2.1. The height process in the discrete case

In this section, we describe a pathwise construction of the height process corresponding to $S^n$ . We do this by considering separately the process before and after it hits its overall infimum. This corresponds to considering the laws of finite and infinite trees in a supercritical branching process separately; this idea can be traced back to Harris [Reference Harris31]. In the descriptions of the two parts, we will make use of a process that is locally absolutely continuous to $S^n$ , which we will denote by $\hat{S}^n$ . We will refer to to $\hat{S}^n$ as ‘ $S^n$ conditioned to die out’. This formalizes, in the random walk framework, the well-known fact that a supercritical Galton–Watson process conditioned to die out is a subcritical Galton–Watson process.

For $t\geq 0$ , define

$${\mathcal L}_{S^n(1)}(\theta)={\mathbb E}\left[\,\!\exp\!({-}\,\theta S^n(1))\right],$$

and set $\phi_n(\theta)=\log {\mathcal L}_{S^n(1)}(\theta)$ . Since ${\mathbb E}[S^n(1)]>0$ , we have ${\mathcal L}_{S^n(1)}'(0)<0$ , so by the convexity of ${\mathcal L}_{S^n(1)}(\theta)$ and the fact that ${\mathcal L}_{S^n(1)}(0)=1$ , there is a unique $\xi_n>0$ such that ${\mathcal L}_{S^n(1)}(\xi_n)=1$ and $\phi_n(\xi_n)=0$ . Let ${\mathbb P}_n$ be the law of $S^n$ , and let

$${\mathcal F}_k^n\,{:\!=}\,\sigma(S^n(m),m\leq k)$$

be the natural filtration of $S^n$ . Let ${\mathbb P}_n^\#$ be locally absolutely continuous with respect to ${\mathbb P}_n$ , with

$$\textrm{d}{\mathbb P}_n^\#|_{{\mathcal F}_k^n}=\exp\!\left({-}\,\xi_n S^n(k)\right) \textrm{d}{\mathbb P}_n|_{{\mathcal F}_k^n},$$

and let $\hat{S}^n$ be a process which under ${\mathbb P}_n$ has the law of $S^n$ under ${\mathbb P}_n^\#$ . Let

$$\tau^n(m)=\inf\left\{k\,{:}\,S^n(k)=-m\right\},$$

so that $\{\tau^n(m)<\infty\}$ may be interpreted as the event that at least m trees in the Galton–Watson forest are finite. The following properties of $\xi_n$ and $\hat{S}^n$ are standard (see e.g. [Reference Athreya and Ney6, Chapter I.12]).

The following lemma is a first reason why $\hat{S}^n$ plays an important rôle in the pathwise construction of $S^n$ .

Proof. Note that the negative of the overall infimum of $S^n$ , which we denote by $-I^{n}_{\infty}$ , equals the number of finite trees in the forest defined by $S^n$ viewed as a Łukasiewicz path. Hence, it is distributed as a geometric random variable with parameter $\exp\!({-}\,\xi_{n})$ . Let $\rho^n$ denote the time when $S^n$ first reaches $I^{n}_{\infty}$ . We have that

$$(S^n(j)\,{:}\,0\leq j \leq \rho^n)\textrm{ under }{\mathbb P}_{n}(\;\cdot\;|I^n_{\infty}=-m) $$

has the same distribution as

$$(S^n(j)\,{:}\,0\leq j \leq \tau^n(m))\textrm{ under }{\mathbb P}_{n}\left(\;\cdot\;|\tau^n(m)<\infty\right),$$

which, by Theorem 3, is equal in distribution to

$$(\hat{S}^{n}(j)\,{:}\,0\leq j \leq \tau^n(m))\textrm{ under }{\mathbb P}_{n}^\#.$$

Combining this with the distribution of $I^{n}_{\infty}$ , we find that for $G_n$ a random variable with the geometric distribution with success probability $\exp\!({-}\,\xi_n)$ independent of $S^n$ , $(S^n(j)\,{:}\,0\leq j \leq \rho^{n})$ under ${\mathbb P}_{n}$ has the same distribution as $(\hat{S}^{n}(j)\,{:}\,0\leq j \leq \tau^n(G^n))$ under ${\mathbb P}_{n}^\#$ .

By Theorem 3, $\hat{S}^n$ is the Łukasiewicz path of a subcritical Galton–Watson forest. We let $\hat{H}^n$ be its height process as defined in (1), i.e.

$$\hat{H}^n(i)=\#\left\{j<i\,{:}\,\hat{S}^n(j)=\min\{\hat{S}^n(k)\,{:}\,j\leq k \leq i\} \right\}. $$

Then Lemma 1 has the following corollary.

Corollary 1. We have that

$$(S^n(m),H^n(m),m\leq \rho^n)\overset{d}{=}(\hat{S}^n(m),\hat{H}^n(m),m\leq \tau^n(G^n)).$$

We will now characterize the post-infimum process and its height process. Let ${\mathbb P}_n^\uparrow$ be the law of $(S^n(k),k\geq 0)$ conditioned to be non-negative for all k. Note that we are conditioning on an event of non-zero probability because $S^n$ is supercritical, so this is well-defined.

The following lemma suggests an adaptation of $S^n$ that has the same height process, and that turns out to be easier to work with.

Proof. Let $H^*$ denote the height process of $X-\underline{\underline{X}}$ . Then, by definition,

$$H^*_n=\#\left\{m< n \,{:}\,X_m-\underline{\underline{X}}_m=\min\{X_j-\underline{\underline{X}}_j , m\leq j \leq n \}\right\}.$$

Set $\underline{\underline{g}}(n)=\max\{ m < n \,{:}\,X_m=\underline{\underline{X}}_m\}$ . Then, for $m\leq \underline{\underline{g}}(n)$ , we have that $\underline{\underline{X}}_m=\min\{ X_j\,{:}\,m\leq j \leq n\}$ and $\min\{X_j-\underline{\underline{X}}_j , m\leq j \leq n \}=0$ . Also, for $m>\underline{\underline{g}}(n)$ , we have $\underline{\underline{X}}_m=\underline{\underline{X}}_n.$ Hence, indeed,

\begin{align*} H^*_n=&\#\left\{m\leq \underline{\underline{g}}(n)\,{:}\, X_m-\underline{\underline{X}}_m=\min\{X_j-\underline{\underline{X}}_j , m\leq j \leq n \}\right\}\\ &+\#\left\{\underline{\underline{g}}(n)<m<n\,{:}\,X_m-\underline{\underline{X}}_m=\min\{X_j-\underline{\underline{X}}_j , m\leq j \leq n \}\right\} \end{align*}

\begin{align*} =&\#\left\{m\leq \underline{\underline{g}}(n) \,{:}\, X_m-\min\{ X_j\,{:}\, m\leq j \leq n\}=0\right\}\\&+\#\left\{\underline{\underline{g}}(n)<m<n \,{:}\, X_m-\underline{\underline{X}}_n=\min\{X_j-\underline{\underline{X}}_n , m\leq j \leq n \}\right\}\\ =&\#\left\{m\leq n \,{:}\, X_m=\min\{X_j , m\leq j \leq n \}\right\} = H_n,\end{align*}

where $H_n$ is the height process of $X_n$ .

Note that since $S^n$ drifts to $+\infty$ almost surely, $S^n-\underline{\underline{S}}^{n}$ under ${\mathbb P}^\uparrow_{n}$ is recurrent in zero. Indeed, if $S^n$ drifts to $+\infty$ , we have $\underline{\underline{S}}^{n}(k)>-\infty$ for all $k>0$ . Then, by the definition of $\underline{\underline{S}}^{n}$ , there is a finite $l>k$ such that $S^n(l)=\underline{\underline{S}}^{n}(l)$ . Moreover, $S^n-\underline{\underline{S}}^{n}$ is non-negative. Combining these facts implies that $S^n-\underline{\underline{S}}^{n}$ is the Łukasiewicz path of an infinite spine with finite trees attached to it only on the left-hand side. $S^n$ describes the same infinite spine with trees on its left-hand side, but also contains information on the total number of children (including children to the right of the spine) of vertices on the spine. See Figure 1. We will use this description to study the distributions of these paths.

