2.1. Examples

In this section, we discuss examples of spatial random graph models that are covered under our set-up.

2.1.1. Product SIRGs.

We begin by discussing a particular type of SIRG, in which the connection has a product form.

Definition 5. (Product SIRGs.) For a SIRG $G(\textbf{X},\textbf{W},\kappa)$ , if the connection function $\kappa$ has the product form

(11) \begin{equation} \kappa(t,x,y)= 1 \wedge f(t)g(x,y), \end{equation}

for some non-negative functions $f\colon \mathbb{R}_+ \to \mathbb{R}_+$ and $g\colon\mathbb{R}^2 \to \mathbb{R}_+$ , where g is symmetric, then we call the SIRG a product SIRG, or PSIRG for short, and we call $\kappa$ a product kernel.

Theorem 1 can be directly adapted to the PSIRG setting under appropriate conditions on the product kernel.

Assumption 4. (PSIRG connection functions.) Let $\kappa\colon \mathbb{R}_+\times \mathbb{R} \times \mathbb{R} \to [0,1]$ be a product kernel $\kappa(t,x,y) = 1\wedge f(t)g(x,y)$ such that the following hold:

1. 1. There exist $\alpha_p > 1$ and $t_1 \in \mathbb{R}_+$ such that for all $t>t_1$ ,

   \begin{align*}f(t)\leq t^{-\alpha_p}.\end{align*}

2. 2. There exist $\beta_p > 0$ and $t_2 \in \mathbb{R}_+$ such that for all $t> t_2$ ,

   \begin{align*}\mathbb{P}\left(g(W^{(1)},W^{(2)})>t\right)\leq t^{-\beta_p},\end{align*}
   where $W^{(1)}, W^{(2)}$ are i.i.d. copies of the limiting weight random variable W in (4).

Let $W^{(1)}$ and $W^{(2)}$ be two i.i.d. copies of W (see Assumption 2). The following lemma derives a useful bound on product kernels that satisfy Assumption 4.

Lemma 1. (Bound on product kernel.) Under Assumption 4, for any $\epsilon>0$ , there exists $t_0=t_0(\epsilon)>0$ such that whenever $t>t_0$ ,

\begin{align*} \mathbb{E}\left[\kappa(t,W^{(1)},W^{(2)})\right] \leq t^{-\min\{\alpha_p,\alpha_p\beta_p\}+\epsilon}. \end{align*}

In other words, if the limiting connection function $\kappa$ in Theorem 2 satisfies Assumption 4, it also satisfies Assumption 3(3), with $\alpha=\min\{\alpha_p,\alpha_p\beta_p\}-\epsilon$ for any $\epsilon>0$ . So, if

\begin{align*} \gamma_p\,{:\!=}\,\min\{\alpha_p,\alpha_p\beta_p\}>d, \end{align*}

then using Lemma 1, we have that $\kappa$ satisfies Assumption 3(3) with some $\alpha>d$ , by choosing $\epsilon>0$ sufficiently small. Hence we obtain the following direct corollary to Theorem 2, whose proof we omit. Recall the vector $\textbf{W}=(W_i)_{i \in \mathbb{N} \cup \{0\}}$ of i.i.d. copies of the limiting weight variable W (see (4)), and $\textbf{Y}=(Y_i)_{i \in \mathbb{N} \cup \{0\}}$ the atoms of $\Gamma_{\infty}$ (see (10)), with $Y_0=\textbf{0}$ .

Corollary 1. (Convergence of PSIRGs locally in probability.) Consider the sequence $(\mathbb{G}_n)_{n\geq 1}=(G(\textbf{Y}^{(n)},\textbf{W}^{(n)},\kappa_n))_{n \geq 1}$ of SIRGs, where, for each n and each $i\in[n]$ , $Y^{(n)}_i=n^{1/d}X^{(n)}_i$ , with $\textbf{X}^{(n)}=(X^{(n)}_i)_{i\in[n]}$ and $\textbf{W}^{(n)}=(W^{(n)}_i)_{i\in[n]}$ satisfying Assumptions 1 and 2, $\kappa_n$ satisfying Assumption 3(1), $\kappa$ satisfying Assumption 4, and

\begin{align*} \gamma_p\,{:\!=}\,\min\{\alpha_p, \alpha_p\beta_p\} > d. \end{align*}

Then $(\mathbb{G}_n)_{n \geq 1}$ converges locally in probability to the infinite rooted SIRG $(\mathbb{G}_{\infty},0)$ , rooted at vertex 0, where $\mathbb{G}_{\infty}=G(\textbf{Y},\textbf{W},\kappa)$ .

Remark 3. (Regularly varying product forms.) Note that if f(t) and $\mathbb{P}\left(g(W^{(1)},W^{(2)})>t\right)$ are regularly varying functions of t outside some compact sets, with respective exponents $\alpha_p>0$ and $\beta_p>0$ with $\min\{\alpha_p,\alpha_p\beta_p\}>d$ , then by Potter’s theorem (see [Reference Potter30]), they respectively satisfy Assumptions 4(1) and 4(2) with some exponents $\overline{\alpha_p}>0$ and $\overline{\beta_p}>0$ , with $\min\{\overline{\alpha_p},\overline{\alpha_p}\overline{\beta_p}\}>d$ . Hence for these kinds of connection functions, Lemma 1, and hence Corollary 1, continue to be true.

Remark 4. (Dominance by PSIRG connections.) Note that even if $\kappa$ is not of product form, but instead is dominated by $1 \wedge f(t)g(x,y)$ , with f and g respectively satisfying Assumptions 4(1) and 4(2), with $\gamma_p>d$ , then Lemma 1, and hence Corollary 1, continue to be true.

Next we discuss several known models which are all examples of PSIRGs, or of SIRGs with connection functions dominated by PSIRG connection functions. The results that follow are presented as corollaries of Theorem 2, and their proofs are in Section 3.7.

2.1.2. Geometric inhomogeneous random graphs.

Geometric inhomogeneous random graphs (GIRGs) [Reference Bringmann, Keusch and Lengler6–Reference Bringmann, Keusch and Lengler8] were motivated as spatial versions of the classic Chung–Lu random graphs [Reference Chung and Lu9, Reference Chung and Lu10]. Although Chung–Lu random graphs with suitable parameters are scale-free and exhibit small-world properties, they fail to capture clustering, a ubiquitous property of real-world networks.

GIRGs have four parameters: the number of vertices n, $\alpha_G \in (1,\infty]$ , $\beta_G>2$ , and the dimension $d\geq 1$ . To each vertex $i \in \mathbb{N}$ , one associates an independent, uniformly distributed random location $X^{(n)}_i$ on I (recall (3)) and a real weight $W^{(n)}_i$ (possibly random) such that (4) holds. Here the limiting weight variable W has a power-law tail with exponent $\beta_G$ : there exists $t_G \in \mathbb{R}_+$ such that

(12) \begin{equation} c_Gz^{1-\beta_G} \leq \mathbb{P}\left(W>z\right)\leq C_Gz^{1-\beta_G}, \end{equation}

whenever $z > t_G$ , for some absolute constants $c_G, C_G>0$ . In particular, we have $\mathbb{E}\left[W\right]< \infty$ . Conditionally on $(X^{(n)}_i)_{i \in [n]}$ and $(W^{(n)}_i)_{i \in [n]}$ , each edge $\{i,j\}$ is included independently with probability $p_{i,j}$ given by

(13) \begin{equation} \begin{split} p_{i,j}= \begin{cases} 1 \wedge \left(\frac{W^{(n)}_iW^{(n)}_j}{\sum_{i\in[n]} W^{(n)}_i}\right)^{\alpha_G}\frac{1}{\|X^{(n)}_i-X^{(n)}_j\|^{d\alpha_G}} &\text{if}\; 1<\alpha_G<\infty, \\ \mathbb{1}_{\left\{\left(\frac{W^{(n)}_iW^{(n)}_j}{\sum_{i\in[n]} W^{(n)}_i}\right)^{1/d}>\|X^{(n)}_i-X^{(n)}_j\|\right\}} & \text{if}\; \alpha_G = \infty. \end{cases} \end{split} \end{equation}

We denote the resulting random graph by

\begin{align*} \textrm{GIRG}_{n,\alpha_G,\beta_G,d}. \end{align*}

Remark 5. (Relationship to GIRGs in [Reference Bringmann, Keusch and Lengler8].) Our formulation of GIRGs is closer to the formulation adopted in [Reference Komjáthy and Lodewijks24] than to that of [Reference Bringmann, Keusch and Lengler8]. In the original definition of GIRGs (see [Reference Bringmann, Keusch and Lengler8]), the connection function is only assumed to be bounded above and below by multiples of (13), and vertex locations are assumed to be uniform on the torus $\mathbb{T}_n$ which is obtained by identifying the boundaries of I. However, to define a local limit, we need the connection function to converge to a limiting function, for which we have taken the explicit form (13). Finally, using the observation that only a negligible proportion of vertex locations fall near the boundary of I, our results can easily be transferred to the torus setting.

Now consider the infinite SIRG $G(\textbf{Y},\textbf{W},\kappa^{\alpha_G})$ , where $\textbf{Y}=(Y_i)_{i \in \mathbb{N}\cup \{0\}}$ is the sequence of atoms of $\Gamma \cup \{\textbf{0}\}$ (with $Y_0=\textbf{0}$ ), $\Gamma$ is a unit-rate homogeneous Poisson point process on $\mathbb{R}^d$ , $\textbf{W}=(W_i)_{i \in \mathbb{N}\cup \{0\}}$ is an i.i.d. collection of limiting weight random variables W, and $\kappa^{(\alpha_G)}$ is the connection function

(14) \begin{equation} \begin{split} \kappa^{(\alpha_G)}(t,x,y)= \begin{cases} 1 \wedge \left(\frac{xy}{\mathbb{E}\left[W\right]}\right)^{\alpha_G}{t^{-d\alpha_G}} &\text{if}\; 1<\alpha_G<\infty, \\ \mathbb{1}_{\left\{\left(\frac{xy}{\mathbb{E}\left[W\right]}\right)^{1/d}>t\right\}} & \text{if}\; \alpha_G = \infty. \end{cases} \end{split} \end{equation}

As a corollary to Theorem 2, we establish the local limit of the GIRG sequence to be $(G(\textbf{Y},\textbf{W},\kappa^{\alpha_G}),0)$ . This answers a question posed in [Reference Komjáthy and Lodewijks24] (see [Reference Komjáthy and Lodewijks24, Section 2.1]) in the affirmative. We call the above infinite SIRG the infinite GIRG, and denote it by $\textrm{GIRG}_{\infty,\alpha_G,\beta_G,d}$ .

Corollary 2. (Convergence of GIRGs locally in probability.) As $n\rightarrow \infty$ , the sequence $(\textrm{GIRG}_{n,\alpha_G,\beta_G,d})_{n \geq 1}$ converges locally in probability to the rooted infinite GIRG $(\textrm{GIRG}_{\infty,\alpha_G,\beta_G,d},0)$ , rooted at 0, where $\alpha_G \in (1,\infty]$ , $\beta_G > 2$ , $d \in \mathbb{N}$ .

2.1.3. Hyperbolic random graphs.

Hyperbolic random graphs (HRGs) were first proposed by Krioukov et al. in 2010 [Reference Krioukov25], as a model that captures the three main properties of most real-world networks: scale-free, small distances, and non-vanishing clustering coefficient.

HRGs have three parameters, namely the number of vertices n, $\alpha_H>\frac{1}{2}$ , and $\nu>0$ , which are fixed constants. Let

(15) \begin{align} R_n\,{:\!=}\,2\log{\frac{n}{\nu}}. \end{align}

The vertex set of the graph is the set of n i.i.d. points $u^{(n)}_1,\ldots,u^{(n)}_n$ on the hyperbolic plane $\mathbb{H}$ , where $u^{(n)}_i=(r^{(n)}_i,\theta^{(n)}_i)$ is the polar representation of $u^{(n)}_i$ . The angular component vector $(\theta^{(n)}_i)_{i=1}^n$ is a vector with i.i.d. coordinates, each coordinate having the uniform distribution on $[-\pi,\pi]$ . The radial component vector $(r^{(n)}_i)_{i=1}^n$ is independent of $(\theta_i)_{i=1}^n$ and has i.i.d. coordinates, with cumulative distribution function

(16) \begin{equation} F^{(n)}_{\alpha_H,\nu}(r)= \begin{cases} 0 &\text{if}\;\;r<0, \\ \dfrac{\cosh{\alpha_H r-1}}{\cosh{\alpha_H R_n -1}} &\text{if}\;\;0\leq r \leq R_n, \\ 1 & \text{if}\;\; r> R_n. \end{cases} \end{equation}

Given $(u^{(n)}_i)_{i=1}^n=((r^{(n)}_i,\theta^{(n)}_i))_{i=1}^n$ , one forms the threshold HRG (THRG) by placing edges between all pairs of vertices $u^{(n)}_i$ and $u^{(n)}_j$ with conditional probability

(17) \begin{align} p^{(n)}_{\textrm{THRG}}\big(u^{(n)}_i,u^{(n)}_j\big)\,{:\!=}\,\mathbb{1}_{\left\{d_{\mathbb{H}}\big(u^{(n)}_i,u^{(n)}_j\big)<R_n\right\}}, \end{align}

where $d_{\mathbb{H}}$ denotes the distance in the hyperbolic plane $\mathbb{H}$ ; i.e., the edge between $u^{(n)}_i$ and $u^{(n)}_j$ is included if and only if $d_{\mathbb{H}}\big(u^{(n)}_i,u^{(n)}_j\big)<R_n$ .

Similarly, one forms a parametrized version of the THRG (see [Reference Krioukov25, Section 6]), which we call the parametrized HRG (PHRG), by placing edges independently between $u^{(n)}_i$ and $u^{(n)}_j$ with conditional probability

(18) \begin{align} {p^{(n)}_{\textrm{PHRG}}\big(u^{(n)}_i,u^{(n)}_j\big)}\,{:\!=}\,\left(1 + \exp{\left(\frac{d_{\mathbb{H}}\big(u^{(n)}_i,u^{(n)}_j\big)-R_n}{2T_H}\right)}\right)^{-1}, \end{align}

where $T_H>0$ is another parameter.

We denote the THRG model with parameters n, $\alpha_H$ , $\nu$ by $\textrm{THRG}_{n,\alpha_H,\nu},$ and the PHRG model with parameters $n,\alpha_H,T_H,\nu$ by $\textrm{PHRG}_{n,\alpha_H,T_H,\nu}.$

Both the THRG and PHRG models can be seen as finite SIRGs, which gives us the local limits for these models, as a corollary to Theorem 2.

Corollary 3. (Convergence of HRGs locally in probability.) Let $\alpha_H> {\tfrac{1}{2}}$ , $0<T_H<1$ , and $n \in \mathbb{N}$ . Let $\textbf{Y}$ be the sequence of atoms of (10). Then there exists a random variable W having a power-law distribution with exponent $2\alpha_H+1$ , such that if $\textbf{W}=(W_i)_{i \in \mathbb{N}\cup\{0\}}$ is a sequence of i.i.d. copies of W, the following hold:

1. (a) The sequence $(\textrm{THRG}_{n,\alpha_H,\nu})_{n \geq 1}$ converges locally in probability to the infinite SIRG $G(\textbf{Y},\textbf{W},\kappa_{\textrm{THRG},\infty})$ , where

   \begin{align*} \kappa_{\textrm{THRG},\infty}(t,x,y)\,{:\!=}\, \mathbb{1}_{\left\{t \leq \frac{\nu xy}{\pi}\right\}}. \end{align*}

2. (b) The sequence $(\textrm{PHRG}_{n,\alpha_H,T_H,\nu})_{n \geq 1}$ converges locally in probability to the infinite SIRG $G(\textbf{Y},\textbf{W},\kappa_{\textrm{PHRG},\infty})$ , where

   \begin{align*} \kappa_{\textrm{PHRG},\infty}(t,x,y)\,{:\!=}\, \left(1+\left(\frac{ \pi t}{\nu xy} \right)^{1/T_H} \right)^{-1}. \end{align*}

2.1.4. Continuum scale-free percolation.

The continuum scale-free percolation (CSFP) model [Reference Deprez and Wüthrich13] was introduced as a continuum analogue of the discrete scale-free percolation (SFP) model [Reference Deijfen, van der Hofstad and Hooghiemstra12], a model that captures power-law degree distributions while preserving non-zero clustering and logarithmic typical distances.

We now formally define the model following [Reference Deprez and Wüthrich13]. The vertex set is the set of points of a homogeneous Poisson point process $(Y_i)_{i \in \mathbb{N}}$ , marked with i.i.d. weights $(W_i)_{i \in \mathbb{N}}$ which have a Pareto distribution with power-law tail parameter $\beta>0$ and scale parameter 1:

(19) \begin{align} \mathbb{P}\left(W>w\right)=w^{- \beta}, \end{align}

whenever $w>1$ . Conditionally on $(Y_i)_{i \in \mathbb{N}}$ and $(W_i)_{i \in \mathbb{N}}$ , each edge $\{Y_i,Y_j\}$ is included independently with probability

\begin{align*} 1-\exp{\left(-\frac{\lambda W_iW_j}{\|Y_i-Y_j\|^{\alpha}}\right)}, \end{align*}

where $\lambda>0$ is a parameter. Here we remark that in the original definition in [Reference Deprez and Wüthrich13], instead of a homogeneous Poisson process $(Y_i)_{i \in \mathbb{N}}$ , a Poisson process with some constant intensity $\nu>0$ was considered. By standard scaling arguments, this does not make any difference in our results.

