+ j^2(\log n)^{j/r}\end{equation}

for $1\leqslant j\leqslant q$ on the event $\mathcal{H}_{q,3}$ . Because the $\widehat P_{j\to v[\tau[q]]}$ are sufficiently small and there are at most $O((\log n)^{q/r})$ such pairs of Bernoulli variables, we can use a union bound to show that the probability that $Y_{j\to v[\tau[q]]}\not =V_{j\to v[\tau[q]]}$ for any such j tends to zero as $n\to\infty$ . We now consider the coupling of $Y_{j\to v[\tau[q]]}$ and $V_{j\to v[\tau[q]]}$ for $\widetilde M_{L[q]}\leqslant j\leqslant \max\Big\{M_{L[q]},k^{(\textrm{u})}_{L[q]}+1\Big\}$ . In Table 1 below, we give the possible combinations of the Bernoulli and Poisson means, and our choice of intermediate means. As indicated in the table, we choose $\widehat P_{j\to v[\tau[q]]}>0$ whenever $P_{j\to v[\tau[q]]}>0$ . When $M_{L[q]}\leqslant v[\tau[q]]$ , we couple $V_{j\to v[\tau[q]]}$ and a Poisson variable with mean zero, while when $M_{L[q]}\geqslant v[\tau[q]]+2$ , $V_{j\to v[\tau[q]]}\,:\!=\,0$ by construction, and it is coupled with a Poisson variable with mean (6.12). However, the number of these pairs is small because $M_{L[q]}\approx v[\tau[q]]+1$ when $a^{(\textrm{i})}_{L[q]}\approx ( v[\tau[q]]/n)^\chi$ . Consequently, another union bound argument shows that the probability that any of these couplings fail tends to zero as $n\to\infty$ .

We are now ready to prove Lemma 12.

Table 1.

Combinations of means for $\widetilde M_{L[q]}\leqslant j \leqslant \max\{M_{L[q]},k^{(\textrm{u})}_{L[q]}+1\}$ . Note that $j=k^{(\textrm{u})}_{L[q]}+1$ when $M_{L[q]}=k^{(\textrm{u})}_{L[q]}+1$ , and in that case, the coupling is the same as that of Lemma 9.

Proof of Lemma 12. Recall the definition of $\overline{\mathcal{H}}_{q,l}$ in (6.5). We bound the right-hand side of

(6.14) \begin{align}&\mathbb{P}_{\textbf{x}}\Big[\overline{\mathcal{H}}_{q,l} \cap \mathcal{H}_{q+1,1}\cap \mathcal{K}^c_{\tau[q],1} \Big]\nonumber\\&\leqslant \mathbb{P}_{\textbf{x}}\Bigg[\overline{\mathcal{H}}_{q,l} \cap \bigcap^3_{j=1} F_{\tau[q],j}\cap \mathcal{H}_{q+1,1}\cap \mathcal{K}^c_{\tau[q],1} \Bigg]+\mathbb{P}_{\textbf{x}}\Bigg[ \overline{\mathcal{H}}_{q,l}\cap\Bigg(\bigcap^3_{j=1} F_{\tau[q],j}\Bigg)^c \Bigg]\nonumber\\&\leqslant \mathbb{P}_{\textbf{x}}\Bigg[\overline{\mathcal{H}}_{q,l}\cap \bigcap^3_{j=1} F_{\tau[q],j}\cap \mathcal{H}_{q+1,1} \cap \mathcal{K}^c_{\tau[q],1} \Bigg] + \sum^3_{j=1} \mathbb{P}_{\textbf{x}}\Big[\overline{\mathcal{H}}_{q,l} \cap F^c_{\tau[q],j}\Big], \end{align}

starting from the last three terms. Using (6.13), a computation similar to that for Lemma 2 shows that there are positive constants $C^\star_q=C^\star_q(x_1,\mu)$ and $c^\star=c^\star(x_1,\mu,q)$ such that