Figure 1. The information captured in $S^n$ and $S^n-\underline{\underline{S}}^{n}$ under $\mathbb{P}_n^\uparrow$ .

Using terminology introduced by Janson in [Reference Janson36], we will call a vertex immortal if it is the root of an infinite tree. Otherwise we will call it mortal. Note that since $S^n$ drifts to $+\infty$ almost surely, every vertex has a positive probability of being immortal. Hence there are almost surely countably infinitely many infinite spines. Note that in every generation, we only visit the vertices to the left of the leftmost immortal vertex. This means that, under ${\mathbb P}^\uparrow_{n}$ , an excursion of $S^n-\underline{\underline{S}}^{n}$ above 0 consists of an increment (of size, say, $B^{(n)}$ ) corresponding to the mortal older brothers of the first immortal vertex, and then an excursion with the law of $S^n$ starting at $B^{(n)}$ conditioned to hit 0 in finite time. This corresponds to sampling first the number of trees to the left of the infinite spine and then the shapes of those trees.

The sample paths of $S^n$ can be constructed from the sample paths of $S^n-\underline{\underline{S}}^{n}$ , by adding the randomness that encodes the number of younger brothers of vertices on the leftmost path to infinity. This corresponds to replacing the jumps of size $B^{(n)}$ by jumps of size $N^{(n)}-1$ , with $N^{(n)}$ being distributed as the total generation size given $B^{(n)}$ . This is illustrated in Figure 1. (A similar decomposition of Galton–Watson trees conditioned on non-extinction is discussed by Lyons and Peres in Section 5.7 of [Reference Lyons and Peres44], where they do not introduce the encoding by a Łukasiewicz path and height process.)

Note that the height at time k in the tree encoded by $S^n$ under ${\mathbb P}^\uparrow_n$ is given by the height of the position along the infinite spine where the finite subtree containing the vertex visited at time k is attached plus the height of the vertex in the finite subtree.

We will now investigate the joint distribution of $N^{(n)}$ and $B^{(n)}$ .

Lemma 3. Let $N^{(n)}$ be the number of children in a set of offspring conditioned to contain at least one immortal vertex, let $B^{(n)}$ be the number of older brothers of the oldest immortal vertex in such a set of offspring, and let $\exp\!({-}\,\xi_n)$ be the probability that, under ${\mathbb P}_{n}$ , a tree dies out. Then for $l>k$ we have

(4) \begin{equation} {\mathbb P}(B^{(n)}=k,N^{(n)}=l)=\exp\!({-}\,k\xi_n){\mathbb P}(S_1^n=l-1).\end{equation}

Proof. Let $\psi_n$ denote the probability generating function of the offspring distribution under ${\mathbb P}_{n}$ (i.e. the law of $Z^n_1$ under ${\mathbb P}_{n}$ ). Let $M_n$ be the random variable representing the number of mortal children of an immortal parent, and let $J_n$ be the random variable representing the number of immortal children of an immortal parent. Then, in [Reference Janson36], Janson gives the generating function of the joint law of $M_n$ and $J_n$ as

(5) \begin{equation} {\mathbb E}\left[x^{M_n} y^{J_n}\right]=\frac{\psi_n\left(\!\exp\!({-}\,\xi_n) x+(1-\exp\!({-}\,\xi_n))y\right)-\psi_n\left(\!\exp\!({-}\,\xi_n) x\right)}{1-\exp\!({-}\,\xi_n)}.\end{equation}

Also, given the number of mortal and immortal children, they appear in a uniformly random order. It is then straightforward that the generating function of the total number of children of an immortal parent is given by

$${\mathbb E}\left[x^{N^{(n)}}\right]=\frac{\psi_n(x)-\psi_n(\!\exp\!({-}\,\xi_n) x)}{1-\exp\!({-}\,\xi_n)},$$

so we obtain that for $k=1,2,\dots$ ,

\begin{equation*}{\mathbb P}(N^{(n)}=k)=\frac{1-\exp\!({-}\,k\xi_n)}{1-\exp\!({-}\,\xi_n)}{\mathbb P}(S_1^n=k-1). \end{equation*}

Using (5) to analyze the generating function of the joint law of $N^{(n)}$ and $J_n$ , we see that, conditional on the value of $N^{(n)}$ , $J_n$ is distributed as a binomial random variable with parameters $N^{(n)}$ and $1-\exp\!({-}\,\xi_n)$ conditioned to be at least 1. Since the mortal and immortal children appear in a uniform order, conditional on the generation size $N^{(n)}$ , the number of mortal older brothers of the first immortal vertex, $B^{(n)}$ , is distributed as a geometric random variable with success parameter $\exp\!({-}\,\xi_n)$ conditioned to be at most $N^{(n)}-1$ . We obtain that

$${\mathbb P}(B^{(n)}=k|N^{(n)}=l)=\frac{\exp\!({-}\,k\xi_n)(1-\exp\!({-}\,\xi_n))}{1-\exp\!({-}\,l\xi_n)},$$

which proves the statement.

These findings on the post-infimum process can be summarized as follows.

Proposition 1. The sample paths of $S^n$ and $S^n-\underline{\underline{S}}^n$ under ${\mathbb P}^\uparrow_n$ can be constructed by concatenating excursions above the future infimum as follows. Sample a countably infinite number of independent copies of $B^{(n)}$ and $N^{(n)}$ according to

$${\mathbb P}(B^{(n)}=k,N^{(n)}=l)=\begin{cases}\exp\!({-}\,k\xi_n){\mathbb P}(S_1^n=l-1), &0\leq k<l, \\ 0 &\textrm{otherwise.}\end{cases}$$

Start an excursion with an increment of size $N^{(n)}-1$ above the previous future infimum, and continue from there as a process with law ${\mathbb P}^\#_n$ . Kill the excursion at $N^{(n)}-B^{(n)}-1$ above the previous future infimum, which will be the new future infimum.

By replacing the jumps of size $N^{(n)}-1$ by jumps of size $B^{(n)}$ we obtain $S^n-\underline{\underline{S}}^{n}$ .

2.1.1. A pathwise construction.

The characterization of the pre- and post-infimum processes justifies the following pathwise construction of $(S^n,S^n-\underline{\underline{S}}^n, H^n)$ under ${\mathbb P}_n$ . This is similar to the pathwise construction of the encoding processes of a Galton–Watson process with immigration given in Section 2.2 of [Reference Duquesne22].

See Figure 2 for a graphical representation of the construction.

Figure 2. The information captured in $S^n$ and $S^n-\underline{\underline{S}}^{n}$ under $\mathbb{P}_n$ . The finite trees are encoded by the pre-infimum process, and the infinite spine and its pendant subtrees are encoded by the post-infimum process.

• Let $\hat{S}^n(k)$ be a random walk with law ${\mathbb P}^\#_n$ , and let $\hat{H}^n(k)$ be the corresponding height process. Set $\hat{I}^n(k)=\min_{m\leq k} \hat{S}^n(k)$ . The trees encoded by $\hat{S}^n(k)$ will be used as the finite trees that are explored before the first infinite tree is encountered, and as the finite subtrees to the left of the leftmost infinite spine which are rooted at the vertices on the infinite spine.