Considering the Palm version $(Y_i)_{i \in \mathbb{N}\cup \{0\}}$ of $(Y_i)_{i \in \mathbb{N}}$ , where $Y_0=\textbf{0} \in \mathbb{R}^d$ , marking $Y_0$ with an independent weight $W_0$ , and rooting the resulting graph at 0, it is immediate that the resulting rooted infinite CSFP model is the rooted SIRG $(G(\textbf{Y},\textbf{W},\kappa),0)$ , where $\textbf{Y}=(Y_i)_{i \in \mathbb{N}\cup \{0\}}$ , $\textbf{W}=(W_i)_{i \in \mathbb{N}\cup\{0\}}$ , and $\kappa(t,x,y)\,{:\!=}\,1-\exp{\left(-\frac{\lambda xy}{t^{\alpha}}\right)} $ .

For each $n \geq 1$ , let $\textbf{X}^{(n)}=(X^{(n)}_i)_{i \in [n]}$ satisfy Assumption 1, and consider the ‘blown-up’ finite CSFP model $G(\textbf{Y}^{(n)},\textbf{W}^{(n)},\kappa_n)$ , where $Y^{(n)}_i=n^{1/d}X^{(n)}_i$ for $i \in [n]$ , $\textbf{W}^{(n)}=(W^{(n)}_i)_{i \in [n]}$ is a vector of i.i.d. weight variables having law (19), and $\kappa_n(t,x,y)=1-\exp{\left(-\frac{\lambda xy}{n^{-\frac{\alpha}{d}}t^{\alpha}}\right)}$ . Then as corollary to Theorem 2, we have that the infinite CSFP is the local limit of finite CSFPs under suitable assumptions.

Corollary 4. (Convergence of CSFPs locally in probability.) Let

\begin{align*} \min\{\alpha,\alpha\beta\}>d. \end{align*}

Then, as $n \to \infty$ , the graph sequence $G(\textbf{Y}^{(n)},\textbf{W}^{(n)},\kappa_n)$ converges locally in probability to the infinite rooted CSFP $(G(\textbf{Y},\textbf{W},\kappa),0)$ .

2.1.5. Weight-dependent random connection models.

Another very general class of spatial random graph model, called the weight-dependent random connection model (WDRCM), was first introduced in [Reference Gracar, Heydenreich, Mönch and Mörters18], motivated by the study of recurrence and transience properties of general geometric graphs, which we briefly discuss.

To construct the graph, one takes a unit-rate Poisson process on $\mathbb{R}^d \times[0,1]$ , conditionally on which edges between pairs of vertices $(\textbf{x},s)$ and $(\textbf{y},t)$ are placed independently with probability

\begin{align*} \rho(h(s,t,\|\textbf{x}-\textbf{y}\|)), \end{align*}

for some profile function $\rho\,{:}\,\mathbb{R}_+ \to [0,1]$ and a suitable kernel $h\,{:}\,[0,1]\times[0,1]\times \mathbb{R}_+ \to \mathbb{R}_+$ . The vertex $(\textbf{x},s)$ is thought of as being located at $\textbf{x} \in \mathbb{R}^d$ , and having a weight of $s^{-1}$ associated to it.

Including the point $\textbf{r}=(\textbf{0},U) \in \mathbb{R}^d \times [0,1]$ , where U is uniformly distributed on [0, 1] and independent of the weights and locations of other vertices, and rooting the graph at $\textbf{r}$ , we obtain the infinite rooted SIRG $(G(\textbf{Y},\textbf{W},\kappa),0)$ , where $\textbf{Y}=(Y_i)_{i \in \mathbb{N}\cup \{0\}}$ are the atoms of (10), $\textbf{W}=(W_i)_{i \in \mathbb{N}\cup\{0\}}$ is a sequence of i.i.d. random weights that are uniform on [0, 1], and $\kappa = \rho \circ h$ .

For $n \geq 1$ , let $(X^{(n)}_i)_{i \in [n]}$ satisfy Assumption 1, let $(W^{(n)}_i)_{i \in [n]}$ be a collection of n i.i.d. weights that are uniform on [0, 1], and let $\kappa_n:\mathbb{R}\times \mathbb{R} \times \mathbb{R}_+ \to [0,1]$ be defined as $\kappa_n(t,x,y)=\rho(h(x,y,n^{-1/d}t))$ , with $\kappa(t,x,y)=\rho(g(x,y,t))$ satisfying Assumption 3(2). Then as a direct consequence of Theorem 2 we have the following corollary, whose proof we omit.

Corollary 5. (Convergence of WDRCMs locally in probability.) Let $\textbf{Y}^{(n)}=(Y^{(n)}_i)_{i \in [n]}=(n^{1/d}X^{(n)}_i)_{i \in [n]}$ . Then, as $n \to \infty$ , the sequence of SIRGS $G(\textbf{Y}^{(n)},\textbf{W}^{(n)},\kappa_n)$ converges locally in probability to the infinite rooted WDRCM $(G(\textbf{Y},\textbf{W},\kappa),0)$ .

2.2. Consequences of local convergence: degrees

Theorem 2 is equivalent to the statement that for any subset $A \subset \mathcal{G}_{\star}$ ,

\begin{align*}\frac{1}{n}\sum_{i=1}^n \mathbb{1}_{\left\{(\mathbb{G}_n,i)\in A\right\}}\end{align*}

converges in probability to $\mathbb{P}\left((\mathbb{G}_{\infty},0)\in A\right)$ (see e.g. [Reference Van der Hofstad22, (2.3.5)]).

In particular, for fixed $k\in \mathbb{N}$ , one can take $A_k$ to be the subset of those rooted graphs (G, o) for which the root o has degree k in G, to conclude that

\begin{align*} \frac{N_k(\mathbb{G}_n)}{n} {{\stackrel{\mathbb{P}}{\rightarrow}}} \mathbb{P}\left(D=k\right)\!, \end{align*}

where $N_k(\mathbb{G}_n)$ is the number of vertices with degree k in $\mathbb{G}_n$ , and where D is the degree of 0 in $\mathbb{G}_{\infty}$ .

Taking expectation and applying dominated convergence, we have that for any $ {k} \in \mathbb{N}$ , $\mathbb{P}\left(D_n=k\right)\to \mathbb{P}\left(D=k\right)$ as $n \to \infty$ , where $D_n$ is the degree of $U_n$ in $\mathbb{G}_n$ , which implies

(20) \begin{equation} D_n {{\stackrel{d}{\rightarrow}}} D, \end{equation}

(where ${{\stackrel{d}{\rightarrow}}}$ means convergence in distribution) as $n \to \infty$ . We next give a description of the random variable D. For this, let $W_0$ be the weight of 0 in $\mathbb{G}_{\infty}$ , and let $W^{(1)}$ be an independent copy of $W_0$ .

Proposition 1. (Degree distribution.) Under the assumptions of Theorem 1, the random variable D is distributed as

\begin{align*} \textrm{Poi}\left(\int_{\mathbb{R}^d} \mathbb{E}\left[\left.\kappa(\|z\|,W_0,W^{(1)})\right | W_0\right]dz \right); \end{align*}

i.e. it has a mixed Poisson distribution with mixing parameter

\begin{align*} \int_{\mathbb{R}^d} \mathbb{E}\left[\left.\kappa(\|z\|,W_0,W^{(1)})\right | W_0\right]dz. \end{align*}

Note that Proposition 1, Assumption 3(3), and Fubini’s theorem imply that

\begin{align*} \mathbb{E}\left[D\right]=\int_{\mathbb{R}^d}\mathbb{E}\left[\kappa(\|x\|,W_0,W^{(1)})\right]dx, \end{align*}

which is finite. Since the expectation of the mixing parameter is finite, the mixing parameter is finite almost surely, and hence Proposition 1 makes sense.

In particular, when the mixing parameter $\int_{\mathbb{R}^d} \mathbb{E}\left[\left.\kappa(\|z\|,W_0,W^{(1)})\right | W_0\right]dz$ is regularly varying with some exponent $\zeta>0$ , the random variable D is also regularly varying with the same exponent $\zeta>0$ . This allows for the existence of power-law degree distributions in spatial random graphs.

Recall that $D_n$ is the degree of the uniform vertex $U_n$ in $\mathbb{G}_n$ .

The proof is given in Section 3.8. This result is of independent interest. Uniform integrability of the typical degree sequence in general does not follow from local convergence, even when the limiting degree distribution has finite mean; see for example [Reference Van der Hofstad22, Exercise 2.14].

Combining Proposition 2 with (20), we note that

\begin{align*} \mathbb{E}\left[D_n\right] \to \mathbb{E}\left[D\right], \end{align*}

as $n \to \infty$ . Note that one cannot directly conclude this from Theorem 1, because the function $\textrm{D}\,{:}\,\mathcal{G}_{\star}\to \mathbb{R}_+$ defined by

\begin{align*} \textrm{D}((G,o))\,{:\!=}\,\text{degree of}\;\;o\;\;\text{in}\;\;G \end{align*}

is continuous, but not necessarily bounded.

2.3. Consequences of local convergence: clustering

In this section, we discuss convergence of various clustering measures of SIRGs.

For any graph $G=(V(G),E(G))$ , we let

(21) \begin{equation} \mathcal{W}_{G}\,{:\!=}\, \sum_{v_1,v_2,v_3 \in V(G)}\mathbb{1}_{\left\{\{v_1,v_2\},\{v_2,v_3\}\in E(G)\right\}}=\sum_{v \in V(G)} d_v(d_v-1) \end{equation}

(where $d_v$ is the degree of v in G) be twice the number of wedges in the graph G, and we let

(22) \begin{equation} \Delta_{G}\,{:\!=}\, \sum_{v_1,v_2,v_3 \in V(G)} \mathbb{1}_{\left\{\{v_1,v_2\},\{v_2,v_3\},\{v_3,v_1\} \in E(G)\right\}} \end{equation}

be six times the number of triangles in the graph G, where the sums in (21) and (22) are over distinct vertices $v_1,v_2,v_3 \in V(G)$ .

Then the global clustering coefficient $\textrm{CC}_{G}$ of the graph G is defined as

(23) \begin{equation} \textrm{CC}_{G}\,{:\!=}\, \frac{\Delta_{G}}{\mathcal{W}_{G}}. \end{equation}

We next discuss a local notion of clustering. For $v \in V(G)$ , define

\begin{align*} \textrm{CC}_{G}(v)\,{:\!=}\, \begin{cases} \frac{\Delta_v(G)}{d_v(d_v-1)} &\mbox{if } d_v \ge 2, \\[3pt] 0 &\mbox{otherwise,} \end{cases} \end{align*}

where

\begin{align*} \Delta_v(G)=\sum_{v_1,v_2 \in V(G)}\mathbb{1}_{\left\{\{v_1,v\},\{v_2,v\},\{v_1,v_2\}\in E(G)\right\}} \end{align*}

is twice the number of triangles in G containing the vertex v, and $d_v$ , as before, is the degree of v in G. The local clustering coefficient $\overline{\textrm{CC}}_{G}$ of G is then defined as

(24) \begin{equation} \overline{\textrm{CC}}_{G}\,{:\!=}\, \frac{1}{n}\sum_{v \in V(G)}\textrm{CC}_{G}(v). \end{equation}

Finally, we discuss a notion of clustering contribution from only vertices of a certain degree. For $k \in \mathbb{N}$ , define the clustering function to be

\begin{align*} k \mapsto \textrm{CC}_{G,k}, \end{align*}

where $\textrm{CC}_{G,k}$ is defined as

(25) \begin{equation} \textrm{CC}_{G,k}\,{:\!=}\, \begin{cases} \frac{1}{N_k(G)}\sum_{v \in v(G),d_v=k} {\frac{\Delta_v(G)}{k(k-1)}} &\text{if}\;N_k(G)>0,\\ \\[-8pt] 0&\text{otherwise}, \end{cases} \end{equation}

where $N_k(G)$ is the total number of vertices in G with degree k. Thus, $\textrm{CC}_{G,k}$ measures the proportion of wedges that are triangles, where one of the participant vertices has degree k.

We now present the results on convergence of these various clustering measures for SIRGs.

Parts 2 and 3 of Corollary 6 are direct consequences of local convergence (see [Reference Van der Hofstad22, Section 2.4.2]). For Corollary 6(1), we need an additional uniform convergence property of the square of the degree of a uniform vertex, which we prove is implied by the condition $\alpha>2d$ .

Recently, in [Reference Fountoulakis, van der Hoorn, Müller and Schepers15], precise results have been obtained concerning convergence of clustering coefficients and scaling of the clustering function as k grows to infinity for hyperbolic random graphs. Also, it was shown in [Reference Dalmau and Salvi11] that under suitable conditions, the CSFP model has non-zero clustering in the limit.

2.4. Consequences of our local convergence proof: distance lower bound

Finally, we provide a result on typical distances in our graphs. Let $U_{n,1}$ and $U_{n,2}$ be two i.i.d. uniformly distributed vertices of $\mathbb{G}_n$ , so that $d_{\mathbb{G}_n}(U_{n,1},U_{n,2})$ is the graph distance in $\mathbb{G}_n$ between $U_{n,1}$ and $U_{n,2}$ . Recall that by convention we let $d_{\mathbb{G}_n}(U_{n,1},U_{n,2})=\infty$ when $U_{n,1}$ and $U_{n,2}$ are not in the same connected component of $\mathbb{G}_n$ , so that $d_{\mathbb{G}_n}(U_{n,1},U_{n,2})$ is a well-defined random variable.

Theorem 3. (Lower bound on typical distances.) Under the assumptions of Theorem 1, for any $C \in \left( 0, \frac{1}{\log(\frac{\alpha}{\alpha-d})}\right)$ ,

\begin{align*} \mathbb{P}\left(d_{\mathbb{G}_n}(U_{n,1},U_{n,2})> C \log \log n\right) \to 1,\end{align*}

as $n \to \infty$ .

In fact, we believe that the limit in the above display holds with C replaced by $\frac{1}{\log(\frac{\alpha}{\alpha-d})}$ , but our method of proof does not allow us to establish this improvement. As we will see, the proof is a direct byproduct of the proof of the local weak limit in Theorem 1. Note that as $\alpha$ approaches d, the lower bound in Theorem 3 becomes trivial.

If, instead of a regularly varying domination as in Assumption 3(2), we have that $\mathbb{E}\left[\kappa(t,W_0,W_1)\right]$ itself is regularly varying in t with exponent $\alpha$ , then it follows from Proposition 1 that the expectation of the limiting degree distribution D is infinite in the regime $\alpha \in (0,d)$ . We conjecture that the distances are of constant order in this regime.

Conjecture 1. (Constant distances for $\alpha \in (0,d)$ .) Let $\mathbb{G}_n=G(\textbf{Y}^{(n)},\textbf{W}^{(n)},\kappa_n)$ satisfy the assumptions of Theorem 1 except for Assumption 3(2), instead of which we assume that the limiting connection function $\kappa$ is such that $\mathbb{E}\left[\kappa(t,W_0,W_1)\right]$ is regularly varying with exponent $\alpha \in (0,d)$ . Then if $U_{n,1}$ and $U_{n,2}$ are two uniformly chosen vertices in the SIRG $\mathbb{G}_n=G(\textbf{Y}^{(n)},\textbf{W}^{(n)},\kappa_n)$ , conditionally on the event that $U_{n,1}$ and $U_{n,2}$ are connected in $\mathbb{G}_n$ ,

\begin{align*} d_{\mathbb{G}_n}(U_{n,1},U_{n,2}) {{\stackrel{\mathbb{P}}{\,\rightarrow}}} K(\alpha,d), \end{align*}

as $n \to \infty$ , where $K(\alpha,d)$ is a constant depending only on the exponent $\alpha$ and dimension d.

For the case $\alpha=d$ , we do not expect universal behavior, and the question then becomes model-dependent. Results for constant distances when the limiting degree distribution has infinite mean are known for lattice models such as long-range percolation (see [Reference Benjamini, Kesten, Peres and Schramm3, Example 6.1]), for scale-free percolation (see [Reference Heydenreich, Hulshof and Jorritsma19, Theorem 2.1]), and for the configuration model [Reference Van den Esker, van der Hofstad, Hooghiemstra and Znamenski14], which is a model without geometry.

Theorem 3 poses the following question: when are the typical distances exactly doubly logarithmic? Interestingly, distances can be larger than doubly logarithmic, even when the limiting degree distribution has infinite second moment, $\mathbb{E}\left[D^2\right]=\infty$ , as was shown in [Reference Gracar, Grauer and Mörters17]; see for example [Reference Gracar, Grauer and Mörters17, Theorem 1.1(a)]. We conjecture that if certain lower-order moments below a (model-dependent) critical threshold are infinite, this implies ultra-small distances. This is also the behavior that the authors of [Reference Gracar, Grauer and Mörters17] observe, for a special class of models, but we believe this behavior is universal.

Conjecture 2. (Ultra-small distances.) Under the assumptions of Theorem 3, there is a constant $\varepsilon_{\ast} \in (0,1)$ , depending on the model parameters, such that for any $\varepsilon>\varepsilon_{\ast}$ , $\mathbb{E}\left[D^{2-\varepsilon}\right]=\infty$ implies that there is a constant $C(\alpha,d)>0$ such that

\begin{align*} \mathbb{P}\left(d_{\mathbb{G}_n}(U_{n,1},U_{n,2})< C(\alpha,d) \log \log n \bigg| U_{n,1}\;\text{and}\;U_{n,2}\;\text{are connected in}\;\mathbb{G}_n\right) \to 1, \end{align*}

as $n \to \infty$ .

3.1. Notation and general proof strategy for Theorems 1 and 2

Recall the SIRGs $\mathbb{G}_n=G(\textbf{Y}^{(n)},\textbf{W}^{(n)},\kappa_n)$ and $\mathbb{G}_{\infty}=G(\textbf{Y},\textbf{W},\kappa)$ from Theorem 1. We first define some notation which we will use throughout. Recall that $\mathbb{G}_n$ has vertex set $V(\mathbb{G}_n)=[n]$ , and $\mathbb{G}_{\infty}$ has vertex set $V(\mathbb{G}_{\infty})=\mathbb{N}\cup\{0\}$ .