(6.15) \begin{equation}\mathbb{P}_{\textbf{x}}\left[{\overline{\mathcal{H}}_{q,l}\cap\left\{\max_{\lceil{\phi(n)}\rceil\leqslant j\leqslant n}\bigg|S_{n,j}{[\tau[q]]}-\bigg(\frac{j}{n}\bigg)^\chi\bigg|\leqslant C^\star_q n^{-\frac{\chi}{12}}\right\}}\right]\leqslant c^\star n^{-\tfrac{\chi}{6}},\end{equation}

which bounds $\mathbb{P}_{\textbf{x}}\Big[ \overline{\mathcal{H}}_{q,l}\cap F^c_{\tau[q],1}\Big]$ . Combining (6.13) and the argument of Lemma 3 also yields

(6.16) \begin{equation}\mathbb{P}_{\textbf{x}}\Big[ \overline{\mathcal{H}}_{q,l} \cap F^c_{\tau[q],3}\Big]\leqslant C \kappa^4 n^{-\chi}\quad\text{and}\quad\mathbb{P}_{\textbf{x}}\Big[ \overline{\mathcal{H}}_{q,l} \cap F^c_{\tau[q],2}\Big]\leqslant C^{\prime}n^{4\gamma-\tfrac{\chi}{3}},\end{equation}

for some positive constants $C\,:\!=\,C(q)$ , $C^{\prime}=C^{\prime}(x_1,\gamma,\mu,q)$ , and $0<\gamma<\chi/12$ . Next, we bound the first term under a suitable coupling of the vertices $k^{(\textrm{u})}_{L[q]}\in \partial \mathcal{B}_q$ and $L[q]\in \partial \mathfrak{B}_q$ , starting from their type-R children. Assume that

\begin{gather*}\Big(\Big(k^{(\textrm{u})}_{\bar w}, k^{(\textrm{i})}_{\bar w}, a^{(\textrm{i})}_{\bar w}\Big), \bar w\in V\Big(B_q(\mathcal{T}_{\textbf{x},n},0)\Big)\Big),\quad\Big(\Big(R^{(\textrm{u})}_{\bar w}, R^{(\textrm{i})}_{\bar w}\Big),\bar w \in V(B_{q-1}(\mathcal{T}_{\textbf{x},n},0))\Big),\\\Big(\Big(\mathcal{Z}_j[\tau[q]], \mathcal{\widetilde Z}_{j}[\tau[q]]\Big),j\in [n]\setminus V(B_{q-1}(\mathcal{T}_{\textbf{x},n},0))\Big)\end{gather*}

are coupled so that $\textrm{E}_q\,:\!=\,\bigcap^3_{j=1}\mathcal{H}_{q,j}\cap \bigcap^3_{j=1}F_{\tau[q],j}$ occurs. In view of Definitions 8 and 7, it follows from Lemma 13 that on the event $\textrm{E}_q$ , there is a coupling such that $R^{(\textrm{u})}_{L[q]}=R^{(\textrm{i})}_{L[q]}$ and $k^{(\textrm{u})}_{L[q],j}=k^{(\textrm{i})}_{L[q],j}$ for $2\leqslant j\leqslant 1+R^{(\textrm{i})}_{L[q]}$ w.h.p. Note that $k_{L[q],j}\geqslant M_{L[q]}$ when $\textbf{Y}_\textrm{Be}^{(v[\tau[q]],n)}$ and $\textbf{V}_\textrm{Po}^{(v[\tau[q]],n)}$ are coupled. Thus, using $M_{L[q]}\geqslant n (a^{(\textrm{i})}_{L[q]})^{1/\chi}$ , a little calculation shows that there is a constant $\widehat c\,:\!=\,\widehat c(\mu)$ such that

\begin{align*}\mathbb{1}\bigg[\bigcap^2_{i=1}\mathcal{H}_{q,i}\bigg] \left[\left(\frac{j}{n}\right)^\chi-\left(\frac{j-1}{n}\right)^\chi\right] \leqslant \frac{\widehat c(\log\log n)^{(1+q)(1-\chi)}}{n},\quad M_{L[q]}\leqslant j\leqslant n,\end{align*}

implying that the ages of the type-R children are close enough. We proceed to couple the type-L child of $k^{(\textrm{u})}_{L[q]}\in \partial \mathcal{B}_q$ and $L[q]\in \partial \mathfrak{B}_q$ on the event $\textrm{E}_q\cap \mathcal{H}_{q+1}$ . Independently from all the variables generated so far, let $U_{L[q],1}\sim\textrm{U}[0,1]$ . Set

\begin{equation*}a^{(\textrm{i})}_{L[q],1}\,:\!=\,a^{(\textrm{i})}_{L[q+1]}=U_{L[q],1} a^{(\textrm{i})}_{L[q]},\end{equation*}

so that $a^{(\textrm{i})}_{L[q],1}$ is the age of vertex $(L[q],1)\in \partial \mathfrak{B}_{q+1}$ . Temporarily defining $f\,:\!=\,k^{(\textrm{u})}_{L[q]}$ and $g\,:\!=\,k^{(\textrm{u})}_{L[q],1}$ , we define g to satisfy