• Let $G^n$ be distributed as a geometric random variable with mean $\exp\!({-}\,\xi_n)$ , which gives the number of finite trees explored by the process. We then let $(B^{(n)}_1,N^{(n)}_1)$ , $(B_2^{(n)},N^{(n)}_2),\dots$ be i.i.d. pairs of random variables, independent of $\hat{S}^n$ , distributed as $(B^{(n)},N^{(n)})$ . Set $Q^n(0)=G_n$ , $Q^n(k)=G_n+\sum\limits_{i=1}^k N_i^{(n)}$ for $k\geq 1$ . Then $N^{(n)}_m$ will be the number of children attached to the mth vertex on the infinite spine. Also, set $R^n(0)=G_n$ and $R^n(k)=G_n+\sum\limits_{i=1}^k B_i^{(n)}$ for $k\geq 1$ . Then $B^{(n)}_m$ will be the number of finite subtrees to the left of the infinite spine rooted at the mth vertex on the infinite spine.

• Define $F^{n}(k)=\inf\{l\leq k-1:-\hat{I}^n(k-1-l)<R^n(l)\}.$ Then $F^{n}(k)$ will be the number of vertices located on the leftmost infinite spine among the first k vertices that we visit in our depth-first exploration.

• Then define

  $$S^n(k)=-G^{(n)}+\hat{S}^n\left(k-{F}^{(n)}(k)\right)+{Q}^n\left({F}^{(n)}(k)\right)-F^{n}(k),$$
  so that
  $$(S^n-\underline{\underline{S}}^{n})(k)= \hat{S}^n(k-{F}^{(n)}(k))+{R}^n\left({F}^{(n)}(k)\right)$$
  and
  (6)

  

By Corollary 1, Proposition 1, and Lemma 2, the constructed process has the desired distribution.

In the next section, we will introduce the continuous counterpart of this construction. In Section 3, we will use Equation (6) to show the joint convergence under rescaling of the discrete processes to the continuous processes.

2.2. The height process of a supercritical Lévy process

Just as in the discrete case, we will obtain a pathwise construction of a supercritical spectrally positive Lévy process and its height process by considering its pre- and post-infimum processes separately. As before, let L be such a Lévy process with Laplace exponent $\phi$ . Define $I_\infty=\inf\{L_t,t\geq 0\}$ , and set $\rho=\sup\{t>0\,{:}\,L_{t}=I_\infty\}$ . The process on $[0,\rho]$ will be referred to as the pre-infimum process, and the process on $[\rho,\infty)$ (shifted up by $-I_\infty$ so that it starts at 0) will be referred to as the post-infimum process. Informally, as in the discrete case, the pre-infimum process encodes the ${\mathbb R}$ -trees without a path of infinite length, and the post-infimum process encodes the metric structure to the left of the leftmost path of infinite length in the first ${\mathbb R}$ -tree that contains such a path.

(In [Reference Bertoin9], Bertoin uses a different strategy, and splits the process at the infimum attained on a compact time interval. We will not use this approach, and so we will not discuss his results here.)

On an infinite time horizon, the following results on spectrally positive Lévy processes are available:

1. 1. By Théorème 2 in Bertoin [Reference Bertoin8], the pre-infimum process of a spectrally positive Lévy process that drifts to $+\infty$ has the same law as the process ‘conditioned to drift to $-\infty$ ’ stopped at an independent exponential level. Informally, this result says that the pre-infimum process encodes ${\mathbb R}$ -trees conditioned to die out.

2. 2. By Lemma 4.1 in Millar [Reference Millar45] (which is rephrased as ‘Théorème (Millar)’ in [Reference Bertoin8]; we will use the latter version of the result), the post-infimum process of a spectrally positive Lévy process that drifts to $+\infty$ has the same law as the process conditioned to stay positive, and is independent of the pre-infimum process. Informally, this result says that the post-infimum process encodes (part of) a single ${\mathbb R}$ -tree conditioned to be infinite.

3. 3. In [Reference Lambert41], Lambert describes the height process corresponding to a class of spectrally positive Lévy processes conditioned to stay positive.

We start with an overview of the results by Bertoin [Reference Bertoin8] that we use. Firstly, since L is supercritical, there is a unique $\xi>0$ such that $\phi(\xi)=0$ . Let ${\mathbb P}$ be the law of L, and set

$${\mathcal F}_t\,{:\!=}\,\sigma(L_s, s\leq t).$$

Then $(\!\exp\!({-}\,\xi L_t),t\geq 0)$ is a ${\mathbb P}$ -martingale. Let ${\mathbb P}^\#$ be locally absolutely continuous with respect to ${\mathbb P}$ , with

(7) \begin{equation}\textrm{d}{\mathbb P}^\#|_{{\mathcal F}_t}=\exp\!({-}\,\xi L_t)\textrm{d}{\mathbb P}|_{{\mathcal F}_t}.\end{equation}

Let $\hat{L}$ be a process which under ${\mathbb P}$ has the law of L under ${\mathbb P}^\#$ . The following analogue of Theorem 3 is a straightforward consequence of the fact that $\exp\!({-}\,\xi L_t)$ is a martingale. See [Reference Bertoin10, Chapter VII].

The following theorem is then proved in [Reference Bertoin8] as Théorème 2.

These observations, together with Proposition 1.4.3 of Duquesne and Le Gall [Reference Duquesne and Le Gall24], imply the following proposition.

Proposition 2. There exists a continuous modification of the height process corresponding to $\hat{L}$ , which we will denote by $\hat{H}$ . Moreover, for

$$\rho=\sup\{t\,{:}\,L_t=\inf_{s\geq 0} L_s\},$$

we have $\rho<\infty$ almost surely; furthermore, there exists a continuous modification of the height process of L up to $\rho$ , which we will refer to as H, and for E an exponential random variable with rate $\xi$ ,

$$(L_t,H_t,t\leq \rho)\overset{d}{=}(\hat{L}_t,\hat{H}_t, t\leq \tau(E)).$$

We will now focus on the post-infimum process. We will use two important results from the literature. Firstly, by Lemma 4.1 in Millar [Reference Millar45], the post-infimum process $(L_{\rho+t}-I_{\infty}, t\geq 0)$ has the same law as L conditioned to stay positive and is independent of the pre-infimum process. Call the law of L conditioned to stay positive ${\mathbb P}^\uparrow$ . For the definition of this process, see [Reference Bertoin8, Reference Chaumont15, Reference Chaumont16]. The height process of L under ${\mathbb P}^\uparrow$ is characterized by Lambert in [Reference Lambert41], and is obtained via the continuous counterpart of the construction in Section 2.1. Indeed, in [Reference Lambert41], Lambert defines

$$\underline{\underline{L}}_t\,{:\!=}\,\inf\{L_s,s\geq t\},$$

and in [Reference Lambert41, Lemma 8ii] he shows that the height processes of $L-\underline{\underline{L}}$ and L are equal. Then, in [Reference Lambert41, Theorem 3], he gives a pathwise construction of $L-\underline{\underline{L}}$ and its local time at 0 under ${\mathbb P}^\uparrow$ , by viewing $L-\underline{\underline{L}}$ as a continuous-time branching process with immigration, an object introduced in [Reference Kawazu and Watanabe40]. Lemma 8ii of [Reference Lambert41] also illustrates how to construct the height process corresponding to $L-\underline{\underline{L}}$ . Translating these results to our setting yields the following proposition, which is the continuous counterpart of the pathwise construction of $S^n-\underline{\underline{S}}^n$ under ${\mathbb P}_n^\uparrow$ described in Proposition 1.