For $r>0$ , define the set

(29) \begin{equation} A^r_n \,{:\!=}\,\left[-\frac{n^{1/d}}{2}+r, \frac{n^{1/d}}{2}-r\right]^d.\end{equation}

Thus, $A^r_n$ is a sub-box of the box $I_n$ , such that for any point in $A^r_n$ the open Euclidean ball of radius r around that point is contained in the box $I_n$ . Hence, the number of points of the binomial process $\Gamma_n$ (recall (9)) falling in this open ball has the same distribution as the number of points of $\Gamma_n$ falling in the open ball of radius r around the origin $\textbf{0} \in \mathbb{R}^d$ . We will use this property of $A^r_n$ in a suitable manner, which we formally explain next.

To this end, for any $x \in \mathbb{R}^d$ , define the ball

(30) \begin{equation} \mathscr{B}^r_x\,{:\!=}\,\{y\in \mathbb{R}^d\,{:}\, \|x-y\|<r\}.\end{equation}

Then if we let $\partial(I_n)=I_n\setminus\text{int}(I_n)$ denote the boundary of the set $I_n$ , where $\text{int}(I_n)$ is the interior of $I_n$ , i.e. the union of all open subsets of $I_n$ , we note that for any vertex $j \in V(\mathbb{G}_n)$ with location $Y^{(n)}_j \in A^r_n$ , the ball $\mathscr{B}^r_{Y^{(n)}_j}$ does not intersect the boundary $\partial(I_n)$ of $I_n$ , i.e. $\mathscr{B}^r_{Y^{(n)}_j} \subset \text{int}(I_n)$ . As a result, the distribution of the number of vertices of $\mathbb{G}_n$ (other than j) with locations in $\mathscr{B}^r_{Y^{(n)}_j}$ does not depend on $Y^{(n)}_j$ , and it follows a $\text{Bin}\left(n-1, \frac{\lambda_d(\mathscr{B}^r_{\textbf{0}})}{n}\right)$ distribution (where $\lambda_d$ denotes the Lebesgue measure on $\mathbb{R}^d$ ).

In particular, since $\frac{\lambda_d(I_n \setminus A^r_n)}{n}\to 0$ as $n \to \infty$ , the location $Y^{(n)}_{U_n}$ of the uniformly chosen vertex $U_n$ of $\mathbb{G}_n$ will with high probability fall in $A^r_n$ . We will condition on this good event, under which the number of points of $\Gamma_n$ in $\mathscr{B}^r_{Y^{(n)}_{U_n}}$ follows a $\text{Bin}\left(n-1, \frac{\lambda_d(\mathscr{B}^r_{\textbf{0}})}{n}\right)$ distribution, and this will simplify our computations.

Similarly, we define $(F^{\mathbb{G}_{\infty}}_0(r),0)$ to be the rooted subgraph of $G_{\infty}$ rooted at 0, induced by the vertices $j \in V(\mathbb{G}_{\infty})$ whose locations $Y_j$ satisfy $Y_j \in \mathscr{B}^r_{Y_0}=\mathscr{B}^r_{\textbf{0}}$ .

For $i \in [n]$ , we will sometimes abbreviate the rooted graph $(F^{\mathbb{G}_n}_i(r),i)$ as simply $F^{\mathbb{G}_n}_i(r)$ , and similarly for $F^{\mathbb{G}_{\infty}}_0(r)$ .

For any graph $G=(V(G), E(G))$ , edge $e=\{v_1,v_2\} \in E(G)$ , and vertex $v \in V(G)$ , by the graph distance of the edge $e {=\{v_1,v_2\}}$ from v we mean the number

(31) \begin{equation} \min\{d_{G}(v_1,v),d_{G}(v_2,v)\},\end{equation}

where $d_G$ is the graph distance on G.

Having introduced the main notation, we next discuss the main ingredients and the proof strategy for Theorems 1 and 2.

Local convergence of Euclidean graph neighborhoods. Recall the graphs $F^{\mathbb{G}_n}_{U_n}(r)$ and $F^{\mathbb{G}_{\infty}}_{0}(r)$ from Definition 6. In Section 3.2, we will prove that the typical local graph structure in any deterministic Euclidean ball around the root location is asymptotically what it should be, i.e., for any r, $(F^{\mathbb{G}_n}_{U_n}(r),U_n)$ is close in distribution to $(F^{\mathbb{G}_{\infty}}_0(r),0)$ .

Proposition 3. (Local convergence of Euclidean graph neighborhoods.) For any fixed rooted finite graph $H_*=(H ,h) \in \mathcal{G}_{\star}$ , and for any $r>0$ ,

\begin{align*}\mathbb{P}\left((F^{\mathbb{G}_n}_{{U_n}}(r),U_n) \cong (H,h)\right)\to \mathbb{P}\left((F^{\mathbb{G}_{\infty}}_{0}(r),0)\cong (H,h)\right),\end{align*}

as $n \to \infty$ , where $U_n$ is uniformly distributed on $V(\mathbb{G}_n)=[n]$ .

Path-counting analysis. Next, in Section 3.3, we do a path-counting analysis. We begin by proving a technical lemma that will help us in implementing this path-counting analysis. To state this lemma, we first introduce some more notation to keep things neat. Recall that $U_n$ is uniformly distributed on [n].

For $n,j \in \mathbb{N}$ , $v_1,\ldots,v_j \in [n]$ , and $\vec{x}= (x_0,\ldots,x_j) \in (\mathbb{R}^d)^{j+1}$ , we define

(32) \begin{equation} \mathbb{W}^{v_1,\ldots,v_j}_{n}(\vec{x}) \,{:\!=}\,\mathbb{E}\left[\kappa_n\left(\|x_1-x_0\|,W^{(n)}_{U_n},W^{(n)}_{v_1}\right)\cdots\kappa_n\left(\|x_j-x_{j-1}\|,W^{(n)}_{v_{j-1}},W^{(n)}_{v_j}\right)\right],\end{equation}

and for $j \in \mathbb{N}$ , $u_1,\dots,u_j \in \mathbb{N}$ , $x_0,\ldots,x_j \in \mathbb{R}^d$ , we define

(33) \begin{equation} \mathbb{W}_j(\vec{x})\,{:\!=}\,\mathbb{E}\left[\kappa\left(\|x_{1}-x_{0}\|,W_0,W_{u_1} \right)\cdots\kappa\left(\|x_{{j}}-x_{{j-1}}\|,W_{u_{j-1}},W_{u_j}\right)\right].\end{equation}

Note that the values of the expectations on the right-hand sides of (32) and (33), respectively, do not depend on the values of the $v_i$ ’s and $u_i$ ’s.

Then our main path-counting tool is the following lemma.

Lemma 2. (Path-counting estimate.) For any $j \geq 1$ , $a>1$ , we have

(34) \begin{equation} \begin{split} \lim_{m \to \infty} \limsup_{n \to \infty} \frac{1}{n}\int_{I_n}\cdots\int_{I_n}& \frac{1}{n^j} \sum_{v_1,\ldots,v_j \in [n]}\mathbb{W}^{v_1,\ldots,v_j}_{n}(\vec{x})\\& {\times}{\left(\prod_{i=0}^{j-2}\mathbb{1}_{\left\{\|x_{i+1}-x_i\|<a^{m^{i+1}}\right\}}\right)}\mathbb{1}_{\left\{\|x_{j}-x_{j-1}\|>a^{m^j}\right\}} dx_{0}\cdots dx_j =0 \end{split} \end{equation}

and

(35) \begin{equation} \begin{split} \lim_{m \to \infty} \int_{\mathbb{R}^d}\ldots \int_{\mathbb{R}^d} &\mathbb{W}_j(\textbf{0},x_1\ldots,x_j)\\& {\times}\mathbb{1}_{\left\{\|x_1\|<a^m\right\}}{\left( \prod_{i=1}^{j-2}\mathbb{1}_{\left\{\|x_{i+1}-x_i\|<a^{m^{i+1}}\right\}}\right)}\mathbb{1}_{\left\{\|x_j-x_{j-1}\|>a^{m^j}\right\}} dx_1\cdots dx_j =0. \end{split} \end{equation}

As a corollary to Lemma 2, we will show that for any fixed $K \in \mathbb{N}$ and $a,m>1$ , if we choose

(36) \begin{equation} r= r(a,m,K)= a^m+a^{m^2}+a^{m^3}+\dots+a^{m^K},\end{equation}

then as $m \to \infty$ , with high probability, the K-neighborhood of the rooted graph $(F^{\mathbb{G}_n}_{U_n}(r),U_n)$ will be the K-neighborhood $B^{\mathbb{G}_n}_{U_n}(K)$ of $(\mathbb{G}_n,U_n)$ , and a similar result holds for $(\mathbb{G}_{\infty},0)$ . This will help us in proving Theorems 1 and 2. To state the result, we again introduce some shorthand notation:

(37) \begin{align} B_n&\,{:\!=}\,B^{\mathbb{G}_n}_{{U_n}}(K),\quad B\,{:\!=}\,B^{\mathbb{G}_{\infty}}_{0}(K),\qquad F_{n,r}\,{:\!=}\,F^{\mathbb{G}_n}_{{U_n}}(r),\quad F_r\,{:\!=}\,F^{\mathbb{G}_{\infty}}_{0}(r), \nonumber\\[3pt] BF_{n,r}&\,{:\!=}\,B^{F_{n,r}}_{{U_n}}(K),\quad BF_{r}\,{:\!=}\,B^{F_{r}}_{0}(K). \end{align}

Remark 7. (Spatial and graph neighborhoods.) At this point, we emphasize that we rely on two kinds of neighborhoods around the root $U_n$ (respectively 0) of $\mathbb{G}_n$ (respectively $\mathbb{G}_{\infty}$ ). These are the graph $F_{n,r}$ (respectively $F_r$ ), which is the subgraph induced by those vertices whose spatial locations are within Euclidean distance r of the root location $Y^{(n)}_{U_n}$ (respectively $Y_0$ ), and the graph $B_n$ (respectively B), which is the graph neighborhood of radius K around the root $U_n$ (respectively 0) (see Figure 1). The difference between these two kinds of neighborhoods should be understood clearly. For example, the graph $F_{n,r}$ may possibly be disconnected, while $B_n$ is always a connected graph. Moreover, $BF_{n, r}$ (respectively $BF_{r}$ ) is the graph neighborhood of radius K, of the rooted graph $(F_{n,r},U_n)$ (respectively $(F_r,0)$ ).

Figure 1.

Illustration to distinguish between the two kinds of neighborhoods. The star in the middle is the location $Y^{(n)}_{U_n}$ of the root $U_n$ , and the big circle around it is the boundary of the Euclidean ball centered at $Y^{(n)}_{U_n}$ of radius r, in $\mathbb{R}^d$ . Diamonds are the vertices of the graph neighborhood $B^{\mathbb{G}_n}_{U_n}(2)$ of radius 2 around the root. Black dots are the vertices which are not in $B^{\mathbb{G}_n}_{U_n}(2)$ . The circled vertices are the vertices of $F^{\mathbb{G}_n}_{U_n}(r)$ . The circled diamonds are the vertices of $BF_{n,r}$ , the graph neighborhood of radius 2 about the root $U_n$ , of the graph $F^{\mathbb{G}_n}_{U_n}(r)$ .

We have the following corollary to Lemma 2.

Corollary 7. (Coupling spatial and graph neighborhoods.) Let $BF_{n,r}$ , $B_n$ , $BF_r$ , B be as in (37), where $r=r(a,m,K)$ is as in (36). Then

(38) \begin{equation} \lim_{m \to \infty} \limsup_{n \to \infty} \mathbb{P}\left(BF_{n,r}\neq B_n\right) =0 \end{equation}

and

(39) \begin{equation} \lim_{m \to \infty} \mathbb{P}\left(BF_{r}\neq B\right) =0. \end{equation}

In proving Corollary 7, we perform a careful path-counting analysis to bound the expected number of K-paths of $\mathbb{G}_n$ which are not K-paths in $F^{\mathbb{G}_n}_{U_n}(r)$ by the integral expression (34), and to obtain the similar bound (35) for $\mathbb{G}_{\infty}$ . Corollary 7 then follows directly using the technical Lemma 2.

In the course of proving Corollary 7, we develop a general path estimate, where for $r=r(a,m,K)$ as in (36), we can have $a=a_n$ , $K=K_n$ be n dependent sequences, while m does not depend on n. This general estimate will be used in the proof of Theorem 3.

Proof strategy for Theorem 1. In Section 3.4, we prove Theorem 1. Recall that, to conclude Theorem 1, we need to show that for any $K \in \mathbb{N}$ , the K-neighborhoods of $(\mathbb{G}_n,U_n)$ and $(\mathbb{G}_{\infty},0)$ are close in distribution in $\mathcal{G}_{\star}$ . Consequently, to conclude Theorem 1, using Corollary 7, it will be enough to show that the K-neighborhoods of $(F^{\mathbb{G}_n}_{U_n}(r),U_n)$ and $(F^{\mathbb{G}_{\infty}}_{0}(r),0)$ are close in distribution. This we will observe to be an easy consequence of Proposition 3.

Proof strategy for Theorem 2. In Section 3.5, we prove Theorem 2. The first step is to show that the empirical Euclidean graph neighborhood distribution concentrates. That is, for $H_*=(H,h) \in \mathcal{G}_{\star}$ , where $h \in V(H)$ , and for any $r>0$ , we define the random variables

(40) \begin{equation} C_{r,n}(H,h)\,{:\!=}\,\mathbb{P}\left(F^{\mathbb{G}_n}_{U_n}(r)\cong (H,h) \bigg| \mathbb{G}_n\right)=\frac{1}{n}\sum_{i=1}^n \mathbb{1}_{\left\{F^{\mathbb{G}_n}_i(r)\cong (H,h)\right\}},\end{equation}

and show these random variables concentrate, as follows.

Lemma 3. (Concentration of empirical Euclidean graph neighborhood measure.) For any $r>0$ and a locally finite rooted graph $H_*= (H,h) \in \mathcal{G}_{\star}$ , the variance of the random variable $C_{r,n} (H,h)$ converges to 0 as $n \to \infty$ , i.e.

\begin{align*} \left|\mathbb{E}\left[C_{r,n} (H,h)^2\right]-\mathbb{E}\left[C_{r,n} (H,h)\right]^2\right|\to0, \end{align*}

as $n \to \infty$ .

The key observation in proving this lemma is that the Euclidean graph neighborhoods around two uniformly chosen vertices of $\mathbb{G}_n$ are asymptotically independent, which is a consequence of the fact that the distance between the locations of two uniformly chosen vertices of $\mathbb{G}_n$ in $\mathbb{R}^d$ diverges in probability, as $n \to \infty$ .

Next, we combine Lemma 3 with Corollary 7 to show that the empirical neighborhood distribution of $\mathbb{G}_n$ also concentrates. That is, for any $K \in \mathbb{N}$ and $G_*=(G,g) \in \mathcal{G}_{\star}$ , if we define random variables

(41) \begin{equation} B_n(G,g)\,{:\!=}\,\mathbb{P}\left(B^{\mathbb{G}_n}_{U_n}(K)\cong (G,g) \bigg| \mathbb{G}_n\right)=\frac{1}{n}\sum_{i=1}^n \mathbb{1}_{\left\{B^{\mathbb{G}_n}_i(K)\cong (G,g)\right\}},\end{equation}

for $n \in \mathbb{N}$ , then these random variables also concentrate. This is achieved by first using Lemma 3 and taking a sum over all rooted graphs $H_*= (H,h)$ with $B^{H}_h(K) \cong (G,g)$ , to show that the random variables

\begin{align*}\frac{1}{n}\sum_{i =1}^n\mathbb{1}_{\left\{ B^{F^{\mathbb{G}_n}_i(r)}_i(K)\cong (G,g)\right\}}\end{align*}

concentrate, and then employing Corollary 7 to obtain the same conclusion for the variables $B_n(G,g)$ . Theorem 2 is then a direct consequence of these observations, combined with Proposition 3.

3.2. Proof of Proposition 3

Let us first discuss the proof strategy informally. We make use of the following two key observations:

• The number of vertices in $F^{\mathbb{G}_n}_{U_n}(r)$ converges in distribution to the number of vertices in $F^{\mathbb{G}_{\infty}}_0(r)$ .

• Conditionally on the number of vertices of $F^{\mathbb{G}_n}_{U_n}(r)$ (respectively $F^{\mathbb{G}_{\infty}}_0(r)$ ) other than the root, their locations are uniform on $\mathscr{B}^r_{Y^{(n)}_{U_n}}$ (respectively $\mathscr{B}^r_{\textbf{0}}$ ). It will then follow that any ‘not so bad’ translation-invariant function, evaluated at the locations of the vertices of $F^{\mathbb{G}_n}_{U_n}(r)$ , should have nice limiting behavior.

In particular, we define functions $\mathcal{F}^{n}_{ (H,h)}$ and $\mathcal{F}^{\infty}_{ (H,h)}$ (see (52)), which for a given rooted graph $ (H,h) \in \mathcal{G}_{\star}$ count the number of rooted isomorphisms between (H, h) and $F^{\mathbb{G}_n}_{U_n}(r)$ , and between (H, h) and $F^{\mathbb{G}_{\infty}}_0(r)$ . We show that for any such rooted graph $ (H,h) \in \mathcal{G}_{\star}$ , the probability of the event $\{\mathcal{F}^{n}_{ (H,h)}>0\}$ converges to that of the event $\{\mathcal{F}^{\infty}_{ (H,h)}>0\}$ . In turn, this implies that the random rooted graph $F^{\mathbb{G}_n}_{U_n}(r)$ converges in distribution to the random rooted graph $F^{\mathbb{G}_{\infty}}_{0}(r)$ , on the space $\mathcal{G}_{\star}$ . We now go into the details.