(6.17) \begin{align}S_{n,g-1}[\tau[q]]\leqslant U_{L[q],1} S_{n,f-1}[\tau[q]]< S_{n,g}[\tau[q]].\end{align}

In light of Definition 8 and Lemma 6, $k^{(\textrm{i})}_{L[q],1}=g$ . To establish $a^{(\textrm{i})}_{L[q],1}\approx (g/n)^\chi$ , we first show that we can substitute $S_{n,j}[\tau[q]]$ with $(j/n)^\chi$ for $j=g-1,g$ at a small enough cost. A straightforward computation shows that on the event $\textrm{E}_q \cap \mathcal{H}_{q+1,1}$ ,

\begin{align*}U_{L[q],1}\geqslant (\log\log n)^{-(q+2)\chi}\quad\text{and}\quad f\geqslant n(\log\log n)^{-(q+1)} - C_q n^{1-\beta_q/\chi},\end{align*}

where $C_q$ is the constant in the event $\mathcal{H}_{q,2}$ . Consequently, there is a constant $C\,:\!=\,C(x_1,\mu,q)$ such that on the event $\textrm{E}_q\cap \mathcal{H}_{q+1,1}$ ,

(6.18) \begin{align}S_{n,g}[\tau[q]]&\geqslant U_{L[q],1} S_{n,f-1}[\tau[q]]\geqslant (\log\log n)^{-(q+2)\chi} \left[\left(\frac{f-1}{n}\right)^\chi - C^\star_q n^{-\frac{\chi}{12}}\right]\nonumber\\&\geqslant C (\log\log n)^{-(2q+3)\chi}. \end{align}

This implies that $g=\Omega(n^\chi)$ , and so $|S_{n,j}[\tau[q]]-(j/n)^\chi|\leqslant C^\star_q n^{-\chi/12}$ for $j=g-1, g$ . Additionally, a direct calculation yields

\begin{align*}\left|\Big(S_{n,f-1}[\tau[q]]/a^{(\textrm{i})}_{L[q]}\Big)-1\right|\leqslant \widetilde Cn^{-\beta_q}(\log \log n)^{(q+1)\chi}\end{align*}

for some $\widetilde C\,:\!=\,\widetilde C(x_1,\mu,q)$ . By replacing $S_{n,j}[\tau[q]]$ , $j=g-1, g$ , in (6.17) with $a^{(\textrm{i})}_{L[q]}$ and $(j/n)^\chi$ at the costs above, and using $\beta_{q+1}<\beta_q$ , we conclude that, on the event $\textrm{E}_q\cap \mathcal{H}_{q+1,1}$ , there is a constant $\widehat C\,:\!=\,\widehat C(x_1,\mu,q)$ such that $|a^{(\textrm{i})}_{L[q],1}-(g/n)^\chi|\leqslant \widehat Cn^{-\beta_{q+1}}$ . Now, pick $C_{q+1}\,:\!=\,\widehat C\vee \widehat c$ for the event $\mathcal{K}_{\tau[q],1}$ and $\mathcal{H}_{q+1,2}$ . By Lemma 13, there is a positive constant $C\,:\!=\,C(x_1,\mu,\kappa,q)$ such that

(6.19) \begin{align}\mathbb{P}_{\textbf{x}}\Bigg[\overline{\mathcal{H}}_{q,l}\cap \bigcap^3_{j=1} F_{\tau[q],j}\cap \mathcal{H}_{q+1,1}\cap \mathcal{K}^c_{\tau[q],1} \Bigg]&\leqslant \mathbb{P}_{\textbf{x}}\Bigg[\!\Big\{\textbf{Y}_\textrm{Be}^{(v[\tau[q]],n)}\not = \textbf{V}_\textrm{Po}^{(v[\tau[q]],n)}\Big\}\cap \bigcap^3_{j=1} F_{\tau[q], j}\cap \overline{\mathcal{H}}_{q,l}\Bigg]\nonumber\\&\leqslant C n^{-d}(\log \log n)^{q+1}(\log n)^{q/r},\end{align}

where $d=\min\{\beta_q,\gamma,1-\chi\}$ . The lemma follows from applying (6.15), (6.16), and (6.19) to (6.14).