Theorem 6. ([Reference Lambert41, Theorem 3, Lemma 8ii].) Recall the definition of ${\mathbb P}^\#$ from (7). Let $\hat{L}$ be a process which under ${\mathbb P}$ has the law of L under ${\mathbb P}^\#$ , and let $\phi^\#$ be its Laplace exponent. Let $\widetilde{R}$ be a subordinator with Laplace exponent $\frac{\phi(\cdot)}{\cdot+\xi}$ independent of $\hat{L}$ . Then $\widetilde{R}$ is a subordinator. For $t\geq 0$ , define the right inverse of $\widetilde{R}$ by

$$\widetilde{R}^{-1}_t\,{:\!=}\,\inf\{s\,{:}\,\widetilde{R}_s>t\}$$

and set

$$\hat{I}_t=\inf_{s\leq t}\hat{L}_s.$$

Then, for $L^*$ defined by

$$L^*_t=\hat{L}_t+\widetilde{R}\left(\widetilde{R}^{-1}({-}\,\hat{I}_t)\right),$$

$L-\underline{\underline{L}}$ under ${\mathbb P}^\uparrow$ has the same law as $L^*$ . Moreover, the local time of $L^*$ at 0 may be defined by

$$\ell^*_t=\widetilde{R}^{-1}({-}\,\hat{I}_t).$$

Finally, suppose that $\hat{H}$ is a continuous modification of the height process corresponding to $\hat{L}$ . Then

$$H^*_t\,{:\!=}\,\ell^*_t+\hat{H}_t$$

is a continuous modification of the height process corresponding to $L^*_t$ .

Combining the above proposition with the characterization of the pre-infimum process in Proposition 2, we obtain the following result.

Proposition 3. Let $\hat{L}$ be a process which under ${\mathbb P}$ has the law of L under ${\mathbb P}^\#$ , satisfying Condition (C3), so that its height process is well-defined and has a continuous modification $\hat{H}$ . Define $\hat{I}_t=\inf\{\hat{L}_s\,{:}\,s\leq t\}$ . Let E be an exponential random variable with rate $\xi$ . Let $\widetilde{D}$ be distributed as in the statement of Theorem 6, independent of $\hat{L}$ and E, and set

$$R_t=\widetilde{R}_t+E$$

and

$${R}^{-1}_t\,{:\!=}\,\inf\{s\,{:}\,{R}_s>t\}.$$

Then the height process of L is well-defined and has a continuous modification H. Moreover, H is also the height process of $L-\underline{\underline{L}}$ , and

$$(L_t-\underline{\underline{L}}_t, H_t,t\geq 0)\overset{d}{=}(\hat{L}_t+R(R^{-1}({-}\,\hat{I}_t)), \hat{H}_t+{R}^{-1}({-}\,\hat{I}_t), t\geq 0).$$

Proof. The existence of $\hat{H}$ follows from Proposition 2. The construction of $L-\underline{\underline{L}}$ follows from Proposition 2 for the pre-infimum process, and from [Reference Millar45, Lemma 4.1] and Theorem 6 for the post-infimum process.

We claim that $(\hat{H}_t+{R}^{-1}({-}\,\hat{I}_t),t\geq 0)$ is the height process corresponding to the process

$$(\hat{L}_t+R(R^{-1}({-}\,\hat{I}_t)),t\geq 0).$$

Firstly, note that for $\rho^*=\sup\{t\,{:}\,\hat{I}_t=-E\}$ ,

$$(\hat{L}_t+R(R^{-1}({-}\,\hat{I}_t)) , \hat{H}_t+{R}^{-1}({-}\,\hat{I}_t), t\in [0,\rho^*])=(\hat{L}_t+E, \hat{H}_t, t\in [0,\rho^*]).$$

The height process is, by definition, not affected by translating the Łukasiewicz path by a constant, so on $[0,\rho^*]$ , the claim follows. On $[\rho^*,\infty)$ , the claim follows from Theorem 6.

Finally, we claim that the height processes of L and $L-\underline{\underline{L}}$ agree. Set $I_\infty=\inf\{L_t,t\geq 0\}$ and $\rho=\sup\{t>0\,{:}\,L_{t}=I_\infty\}$ , and observe that

$$(L_t-\underline{\underline{L}}_t , t\in [0,\rho])=(L_t-I_\infty, t\in [0,\rho]).$$

Again because the height process is not affected by adding a constant to the Łukasiewicz path, the claim follows on $[0,\rho]$ . On $[\rho,\infty)$ , the claim follows from Lemma 8ii in [Reference Lambert41].

4.1. Model and result

We use the configuration model to construct a uniform graph with a random degree sequence. The model we consider is as follows.

Fix $\lambda\in{\mathbb R}$ . Most quantities that will be defined depend on $\lambda$ . To avoid overcomplicating the notation, this will not be made explicit unless necessary to avoid confusion. For each $n \in {\mathbb N}$ , let $D_1^{n}, D_2^{n}, \dots, D_n^{n} \geq 1 $ be an i.i.d. degree sequence satisfying the following properties (which are labeled by ‘CM’ for ‘configuration model’):

1. (CM1) For $\mu_n \,{:\!=}\, {\mathbb E}[D_1^{n}] $ , we have $\mu_n\rightarrow \mu$ as $n \rightarrow \infty$ , with $\mu$ not depending on $\lambda$ .

2. (CM2) We have ${\mathbb E} \left[\left(D_1^{n}\right)^2\right]=\left(2+\lambda n ^{-\tfrac{\alpha-1}{\alpha+1}}\right) {\mathbb E}[D_1^{n}]$ for some $\alpha \in (1,2)$ .

3. (CM3) We have ${\mathbb P} (D_1^{n}=k)\sim c_n k^{-(\alpha+2)}$ as $k\rightarrow \infty$ , with $c_n>0$ for all n and $c_n\to c$ as $n\to \infty$ .

4. (CM4) For $Z^n$ a random variable such that ${\mathbb P}(Z^n=k)=k{\mathbb P}(D_1^{n}=k)/\mu_n$ , and $(S^n(k),k\geq 0)$ a random walk with steps distributed as $Z^n-2$ , let $Y^n_m$ be the number of vertices at height m in the first $\lfloor n^{1/(\alpha+1)}\rfloor$ trees of the forest encoded by $(S^n(k),k\geq 0)$ . Then, for every $\delta>0$ ,

   \begin{equation*}\liminf\limits_{n\to\infty}{\mathbb P}\left[Y^n_{\lfloor \delta n^{\tfrac{\alpha-1}{\alpha+1}}\rfloor}=0\right]>0.\end{equation*}

Let $G_1^n, G_2^n,\dots$ be the components of a uniformly random graph with i.i.d. degrees that are distributed as $D_1^n$ , with the components listed in decreasing order of size. View each $G_i^n$ as a compact measured metric spaces by equipping it with the graph distance $d_i^n$ , and the counting measure on its vertices, $\mu_i^n$ . More generally, a compact measured metric space will be denoted by a triple $(G,d,\mu)$ , for (G, d) a compact metric space and $\mu$ a finite Borel measure on (G, d). Formally, each $G_i^n$ is an element of the Polish space of isometry-equivalence classes of measured metric spaces, endowed with the Gromov–Hausdorff–Prokhorov distance. For a discussion of the topology, we refer the reader to [Reference Addario-Berry, Broutin, Goldschmidt and Miermont3, Section 2]. We will prove the following theorem.

Theorem 9. There exists a sequence of random compact measured metric spaces

$$((\mathcal{G}_1,d_1,\mu_1),(\mathcal{G}_2,d_2,\mu_2),\dots)$$

such that, as $n\to \infty$ ,

$$\left(\left(G_i^n,n^{-\tfrac{\alpha-1}{\alpha+1}}d_i^n, n^{-\alpha/(\alpha-1)}\mu_i^n\right)\!,i\geq 1\right)\overset{d}{\to}((\mathcal{G}_i,d_i,\mu_i),i\geq 1)$$

in the sense of the product Gromov–Hausdorff–Prokhorov topology.