Proof of Proposition 3. The random variable $Y^{(n)}_{U_n}$ is uniformly distributed on $I_n$ . We write

\begin{align*} &\mathbb{P}\left((F^{\mathbb{G}_n}_{{U_n}}(r),U_n) \cong (H,h)\right) \\ &= \mathbb{P}\left((F^{\mathbb{G}_n}_{{U_n}}(r),U_n) \cong (H,h), Y^{(n)}_{U_n} \in A^r_n\right) +\mathbb{P}\left((F^{\mathbb{G}_n}_{{U_n}}(r),U_n) \cong (H,h), Y^{(n)}_{U_n} \notin A^r_n\right) \end{align*}

and observe that, as $n \to \infty$ ,

(42) \begin{align} \mathbb{P}\left((F^{\mathbb{G}_n}_{{U_n}}(r),U_n) \cong (H,h), Y^{(n)}_{U_n} \notin A^r_n\right) & \leq \mathbb{P}\left(Y^{(n)}_{U_n} \notin A^r_n\right) \leq \frac{2d\left(r\left(n^{(d-1)/d}\right)\right)}{n}\to0. \end{align}

Therefore, it is enough to show that, as $n \to \infty$ ,

(43) \begin{equation} \mathbb{P}\left((F^{\mathbb{G}_n}_{{U_n}}(r),U_n) \cong (H,h), Y^{(n)}_{U_n} \in A^r_n\right)\to \mathbb{P}\left((F^{\mathbb{G}_{\infty}}_{0}(r),0) \cong (H,h)\right). \end{equation}

Note that

\begin{align*} &\mathbb{P}\left((F^{\mathbb{G}_n}_{{U_n}}(r),U_n) \cong (H,h), Y^{(n)}_{U_n} \in A^r_n\right) \\& \qquad\quad =\mathbb{P}\left((F^{\mathbb{G}_n}_{{U_n}}(r),U_n) \cong (H,h), Y^{(n)}_{U_n} \in A^r_n, |V(F^{\mathbb{G}_n}_{U_n}(r))|=|V(H)|\right), \end{align*}

and so we can repeatedly condition to rewrite

(44) \begin{equation} \begin{split} &\mathbb{P}\left((F^{\mathbb{G}_n}_{{U_n}}(r),U_n) \cong (H,h), Y^{(n)}_{U_n} \in A^r_n\right) \\& \qquad= \mathbb{P}\left((F^{\mathbb{G}_n}_{{U_n}}(r),U_n) \cong (H,h) \bigg| |V(F^{\mathbb{G}_n}_{{U_n}}(r))|=|V(H)|, Y^{(n)}_{U_n} \in A^r_n\right) \\& \qquad\quad \times \mathbb{P}\left(|V(F^{\mathbb{G}_n}_{{U_n}}(r))|=|V(H)| \bigg| Y^{(n)}_{U_n} \in A^r_n\right)\mathbb{P}\left(Y^{(n)}_{U_n} \in A^r_n\right). \end{split} \end{equation}

Using (42), the last term in the right-hand side of (44) tends to 1 as $n \to \infty$ . Observe that

\begin{align*}\mathbb{P}\left(|V(F^{\mathbb{G}_n}_{{U_n}}(r))| =|V(H)| \bigg| Y^{(n)}_{U_n} \in A^r_n\right)=\mathbb{P}\left(\mathcal{Y}_n=|V(H)|-1\right), \end{align*}

where $\mathcal{Y}_n$ follows a $\text{Bin}(n-1, \lambda_d(\mathscr{B}^r_{\textbf{0}})/n)$ distribution.

Since $\mathcal{Y}_n$ converges in distribution to $\mathcal{Y}\sim \text{Poi}(\lambda_d(\mathscr{B}^r_{\textbf{0}}))$ , and since $\mathcal{Y}$ is equal in distribution to $\Gamma(\mathscr{B}^r_{\textbf{0}})$ (recall $\Gamma$ from (10)), it follows that $\mathcal{Y}_n {{\stackrel{d}{\rightarrow}}} \Gamma(\mathscr{B}^r_{\textbf{0}})$ , as $n \to \infty$ .

Observe that $\Gamma(\mathscr{B}^r_{\textbf{0}})\stackrel{d}{=}|V(F^{\mathbb{G}_{\infty}}_{\textbf{0}}(r))|-1$ , so that

\begin{align*} \lim_{n \to \infty} \mathbb{P}\left(|V(F^{\mathbb{G}_n}_{{U_n}}(r))|=|V(H)| \bigg| Y^{(n)}_{U_n} \in A^r_n\right) &=\lim_{n \to \infty} \mathbb{P}\left(\mathcal{Y}_n=|V(H)|-1\right) \\&= \mathbb{P}\left(|V(F^{\mathbb{G}_{\infty}}_{0}(r))|-1=|V(H)|-1\right) \\&=\mathbb{P}\left(|V(F^{\mathbb{G}_{\infty}}_{0}(r))|=|V(H)|\right). \end{align*}

Hence, from (44), we note that to conclude (43), it is enough to show that

(45) \begin{equation} \begin{split} \lim_{n \to \infty}\mathbb{P}\left((F^{\mathbb{G}_n}_{{U_n}}(r),U_n) \cong (H,h) \bigg| |V(F^{\mathbb{G}_n}_{{U_n}}(r))|=|V(H)|, Y^{(n)}_{U_n} \in A^r_n\right) \\&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!=\mathbb{P}\left((F^{\mathbb{G}_{\infty}}_{0}(r),0) \cong (H,h) \bigg| |V(F^{\mathbb{G}_{\infty}}_{0}(r))|=|V(H)|\right). \end{split} \end{equation}

For the remainder of the proof, we assume $|V(H)|=l+1$ . We continue by making some observations on the locations and weights of the vertices of the graph $F^{\mathbb{G}_n}_{U_n}(r)$ (respectively $F^{\mathbb{G}_\infty}_{\textbf{0}}(r)$ ), conditionally on $\{|V(F^{\mathbb{G}_n}_{U_n}(r))|=l+1, Y^{(n)}_{U_n} \in A^r_n\}$ (respectively { $|V(F^{\mathbb{G}_\infty}_{\textbf{0}}(r))|=l+1$ }).

Locations of $F^{\mathbb{G}_n}_{U_n}(r)$ . Since $(Y^{(n)}_i)_{i\in[n]}$ is an i.i.d. collection of uniform random variables on $I_n$ , conditionally on the event $\{Y^{(n)}_{U_n}\in A^r_n, |V(F^{\mathbb{G}_n}_{{U_n}}(r))|=l+1\}$ , the locations $P_1,\ldots,P_l$ of the l vertices of $\mathbb{G}_n$ (in some order) falling in $\mathscr{B}^r_{Y^{(n)}_{U_n}}$ , other than $Y^{(n)}_{U_n}$ , are at independent uniform locations in the ball $\mathscr{B}^r_{Y^{(n)}_{U_n}}$ , given $Y^{(n)}_{U_n}$ . Hence, conditionally on $\{Y^{(n)}_{U_n}\in A^r_n, |V(F^{\mathbb{G}_n}_{{U_n}}(r))|=l+1\}$ , the random variables $P_1-Y^{(n)}_{U_n},\ldots,P_l-Y^{(n)}_{U_n}$ are independently, uniformly distributed on the ball $\mathscr{B}^r_{\textbf{0}}$ .

Conditionally on $\{Y^{(n)}_{U_n}\in A^r_n, |V(F^{\mathbb{G}_n}_{{U_n}}(r))|=l+1\}$ , let the locations of all the vertices of $\mathbb{G}_n$ falling in $\mathscr{B}^r_{Y^{(n)}_{U_n}}$ (including $Y^{(n)}_{U_n}$ ) be $P_0, P_1, \ldots, P_l$ , where $Y^{(n)}_{U_n}=P_0$ . So, conditionally on $\{Y^{(n)}_{U_n}\in A^r_n, |V(F^{\mathbb{G}_n}_{{U_n}}(r))|=l+1\}$ , the random matrix $(\|P_i-P_j\|)_{0\leq i,j \leq l; i\neq j}$ is equal in distribution to $(\|Y_i-Y_j\|)_{0\leq i,j \leq l; i\neq j}$ , where the set $\{Y_i\colon 1\leq i\leq l\}$ consists of l i.i.d. uniform points in $\mathscr{B}^r_{\textbf{0}}$ (independent of $\textbf{Y}^{(n)}$ , $\textbf{W}^{(n)}$ , $\textbf{Y}$ , $\textbf{W}$ ) and $Y_0=\textbf{0}$ .

Locations of $F^{\mathbb{G}_{\infty}}_0(r)$ . Again, conditionally on the event $\{\Gamma_{\infty}(\mathscr{B}^r_{\textbf{0}})=l+1\}=\{|V(F^{\mathbb{G}_{\infty}}_{0}(r))|=l+1\}$ , the locations of the l vertices of $\mathbb{G}_{\infty}$ in $\mathscr{B}^r_{\textbf{0}}$ other than 0 are i.i.d. uniform on $\mathscr{B}^r_{\textbf{0}}$ (since $\Gamma$ is a homogeneous Poisson point process). So if $Z_0,\ldots,Z_l$ are the locations of the $l+1$ vertices of $\mathbb{G}_{\infty}$ in $\mathscr{B}^r_{\textbf{0}}$ (in some order), where $Z_0=\textbf{0}$ , the random matrix $(\|Z_i-Z_j\|)_{0\leq i,j \leq l; i\neq j}$ is also equal in distribution to $(\|Y_i-Y_j\|)_{0\leq i,j \leq l; i\neq j}$ , where the set $\{Y_i\colon 1\leq i\leq l\}$ consists of l i.i.d. uniform points in $\mathscr{B}^r_{\textbf{0}}$ (independent of $\textbf{Y}^{(n)}$ , $\textbf{W}^{(n)}$ , $\textbf{Y}$ , $\textbf{W}$ ) and $Y_0=\textbf{0}$ .

We conclude that

(46) \begin{align} &(\|P_i-P_j\|)_{0\leq i,j \leq l; i\neq j}\bigg|{\{Y^{(n)}_{U_n}\in A^r_n, |V(F^{\mathbb{G}_n}_{{U_n}}(r))|=l+1\}} \nonumber\\&\qquad\quad \stackrel{d}{=}(\|Y_i-Y_j\|)_{1\leq i,j \leq l+1; i\neq j} \nonumber\\&\qquad\quad \stackrel{d}{=} (\|Z_i-Z_j\|)_{0\leq i,j \leq l; i\neq j}\bigg|{\{\Gamma_{\infty}(\mathscr{B}^r_{\textbf{0}})=l+1\}}. \end{align}

Weights of $F^{\mathbb{G}_n}_{U_n}(r)$ . Next, conditionally on $\{Y^{(n)}_{U_n}\in A^r_n, |V(F^{\mathbb{G}_n}_{{U_n}}(r))|=l+1\}$ , for $0 \leq i \leq l$ , let $W_{n,i}$ denote the weight of the vertex of $F^{\mathbb{G}_n}_{U_n}(r)$ with location $P_i$ . Then $(W_{n,0},\ldots,W_{n,l})$ has the distribution of $l+1$ uniformly chosen weights from the weight set $\{W^{(n)}_1,\ldots,W^{(n)}_n\}$ without replacement, in some arbitrary order. This is because for any $i_0,\ldots,i_l \in [n]$ ,

(47) \begin{align} & \mathbb{P}\left((W_{n,0},\ldots,W_{n,l})=(W^{(n)}_{i_0},\ldots,W^{(n)}_{i_l}) \bigg| Y^{(n)}_{U_n}\in A^r_n, |V(F^{\mathbb{G}_n}_{{U_n}}(r))|=l+1\right)\nonumber\\ & = \frac{\mathbb{P}\left((W_{n,0},\ldots,W_{n,l})=(W^{(n)}_{i_0},\ldots,W^{(n)}_{i_l}) \bigg| Y^{(n)}_{U_n}\in A^r_n\right)}{\mathbb{P}\left(|V(F^{\mathbb{G}_n}_{{U_n}}(r))|=l+1 \bigg| Y^{(n)}_{U_n}\in A^r_n\right)}, \end{align}

where in the second step we use the fact that the event that the vector of weights of the vertices of $F^{\mathbb{G}_n}_{U_n}(r)$ is of length $l+1$ is contained in the event that $\{|V(F^{\mathbb{G}_n}_{{U_n}}(r))|=l+1\}$ . The numerator in (47) is

\begin{align*}\frac{1}{n} \times \mathbb{P}\left(Y^{(n)}_{i_p}\in \mathscr{B}^{r}_{Y^{(n)}_{i_0}}\;\forall\;p \in [l],\; Y^{(n)}_{i_q}\notin \mathscr{B}^{r}_{Y^{(n)}_{i_0}}\;\forall\; i_q \in [n]\setminus\{i_0,i_1,\ldots,i_l\} \bigg| Y^{(n)}_{i_0}\in A^r_n\right)\times \frac{1}{l!},\end{align*}

where the term $ {1/n}$ is just the probability that $U_n = i_0$ , and the term $\frac{1}{l!}$ accounts for the choice of the ordering $P_k=Y^{(n)}_{i_k}$ for $1 \leq k \leq l$ among all $l!$ possible labelings. This evaluates to

\begin{align*} \frac{1}{n} \times \left(\frac{\lambda_d(\mathscr{B}^r_{\textbf{0}})}{n}\right)^{l}\times \left(1-\frac{\lambda_d(\mathscr{B}^r_{\textbf{0}})}{n}\right)^{n-1-l} \times \frac{1}{l!}. \end{align*}

The denominator in (47) is just the probability that a $\text{Bin}\left(n-1, \frac{\lambda_d(\mathscr{B}^r_{\textbf{0}})}{n}\right)$ random variable takes the value l, which evaluates to

$$\binom{n-1}{l}\times \left(\frac{\lambda_d(\mathscr{B}^r_{\textbf{0}})}{n}\right)^{l}\times \left(1-\frac{\lambda_d(\mathscr{B}^r_{\textbf{0}})}{n}\right)^{n-1-l}.$$

Combining, we get that

(48) \begin{align} &\mathbb{P}\left((W_{n,0},\ldots,W_{n,l})=(W^{(n)}_{i_0},\ldots,W^{(n)}_{i_l}) \bigg| Y^{(n)}_{U_n}\in A^r_n, |V(F^{\mathbb{G}_n}_{{U_n}}(r))|=l+1\right)\nonumber \\&=\frac{1}{n(n-1)\cdots(n-l)}. \end{align}

We also make the observation that the random vector $(W_{n,0},\ldots,W_{n,l})$ is permutation-invariant: for any $\sigma \in S(l+1)$ , where $S(l+1)$ denotes the set of all permutations of $\{0,\ldots,l\}$ ,

(49) \begin{equation} (W_{n,0},\ldots,W_{n,l}) \stackrel{d}{=} (W_{n,\sigma(0)},\ldots,W_{n,\sigma(l)}). \end{equation}

Weights of $F^{\mathbb{G}_{\infty}}_0(r)$ . Again, conditionally on $\{\Gamma_{\infty}(\mathscr{B}^r_{\textbf{0}})=l+1\}$ , for $0 \leq i \leq l$ , let $W_{\infty,i}$ denote the weight of the vertex of $F^{\mathbb{G}_{\infty}}_{0}(r)$ with location $Z_i$ . Since the weights in the limiting graph $\mathbb{G}_{\infty}$ are i.i.d., it immediately follows that the random weight vector $(W_{\infty,0},\ldots,W_{\infty,l})$ is permutation-invariant, i.e. for any $\sigma \in S(l+1)$ ,

(50) \begin{equation} (W_{\infty,0},\ldots,W_{\infty,l})\stackrel{d}{=}(W_{\infty,\sigma(0)},\ldots,W_{\infty,\sigma(l)}), \end{equation}

and that given $\{\Gamma_{\infty}(\mathscr{B}^r_{\textbf{0}})=l+1\}$ , the vector $(W_{\infty,0},\ldots,W_{\infty,l})\stackrel{d}{=}(\textrm{W}_0,\ldots,\textrm{W}_l)$ , where each entry of the vector $(\textrm{W}_0,\ldots,\textrm{W}_l)$ is an i.i.d. copy of the limiting weight variable W (recall (4)).

We now continue with the proof.