The following analogue of Lemma 10 shows that under the graph coupling, w.h.p. $v[\tau[q]]$ has at most $(\log n)^{1/r}$ children. We omit the proof as it is similar to that of Lemma 10.

Lemma 14. Retaining the assumptions and the notation in Lemma 11, let $\mathcal{K}_{\tau[q],j}$ and $\overline{\mathcal{H}}_{q,j}$ be as in (6.2) and (6.5). Then, given positive integers $r<\infty$ and $q<r$ , there is a coupling of $\big(G_n,k^{(\textrm{u})}_0\big)$ and $(\mathcal{T}_{\textbf{x},n},0)$ , with an integer $p>0$ and a constant $C\,:\!=\,C(\kappa,p)>0$ such that

\begin{equation*}\mathbb{P}_{\textbf{x}}\Big[\overline{\mathcal{H}}_{q,l}\cap \mathcal{K}_{q,1}\cap\mathcal{K}^c_{q,2}\Big]\leqslant C (\log n)^{-\frac{p}{r}}(\log \log n)^{\frac{p(q+1)}{\mu+1}}\quad for\ all\ n\geqslant 3. \end{equation*}

We now apply Lemmas 12 and 14 to prove Lemma 11.

Proof of Lemma 11. We begin by stating the type-R analogues of Lemmas 12 and 14. Suppose that $\big(G_n,k^{(\textrm{u})}_0\big)$ and $(\mathcal{T}_{\textbf{x},n},0)$ are coupled so that for a type-R child $v[t]=k^{(\textrm{u})}_{\bar w}\in \partial \mathcal{B}_q$ , the event

(6.20) \begin{align}\mathcal{H}_{q+1,1}\cap \mathcal{J}_{q,l,t}\,:\!=\,\mathcal{H}_{q+1,1}\cap \overline{\mathcal{H}}_{q,l}\cap \bigcap_{\{s<t\,:\,v[s]\in \partial \mathcal{B}_q\}} (\mathcal{K}_{s,1}\cap\mathcal{K}_{s,2})\end{align}

has occurred for some $l\geqslant2$ , where $ \mathcal{H}_{q+1,1}$ , $\overline{\mathcal{H}}_{q,l}$ , $\mathcal{K}_{s,1}$ , and $\mathcal{K}_{s,2}$ are as in (6.1), (6.5), and (6.2). Let $\Big(\Big(\mathcal{Z}_j[t], \mathcal{\widetilde Z}_{j-1}[t]\Big),j \in \mathcal{A}_{t-1}\cup \mathcal{N}_{t-1}\Big)$ and $\left(S_{n,j}[t],1\leqslant j\leqslant n\right)$ be as in (3.7), (3.8), and (3.10). We use these variables to generate the type-R children of $k^{(\textrm{u})}_{\bar w}$ , and to sample the points $\Big(a^{(\textrm{i})}_{\bar w,i},1\leqslant i\leqslant R^{(\textrm{i})}_{\bar w}\Big)$ of the mixed Poisson process with intensity (6.4), where $\zeta_q$ and $a^{(\textrm{i})}_{L[q]}$ are replaced with $\mathcal{Z}_{k^{(\textrm{i})}_{\bar w}}[t]\sim \textrm{Gamma}\big(x_{k^{(\textrm{i})}_{\bar w}},1\big)$ and $a^{(\textrm{i})}_{\bar w}$ . Note that $v[t]>k^{(\textrm{u})}_{L[q]}$ , and on the event $\mathcal{J}_{q,l,t}$ , the number of discovered vertices up to time t can be bounded as

\begin{align*}|\mathcal{A}_{t-1}\cup \mathcal{P}_{t-1}|\leqslant 2+q+(q+1)^2(\log n)^{(q+1)/r},\end{align*}

as implied by (6.13). Hence, we can proceed similarly as for Lemmas 12 and 13. In particular, there is a coupling of $\big(G_n,k^{(\textrm{u})}_0\big)$ and $(\mathcal{T}_{\textbf{x},n},0)$ , with positive constants $C\,:\!=\,C(x_1,\mu,\kappa,q)$ and $c\,:\!=\,c(\kappa,p)$ such that for $n\geqslant 3$ ,