In the case where $\lambda=0$ , and the degree distribution does not depend on n, this result was already known from [Reference Conchon-Kerjan and Goldschmidt17, Theorem 1.1]; this is known as the critical case. Intuitively, criticality entails that for large n, and for $(V_n,W_n)$ an edge chosen uniformly at random from the graph, the expected degree of $V_n$ is roughly 2. Our contribution is then to prove the theorem for all $\lambda\in{\mathbb R}$ and for degree distributions depending on n. This is known as the critical window, in which, for large n, and for $(V_n,W_n)$ an edge chosen uniformly at random from the graph, the expected degree of $V_n$ is roughly $2+ \lambda n ^{-\tfrac{\alpha-1}{\alpha+1}}$ .

4.2. Related work

Most results on the configuration model have been obtained for models with a deterministic degree sequence. The phase transition for the undirected setting was shown in [Reference Janson and Łuczak37, Reference Molloy and Reed46, Reference Molloy and Reed47]. The limiting law of the rescaled component sizes at criticality and in the critical window were obtained by Riordan [Reference Riordan49] under the assumption that the degrees are bounded. Dhara, van der Hofstad, van Leeuwaarden, and Sen showed convergence under rescaling of the component sizes and surpluses in the critical window in the finite-third-moment setting [Reference Dhara, van der Hofstad, van Leeuwaarden and Sen19] and in the heavy-tailed regime [Reference Dhara, van der Hofstad, van Leeuwaarden and Sen20]. Bhamidi, Dhara, van der Hofstad, and Sen obtained metric space convergence in the critical window in [Reference Bhamidi, Dhara, van der Hofstad and Sen11], a result that the authors later improved to a stronger topology in [Reference Bhamidi, Dhara, van der Hofstad and Sen12]; both of these results also hold conditional on the constructed multigraph being simple. We will discuss the results of Bhamidi, Dhara, van der Hofstad, and Sen further at the end of this section.

Configuration models with a random degree sequence are considered in [Reference Conchon-Kerjan and Goldschmidt17, Reference Joseph38]. Joseph [Reference Joseph38] showed convergence of the component sizes and surpluses of the large components under rescaling at criticality, both for degree distributions with finite third moments and for the heavy-tailed regime. Conchon-Kerjan and Goldschmidt [Reference Conchon-Kerjan and Goldschmidt17] showed product Gromov–Hausdorff–Prokhorov convergence of the large components at criticality in these two regimes, and showed that the results also hold conditioned on the resulting graph being simple.

In recent work [Reference Donderwinkel and Xie21], the author and Xie showed metric space convergence under rescaling of the strongly connected components of the directed configuration model at criticality.

4.2.1. Results in [Reference Bhamidi, Dhara, van der Hofstad and Sen11, Reference Bhamidi, Dhara, van der Hofstad and Sen12].

We now describe the model that is considered in [Reference Bhamidi, Dhara, van der Hofstad and Sen11, Reference Bhamidi, Dhara, van der Hofstad and Sen12] to provide a comparison with our result. Our description of the results in [Reference Bhamidi, Dhara, van der Hofstad and Sen11] closely follows the presentation in [Reference Conchon-Kerjan and Goldschmidt17].

Let $\mathfrak{d}_1^n\geq \cdots\geq \mathfrak{d}_n^n$ be a family of deterministic degree sequences, such that $\sum_{i=1}^n\mathfrak{d}_i^n$ is even, and if $\mathfrak{D}_n$ denotes the degree of a vertex chosen uniformly at random, the following conditions hold (they are labeled by ‘DD’ for ‘deterministic degrees’):

1. (DD1) We have $n^{-1/(\alpha+1)}\mathfrak{d}_i^n\to \mathfrak{\vartheta}_i$ as $n\to\infty$ for each $i\geq 1$ , where $\vartheta_1\geq \vartheta_2\geq \cdots \geq 0 $ is such that $\sum_{i\geq 1} \vartheta^3<\infty$ , but $\sum_{i\geq 1}\vartheta_i^2=\infty$ .

2. (DD2) We have $\mathfrak{D}_n\overset{d}{\to}\mathfrak{D}$ , along with the convergence of its first two moments, for some random variable $\mathfrak{D}$ with ${\mathbb P}(\mathfrak{D}=1)>0$ , ${\mathbb E}[\mathfrak{D}]=\mu$ , and ${\mathbb E}[\mathfrak{D}(\mathfrak{D}-1)]/{\mathbb E}[\mathfrak{D}]=\theta>1$ , and

   $$\lim_{K\to\infty}\limsup_{n\to\infty}n^{-3/(\alpha+1)} \sum_{i\geq K+1}\left(\mathfrak{d}_i^n\right)^3=0.$$

Let $\theta_n={\mathbb E}\left[\mathfrak{D}_n(\mathfrak{D}_n-1)\right]/{\mathbb E}\left[\mathfrak{D}_n\right]$ . The authors of [Reference Bhamidi, Dhara, van der Hofstad and Sen11, Reference Bhamidi, Dhara, van der Hofstad and Sen12] sample a uniform graph with this degree sequence and perform percolation at parameter

$$p_n(\lambda)=\frac{1}{\theta_n}+\tilde{\lambda} n^{-\tfrac{\alpha-1}{\alpha+1}},$$

for some $\tilde{\lambda}\in {\mathbb R}$ , which yields a graph in the critical window. Call the resulting degree sequence $(\mathfrak{D}_1^{n,\tilde{\lambda}}, \dots ,\mathfrak{D}_n^{n,\tilde{\lambda}})$ . In this setting, [Reference Bhamidi, Dhara, van der Hofstad and Sen11, Theorem 2.2] is the precise analogue of our Theorem 9 in the Gromov-weak topology. In [Reference Bhamidi, Dhara, van der Hofstad and Sen12], this result is strengthened to convergence in the Gromov–Hausdorff–Prokhorov topology, under the following additional assumptions:

1. (DD3) For $\mathfrak{D}^*_n$ the degree of a size-biased pick from $\mathfrak{d}_1^n,\dots,\mathfrak{d}_n^n$ , there exists $c_0>0$ such that for all $1\leq l\leq \mathfrak{d}^n_1$ and $n\geq 1$ ,

   $${\mathbb P}\left(\mathfrak{D}^*_n\geq l\right)\geq \frac{c_0}{l^\alpha}.$$

2. (DD4) For $\beta_i^n=n^{-2/(\alpha+1)}\sum_{j=1}^{i-1}\left(\mathfrak{d}^n_j\right)^2$ , there exists a sequence $(k_n)_{n\geq 1}$ with $k_n\to \infty$ and $k_n=o\left(n^{1/(\alpha+1)}\right)$ such that $\beta_{k_n}^n/\log n\to \infty$ as $n\to\infty$ .

3. (DD5) For all $\epsilon>0$ ,

   $$\lim_{k\to \infty}\limsup_{n\to\infty} n^{-1/(\alpha+1)}\sum_{i\in (k,kn)}e^{-\epsilon \beta_i^n}\mathfrak{d}^n_i=0.$$

These extra assumptions allow the authors of [Reference Bhamidi, Dhara, van der Hofstad and Sen12] to show that the components in their graph model satisfy the global lower mass-bound property [Reference Bhamidi, Dhara, van der Hofstad and Sen12, Theorems 1.2 and 1.3], which allows them to extend the results in [Reference Bhamidi, Dhara, van der Hofstad and Sen11] to convergence in the Gromov–Hausdorff–Prokhorov topology using results from [Reference Athreya, Löhr and Winter7].

The limit object in [Reference Bhamidi, Dhara, van der Hofstad and Sen11, Reference Bhamidi, Dhara, van der Hofstad and Sen12] is constructed by making vertex identifications in tilted inhomogeneous continuum random trees. The scaling limit of the depth-first walk that Bhamidi, Dhara, van der Hofstad, and Sen consider is a thinned Lévy process, whereas we show convergence to a measure-changed Lévy process. The connection between our results and theirs will become clear in the following section.