Convergence of weights of $F^{\mathbb{G}_n}_{U_n}(r)$ to the weights of $F^{\mathbb{G}_{\infty}}_0(r)$ . We also note that

(51) \begin{align} &(W_{n,0},\ldots,W_{n,l}) \bigg| \{Y^{(n)}_{U_n}\in A^r_n, |V(F^{\mathbb{G}_n}_{{U_n}}(r))|=l+1\} \nonumber\\[3pt]& {{\stackrel{d}{\rightarrow}}} (W_{\infty,0},\ldots,W_{\infty,l}) \bigg| \{\Gamma_{\infty}(\mathscr{B}^r_{\textbf{0}})=l+1\}. \end{align}

This is because for any continuity set $A_0 \times\dots\times A_l$ of $(\textrm{W}_0,\ldots,\textrm{W}_l)$ , by (48),

\begin{align*} & \mathbb{P}\left((W_{n,0},\ldots,W_{n,l})\in A_0 \times\dots\times A_l \bigg| Y^{(n)}_{U_n}\in A^r_n, |V(F^{\mathbb{G}_n}_{{U_n}}(r))|=l+1\right) \\ & \geq \mathbb{E}\left[\sum_{j_0,\ldots,j_l}\frac{1}{n^l}\prod_{k=0}^l{\mathbb{1}_{\left\{W^{(n)}_{j_k}\in A_k\right\}}}\right]= \mathbb{E}\left[\prod_{k=0}^l \left(\frac{1}{n}\sum_{i=1}^n {\mathbb{1}_{\left\{W^{(n)}_i \in A_k\right\}}} \right)\right] \end{align*}

and

\begin{align*} & \mathbb{P}\left((W_{n,0},\ldots,W_{n,l})\in A_0 \times\cdots\times A_l \bigg| Y^{(n)}_{U_n}\in A^r_n, |V(F^{\mathbb{G}_n}_{{U_n}}(r))|=l+1\right) \\ & \leq \mathbb{E}\left[\sum_{j_0,\ldots,j_l}\frac{1}{(n-l)^l}\prod_{k=0}^l{\mathbb{1}_{\left\{W^{(n)}_{j_k}\in A_k\right\}}}\right] =\mathbb{E}\left[\prod_{k=0}^l \left(\frac{1}{n-l}\sum_{i=1}^n {\mathbb{1}_{\left\{W^{(n)}_i \in A_k\right\}}} \right)\right], \end{align*}

where the sums in the last two displays are taken over all cardinality- $(l+1)$ subsets $\{j_0,\ldots,j_l\}$ of [n]. Since the right-hand sides of the last two displays are bounded from above by 1, we use dominated convergence and apply (4) to conclude that both right-hand sides converge to

\begin{align*} \prod_{k=0}^l \mathbb{P}\left(W \in A_k\right) & = \mathbb{P}\left((\textrm{W}_0,\ldots,\textrm{W}_l)\in A_0\times \cdots \times A_l\right)\\ & = \mathbb{P}\left((W_{\infty,0},\ldots,W_{\infty,l})\in A_0\times \cdots \times A_l \bigg| \Gamma_{\infty}(\mathscr{B}^r_{\textbf{0}})=l+1\right). \end{align*}

Functions counting rooted isomorphisms, and their symmetry properties. Now we proceed to define the functions we use to count the number of isomorphisms between the given rooted graph (H, h) and the Euclidean graph neighborhoods $F^{\mathbb{G}_n}_{U_n}(r)$ , $F^{\mathbb{G}_{\infty}}_0(r)$ . Let the vertices of (H, h) be $v_0,\ldots,v_l$ , where $h=v_0$ . Let us denote the subset of the set of permutations $S(l+1)$ of $\{0,1,\ldots,l\}$ that fix 0 by $S_0(l+1)$ ; i.e., for all $\sigma \in S_0(l+1)$ , $\sigma(0)=0$ .

Let $M^S_{l+1}(\mathbb{R})$ denote the space of all symmetric square matrices of order $l+1$ with entries in $\mathbb{R}$ . For each $n \in \mathbb{N}$ , define the function $\mathcal{F}^n_{ (H,h)}\,{:}\, \left(\mathbb{R}^d\right)^{l+1}\times \left(\mathbb{R}^d\right)^{l+1} \times M^S_{l+1}(\mathbb{R}) \to \mathbb{R}$ by

(52) \begin{equation} \begin{split} \mathcal{F}^n_{ (H,h)}\left(\vec{x},\vec{y},(a_{ij})_{i,j=0}^l\right)\\ \,{:\!=}\,\sum_{\pi \in S_0(l+1)} & \prod_{\{v_i,v_j\}\in E(H)} \left({\mathbb{1}_{\left\{a_{\pi(i)\pi(j)}<\kappa_n\left(\|x_{\pi(i)}-x_{\pi(j)}\|,y_{\pi(i)},y_{\pi(j)}\right)\right\}}}\right) \\& \times \prod_{\{v_i,v_j\}\notin E(H)}\left({\mathbb{1}_{\left\{a_{\pi(i)\pi(j)}>\kappa_n\left(\|x_{\pi(i)}-x_{\pi(j)}\|,y_{\pi(i)},y_{\pi(j)}\right)\right\}}} \right), \end{split} \end{equation}

for $\vec{x}=(x_0,\dots,x_l), \vec{y}=(y_0,\dots,y_l) \in (\mathbb{R}^d)^{l+1}$ and $(a_{ij})_{i,j=0}^l \in M^S_{l+1}(\mathbb{R})$ . Similarly define the function $\mathcal{F}^{\infty}_{ (H,h)}\colon \left(\mathbb{R}^d\right)^{l+1}\times \left(\mathbb{R}^d\right)^{l+1} \times M^S_{l+1}(\mathbb{R})\to \mathbb{R}$ , where $\mathcal{F}^{\infty}_{ (H,h)}$ is just $\mathcal{F}^n_{ (H,h)}$ with $\kappa_n$ replaced by $\kappa$ .

Heuristically, we want the indicators

$$\mathbb{1}_{\left\{a_{\pi(i)\pi(j)}<\kappa_n\left(\|x_{\pi(i)}-x_{\pi(j)}\|,y_{\pi(i)},y_{\pi(j)}\right)\right\}}$$

to be the indicators of the events $\{\text{the edge}\; \{\pi(i),\pi(j)\}\; \text{is present}\}$ . Since in our graphs these events occur independently, each with probability $\kappa_n\left(\|x_{\pi(i)}-x_{\pi(j)}\|,y_{\pi(i)},y_{\pi(j)}\right)$ when the locations and weights of the vertices $\pi(i)$ and $\pi(j)$ are respectively $(x_{\pi(i)},y_{\pi(i)})$ and $(x_{\pi(j)},y_{\pi(j)})$ , we will take the matrix $(a_{ij})_{i,j,=0}^l$ to be a symmetric i.i.d. uniform matrix. Before that, we first discuss some symmetry properties of the functions $\mathcal{F}^n_{ (H,h)}$ and $\mathcal{F}^{\infty}_{ (H,h)}$ .

Observe the following symmetry: for any permutation $\pi \in S_0(l+1)$ , and for $ {\bullet}$ being either n or $\infty$ ,

\begin{align*} & \mathcal{F}^{ {\bullet}}_{ (H,h)}\left((x_0,\ldots,x_l),(y_0,\ldots,y_l),(a_{ij})_{i,j=0}^l\right) \\ & = \mathcal{F}^{ {\bullet}}_{ (H,h)}\left((x_{\pi(0)},\ldots,x_{\pi(l)}),(y_{\pi(0)},\ldots,y_{\pi(l)}),(a_{\pi(i)\pi(j)})_{i,j=0}^l\right). \end{align*}

Recall the permutation-invariance of the weights of $F^{\mathbb{G}_n}_{U_n}(r)$ and $F^{\mathbb{G}_{\infty}}_{0}(r)$ from (49) and (50), and note that for $(x_0,\ldots,x_l), (z_0,\ldots,z_l)\in \left(\mathbb{R}^d\right)^{l+1}$ , if there exists a permutation $\pi \in S_0(l+1)$ such that

\begin{align*} (\|x_i-x_j\|)_{i,j=0}^l=(\|z_{\pi(i)}-z_{\pi(j)}\|)_{i,j=0}^l\;\; \text{(entrywise)}, \end{align*}

then for any $(a_{ij})_{i,j=0}^l\in M^S_{l+1}(\mathbb{R})$ , conditionally on $\{Y^{(n)}_{U_n}\in A^r_n, |V(F^{\mathbb{G}_n}_{{U_n}}(r))|=l+1\}$ ,

(53) \begin{align} & \mathcal{F}^n_{ (H,h)}\left((x_0,\ldots,x_l),(W_{n,0},\ldots,W_{n,l}),(a_{ij})_{i,j=0}^l\right) \nonumber \\& \stackrel{d}{=}\mathcal{F}^n_{ (H,h)}\left((z_0,\ldots,z_l),(W_{n,0},\ldots,W_{n,l}),(a_{ij})_{i,j=0}^l)\right), \end{align}

and a similar distributional equality holds for $\mathcal{F}^{\infty}_{ (H,h)}$ , conditionally on $\{\Gamma_{\infty}(\mathscr{B}^r_{\textbf{0}})=l+1\}$ .

Simplifying the events $\{F^{\mathbb{G}_n}_{{U_n}}(r)\cong (H,h)\}$ and $\{F^{\mathbb{G}_{\infty}}_{0}(r) \cong (H,h)\}$ . Now, let $(U_{ij})_{i,j=0}^l$ and $(U^{\prime}_{ij})_{i,j=0}^l$ be two i.i.d. random elements of $M^S_{l+1}(\mathbb{R})$ which are also independent from $\textbf{X}^{(n)}$ , $\textbf{W}^{(n)}$ , $\textbf{X}$ , $\textbf{W}$ , and $\{Y_i\,{:}\,0\leq i\leq l\}$ , where for each $i<j$ , $U_{ij}\sim \text{U}([0,1])$ , all the entries above the diagonal of the random matrix $(U_{ij})_{i,j=1}^{l+1}$ are independent, and for each i, $U_{ii}\,{:\!=}\,0$ (the diagonal elements will not come into the picture and can be defined arbitrarily).

Let

(54) \begin{equation} \mathcal{A}_n \,{:\!=}\, {\mathbb{1}_{\left\{F^{\mathbb{G}_n}_{{U_n}}(r)\cong (H,h)\right\}}}\bigg|{\{Y^{(n)}_{U_n} \in A^r_n, |V(F^{\mathbb{G}_n}_{{U_n}}(r))|=l+1\}} \end{equation}

and

(55) \begin{equation} \mathcal{A}_{\infty}\,{:\!=}\, {\mathbb{1}_{\left\{F^{\mathbb{G}_{\infty}}_{0}(r) \cong (H,h)\right\}}}\bigg|{\{|V(F^{\mathbb{G}_{\infty}}_{0}(r))|=l+1\}}. \end{equation}

Note from (45) that our target is to show that $\mathbb{E}\left[\mathcal{A}_n\right]\to \mathbb{E}\left[\mathcal{A}_{\infty}\right]$ , as $n \to \infty$ .

Recall (46). Observe that

(56) \begin{equation} \mathcal{A}_n \stackrel{d}{=} {\mathbb{1}_{\left\{\mathcal{F}^n_{(H,h)}\left((P_0,\ldots,P_l), (W_{n,0},\ldots,W_{n,l}), (U_{ij})_{i,j=0}^l\right)>0\right\}}}\bigg|{\{Y^{(n)}_{U_n} \in A^r_n, |V(F^{\mathbb{G}_n}_{{U_n}}(r))|=l+1\}}, \end{equation}

since, conditionally on the event $\{Y^{(n)}_{U_n} \in A^r_n, |V(F^{\mathbb{G}_n}_{{U_n}}(r))|=l+1\}$ , if for some $\pi \in S_0(l+1)$ the corresponding term in the sum in $\mathcal{F}^n_{ (H,h)}$ is positive, then $v_i \mapsto P_{\pi(i)}$ gives a rooted isomorphism (note that the fact that every permutation in $S_0(l+1)$ fixes 0 ensures that the isomorphism is rooted), and if there is a rooted isomorphism $\phi\colon G\to F^{G_n}_{U_n}(r)$ , then the term in the sum corresponding to the permutation $\sigma \in S_0(l+1)$ is positive, where $\phi(v_i)=P_{\sigma(i)}$ .

Using the symmetry (53) with (46), we obtain

(57) \begin{align} \mathcal{A}_n &\stackrel{d}{=} {\mathbb{1}_{\left\{\mathcal{F}^n_{ (H,h)}\left((Y_0,\ldots,Y_l), (W_{n,0},\ldots,W_{n,l}), (U_{ij})_{i,j=0}^l\right)>0\right\}}}\bigg|{\{Y^{(n)}_{U_n} \in A^r_n, |V(F^{\mathbb{G}_n}_{{U_n}}(r))|=l+1\}}\nonumber \\&\,{=\!:}\,\mathscr{A}_n. \end{align}

Using exactly similar arguments for the random variable $\mathcal{A}_{\infty}$ and the function $\mathcal{F}^{\infty}_{ (H,h)}$ , we obtain

(58) \begin{equation} \mathcal{A}_{\infty} \stackrel{d}{=} {\mathbb{1}_{\left\{\mathcal{F}^{\infty}_{ (H,h)}\left((Y_0,\ldots,Y_l), (W_{\infty,0},\ldots,W_{\infty,l}), (U^{\prime}_{ij})_{i,j=0}^l\right)>0\right\}}}\bigg|{\{|V(F^{\mathbb{G}_{\infty}}_{0}(r))|=l+1\}}. \end{equation}

Finally, using that $(U^{\prime}_{ij})_{i,j=0}^l\stackrel{d}{=}(U_{ij})_{i,j=0}^l$ , and that both the matrices $U_{ij}$ and $U^{\prime}_{ij}$ are independent of everything else, it is easy to see that

(59) \begin{equation} \mathcal{A}_{\infty} \stackrel{d}{=} {\mathbb{1}_{\left\{\mathcal{F}^{\infty}_{ (H,h)}\left((Y_0,\ldots,Y_l), (W_{\infty,0},\ldots,W_{\infty,l}), (U_{ij})_{i,j=0}^l\right)>0\right\}}}\bigg|{\{|V(F^{\mathbb{G}_{\infty}}_{0}(r))|=l+1\}}\,{=\!:}\,\mathscr{A}_{\infty}. \end{equation}

Conclusion. Using (51), and using the convergence of connection functions (5), for any $(x_0,\ldots,x_l) \in \left( \mathbb{R}^d \right)^{l+1}$ such that the collection of positive reals $\{\|x_i-x_j\|\,{:}\,1\leq i,j, \leq l,i\neq j\}$ avoids some set of measure zero, and for any $(a_{ij})_{i,j=0}^l \in M^S_{l+1}([0,1])$ , as $n\rightarrow \infty$ ,

(60) \begin{equation} \begin{split} & \mathcal{F}^n_{ (H,h)}\left((x_0,\ldots,x_l),(W_{n,0},\ldots,W_{n,l}),(a_{ij})_{i,j=0}^l \right)\bigg|{\{Y^{(n)}_{U_n} \in A^r_n, |V(F^{\mathbb{G}_n}_{{U_n}}(r))|=l+1\}} \\ & {{\stackrel{d}{\rightarrow}}} \mathcal{F}^{\infty}_{ (H,h)}\left((x_0,\ldots,x_l), (W_{\infty,0},\ldots,W_{\infty,l}), (a_{ij})_{i,j=0}^l\right)\bigg|{\{|V(F^{\mathbb{G}_{\infty}}_{0}(r))|=l+1\}}, \end{split} \end{equation}

which implies that, as $n\rightarrow \infty$ , writing $\vec{x}=(x_0,x_1\dots,x_n)$ ,

(61) \begin{equation} \begin{split} & \mathbb{P}\left(\mathcal{F}^n_{ (H,h)}\left({\vec{x}},(W_{n,0},\ldots,W_{n,l}),(a_{ij})_{i,j=0}^l \right)>0 \bigg| \{Y^{(n)}_{U_n} \in A^r_n, |V(F^{\mathbb{G}_n}_{{U_n}}(r))|=l+1\}\right) \\ & \to \mathbb{P}\left(\mathcal{F}^{\infty}_{ (H,h)}\left({\vec{x}}, (W_{\infty,0},\ldots,W_{\infty,l}), (a_{ij})_{i,j=0}^l\right)>0 \bigg| \{|V(F^{\mathbb{G}_{\infty}}_{0}(r))|=l+1\}\right). \end{split}\end{equation}

Combining (61) with the fact that both $(Y_0,\ldots, Y_l)$ and $(U_{ij})_{i,j=0}^l$ are independent of $\textbf{W}^{(n)}$ and $\textbf{W}$ , and the fact that the random variables $\|Y_i-Y_j\|$ for $1\leq i,j \leq l$ with $i \neq j$ are continuous random variables and hence almost surely avoid sets of measure 0, we have (recalling $\mathscr{A}_n$ and $\mathscr{A}_{ {\infty}}$ from (57) and (59))

\begin{align*}\mathbb{E}\left[\left.\mathscr{A}_n\right | (Y_0,\ldots, Y_l),(U_{ij})_{i,j=0}^l\right]\stackrel{\text{a.s.}}{\to} \mathbb{E}\left[\left.\mathscr{A}_{ {\infty}}\right | (Y_0,\ldots, Y_l),(U_{ij})_{i,j=0}^l\right],\end{align*}

which implies, using dominated convergence (note that domination by 1 works), that

\begin{align*}\mathbb{E}\left[\mathscr{A}_n\right] \to \mathbb{E}\left[\mathscr{A}_{ {\infty}}\right],\end{align*}

as $n \to \infty$ .

Hence, again using (57) and (59), we have $\mathbb{E}\left[\mathcal{A}_n\right] \to \mathbb{E}\left[\mathcal{A}_{ {\infty}}\right]$ , which is just (45), as can be seen by recalling the definition of $\mathcal{A}_n$ from (54) and the definition of $\mathcal{A}_{ {\infty}}$ from (55). This completes the proof of Proposition 3. □

3.3. Proofs of path-counting results: Lemma 2 and Corollary 7

Proof of Lemma 2 . Recall the notation $\mathscr{B}^r_x$ from (30), which denotes the open Euclidean ball of radius r in $\mathbb{R}^d$ centered at $x \in \mathbb{R}^d$ . We first prove (35).