(6.21) \begin{align}&\mathbb{P}_{\textbf{x}}\left[\mathcal{K}^c_{t,1}\cap \mathcal{J}_{q,l,t}\right]\leqslant C n^{-d}(\log \log n)^{q+1} (\log n)^{\frac{q+1}{r}},\nonumber\\&\mathbb{P}_{\textbf{x}}\left[\mathcal{K}^c_{t,2}\cap \mathcal{K}_{t,1}\cap \mathcal{J}_{q,l,t}\right]\leqslant c (\log n)^{-\frac{p}{r}}(\log \log n)^{\frac{p(q+1)}{\mu+1}},\end{align}

where $d=\min\{\chi/3-4\gamma,1-\chi,\gamma,\beta_q\}$ . Define $\widetilde{\mathcal{H}}_q\,:\!=\,\bigcap^3_{j=1}{\mathcal{H}}_{q,j}$ . To bound ${\mathbb{P}}_{\textbf{x}}\big[\widetilde{\mathcal{H}}_{q+1}^c\big]$ using Lemma 12, Lemma 14, and (6.21), we use

(6.22) \begin{align}&\mathbb{P}_{\textbf{x}}\big[\big(\widetilde{\mathcal{H}}_{q+1}\big)^c\big]\leqslant \mathbb{P}_{\textbf{x}}\Big[\widetilde{\mathcal{H}}_{q+1}^c\cap \widetilde{\mathcal{H}}_{q} \Big] + \mathbb{P}_{\textbf{x}}\big[\widetilde{\mathcal{H}}_{q}^c\big]\nonumber\\&=\mathbb{P}_{\textbf{x}}\Bigg[\Bigg(\bigcap^{3}_{j=2}\mathcal{H}_{q+1,j}\Bigg)^c \cap \mathcal{H}_{q+1,1}\cap \widetilde{\mathcal{H}}_{q} \Bigg] + \mathbb{P}_\textbf{x}\Big[\mathcal{H}^c_{q+1,1}\cap \widetilde{\mathcal{H}}_{q}\Big]+ \mathbb{P}_{\textbf{x}}\big[\widetilde{\mathcal{H}}_{q}^c\big]\nonumber\\&=\mathbb{P}_{\textbf{x}}\Bigg[\Bigg(\bigcap_{\{s\,:\,v[s]\in \partial\mathcal{B}_q \}}(\mathcal{K}_{s,1}\cap\mathcal{K}_{s,2})\Bigg)^c\cap \mathcal{H}_{q+1,1}\cap \widetilde{\mathcal{H}}_{q}\Bigg]\end{align}

(6.23) \begin{align}&\hspace{5cm}+\mathbb{P}_\textbf{x}\Big[\widetilde{\mathcal{H}}_{q}\cap \mathcal{H}^c_{q+1,1}\Big]+\mathbb{P}_\textbf{x}\big[\widetilde{\mathcal{H}}^c_{q}\big],\end{align}

where the last equality follows from

\begin{equation*}\Bigg(\bigcap^{3}_{j=2}\mathcal{H}_{q+1,j}\Bigg)^c \cap \widetilde{\mathcal{H}}_{q}= \Bigg(\bigcap_{\{s\,:\,v[s]\in \partial\mathcal{B}_q \}}(\mathcal{K}_{s,1}\cap\mathcal{K}_{s,2})\Bigg)^c \cap \widetilde{\mathcal{H}}_{q}.\end{equation*}

The lemma is proved once we show that the probabilities in (6.22) and (6.23) are of order at most $(\log\log n)^{-\chi}$ . By assumption, there is a constant $C\,:\!=\,C(x_1,\mu,\kappa,q)$ such that $\mathbb{P}_{\textbf{x}}\big[\widetilde{\mathcal{H}}_{q}^c\big] \leqslant C (\log\log n)^{-\chi}$ . Recall that $a^{(\textrm{i})}_{L[q],1}=U_{L[q],1}a^{(\textrm{i})}_{L[q]}$ , where $U_{L[q],1}\sim\textrm{U}[0,1]$ is independent of $a^{(\textrm{i})}_{L[q]}$ . Thus, similarly to (5.13),

\begin{align*}\mathbb{P}_{\textbf{x}}\Big[\widetilde{\mathcal{H}}_{q}\cap \mathcal{H}^c_{q+1,1}\Big]\leqslant (\log \log n)^{-\chi}.\end{align*}