4.2.2. Relationship to percolation.

We will illustrate that the law of a degree sequence that is obtained by bond percolation on the half-edges of a sequence of vertices with i.i.d. degrees in the supercritical regime satisfies the conditions of Theorem 9. This is approximately the degree distribution after bond percolation on the edges of a uniform random graph with such a supercritical random degree sequence, although we ignore dependence between the degrees of different vertices. Using results of Janson [Reference Janson34], such mild dependence can be shown to have a negligible effect on the properties of the graph, but we omit the straightforward details. We also show how our results are related to the results in [Reference Bhamidi, Dhara, van der Hofstad and Sen11, Reference Bhamidi, Dhara, van der Hofstad and Sen12].

Let D be a random variable in ${\mathbb N}$ that satisfies ${\mathbb E} [D]=\mu$ , ${\mathbb E} [D^2]=\rho>2\mu$ and ${\mathbb P} (D=k)\sim ck^{-(\alpha+2)}$ for $\alpha \in (1,2)$ as $k\rightarrow \infty$ . Define $C_{\alpha}=\frac{c\Gamma(2-\alpha)}{\alpha(\alpha-1)}$ , so that the Laplace transform ${\mathcal L}_D$ of D satisfies

$${\mathcal L}_D(\theta)={\mathbb E} [\,\!\exp ({-}\,\theta D)]=1-\mu \theta +\frac{\rho}{2}\theta^2-\frac{C_{\alpha}}{\alpha+1} \theta^{\alpha+1}+o(\theta^{\alpha+1})$$

for $\theta \rightarrow 0$ . View D as the degree distribution. Then, keep every half-edge with probability

$$p(\lambda,n)=\frac{1+\lambda n^{-\tfrac{\alpha-1}{\alpha+1}}}{\frac{\rho}{\mu}-1},$$

and call the resulting degree distribution $B^{(\lambda,n)}$ .

In the next paragraph, we will show that the conditions of Theorem 9 are satisfied for $B^{(\lambda,n)}$ . Then, for $\mathfrak{d}_1^n,\dots,\mathfrak{d}_n^n$ denoting a sample of i.i.d. random variables $\mathfrak{D}_1,\dots, \mathfrak{D}_n$ with the same distribution as D, Conditions (DD1) and (DD2) are satisfied almost surely for some sequence of random variables $\mathfrak{\vartheta}_i$ [Reference Dhara, van der Hofstad, van Leeuwaarden and Sen20, Section 2.2]. Moreover, the order statistics of the percolated degree sequence with $\tilde{\lambda}=\lambda/(\rho/\mu-1)$ closely resemble an ordered sample of i.i.d. random variables distributed as $B^{(\lambda,n)}$ . Therefore, it should be the case that, in this particular set-up, the limit of Bhamidi, Dhara, van der Hofstad, and Sen corresponds to the limit in Theorem 9.

We will now verify the conditions of Theorem 9 for $B^{(\lambda,n)}$ . Note that the Laplace transform ${\mathcal L}_{B^{(\lambda,n)}}$ of $B^{(\lambda,n)}$ satisfies

(22) \begin{equation} {\mathcal L}_{B^{(\lambda,n)}}(\theta)={\mathbb E} [\,\!\exp\!(D\log ( 1-p(\lambda,n)+p(\lambda,n)e^{-\theta}))],\end{equation}

so that

\begin{align*} {\mathcal L}_{B^{(\lambda,n)}}(\theta) =&1-p(\lambda,n)\mu\theta+p(\lambda,n)\mu\left(2+\lambda n^{-\tfrac{\alpha-1}{\alpha+1}}\right)\frac{\theta^2}{2}-\frac{C_{\alpha}}{\alpha+1}p(\lambda,n)^{\alpha+1}\theta^{\alpha+1}\\ &+o(\theta^{\alpha+1})\end{align*}

as $\theta\to 0$ . This implies that Conditions (CM1), (CM2), and (CM3) are satisfied with $\mu_n=p(\lambda,n)\mu\to\mu/(\frac{\rho}{\mu}-1)$ and $c_n=cp(\lambda,n)^{\alpha+1}\to c/(\frac{\rho}{\mu}-1)^\alpha$ as $n\to \infty$ .

We now check Condition (CM4). We will drop the dependency on $\lambda$ from the notation, unless it is necessary to avoid confusion. Let $\tilde{D}$ be a random variable with the size-biased distribution of D, i.e. for all $k\in {\mathbb N}$ ,

$${\mathbb P}(\tilde{D}=k)=\frac{k{\mathbb P}(D=k)}{{\mathbb E}[D]},$$

and similarly, let $\tilde{B}_n$ be a random variable with the size-biased distribution of ${B}^{(n,\lambda)}$ , i.e.

$${\mathbb P}(\tilde{B}_n=k)=\frac{k{\mathbb P}({B}^{(n,\lambda)}=k)}{{\mathbb E}\left[{B}^{(n,\lambda)}\right]}.$$

Let $g_{\tilde{D}-1}(x)$ and $g_{{\tilde{B}}_n-1}(x)$ be the probability generating functions of $\tilde{D}-1$ and $\tilde{B}_n-1$ respectively. Then (22) implies that

$$g_{{\tilde{B}}_n-1}(x)=g_{\tilde{D}-1}\left(1-p(\lambda,n)+p(\lambda,n)x\right)\!.$$

Letting $g_{{\tilde{B}}_n-1}^{\circ k }$ denote the kth iterate of $g_{{\tilde{B}}_n-1}$ , note that Condition (CM4) is equivalent to

(23) \begin{equation}\liminf_{n\to\infty}\left(g_{{\tilde{B}}_n-1}^{\circ \lfloor \delta n^{\tfrac{\alpha-1}{\alpha+1}}\rfloor }(0)\right)^{\lfloor n^{1/(\alpha+1)}\rfloor}>0.\end{equation}

(See for instance the discussion below Theorem 2.3.1 in [Reference Duquesne and Le Gall24].) As in the proof of [Reference Broutin, Duquesne and Wang14, Proposition 5.25], it is sufficient to show that, for $t\in (0,\infty)$ and $r_n(t)$ the value such that

$$\int_{r_n(t)}^1 \frac{dr}{g_{{\tilde{B}}_n-1}(1-r)-1+r}=n^{\tfrac{\alpha-1}{\alpha+1}}t,$$

we have

$$\limsup_{n\to\infty}n^{1/(\alpha+1)}r_n(t)<\infty.$$

Note that

\begin{align*} \int_{r_n(t)}^1 \frac{dr}{g_{{\tilde{B}}_n-1}(1-r)-1+r}&=\int_{r_n(t)}^1 \frac{dr}{g_{\tilde{D}-1}\left(1-p(\lambda,n)r\right)-1+r}\\ &=\int_{p(\lambda,n)r_n(t)}^{p(\lambda,n)} \frac{ds}{p(\lambda,n)g_{\tilde{D}-1}\left(1-s\right)-p(\lambda,n)+s},\end{align*}

where we change variable to $s=p(\lambda,n)r$ . An elementary calculation yields that

$$g_{\tilde{D}-1}\left(1-s\right)=1-\left(\rho/\mu-1\right)s+\frac{C_\alpha}{\mu}s^\alpha+o(s^\alpha)$$

as $s\to 0$ , which implies that

$$p(\lambda,n)g_{\tilde{D}-1}\left(1-s\right)-p(\lambda,n)+s=-\lambda n^{-\tfrac{\alpha-1}{\alpha+1}}s+p(\lambda,n)\frac{C_\alpha}{\mu}s^\alpha+p(\lambda,n)o(s^\alpha)$$

as $s\to 0$ . Then, setting $v=n^{1/(\alpha+1)}s$ implies that

\begin{align*}&\int_{r_n(t)}^1 \frac{dr}{g_{{\tilde{B}}_n-1}(1-r)-1+r}=n^{\tfrac{\alpha-1}{\alpha+1}}\int_{n^{1/(\alpha+1)}p(\lambda,n)r_n(t)}^{n^{1/(\alpha+1)}p(\lambda,n)} \frac{dv}{-\lambda v +\frac{C_\alpha}{\mu}v^\alpha+o(v^\alpha)},\end{align*}

so that, since $p(n,\lambda)=\Theta(1)$ as $n\to \infty$ , we obtain

$$\limsup_{n\to\infty}n^{1/(\alpha+1)}r_n(t)<\infty$$

as required. This implies that Condition (CM4) is satisfied and so, indeed, performing bond percolation on the half-edges of a sequence of vertices with i.i.d. degrees in the supercritical regime yields a degree distribution that satisfies the conditions of Theorem 9.