We first bound $\mathbb{W}_j(\textbf{0},x_1,\ldots,x_j)$ from above by $\mathbb{E}\left[\kappa(\|x_j-x_{j-1}\|,W_0,W_1)\right]$ , where $W_0$ and $W_1$ are i.i.d. copies of the limiting weight distribution W (recall (4)). Then we apply the change of variables

\begin{align*} z_i=x_i-x_{i-1},\qquad 1\leq i \leq j, \end{align*}

where $x_0=\textbf{0}$ , and apply Fubini’s theorem to obtain

(62) \begin{align} & \int_{\mathbb{R}^d}\cdots \int_{\mathbb{R}^d} \mathbb{W}_j(\textbf{0},x_1\ldots,x_j) \nonumber\\&\hspace{50 pt} \mathbb{1}_{\left\{\|x_1\|<a^m\right\}}\prod_{i=1}^{j-2}\mathbb{1}_{\left\{\|x_{i+1}-x_i\|<a^{m^{i+1}}\right\}}\mathbb{1}_{\left\{\|x_j-x_{j-1}\|>a^{m^j}\right\}} dx_1\cdots dx_j \nonumber\\ & \leq \int_{\mathscr{B}^{a^m}_{\textbf{0}}}\cdots \int_{\mathscr{B}^{a^{m^{j-1}}}_{\textbf{0}}}\int_{\mathbb{R}^d \setminus \mathscr{B}^{a^{m^j}}_{\textbf{0}}} \mathbb{E}\left[\kappa(\|z_j\|,W_0,W_1)\right] dz_jdz_{j-1}\cdots dz_1 \nonumber\\ & \leq C_0 (a^m)^d \cdots (a^{m^{j-1}})^d \int_{\mathbb{R}^d \setminus \mathscr{B}^{a^{m^j}}_{\textbf{0}}} \mathbb{E}\left[\kappa(\|z\|,W_0,W_1)\right] dz, \end{align}

for some constant $C_0>0$ . Recall the polynomial domination from (6) and the assumption $\alpha>d$ in Theorem 1.

We note, by first making a change of variables to bring the integral

\begin{align*} \int_{\mathbb{R}^d \setminus \mathscr{B}^{a^{m^j}}_{\textbf{0}}} \mathbb{E}\left[\kappa(\|z\|,W_0,W_1)\right] dz \end{align*}

down to an integral on $\mathbb{R}$ , and then using (6), that for any m sufficiently large so that $a^{m^j}>t_0$ ,

(63) \begin{align} \int_{\mathbb{R}^d \setminus \mathscr{B}^{a^{m^j}}_{\textbf{0}}} \mathbb{E}\left[\kappa(\|z\|,W_0,W_1)\right] dz \leq C_1 \frac{1}{a^{(\alpha-d)m^j}}, \end{align}

for some constant $C_1>0$ .

Combining (63) with (62), we note that for some constant $C_2>0$ ,

(64) \begin{align} &\int_{\mathbb{R}^d}\cdots \int_{\mathbb{R}^d} \mathbb{W}_j(\textbf{0},x_1,\ldots,x_j)\nonumber\\&\hspace{50 pt} \mathbb{1}_{\left\{\|x_1\|<a^m\right\}}\prod_{i=1}^{j-2}\mathbb{1}_{\left\{\|x_{i+1}-x_i\|<a^{m^{i+1}}\right\}}\mathbb{1}_{\left\{\|x_j-x_{j-1}\|>a^{m^j}\right\}} dx_1\cdots dx_j\nonumber \\& \leq C_2 \frac{1}{a^{dm^j\left(\frac{\alpha}{d}-1-\frac{1}{m}-\cdots-\frac{1}{m^{j-1}} \right)}} \to 0, \end{align}

as $m \to \infty$ , since $\frac{\alpha}{d}>1$ . This finishes the proof of (35).

We next go into the proof of (34). Recall the definition of $\mathbb{W}^{v_1,\ldots,v_j}_{n}(x_0,x_1,\ldots,x_j)$ from (32), where $v_0=U_n$ is the uniformly chosen vertex of $\mathbb{G}_n$ .

We use the notation

\begin{align*} \vec{v}=v_1,\dots,v_j; \;\; \vec{x}=(x_0,x_1,\dots,x_j). \end{align*}

For fixed $x_0,\ldots,x_j \in \mathbb{R}^d$ , we define the function $\mathcal{W}^{\vec{x}}_n\,{:}\,\mathbb{R}^{j+1} \to \mathbb{R}$ by

(65) \begin{align} \mathcal{W}^{\vec{x}}_n(\vec{t})\,{:\!=}\, \kappa_n\left(\|x_1-x_0\|,t_0,t_1\right)\cdots\kappa_n\left(\|x_j-x_{j-1}\|,t_{j-1},t_j\right), \end{align}

for $\vec{t}=(t_0,t_1,\dots,t_j) \in \mathbb{R}^{j+1}$ .

Note that

(66) \begin{align} & \frac{1}{n^j}\sum_{v_1,\ldots,v_j \in [n]} \mathbb{W}^{\vec{v}}_{n}(\vec{x}) = \mathbb{E}\left[\frac{1}{n^{j+1}} \sum_{i_0,i_1,\ldots,i_j} \mathcal{W}^{\vec{x}}_{n}(W^{(n)}_{i_0},\ldots,W^{(n)}_{i_j})\right], \end{align}

where we have used (32) and the fact that $v_0=U_n$ is uniformly distributed over n.

Since clearly

\begin{align*} \mathbb{E}\left[\frac{1}{n^{j+1}} \sum_{i_0,i_1,\ldots,i_j} \mathcal{W}^{\vec{x}}_{n}(W^{(n)}_{i_0},\ldots,W^{(n)}_{i_j})\right] = \mathbb{E}\left[\mathcal{W}^{\vec{x}}_{n}(W^{(n)}_{U_{n,0}},\ldots,W^{(n)}_{U_{n,j}})\right], \end{align*}

where $U_{n,0},\ldots,U_{n,j}$ is an i.i.d. collection of uniformly distributed random variables on $[n]=\{1,\ldots,n\}$ , we can take $h_n$ in (8) to be $\mathcal{W}^{\vec{x}}_{n}$ and conclude (recalling the definition of $\mathbb{W}_j(\vec{x})=\mathbb{W}_j(x_0,x_1\cdots,x_j)$ from (33)) that

(67) \begin{align} \frac{1}{n^j}\sum_{v_1,\ldots,v_j \in [n]} \mathbb{W}^{\vec{v}}_{n}(\vec{x}) = \mathbb{E}\left[\mathcal{W}^{\vec{x}}_{n}(W^{(n)}_{U_{n,0}},\ldots,W^{(n)}_{U_{n,j}})\right] \to \mathbb{W}_j(\vec{x}), \end{align}

as $n \to \infty$ . Now, (34) can be concluded using (67), a routine change of variables, Fatou’s lemma, and (35). □

Remark 8. (Efficacy of our bounds.) In the proof of Lemma 2, we have bounded $\mathbb{W}_j(\textbf{0},x_1,\ldots,x_j)$ from above by

\begin{align*} \mathbb{E}\left[\kappa(\|x_j-x_{j-1}\|,W_0,W_1)\right]. \end{align*}

That is, we have bounded all except the last term in the product inside the expectation $\mathbb{W}_j(\textbf{0},x_1,\ldots,x_j)$ by 1. This is usually a poor bound. However, as we see in the proof, this loss is well compensated by the strong double-exponential growth of $r=r(a,m,K)$ (recall (36)). In particular, for our purpose, we have been able to avoid the question of how correlated the random variables $\kappa(\|x_0-x_1\|,W^{(0)},W^{(1)})$ and $\kappa(\|x_1-x_2\|,W^{(1)},W^{(2)})$ are, where $x_0,x_1,x_2 \in \mathbb{R}^d$ , and $W^{(0)},W^{(1)},W^{(2)}$ are i.i.d. copies of the limiting weight distribution. We believe this question to be hard to tackle in general, under our general assumptions on $\kappa$ as formulated in Assumption 3.

Next we go into the proof of Corollary 7.

Proof of Corollary 7 . Recall the definition of the distance of an edge from a vertex from (31). Also recall the abbreviations in (37).

We begin by analyzing the event $\{BF_{n,r}\neq B_n\}$ . Note that if $\|Y^{(n)}_i-Y^{(n)}_j\|<a^{m^{L+1}}$ for every edge $\{i,j\}$ in $B_n$ that is at graph distance L ( $0\leq L \leq K-1$ ) from the root $U_n$ , then $B_n$ is a subgraph of $F_{n,r}$ with the same root ${U_n}$ , which implies that $BF_{n,r}=B_n$ . Hence, the event $\{BF_{n,r}\neq B_n\}$ implies the event

(68) \begin{equation} \textbf{Bad}_{r,n}\,{:\!=}\, \{\text{there is some } {bad \,edge} {\text { in}}\;B_n\}, \end{equation}

where a bad edge is an edge $\{i,j\}$ in $B_n$ with $\|Y^{(n)}_i-Y^{(n)}_j\|>a^{m^{L+1}}$ , where $0\leq L \leq K-1$ is the graph distance of the edge $\{i,j\}$ from the root $U_n$ of $B_n$ .

Figure 2.

Illustration demonstrating bad edges. The star is the root. The dashed edges are bad: each of them connects a pair of vertices whose locations are at least $a^{m^{L+1}}$ apart, where L is the distance of the edge from the root. The solid edges are good.

Therefore,

(69) \begin{equation} \mathbb{P}\left(BF_{n,r}\neq B_n\right)\leq \mathbb{P}\left(\textbf{Bad}_{r,n}\right). \end{equation}

By a similar argument,

(70) \begin{equation} \mathbb{P}\left(BF_{r}\neq B\right)\leq \mathbb{P}\left(\textbf{Bad}_r\right), \end{equation}

where the event $\textbf{Bad}_r$ is similarly defined for the rooted graph $(\mathbb{G}_{\infty},0)$ .

Define the event $\mathcal{I}_{n,j}$ , for $n \in \mathbb{N},j \in [n]$ , as

(71) \begin{align} \mathcal{I}_{n,j}\,{:\!=}\, & \{\exists\;v_1,\ldots,v_j \in V(\mathbb{G}_n)=[n]\,{:}\,\{v_0,v_1,\ldots,v_j\}\;\text{is a}\;j\text{-path in}\;(\mathbb{G}_n,U_n)\; \text{starting from} \nonumber\\[3pt] & \text{the root}\;v_0=U_n,\; \|Y^{(n)}_{v_{i-1}}-Y^{(n)}_{v_i}\|<a^{m^{i}} \forall {i \in[j-1]}, \|Y^{(n)}_{v_{j-1}}-Y^{(n)}_{v_j}\|>a^{m^{j}} \}. \end{align}

A simple union bound gives

(72) \begin{equation} \mathbb{P}\left(\textbf{Bad}_{r,n}\right)\leq \sum_{j=1}^K\mathbb{P}\left( \mathcal{I}_{n,j}\right). \end{equation}

Similarly,

(73) \begin{equation} \mathbb{P}\left(\textbf{Bad}_r\right) \leq \sum_{j=1}^K \mathbb{P}\left(\mathcal{I}_j\right), \end{equation}

where the event $\mathcal{I}_j$ for $j \in \mathbb{N}$ is similarly defined for the rooted graph $(\mathbb{G}_{\infty},0)$ .

Note that since K is fixed, it suffices to prove that $\lim_{m \to \infty} \limsup_{n \to \infty} \mathbb{P}\left(\mathcal{I}_{n,j}\right) = 0$ and $\lim_{m \to \infty} \limsup_{n \to \infty} \mathbb{P}\left(\mathcal{I}_{j}\right) = 0$ for $1 \le j \le K$ .

We proceed by bounding the probabilities $\mathbb{P}\left(\mathcal{I}_{n,j}\right)$ and $\mathbb{P}\left(\mathcal{I}_{j}\right)$ . Recall that we use $v_0$ to denote the typical vertex $U_n$ of $\mathbb{G}_n$ from the definition of the event $\mathcal{I}_{n,j}$ from (71). Note that, by Markov’s inequality, using the notation $\vec{v}=(v_0,v_1,\ldots,v_n)$ ,

(74) \begin{equation}\begin{split} & \mathbb{P}\left(\mathcal{I}_{n,j}\right) \leq \\& \sum_{{v_1},\ldots,{v_j} \in [n]} \mathbb{P}\left(\vec{v}\;\text{is a}\;j\text{-path},\; \|Y^{(n)}_{v_{i-1}}-Y^{(n)}_{v_i}\|\lt a^{m^{i}}\; \forall {i \in[j-1]}, \|Y^{(n)}_{v_{j-1}}-Y^{(n)}_{v_j}\|\gt a^{m^{j}}\right). \end{split} \end{equation}

For convenience, for $v_0,\ldots,v_j \in [n]$ , we define the event

(75) \begin{equation} \mathcal{E}_{n,j}(\vec{v})\,{:\!=}\,\{\|Y^{(n)}_{v_{i-1}}-Y^{(n)}_{v_i}\|<a^{m^{i}}\; \forall {i \in[j-1]}, \|Y^{(n)}_{v_{j-1}}-Y^{(n)}_{v_j}\|>a^{m^{j}}\}. \end{equation}

We compute

(76) \begin{align} & \mathbb{P}\left(\{v_0,v_1,\ldots,v_j\}\;j\text{-path},\; \|Y^{(n)}_{v_{i-1}}-Y^{(n)}_{v_i}\|<a^{m^{i}}\; \forall {i \in[j-1]}, \|Y^{(n)}_{v_{j-1}}-Y^{(n)}_{v_j}\|>a^{m^{j}}\right) \nonumber \\& = \mathbb{E}\left[\mathbb{1}_{\mathcal{E}_{n,j}(\vec{v})}\;\mathbb{P}\left(\{v_0,v_1,\ldots,v_j\}\;j\text{-path} \bigg| Y^{(n)}_{v_0},\ldots,Y^{(n)}_{v_j},W^{(n)}_{v_0},\ldots,W^{(n)}_{v_j}\right)\right] \nonumber \\& = \mathbb{E}\left[\mathbb{1}_{\mathcal{E}_{n,j}(\vec{v})}\;\kappa_n\left(\|Y^{(n)}_{v_0}-Y^{(n)}_{v_1}\|,W^{(n)}_{v_0},W^{(n)}_{v_1}\right)\cdots\kappa_n\left(\|Y^{(n)}_{v_{j-1}}-Y^{(n)}_{v_j}\|,W^{(n)}_{v_{j-1}},W^{(n)}_{v_j}\right)\right]. \end{align}

Recall the notation $\mathbb{W}^{v_1,\ldots,v_j}_{n}$ from (32). Using Fubini’s theorem and the fact that $\{Y^{(n)}_{v_0},Y^{(n)}_{v_1},\ldots,Y^{(n)}_{v_j}\}$ is an i.i.d. collection of $j+1$ uniform random variables on $I_n$ , recalling (75), and using the notation $\vec{x}=(x_0,x_1,\dots,x_j)$ , we find that (76) becomes

(77) \begin{equation} \begin{split} \frac{1}{n^{j+1}} \int_{I_n}\cdots\int_{I_n} \mathbb{W}^{v_1,\ldots,v_j}_{n}(\vec{x}) \prod_{i=0}^{j-2}\mathbb{1}_{\left\{\|x_{i+1}-x_i\|<a^{m^{i+1}}\right\}}\mathbb{1}_{\left\{\|x_{j}-x_{j-1}\|>a^{m^j}\right\}} dx_{0}\cdots dx_j. \end{split} \end{equation}

Using (74) and (77), we have

(78) \begin{align} \mathbb{P}\left(\mathcal{I}_{n,j}\right) &\leq \frac{1}{n^{j+1}}\int_{I_n}\cdots\int_{I_n} \sum_{v_1,\ldots,v_j \in [n]}\mathbb{W}^{v_1,\ldots,v_j}_{n}(\vec{x}) \nonumber\\ &\quad \times \prod_{i=0}^{j-2}\mathbb{1}_{\left\{\|x_{i+1}-x_i\|<a^{m^{i+1}}\right\}}\mathbb{1}_{\left\{\|x_{j}-x_{j-1}\|>a^{m^j}\right\}} dx_{0}\cdots dx_j. \end{align}

Recall the notation $\mathbb{W}_j$ from (33). Similarly, by the multivariate Mecke formula for Poisson processes [Reference Last and Penrose27, Theorem 4.4],

(79) \begin{align} \mathbb{P}\left(\mathcal{I}_j\right) &\leq \int_{\mathbb{R}^d}\cdots \int_{\mathbb{R}^d} \mathbb{W}_j(\textbf{0},x_1,\ldots,x_j) \mathbb{1}_{\left\{\|x_1\|<a^m\right\}}\nonumber\\ &\quad \times \prod_{i=1}^{j-2}\mathbb{1}_{\left\{\|x_{i+1}-x_i\|<a^{m^{i+1}}\right\}}\mathbb{1}_{\left\{\|x_j-x_{j-1}\|>a^{m^j}\right\}} dx_1\dots dx_j. \end{align}

Now we apply Lemma 2 for the bounds (79) and (77), and use the bounds (69), (70), (72), and (73) to conclude Corollary 7. □

Remark 9. (A general estimate.) Recall the notation (37), and recall r(a, m, K) from (36). Note that the bound (69) is true even when one has $a=a_n$ and $K=K_n$ in $r=r(a_n,m,K_n)$ , and so is the simple union bound (72). The integral bound (78) also works in this generality. Recalling the definition of the function $\mathcal{W}^{(x_0,\ldots,x_j)}_n$ from (65), the equality (66), and the display below it, bounding $\mathbb{E}\left[\mathcal{W}^{(x_0,\ldots,x_j)}_{n}(W^{(n)}_{U_{n,0}},\ldots,W^{(n)}_{U_{n,j}})\right]$ from above by $\mathbb{E}\left[\kappa_n(\|x_j-x_{j-1}\|,W^{(n)}_{U_{n,j}},W^{(n)}_{U_{n,j-1}})\right]$ (where $U_{n,0},\dots,U_{n,j}$ are i.i.d. uniformly distributed random variables on [n]), and making an easy change of variable, we obtain the general bound

(80) \begin{align} & \mathbb{P}\left(B^{F^{\mathbb{G}_n}_{U_{n}}(r_n)}_{U_{n}}(K_n) \neq B^{\mathbb{G}_n}_{U_{n}}(K_n)\right) \nonumber\\ & \leq \sum_{j=1}^{K_n} \int_{\mathscr{B}^{a_n^m}_{\textbf{0}}}\cdots\int_{\mathscr{B}^{a_n^{m^{j-1}}}_{\textbf{0}}}\int_{\mathbb{R}^d \setminus \mathscr{B}^{a_n^{m^j}}_{\textbf{0}}} \mathbb{E}\left[\kappa_n(\|z_j\|,W^{(n)}_{\textrm{U}^{\prime}_{n,1}},W^{(n)}_{\textrm{U}^{\prime}_{n,2}})\right] dz_j\cdots dz_1, \end{align}

where $\textrm{U}^{\prime}_{n,1}$ and $\textrm{U}^{\prime}_{n,2}$ are two i.i.d. uniform elements in [n]. Below, we will use the bound (80) with suitable choices of $a=a_n$ and $K=K_n$ to prove Theorem 3.