For (6.22), define

\begin{equation*}\widetilde{\mathcal{K}}_{q,s,j}=\mathcal{K}_{\Big|V\Big(B_q\big(G_n,k^{(\textrm{u})}_0\big)\Big)\Big|+s,j}\end{equation*}

for $j=1,2$ and $1\leqslant s\leqslant |\partial\mathcal{B}_q|$ . Note that $|\partial \mathcal{B}_q|\leqslant c[n,q]\,:\!=\, \lfloor{1+q(\log n)^{\frac{q}{r}}}\rfloor$ on the event $\mathcal{H}_{q,3}$ , and so

\begin{align*}&\mathbb{P}_{\textbf{x}}\Bigg[\Bigg(\bigcap_{\{s\,:\,v[s]\in \partial\mathcal{B}_q \}}(\mathcal{K}_{s,1}\cap\mathcal{ K}_{s,2})\Bigg )^c\cap \mathcal{H}_{q+1,1}\cap \widetilde{\mathcal{H}}_{q}\Bigg]\\&\qquad = \mathbb{P}_{\textbf{x}}\left[{\bigcup^{c[n,q]}_{l=1}\left\{{\{|\partial\mathcal{B}_q|=l\}\cap \left({\bigcap^l_{s=1}\Big(\mathcal{\widetilde K}_{q,s,1}\cap\mathcal{\widetilde K}_{q,s,2}\Big)}\right)^c}\right\}\cap \mathcal{H}_{q+1,1}\cap \widetilde{\mathcal{H}}_{q}}\right].\end{align*}

By a union bound, and recalling the definition of $\overline{\mathcal{H}}_{q,l}$ in (6.5), the probability above is at most

(6.24) \begin{align}&\sum^{c[n,q]}_{l=1}\bigg\{\mathbb{P}_{\textbf{x}}\Big[\widetilde{\mathcal{K}}^c_{q,1,2}\cap \widetilde{\mathcal{K}}_{q,1,1}\cap\mathcal{H}_{q+1,1}\cap\overline{\mathcal{H}}_{q,l}\Big]+ \mathbb{P}_{\textbf{x}}\Big[\widetilde{\mathcal{K}}^c_{q,1,1}\cap\mathcal{H}_{q+1,1}\cap \overline{\mathcal{H}}_{q,l}\Big]\nonumber\\&\quad+\sum^l_{s=2} \mathbb{P}_{\textbf{x}}\Big[\widetilde{\mathcal{K}}^c_{q,s,2}\cap \widetilde{\mathcal{K}}_{q,s,1}\cap\mathcal{H}_{q+1,1}\cap \mathcal{J}_{q,l,s}\Big]+\mathbb{P}_{\textbf{x}}\Big[\widetilde{\mathcal{K}}^c_{q,s,1}\cap\mathcal{H}_{q+1,1}\cap \mathcal{J}_{q,l,s}\Big]\bigg\}. \end{align}

We bound (6.24) using (6.21), Lemma 12, and Lemma 14, where we choose $p>2(r-1)$ in Lemma 14 so that

\begin{equation*}\sum^{c[n,q]}_{l=1}\sum^l_{s=2} \mathbb{P}_{\textbf{x}}\Big[\widetilde{\mathcal{K}}^c_{q,s,2}\cap \widetilde{\mathcal{K}}_{q,s,1}\cap \mathcal{J}_{q,l,s}\Big]\to 0\end{equation*}

as $n\to\infty$ . It follows that there is a constant $C\,:\!=\,C(x_1,\mu,\kappa,q)$ such that (6.24) is bounded by $C(\log\log n)^{-\chi}$ .

7. Completion of the local weak limit proof

Equipped with Corollary 2, we can prove Theorem 1.