4.3. Adapting the methods in [Reference Conchon-Kerjan and Goldschmidt17] to the critical window

The largest barrier to generalizing the methods in [Reference Conchon-Kerjan and Goldschmidt17] is showing the convergence under rescaling of S, jointly with its height process. The results proved in Section 3 allow us to do this. After that, it is trivial to extend most of the arguments in [Reference Conchon-Kerjan and Goldschmidt17] to the critical window.

The convergence under rescaling of S is the content of Proposition 4. We then discuss the results in [Reference Conchon-Kerjan and Goldschmidt17] that are not trivially extended to the critical window and need some further justification.

Let $D_1^n,\dots, D_n^n$ be i.i.d. with a degree distribution as specified in 4.1. Recall that the degree distribution depends on both $\lambda$ and n, but that we have dropped the dependency on $\lambda$ in the notation.

We consider the configuration model executed in depth-first order on vertices with degrees $D_1^n,\dots, D_n^n$ . Let $(\hat{D}_1^n,\dots,\hat{D}_n^n)$ denote the degrees in order of discovery, so that $(\hat{D}_1^n,\dots,\hat{D}_n^n)$ is distributed as a size-biased random ordering of $D_1^n,\dots,D_n^n$ . This is defined as follows.

Let $\Sigma$ be a random permutation of $\{1,\dots,n\}$ such that

$${\mathbb P}(\Sigma=\sigma|D_1^n,\dots,D_n^n)=\frac{D^n_{\sigma(1)}}{\sum_{j=1}^n D^n_{\sigma(j)}}\frac{D^n_{\sigma(2)}}{\sum_{j=2}^n D^n_{\sigma(j)}}\cdots \frac{D^n_{\sigma(n)}}{D^n_{\sigma(n)}}.$$

Then, by Proposition 3.2 of [Reference Conchon-Kerjan and Goldschmidt17],

\begin{align*}&{\mathbb P} (\hat{D}_1^n=k_1, \hat{D}_2^n=k_2,\dots, \hat{D}_n^n=k_n)\\&=n! k_1 {\mathbb P}(D_1^n=k_1) k_2 {\mathbb P}(D_1^n=k_2) \cdots k_n {\mathbb P}(D_1^n=k_n) \prod\limits_{i=1}^n \frac{1}{\sum_{j=i}^n k_j}.\end{align*}

Now let $0\leq m\leq n$ and $k_1,k_2,\dots,k_m\geq 1$ , and define $\Xi^n_{n-m}$ to be a random variable having the same law as $D^n_{m+1}+\cdots+D^n_{n}$ . Then, for

$$\phi_m^n(k_1,k_2,\dots,k_m)\,{:\!=}\,\frac{n!\mu^m}{(n-m)!}{\mathbb E}\left[\prod \limits_{i=1}^m \frac{1}{\sum_{j=i}^m k_j+\Xi^n_{n-m}}\right],$$

Proposition 3.2 in [Reference Conchon-Kerjan and Goldschmidt17] yields that for $Z^n_1,\dots,Z^n_n$ i.i.d. random variables with the size-biased degree of $D_1^n$ , i.e.

$${\mathbb P}(Z^n_1=k)=\frac{k{\mathbb P}(D^n_1=k)}{{\mathbb E}[D^n_1]},$$

for any test-function $g\,{:}\,{\mathbb N}^m\rightarrow {\mathbb R},$ we have

$${\mathbb E}[g(\hat{D}_1^n,\hat{D}_2^n,\dots,\hat{D}_m^n)]={\mathbb E}[\phi_m^n(Z^n_1,Z^n_2,\dots,Z^n_m)g(Z^n_1,Z^n_2,\dots,Z^n_m)];$$

that is, $\phi_m^n$ defines a measure change to get from a vector of size-biased distributed random variables to a vector of size-biased ordered random variables. We note that

$$\tilde{S}^n(k)\,{:\!=}\,\sum_{i=1}^k\left(\hat{D}^n_i-2\right)$$

is the Łukasiewicz path of a forest that is closely related to the depth-first spanning forest of our graph of interest, because it encodes the degrees in order of discovery. Therefore, the existence of the measure change motivates the study of the limit under rescaling of

$${S}^n(k)\,{:\!=}\,\sum_{i=1}^k\left(Z^n_i-2\right)$$

and its corresponding height process. This is the content of the following proposition.

Proposition 4. Let L be a spectrally positive $\alpha$ -stable Lévy process with Lévy measure $\pi(dx)=\frac{c}{\mu}x^{-(\alpha+1)}dx$ . Then, for any $\lambda\in {\mathbb R}$ , there exists a continuous modification of the height process of $(L_t+\lambda t, t\geq 0)$ , which we will denote by $H^\lambda$ . Moreover, for $H^n$ the height process corresponding to $S^n$ , we have that, as $n\rightarrow \infty$ ,

\begin{align*}&\left( n^{-1/(\alpha+1)}S^n\left(\lfloor tn^{{\alpha/(\alpha+1)}}\rfloor\right)\!, n^{-\tfrac{\alpha-1}{\alpha+1}}H^n\left(\lfloor tn^{{\alpha/(\alpha+1)}}\rfloor\right)\!, t\geq 0\right)\\&\quad\xrightarrow{d} (L_t+\lambda t, H^\lambda_t, t\geq 0)\end{align*}

in ${\mathbb D}({\mathbb R}_+,{\mathbb R})^2$ .

We will use Theorem 2 to prove Proposition 4. We will first study the Laplace transform of $Z_1^n$ , which is the content of the following lemma.

Lemma 9. Define ${\mathcal L}_{Z_1^{n}}(s)={\mathbb E}[\,\!\exp\!({-}\,s Z_1^{n})] $ . Then

$${\mathcal L}_{Z_1^{n}}(s)=1-\left(2+\lambda n ^{-\tfrac{\alpha-1}{\alpha+1}}\right)s+\frac{C^{(n)}}{\mu_n}s^{\alpha}+o(s^{\alpha})$$

as $s\to 0$ .

Proof. Note that

(24) \begin{align}\begin{split}{\mathcal L}_{Z_1^{n}}''(s)&={\mathbb E} \left[(Z_1^{n})^2\exp\!({-}\,s Z_1^{n})\right]\\&=\sum\limits_{k=1}^{\infty}\frac{{\mathbb P} (D_1^{n}=k)k^3 e^{-sk}}{\mu_n}\\&=\frac{c_n}{\mu_n}s^{\alpha-2}\Gamma(2-\alpha)+o(s^{\alpha-2})\end{split}\end{align}

for $s\rightarrow 0$ , where the last equality follows from the Euler–Maclaurin formula, using that ${\mathbb P}( Z_1^{n}= k)\sim \frac{c_n}{\mu_n}k^{-(\alpha+1)}$ as $k\rightarrow \infty$ . Then, because ${\mathbb E}[Z_1^{n} ]=2+\lambda n ^{-\tfrac{\alpha-1}{\alpha+1}}$ , integrating twice gives the result.