3.4. Proof of Theorem 1

Proposition 3 implies that the random rooted graph $(F^{\mathbb{G}_n}_{U_n}(r),U_n)$ converges in distribution to the random rooted graph $(F^{\mathbb{G}_{\infty}}_0(r),0)$ in the space $\mathcal{G}_{\star}$ . The proof of this fact can be carried out in the same manner as [Reference Van der Hofstad22, Theorem 2.13] is proved assuming [Reference Van der Hofstad22, Definition 2.10], and so we leave this for the reader to check. In particular, as a consequence of Proposition 3,

(81) \begin{equation} \mathbb{P}\left((F^{\mathbb{G}_n}_{U_n}(r),U_n) \in A\right) \to \mathbb{P}\left((F^{\mathbb{G}_{\infty}}_{0}(r),U_n) \in A\right),\end{equation}

for any subset $A \subset \mathcal{G}_{\star}$ .

Proof of Theorem 1 . Recall the definition of local weak convergence from Definition 3. Recall the abbreviations in (37). Let $G_*=(G,g) \in \mathcal{G}_\star$ .

Fix $K \in \mathbb{N}$ and $\varepsilon >0$ . To conclude Theorem 1, we need to find an $N \in \mathbb{N}$ such that for all $n>N$ ,

(82) \begin{equation} \left|\mathbb{P}\left(B^{\mathbb{G}_n}_{{U_n}}(K)\cong (G,g)\right)-\mathbb{P}\left(B^{\mathbb{G}_{\infty}}_{0}(K)\cong (G,g)\right)\right| < \varepsilon, \end{equation}

that is

\begin{align*} \left| \mathbb{P}\left(B_n \cong (G,g)\right) - \mathbb{P}\left(B \cong (G,g)\right) \right|< \varepsilon. \end{align*}

Note that

(83) \begin{align} &\left|\mathbb{P}\left(B_n\cong (G,g)\right)-\mathbb{P}\left(B\cong (G,g)\right)\right| \nonumber\\&\leq \left|\mathbb{P}\left(BF_{n,r}\cong (G,g)\right)-\mathbb{P}\left(BF_r \cong (G,g)\right)\right| +\left|\varepsilon_{n,r}\right| +\left|\varepsilon_r\right|, \end{align}

where

(84) \begin{equation} \varepsilon_{n,r}= \mathbb{P}\left(B_n \cong (G,g), BF_{n,r}\neq B_n\right) -\mathbb{P}\left(BF_{n,r} \cong (G,g), BF_{n,r}\neq B_n\right) \end{equation}

and

(85) \begin{equation} \varepsilon_{r}= \mathbb{P}\left(B \cong (G,g), BF_{r}\neq B\right) -\mathbb{P}\left(BF_{r} \cong (G,g), BF_{r}\neq B\right). \end{equation}

Clearly,

(86) \begin{equation} |\varepsilon_{n,r}|\leq \mathbb{P}\left(BF_{n,r}\neq B_n\right) \end{equation}

and

(87) \begin{equation} |\varepsilon_r|\leq \mathbb{P}\left(BF_{r}\neq B\right). \end{equation}

For the rest of the proof, we fix $m>0$ and $n_0 \in \mathbb{N}$ such that, for all $n \geq n_0$ ,

(88) \begin{equation} |\varepsilon_{n,r}|+|\varepsilon_r|< \varepsilon/2, \end{equation}

which is possible by Corollary 7.

Note that

\begin{align*} \mathbb{P}\left(BF_{n,r}\cong (G,g)\right)=\mathbb{P}\left(F_{n,r} \in A(K,(G,g))\right), \end{align*}

where $A(K,(G,g))\subset \mathcal{G}_{\star}$ is defined as

(89) \begin{align} A(K,(G,g))\,{:\!=}\,\{ (H,h) \in \mathcal{G}_{\star}\,{:}\, B^{H}_h(K) \cong (G,g)\}. \end{align}

By (81), as $n \to \infty$ ,

(90) \begin{equation} \mathbb{P}\left(F_{n,r} \in A(K,(G,g))\right) \to \mathbb{P}\left(F_{r} \in A(K,(G,g))\right)=\mathbb{P}\left(BF_{r}\cong (G,g)\right). \end{equation}

Combining (90) with (88) and (83), we can choose $n_1 \in \mathbb{N}$ such that for all $n>N=\max\{n_1,n_0\}$ , (82) holds. This completes the proof of Theorem 1. □

3.5. Proof of Theorem 2

For $ (H,h) \in \mathcal{G}_{\star}$ and $r>0$ , recall the empirical Euclidean graph neighborhood distribution as defined in (40). We first give the proof of Lemma 3.

Proof of Lemma 3 . To ease notation, let us write $C_{r,n}$ for $C_{r,n} (H,h)$ .

Note that

\begin{align*} C_{r,n}^2 & = \frac{1}{n^2}\sum_{i,j=1}^n\mathbb{1}_{\left\{F^{\mathbb{G}_n}_i(r)\cong (H,h)\right\}}\mathbb{1}_{\left\{F^{\mathbb{G}_n}_j(r)\cong (H,h)\right\}} \\ & = \mathbb{P}\left(F^{\mathbb{G}_n}_{U_{n,1}}(r)\cong (H,h),F^{\mathbb{G}_n}_{U_{n,2}}(r)\cong (H,h) \bigg| \mathbb{G}_n\right), \end{align*}

where $U_{n,1},U_{n,2}$ are i.i.d. uniformly distributed random variables on [n]. Therefore,

(91) \begin{equation} \mathbb{E}\left[C_{r,n}^2\right]=\mathbb{P}\left(F^{\mathbb{G}_n}_{U_{n,1}}(r)\cong (H,h),F^{\mathbb{G}_n}_{U_{n,2}}(r)\cong (H,h)\right). \end{equation}

We introduce the following abbreviations, which we will use throughout this proof to keep notation concise (recall the point process $\Gamma_n$ of the locations of the vertices of $\mathbb{G}_n$ from (9), the set $A^r_n$ from (29), and the ball $\mathscr{B}^r_x$ from (30)):

(92) \begin{align} \mathscr{E}&\,{:\!=}\,\left\{F^{\mathbb{G}_n}_{U_{n,1}}(r)\cong (H,h)\right\},\quad \mathscr{F}\,{:\!=}\,\left\{F^{\mathbb{G}_n}_{U_{n,2}}(r)\cong (H,h)\right\},\nonumber\\ \mathscr{U}&\,{:\!=}\,\left\{\Gamma_n\left(\mathscr{B}^{r}_{Y^{(n)}_{U_{n,1}}}\right)=|V(H)|\right\},\quad \mathscr{V}\,{:\!=}\,\left\{\Gamma_n\left(\mathscr{B}^{r}_{Y^{(n)}_{U_{n,2}}}\right)=|V(H)|\right\}, \nonumber\\ \mathscr{J}&\,{:\!=}\,\left\{\mathscr{B}^r_{Y^{(n)}_{U_{n,1}}}\cap \mathscr{B}^r_{Y^{(n)}_{U_{n,2}}}=\varnothing\right\},\quad \mathscr{W}\,{:\!=}\,\left\{U_{n,1},U_{n,2} \in A^{r}_n\right\}. \end{align}

Recall that the target is to show

(93) \begin{equation} |\mathbb{P}\left(\mathscr{E} \cap \mathscr{F}\right)-\mathbb{P}\left(\mathscr{E}\right)\mathbb{P}\left(\mathscr{F}\right)|=|\mathbb{P}\left(\mathscr{E}\cap \mathscr{F}\right)-\mathbb{P}\left(\mathscr{E}\right)^2| \to0, \end{equation}

as $n \to \infty$ (note that conditionally on $\mathbb{G}_n$ , the random variables $\mathbb{1}_{\mathscr{E}}$ and $\mathbb{1}_{\mathscr{F}}$ are identically distributed, so that they have the same conditional expectation, and hence the same expectation).

First we write

(94) \begin{align} \mathbb{P}\left(\mathscr{E}\cap \mathscr{F}\right)&=\mathbb{P}\left(\mathscr{E}\cap \mathscr{F} \bigg| \mathscr{J}\cap \mathscr{U}\cap \mathscr{V} \cap \mathscr{W}\right)\mathbb{P}\left(\mathscr{J}\cap \mathscr{U}\cap \mathscr{V}\cap \mathscr{W}\right) \end{align}

(95) \begin{align} &\qquad+\mathbb{P}\left(\{\mathscr{E}\cap \mathscr{F}\}\cap \{\mathscr{J}^c\cup \mathscr{U}^c \cup \mathscr{V}^c \cup \mathscr{W}^c\}\right) . \end{align}

Note that the term in (95) is bounded from above by

\begin{align*}\mathbb{P}\left(\{\mathscr{E}\cap \mathscr{F}\}\cap \{\mathscr{U}^c\cup \mathscr{V}^c\}\right)+\mathbb{P}\left(\mathscr{J}^c\right)+\mathbb{P}\left(\mathscr{W}^c\right),\end{align*}

and it is easily observed that the first term is equal to 0, while the last term tends to 0 as $n \to \infty$ by (42). Also note that

\begin{align*}\mathbb{P}\left(\mathscr{J}^c\right)\leq \mathbb{P}\left(Y^{(n)}_{U_{n,1}} \in \mathscr{B}^{2r}_{Y^{(n)}_{U_{n,2}}} \cap I_n\right).\end{align*}

Clearly,

\begin{align*}\mathbb{P}\left(Y^{(n)}_{U_{n,1}} \in \mathscr{B}^{2r}_{Y^{(n)}_{U_{n,2}}} \cap I_n \bigg| Y^{(n)}_{U_{n,2}}\right)\stackrel{\text{a.s.}}{\leq} \frac{\lambda_d(\{y \in \mathbb{R}^d\,{:}\, \|y\|< 2r\})}{n},\end{align*}

which tends to 0 as $n \to \infty$ . Hence, taking expectations of both sides in the last display and letting $n \to \infty$ , we get $\mathbb{P}\left(\mathscr{J}^c\right)\to0$ . Hence the term in (95) tends to 0 as $n \to \infty$ .

To analyze the term in (94), we observe that, conditionally on $\mathscr{J}\cap \mathscr{U}\cap \mathscr{V}\cap \mathscr{W}$ , the random variables $\mathbb{1}_{\mathscr{E}}$ and $\mathbb{1}_{\mathscr{F}}$ are independent, since they are just functions of the locations of the $|V(H)|$ points falling in $\mathscr{B}^r_{Y^{(n)}_{U_{n,1}}}$ and $\mathscr{B}^r_{Y^{(n)}_{U_{n,2}}}$ , and these locations are independent (since the locations of different vertices of $\mathbb{G}_n$ are independent). Hence,

\begin{align*}(94)=\mathbb{P}\left(\mathscr{E} \bigg| \mathscr{J}\cap \mathscr{U}\cap \mathscr{V}\cap \mathscr{W}\right)\mathbb{P}\left(\mathscr{F} \bigg| \mathscr{J}\cap \mathscr{U}\cap \mathscr{V}\cap \mathscr{W}\right) \mathbb{P}\left(\mathscr{J}\cap \mathscr{U}\cap \mathscr{V}\cap \mathscr{W}\right).\end{align*}

Now note that $\mathscr{E}$ is independent of $\mathscr{V}$ , conditionally on $\{\mathscr{J}\cap \mathscr{U}\cap \mathscr{W}\}$ . Similarly, $\mathscr{F}$ is independent of $\mathscr{U}$ , conditionally on $\{\mathscr{J}\cap \mathscr{V}\cap \mathscr{W}\}$ .

Hence,

\begin{align*}(94)=\mathbb{P}\left(\mathscr{E} \bigg| \mathscr{J}\cap \mathscr{U}\cap \mathscr{W}\right)\mathbb{P}\left(\mathscr{F} \bigg| \mathscr{J}\cap \mathscr{V}\cap \mathscr{W}\right) \mathbb{P}\left(\mathscr{J}\cap \mathscr{U}\cap \mathscr{V}\cap \mathscr{W}\right).\end{align*}

As argued earlier, $\mathbb{P}\left(\mathscr{J}^c\right)\to0$ , so $\mathbb{P}\left(\mathscr{J}\right)\to1$ , and it is easy to observe that $\mathbb{P}\left(\mathscr{W}\right)\to1$ using (42). Hence we can forget about the ‘almost’ certain events $\mathscr{W}$ and $\mathscr{J}$ for the first two terms from the last display, and condition on $\mathscr{J}\cap \mathscr{W}$ for the third term, to conclude that (recall (94))

(96) \begin{align} & \left|\mathbb{P}\left(\mathscr{E}\cap \mathscr{F} \bigg| \mathscr{J}\cap \mathscr{U}\cap \mathscr{V} \cap \mathscr{W}\right)\mathbb{P}\left(\mathscr{J}\cap \mathscr{U}\cap \mathscr{V}\cap \mathscr{W}\right) \right. \nonumber \\ &\quad \left. - \mathbb{P}\left(\mathscr{E} \bigg| \mathscr{U}\right)\mathbb{P}\left(\mathscr{F} \bigg| \mathscr{V}\right)\mathbb{P}\left(\mathscr{U}\cap \mathscr{V} \bigg| \mathscr{J}\cap \mathscr{W}\right) \right|\to 0; \end{align}

that is, the difference between (94) and $\mathbb{P}\left(\mathscr{E} \bigg| \mathscr{U}\right)\mathbb{P}\left(\mathscr{F} \bigg| \mathscr{V}\right)\mathbb{P}\left(\mathscr{U}\cap \mathscr{V} \bigg| \mathscr{J}\cap \mathscr{W}\right)$ tends to 0.

We claim that to conclude the proof, it suffices to check that

(97) \begin{equation} \left|\mathbb{P}\left(\mathscr{U}\cap \mathscr{V} \bigg| \mathscr{J}\cap \mathscr{W}\right)-\mathbb{P}\left(\mathscr{U}\right)\mathbb{P}\left(\mathscr{V}\right) \right|\to0. \end{equation}

This is because (97), combined with (96) and the observations that $\mathscr{E} \subset \mathscr{U}$ and $\mathscr{F} \subset \mathscr{V}$ , implies that the difference between the expression (94) and $\mathbb{P}\left(\mathscr{E}\right)\mathbb{P}\left(\mathscr{F}\right)$ goes to 0. Combining this with the fact that the expression in (95) goes to 0, we obtain (93).

We now show (97). Recall that $|V(H)|$ denotes the size of the vertex set of the graph H. Observe that

\begin{align*}\mathbb{P}\left(\mathscr{U}\cap \mathscr{V} \bigg| \mathscr{J}\cap \mathscr{W}\right)=\mathbb{P}\left((M_1,M_2,M_3)=(|V(H)|-1,|V(H)|-1,n-2|V(H)|)\right),\end{align*}

where $\textbf{M}=(M_1,M_2,M_3)$ is a multinomial vector with parameters

$$\left(n-2;\frac{\lambda_d(\mathscr{B}^{r}_{\textbf{0}})}{n},\frac{\lambda_d(\mathscr{B}^{r}_{\textbf{0}})}{n},1-2\frac{\lambda_d(\mathscr{B}^{r}_{\textbf{0}})}{n} \right).$$

Hence,

\begin{align*} & \mathbb{P}\left(\mathscr{U}\cap \mathscr{V} \bigg| \mathscr{J}\cap \mathscr{W}\right)\\ & = \frac{n!}{(|V(H)|-1)!\;(|V(H)|-1)!\;(n-2|V(H)|)!}\\ & \times \left(\frac{\lambda_d(\mathscr{B}^{r}_{\textbf{0}})}{n}\right)^{|V(H)|-1}\left(\frac{\lambda_d(\mathscr{B}^{r}_{\textbf{0}})}{n}\right)^{|V(H)|-1}\left(1-2\frac{\lambda_d(\mathscr{B}^{r}_{\textbf{0}})}{n}\right)^{n-2|V(H)|}. \end{align*}

It is an easy analysis to check that this converges to $\mathbb{P}\left(Y=|V(H)|-1\right)^2$ , where $Y\sim \text{Poi}(\lambda_d(\mathscr{B}^{r}_{\textbf{0}}))$ .

Again,

\begin{align*}\mathbb{P}\left(\mathscr{U} \bigg| Y^{(n)}_{U_{n,1}}\in A^{r}_n\right)\mathbb{P}\left(\mathscr{V} \bigg| Y^{(n)}_{U_{n,2}}\in A^{r}_n\right)=\mathbb{P}\left(Y^{\prime}_n=|V(H)|-1\right)^2,\end{align*}

where $Y^{\prime}_n \sim \text{Bin}(n-1, \frac{\lambda_d(\mathscr{B}^{r}_{\textbf{0}})}{n})$ .