Proof of Theorem 1. In view of (1.1), it is enough to establish (1.5) of Theorem 1. Let $G^{\prime}_n\sim \textrm{PA}(\pi,X_1)_n$ (Definition 1), let $k_0$ be its randomly chosen vertex, let $(\mathcal{T}_{\textbf{X},n},0)$ be the intermediate Pólya point tree $(\mathcal{T}_{\textbf{x},n},0)$ randomised over $\textbf{X}$ , and let $(\mathcal{T},0)$ be the $\pi$ -Pólya point tree in Definition 3. By the triangle inequality for the total variation distance, for any $r<\infty$ we have

\begin{align*}&d_{\textrm{TV}}\left(\mathscr{L}\left(\left(B_r\left(G^{\prime}_n,k_0\right),k_0\right)\right), \mathscr{L}\left((B_r(\mathcal{T},0),0)\right)\right)\nonumber\\&\quad \leqslant d_{\textrm{TV}}\left(\mathscr{L}\left(\left(B_r\left(G^{\prime}_n,k_0\right),k_0\right)\right), \mathscr{L}\left(\left(B_r(\mathcal{T}_{\textbf{X},n},0),0\right)\right)\right)\\&\hspace{5cm}+ d_{\textrm{TV}}\left(\mathscr{L}\left(\left(B_r(\mathcal{T}_{\textbf{X},n},0),0\right)\right), \mathscr{L}\left((B_r(\mathcal{T},0),0)\right)\right).\end{align*}

Let $A_n\,:\!=\,A_{2/3,n}$ be as in (2.1). Applying Jensen’s inequality to the total variation distance, the above is bounded by

\begin{align*}&\mathbb{E} \left[d_{\textrm{TV}}\left(\mathscr{L}\left(\left(B_r\left(G^{\prime}_n,k_0\right),k_0\right)|\textbf{X}\right),\mathscr{L}\left((B_r(\mathcal{T}_{\textbf{X},n},0),0)|\textbf{X}\right)\right)\right]\\&\hspace{3cm}+ d_{\textrm{TV}}\left(\mathscr{L}\left((B_r(\mathcal{T}_{\textbf{X},n},0),0)\right), \mathscr{L}\left((B_r(\mathcal{T},0),0)\right)\right)\\&\leqslant \mathbb{E} \left[\mathbb{1}[ A_n]d_{\textrm{TV}}\left(\mathscr{L}\left(\left(B_r\left(G^{\prime}_n,k_0\right),k_0\right)|\textbf{X}\right),\mathscr{L}\left((B_r(\mathcal{T}_{\textbf{X},n},0),0)|\textbf{X}\right)\right)\right] \\&\hspace{3cm}+\mathbb{P}\left[ A^c_n\right] + d_{\textrm{TV}}\left(\mathscr{L}\left(\left(B_r\left(\mathcal{T}_{\textbf{X},n},0\right),0\right)\right), \mathscr{L}\left((B_r(\mathcal{T},0),0)\right)\right).\end{align*}

We prove that each term on the right-hand side is of order at most $(\log\log n)^{-\chi}$ , starting from the expectation. Let $\mathcal{H}_{r,j}$ , $j=1,2,3$ , be as in (6.1). For the $(\textbf{x},n)$ -Pólya urn tree $G_n$ with $\textbf{x}\in A_n$ , Corollary 2 implies that there is a coupling of $\big(G_n,k^{(\textrm{u})}_0\big)$ and $(\mathcal{T}_{\textbf{x},n},0)$ , with a positive constant $C\,:\!=\,C(x_1,\mu,\kappa,r)$ such that

(7.1) \begin{align}\mathbb{P}_{\textbf{x}}\left[B_r(\mathcal{T}_{\textbf{x},n},0),0\not \cong \Big(B_r\Big(G_n,k^{(\textrm{u})}_0\Big),k^{(\textrm{u})}_0\Big)\right]\leqslant \mathbb{P}_{\textbf{x}}\Bigg[\Bigg(\bigcap^{3}_{j=1}\mathcal{H}_{r,j}\Bigg)^c\Bigg] \leqslant C (\log \log n)^{-\chi} ;\end{align}