Since $\mathbb{P}\left(Y^{\prime}_n=|V(H)|-1\right)^2\to\mathbb{P}\left(Y=|V(H)|-1\right)^2$ , and both

\begin{align*}\mathbb{P}\left(Y^{(n)}_{U_{n,1}}\in A^{r}_n\right), \mathbb{P}\left(Y^{(n)}_{U_{n,2}}\in A^{r}_n\right) \geq \mathbb{P}\left(\mathscr{W}\right)\to 1\end{align*}

(using (42)), we have shown that

\begin{align*} &|\mathbb{P}\left(\mathscr{U}\cap \mathscr{V} \bigg| \mathscr{J}\cap \mathscr{W}\right)-\mathbb{P}\left(\mathscr{U}\right)\mathbb{P}\left(\mathscr{V}\right)|\\ &\leq \left|\mathbb{P}\left(\mathscr{U}\cap \mathscr{V} \bigg| \mathscr{J}\cap \mathscr{W}\right)-\mathbb{P}\left(\mathscr{U} \bigg| Y^{(n)}_{U_{n,1}}\in A^{r}_n\right)\mathbb{P}\left(\mathscr{V} \bigg| Y^{(n)}_{U_{n,2}}\in A^{r}_n\right) \right|\\ &+\left|\mathbb{P}\left(\mathscr{U} \bigg| Y^{(n)}_{U_{n,1}}\in A^{r}_n\right)\mathbb{P}\left(\mathscr{V} \bigg| Y^{(n)}_{U_{n,2}}\in A^{r}_n\right)-\mathbb{P}\left(\mathscr{U}\right)\mathbb{P}\left(\mathscr{V}\right) \right|\to0. \end{align*}

This completes the proof of (97) and hence of Lemma 3. □

Note that using Lemma 3 with Proposition 3, a direct application of Chebyshev’s inequality gives, for any $ (H,h) \in \mathcal{G}_{\star}$ ,

\begin{align*}C_{r,n} (H,h){{\stackrel{\mathbb{P}}{\rightarrow}}} \mathbb{P}\left(F^{\mathbb{G}_{\infty}}_{0}(r)\cong (H,h)\right).\end{align*}

In particular, this implies that the empirical Euclidean graph neighborhood measure of $\mathbb{G}_n$ converges in probability to the measure induced by the random element $(F^{\mathbb{G}_{\infty}}_0(r),0)$ in $\mathcal{G}_{\star}$ : for any subset $A \subset \mathcal{G}_{\star}$ and for any $r>0$ (recall the Euclidean graph neighborhoods from Definition 6),

(98) \begin{equation} \frac{1}{n}\sum_{i=1}^n \mathbb{1}_{\left\{(F^{\mathbb{G}_n}_i(r),i) \in A\right\}} {{\stackrel{\mathbb{P}}{\rightarrow}}} \mathbb{P}\left((F^{\mathbb{G}_{\infty}}_{0}(r),0) \in A\right),\end{equation}

as $n \to \infty$ . We now use (98) to prove Theorem 2.

Proof of Theorem 2. For $K \in \mathbb{N}$ and $(G,g) \in \mathcal{G}_{\star}$ , recall the random variables $B_n(G,g)$ as defined in (41). Furthermore, recall the definition of local convergence in probability from Definition 4.

Fix $\varepsilon>0$ . Note that the target is to show that, for every $\varepsilon>0$ , there exists $N \in \mathbb{N}$ such that for all $n > N$ ,

(99) \begin{equation} {\mathbb{P}\left(\Big|B_n(G,g)-\mathbb{P}\left(B^{\mathbb{G}_{\infty}}_{0}(K)\cong (G,g)\right)\Big|>\varepsilon\right)< \varepsilon.} \end{equation}

We abbreviate the Euclidean graph neighborhoods $F^{n,r,i}\,{:\!=}\,F^{\mathbb{G}_n}_i(r)$ , with $r=r(a,m,K)$ as in (36), and $i \in V(\mathbb{G}_n)=[n]$ . We note that

(100) \begin{align} B_n(G,g) & =\frac{1}{n}\sum_{i=1}^n \mathbb{1}_{\left\{B^{\mathbb{G}_n}_i(K)\cong (G,g)\right\}} \nonumber \\ & =\frac{1}{n}\sum_{i=1}^n \mathbb{1}_{\left\{B^{\mathbb{G}_n}_i(K)\cong (G,g)\right\}}\mathbb{1}_{\left\{B^{\mathbb{G}_n}_i(K)=B^{F_{n,r,i}}_i(K)\right\}} \end{align}

(101) \begin{align} & \qquad\qquad\quad\quad+\frac{1}{n}\sum_{i=1}^n \mathbb{1}_{\left\{B^{\mathbb{G}_n}_i(K)\cong (G,g)\right\}}\mathbb{1}_{\left\{B^{\mathbb{G}_n}_i(K)\neq B^{F_{n,r,i}}_i(K)\right\}}. \end{align}

Writing the first term on the right-hand side of (101) as

(102) \begin{equation} \frac{1}{n}\sum_{i=1}^n \mathbb{1}_{\left\{B^{F_{n,r,i}}_i(K)\cong (G,g)\right\}}-\frac{1}{n}\sum_{i=1}^n \mathbb{1}_{\left\{B^{F_{n,r,i}}_i(K)\cong (G,g)\right\}}\mathbb{1}_{\left\{B^{\mathbb{G}_n}_i(K)\neq B^{F_{n,r,i}}_i(K)\right\}}, \end{equation}

we note from (101) that

(103) \begin{equation} B_n(G,g)=\frac{1}{n}\sum_{i=1}^n \mathbb{1}_{\left\{B^{F_{n,r,i}}_i(K)\cong (G,g)\right\}}+\varepsilon^{\prime}_{n,r}, \end{equation}

where (recall the event $\textbf{Bad}_{r,n}$ from (68)) the expectation $\mathbb{E}\left[{|\varepsilon^{\prime}_{n,r}|}\right]$ of the absolute value of the error $\varepsilon^{\prime}_{n,r}$ is bounded from above by $2\mathbb{P}\left(\textbf{Bad}_{r,n}\right)$ .

It can be shown using a similar argument that (recall the notation (37))

(104) \begin{equation} \mathbb{P}\left(B^{\mathbb{G}_{\infty}}_{0}(K)\cong (G,g)\right)=\mathbb{P}\left(BF_r\cong (G,g)\right)+\varepsilon^{\prime}_r, \end{equation}

where $\mathbb{E}\left[{|\varepsilon^{\prime}_r|}\right]$ is bounded from above by $2\mathbb{P}\left(\textbf{Bad}_r\right)$ .

Using Corollary 7, it is easy to see that there exists $n_0 \in \mathbb{N}$ such that for $m>0$ sufficiently large (recall $r=r(a,m,K)$ ), for all $n>n_0$ ,

(105) \begin{equation} 2\mathbb{P}\left(\textbf{Bad}_{{r,n}}\right)+2\mathbb{P}\left(\textbf{Bad}_r\right)<\varepsilon/2. \end{equation}

Recall the subset $A(K,(G,g)) \subset \mathcal{G}_{\star}$ from (89). Note that using (98),

\begin{align*} \frac{1}{n}\sum_{i=1}^n \mathbb{1}_{\left\{B^{F_{n,r,i}}_{i}(K)\cong (G,g)\right\}} &=\frac{1}{n}\sum_{i=1}^n \mathbb{1}_{\left\{(F_{n,r,i},i) \in A(K,(G,g))\right\}} \\ & {{\stackrel{\mathbb{P}}{\rightarrow}}} \mathbb{P}\left((F^{\mathbb{G}_{\infty}}_0(r),0)\in A(K,(G,g))\right)\\ &=\mathbb{P}\left(BF_r \cong (G,g)\right), \end{align*}

and so there exists $n_1 \in \mathbb{N}$ such that for all $n>n_1$ we have

(106) \begin{equation} {\mathbb{P}\left(\Big|\frac{1}{n}\sum_{i=1}^n {\mathbb{1}_{\left\{B^{F_{n,r,i}}_{i}(K)\cong (G,g)\right\}}}-\mathbb{P}\left(BF_r \cong (G,g)\right)\Big|>\varepsilon/2\right)< \varepsilon/2.} \end{equation}

Hence, for all $n>N=\max\{n_0,n_1\}$ , we note that by (105) and (106), (99) holds. This completes the proof of Theorem 2. □

3.6. Proof of Theorem 3

The main idea in this proof is that the first result of Corollary 7 can be pushed to the case when $K=K_n$ is allowed to grow in a doubly logarithmic manner, instead of being fixed. For this, we need a finer analysis of the error terms we encounter while proving Lemma 2. We now go into the formal argument.

Proof. Recall $C \in \left( 0, \frac{1}{\log(\frac{\alpha}{\alpha-d})}\right)$ . Fix some $\overline{C} \in \left( C, \frac{1}{\log(\frac{\alpha}{\alpha-d})}\right)$ , and let

(107) \begin{equation} m\,{:\!=}\, e^{1/\overline{C}}. \end{equation}

Since $\overline{C}< \frac{1}{\log(\frac{\alpha}{\alpha-d})}$ ,

\begin{align*} \frac{\alpha}{d}> \frac{m}{m-1}. \end{align*}

Let

(108) \begin{equation} a_n=\log n \end{equation}

and

(109) \begin{equation} K_n = \frac{\log \log n + \log(\frac{1}{d}-\delta) - \log \log a_n}{\log m} = \frac{\log \log n + \log(\frac{1}{d}-\delta) - \log \log \log n}{\log m}, \end{equation}

where $\delta>0$ is such that $\frac{1}{d}-\delta >0$ . Note that by the choice of $K_n$ , we have

(110) \begin{equation} a_n^{m^{K_n}}=n^{\frac{1}{d}-\delta}. \end{equation}

We also let $r_n=r(a_n,m,K_n)$ be as in (36).

Note that, since $C < \frac{1}{\log m}$ by the choice of m, for all large n,

\begin{align*} C \log \log n \leq K_n. \end{align*}

Then for $U_{n,1}$ and $U_{n,2}$ two uniformly chosen vertices of $\mathbb{G}_n$ , for all large n,

(111) \begin{align} & \mathbb{P}\left(d_{\mathbb{G}_n}(U_{n,1},U_{n,2})\leq C\log \log n\right)\nonumber\\ & \leq \mathbb{P}\left(d_{\mathbb{G}_n}(U_{n,1},U_{n,2})\leq K_n\right)\nonumber\\ &= \mathbb{P}\left(U_{n,2}\in B^{\mathbb{G}_n}_{U_{n,1}}(K_n)\right)\nonumber\\ & \leq \mathbb{P}\left(U_{n,2} \in B^{F^{\mathbb{G}_n}_{U_{n,1}}(r_n)}_{U_{n,1}}(K_n)\right) + \mathbb{P}\left(B^{F^{\mathbb{G}_n}_{U_{n,1}}(r_n)}_{U_{n,1}}(K_n) \neq B^{\mathbb{G}_n}_{U_{n,1}}(K_n)\right) \nonumber\\ & \leq \mathbb{P}\left(U_{n,2} \in F^{\mathbb{G}_n}_{U_{n,1}}(r_n)\right)+ \mathbb{P}\left(B^{F^{\mathbb{G}_n}_{U_{n,1}}(r_n)}_{U_{n,1}}(K_n) \neq B^{\mathbb{G}_n}_{U_{n,1}}(K_n)\right)\nonumber\\ & =\mathbb{P}\left(Y^{(n)}_{U_{n,2}}\in \mathscr{B}^{r_n}_{Y^{(n)}_{U_{n,1}}}\right) + \mathbb{P}\left(B^{F^{\mathbb{G}_n}_{U_{n,1}}(r_n)}_{U_{n,1}}(K_n) \neq B^{\mathbb{G}_n}_{U_{n,1}}(K_n)\right), \end{align}

where we recall the Euclidean graph neighborhoods $F^{\mathbb{G}_n}_i(r)$ from Definition 6.

Note that to conclude Theorem 3, it is sufficient to establish that the right-hand side of (111) tends to 0 as $n \to \infty$ .

Via a simple conditioning on $Y^{(n)}_{U_{n,1}}$ , and using that $Y^{(n)}_{U_{n,2}}$ is uniformly distributed on $I_n$ and is independent of $Y^{(n)}_{U_{n,1}}$ , we have

\begin{align*} & \mathbb{P}\left(Y^{(n)}_{U_{n,2}}\in \mathscr{B}^{r_n}_{Y^{(n)}_{U_{n,1}}}\right) \leq \frac{\lambda_d(\mathscr{B}^{r_n}_{\textbf{0}})}{n}. \end{align*}

Furthermore (recalling r(a, m, K) from (36)), note that using the upper bound

\begin{align*} r(a_n,m,K_n)\leq K_n a_n^{m^{K_n}}, \end{align*}

we have

\begin{align*} \frac{\lambda_d(\mathscr{B}^{r_n}_{\textbf{0}})}{n} \leq \frac{w r_n^d}{n} \leq \frac{w K_n^d n^{d\delta-1}}{n}\to 0, \end{align*}

as $n \to \infty$ , by the choice of $K_n$ as in (109), where $w>1$ is some constant upper bound on $\lambda_d(\mathscr{B}^1_{\textbf{0}})$ , and in the last inequality we have used (110). So the first term on the right-hand side of (111) tends to 0 as $n \to \infty$ , and it remains for us to show that

(112) \begin{equation} \mathbb{P}\left(B^{F^{\mathbb{G}_n}_{U_n}(r_n)}_{U_n}(K_n) \neq B^{\mathbb{G}_n}_{U_n}(K_n)\right) \to 0, \end{equation}

as $n \to \infty$ , where $U_n$ is uniformly distributed on $V(\mathbb{G}_n)=[n]$ .

We now recall the general estimate from Remark 9:

(113) \begin{align} & \mathbb{P}\left(B^{F^{\mathbb{G}_n}_{U_n}(r_n)}_{U_n}(K_n) \neq B^{\mathbb{G}_n}_{U_n}(K_n)\right) \nonumber \\ & \leq \sum_{j=1}^{K_n} \int_{\mathscr{B}^{a_n^m}_{\textbf{0}}}\cdots\int_{\mathscr{B}^{a_n^{m^{j-1}}}_{\textbf{0}}}\int_{\mathbb{R}^d \setminus \mathscr{B}^{a_n^{m^j}}_{\textbf{0}}} \mathbb{E}\left[\kappa_n(\|z_j\|,W^{(n)}_{{U}^{\prime}_{n,1}},W^{(n)}_{{U}^{\prime}_{n,2}})\right] dz_j\cdots dz_1, \end{align}

where $a=a_n$ is as in (108), m is as in (107), $K=K_n$ is as in (109), $r_n=r(a_n,m,K_n)$ is as in (36), and ${U}^{\prime}_{n,1}, {U}^{\prime}_{n,2}$ are two independent uniformly distributed random variables on [n].

Using (113) and Assumption 3(3), taking $h_n\colon\mathbb{R}^2 \to \mathbb{R}$ in (8) as being equal to $h_n(s,t)=\kappa_n(\|z_j\|,s,t,)$ , we note that for n sufficiently large so that $a_n^m > t_0$ and

\begin{align*} \mathbb{E}\left[\kappa_n(\|z_j\|,W^{(n)}_{{U}^{\prime}_{n,1}},W^{(n)}_{{U}^{\prime}_{n,2}})\right] \leq \|z_j\|^{-\alpha}, \end{align*}

we have

(114) \begin{align} \mathbb{P}\left(B^{F^{\mathbb{G}_n}_{U_n}(r_n)}_{U_n}(K_n) \neq B^{\mathbb{G}_n}_{U_n}(K_n)\right) & \leq \sum_{j=1}^{K_n} \int_{\mathscr{B}^{a_n^m}_{\textbf{0}}}\cdots\int_{\mathscr{B}^{a_n^{m^{j-1}}}_{\textbf{0}}}\int_{\mathbb{R}^d \setminus \mathscr{B}^{a_n^{m^j}}_{\textbf{0}}} \frac{1}{\|z_j\|^{\alpha}} dz_j\cdots dz_1.\nonumber\\ &= \sum_{j=1}^{K_n} \frac{w^{j}}{a_n^{dm^j\left(\frac{\alpha}{d}-1-\frac{1}{m}-\cdots-\frac{1}{m^{j-1}} \right)}}, \end{align}

where $w>1$ is some constant upper bound on $\lambda_d(\mathscr{B}^1_{\textbf{0}})$ .

Since $\frac{\alpha}{d}> \frac{m}{m-1}$ , and if we let $C_0=d \left(\frac{\alpha}{d}- \frac{m}{m-1}\right)>0$ , it follows from (114) that for all n large enough so that $\frac{w}{a_n^{C_0}}<1$ , and for some $J > 1$ sufficiently large so that for all $j>J$ we have $j^{1/j}<m$ (note that such a J exists since $m>1$ ), we can write (assuming n is large so that $K_n>J+1$ )

(115) \begin{align} \mathbb{P}\left(B^{F^{\mathbb{G}_n}_{U_n}(r_n)}_{U_n}(K_n) \neq B^{\mathbb{G}_n}_{U_n}(K_n)\right) \leq \sum_{j=1}^{K_n} \frac{w^{j}}{a_n^{C_0m^j}} {=\sum_{j=1}^{J} \frac{w^{j}}{a_n^{C_0m^j}}}{+\sum_{j=J+1}^{K_n}\frac{w^{j}}{a_n^{C_0m^j}}.} \end{align}

Note that the first term on the right-hand side of (115) clearly converges to 0 as $n \to \infty$ since $a_n \to \infty$ . For the second term on the right-hand side of (115), we note that since for all $j \geq J+1$ , $m^j>j$ , and since $\frac{w}{a_n^{C_0}}<1$ , we have

\begin{align*} {\sum_{j=J+1}^{K_n}\frac{w^{j}}{a_n^{C_0m^j}}} {\leq \sum_{j=J+1}^{K_n}\frac{w^{m^j}}{a_n^{C_0m^j}}} {\leq \sum_{j=J+1}^{K_n}\left(\frac{w}{a_n^{C_0}}\right)^{j}} {\leq \sum_{j=1}^{\infty}\left(\frac{w}{a_n^{C_0}}\right)^{j}} {=\frac{w}{a_n^{C_0}} \frac{1}{1-\frac{w}{a_n^{C_0}}} \to 0,} \end{align*}

as $n \to \infty$ . This implies (112), completing the proof of Theorem 3.