following from the definition (1.4) of the total variation distance, the expectation is at most $C(\log \log n)^{-\chi}$ . The probability $\mathbb{P}[ A^c_n]$ can be bounded using Lemma 1 (with $\alpha=2/3$ ). We now couple $(B_r(\mathcal{T},0),0)$ and $(B_r(\mathcal{T}_{\textbf{X},n},0),0)$ to bound the last term. For this coupling, denote by E the event that the PA labels in $B_r(\mathcal{T}_{\textbf{X},n},0)$ are distinct, that is, $k^{(\textrm{i})}_{\bar w}\not= k^{(\textrm{i})}_{\bar v}$ for any $\bar w\not =\bar v$ . We first construct $B_r(\mathcal{T}_{\textbf{X},n},0)$ ; then on the event E, we set $B_r(\mathcal{T},0)$ as $B_r(\mathcal{T}_{\textbf{X},n},0)$ , inheriting the Ulam–Harris labels, ages, types, and fitness from the latter. If the PA labels are not distinct, we generate $B_r(\mathcal{T},0)$ independently from $B_r(\mathcal{T}_{\textbf{X},n},0)$ . Under this coupling,

(7.2) \begin{align}\mathbb{P}[(B_r(\mathcal{T}_{\textbf{X},n},0),0)\not \cong (B_r(\mathcal{T},0),0)]= \mathbb{P}[\{(B_r(\mathcal{T}_{\textbf{X},n},0),0)\not\cong (B_r(\mathcal{T},0),0)\}\cap\textrm{E}^c]. \end{align}

From the definition of $\mathcal{H}_{r,2}$ in (6.1), we can use (7.1) and Lemma 1 to bound the second term in (7.2) by $\mathbb{P}[\textrm{E}^c]\leqslant \mathbb{P}\left[\left(\bigcap^3_{j=1}\mathcal{H}_{r,j}\right)^c\right]\leqslant C(\log\log n)^{-\chi}$ . So, once again by (1.4),

\begin{equation*}d_{\textrm{TV}}\left(\mathscr{L}\left((B_r(\mathcal{T}_{\textbf{X},n},0),0)\right), \mathscr{L}\left((B_r(\mathcal{T},0),0)\right)\right)\leqslant C(\log\log n)^{-\chi},\end{equation*}

which concludes the proof.

8. Remarks on the limiting distributions of the degree statistics

In this section, we discuss the connection of Corollary 1 and Theorem 2 to [Reference Berger, Borgs, Chayes and Saberi4, Reference Bollobás, Riordan, Spencer and Tusnády7, Reference Van der Hofstad20, Reference Lodewijks and Ortgiese25, Reference Peköz, Röllin and Ross30]. To this end, we state the probability mass function of the limiting distributions of the degrees of the uniformly chosen vertex $k_0$ in PA $(\pi,X_1)_n$ and its type-L neighbour $k_{L[1]}$ . These results are direct consequences of Corollary 1 and Theorem 2 (see [Reference Lo24] for a detailed proof). We also show that both probability mass functions also exhibit power-law behaviour, which follows from a dominated convergence argument (see [Reference Lodewijks and Ortgiese25]). Below we write $a_n\sim b_n$ to indicate $\lim_{n\to\infty} a_n/b_n=1$ . We start with the degree of the uniformly chosen vertex.

Proposition 1. Retaining the assumptions and the notation in Theorem 2, let $\xi_0$ be as in the theorem. The probability mass function of the distribution of $\xi_0$ is

(8.1) \begin{align}p_{\pi}(j)= (\mu+1)\int^\infty_0 \frac{\Gamma(x+j-1)\Gamma(x+\mu+1)}{\Gamma(x)\Gamma(x+\mu+j+1)} d\pi(x),\qquad j\geqslant 1.\end{align}

Furthermore, if $\mathbb{E} X^{\mu+1}_2<\infty$ , then as $j\to\infty$ ,

(8.2) \begin{equation}p_{\pi}(j)\sim C_{\pi} j^{-(\mu+2)},\qquad C_{\pi}\,:\!=\,(\mu+1)\int^\infty_0 \frac{\Gamma(x+\mu+1)}{\Gamma(x)} d\pi(x).\end{equation}

In the following, we give the limiting probability mass function of the degree of the type-L neighbour $k_{L[1]}$ of the uniformly chosen vertex $k_0$ , which shows that the distribution also exhibits power-law behaviour. When the PA tree has constant initial attractiveness, the proposition is a special case of [Reference Berger, Borgs, Chayes and Saberi4, Lemma 5.2] and [Reference Van der Hofstad20, Lemma 5.9].