Proof of Proposition 4. We will first prove that $S^n$ converges in distribution under rescaling. Set

$$M^{n}(k)=\sum\limits_{i=1}^{k}\left(Z_i^{n}-2-\lambda n ^{-\tfrac{\alpha-1}{\alpha+1}}\right)$$

and

$$A^{n}(k)= k\lambda n ^{-\tfrac{\alpha-1}{\alpha+1}},$$

so that

$$S^n(k)=M^{n}(k)+A^{n}(k)$$

is the Doob–Meyer decomposition of $S^n$ . Observe that

$$\left(n^{-1/(\alpha+1)} A^{n}\left(\lfloor tn^{{\alpha/(\alpha+1)}}\rfloor\right)\!,t \geq 0\right) \rightarrow (\lambda t,t \geq 0) $$

in ${\mathbb D}({\mathbb R}_+,{\mathbb R})$ as $n\to\infty$ .

We claim that for every $t\geq 0$ ,

(25) \begin{equation} n^{-1/(\alpha+1)} M^{n}\left(\lfloor tn^{{\alpha/(\alpha+1)}}\rfloor\right)\xrightarrow{d} L_t .\end{equation}

Firstly, observe that for all $\theta>0$ ,

$${\mathbb E}\left[\,\!\exp\!({-}\,\theta L_t)\right]=\exp\!\left(t\frac{c\Gamma(2-\alpha)}{\mu \alpha (\alpha-1)}\theta^\alpha\right).$$

Recall that $C_{\alpha}=\frac{c\Gamma(2-\alpha)}{\alpha(\alpha-1)}$ , and set $C^{(n)}=\frac{c_n\Gamma(2-\alpha)}{\alpha(\alpha-1)}$ , so that $C^{(n)}\to C_\alpha$ as $n\to\infty$ . We will show that for every $\theta>0$ ,

$${\mathbb E}\left[\,\!\exp\!\left({-}\,\theta n^{-1/(\alpha+1)} M^{n}\left(\lfloor tn^{{\alpha/(\alpha+1)}}\rfloor\right) \right)\right] \rightarrow \exp\!\left(t\frac{C_\alpha}{\mu }\theta^\alpha\right)$$

as $n\rightarrow \infty$ , which will prove the claim.

Note that

(26) \begin{align}\begin{split} &{\mathbb E}\left[\,\!\exp\!({-}\,\theta n^{-1/(\alpha+1)} M^{n}(\lfloor tn^{{\alpha/(\alpha+1)}}\rfloor)\right ]\\&=\left[{\mathcal L}_{Z_1^{n}}(\theta n^{-1/(\alpha+1)})\exp\!\left(\theta n^{-1/(\alpha+1)}\left(2+\lambda n ^{-\tfrac{\alpha-1}{\alpha+1}}\right)\right) \right]^{\lfloor tn^{{\alpha/(\alpha+1)}}\rfloor }.\end{split}\end{align}

Then, Lemma 9 implies that

\begin{align*} {\mathcal L}_{Z_1^{n}}(s)=\exp\!\left({-}\,\left(2+\lambda n ^{-\tfrac{\alpha-1}{\alpha+1}}\right)s+\frac{C^{(n)}}{\mu_n}s^{\alpha}+o(s^{\alpha})\right)\end{align*}

as $s\rightarrow 0$ . Plugging this into (26), we find that as $n\rightarrow \infty$ ,

$${\mathbb E}\left[\,\!\exp\!\left({-}\,\theta n^{-1/(\alpha+1)} M^{n}(\lfloor tn^{{\alpha/(\alpha+1)}}\rfloor)\right)\right ]\rightarrow \exp\!\left(t\frac{C_{\alpha}}{\mu }\theta^\alpha\right)\!,$$

which proves (25).

Now, using [Reference Kallenberg39, Theorem 16.14], we may deduce that as $n\rightarrow \infty$ ,

(27) \begin{equation} \left( n^{-1/(\alpha+1)} S^n\left(\lfloor tn^{{\alpha/(\alpha+1)}}\rfloor\right)\!, t\geq 0\right)\xrightarrow{d} (L_t+\lambda t, t\geq 0)\end{equation}

in ${\mathbb D}({\mathbb R}_+,{\mathbb R})$ .

In order to also obtain the convergence of the height process under rescaling, we note that for $\lambda\leq 0$ , we can directly apply [Reference Duquesne and Le Gall24, Theorems 1.4.3 and 2.2.1], stated in this work as Theorem 1. In the case of $\lambda>0$ , we apply Theorem 2. In both results, we use the scaling parameters $p=n^{1/\alpha+1}$ and $\gamma_p=n^{\tfrac{\alpha-1}{\alpha+1}}$ . The conditions of the theorems then follow directly from our assumptions on the degree distributions and (27). This implies the existence of a continuous modification of the height process and the claimed convergence.

We will now prove that in the cases we consider, $\phi^n_m$ behaves well under rescaling. This is the content of the following proposition, which is a generalization of the proof of [Reference Conchon-Kerjan and Goldschmidt17, Proposition 4.3].

Proposition 5. Set

$$\phi(t)=\exp\!\left(\frac{1}{\mu} \int_0^t s dL_s -\frac{C_{\alpha}}{(\alpha+1) \mu^{\alpha+1}}t^{\alpha+1} \right). $$

Then

$$\phi(n,t)\xrightarrow{d}\phi(t)$$

as $n\rightarrow \infty$ , and the sequence $\{\phi(n,t), n\in{\mathbb N}\}$ is uniformly integrable.

The proof will follow the structure of the proof of [Reference Conchon-Kerjan and Goldschmidt17, Proposition 4.3], but we will need to adapt the technical lemmas presented there to our more general setting.

Proof. The following technical lemmas need justification in the more general setting:

• For $\lambda=0$ and a degree distribution that does not depend on n, for any $t\geq 0$ , by [Reference Conchon-Kerjan and Goldschmidt17, Lemma 4.4] it holds that, as $n\to\infty$ ,

  $$\frac{1}{n}\sum\limits_{k=0}^{\lfloor tn^{{\alpha/(\alpha+1)}}\rfloor -1} S^n(k) \xrightarrow{d} \int_0^tL_sds.$$
  Using the continuous mapping theorem (see e.g. [Reference Durrett27, Theorem 3.2.4]), we find that for general $\lambda$ and for the degree distribution depending on n, for any $t\geq 0$ ,
  (28) \begin{equation}\frac{1}{n}\sum\limits_{k=0}^{\lfloor tn^{{\alpha/(\alpha+1)}}\rfloor -1} S^n(k) \xrightarrow{d} \int_0^tL_sds+\frac{\lambda t^2}{2}.\end{equation}

• Define ${\mathcal L}_{D_1^n}(s)\,{:\!=}\,{\mathbb E}[\,\!\exp\!({-}\,sD_1^{n})]$ . Then [Reference Conchon-Kerjan and Goldschmidt17, Lemma 4.5a] and the argument thereafter state that for $\lambda=0$ and a degree distribution that does not depend on n,

  \begin{align*}{\mathcal L}_{D_1^n}(s)&=1-\mu_ns+\mu_ns^2-\frac{C_{\alpha}}{\alpha+1}s^{\alpha+1}+o(s^{\alpha+1})\\&=\exp\!\left({-}\,\mu_n s + \frac{\mu_n(2-\mu_n)}{2}s^2-\frac{C_{\alpha}}{\alpha+1}s^{\alpha+1}+o(s^{\alpha+1})\right).\end{align*}
  In our set-up, we find, similarly to how we obtained ${\mathcal K}_{n}^{\prime\prime}(s)$ , that
  $${\mathcal L}_{D_1^n}^{\prime\prime\prime}(s)=-c\Gamma(2-\alpha)s^{\alpha-2}+o(s^{\alpha-2})$$
  as $s\rightarrow 0$ , and using ${\mathbb E}[D_1^{n}]=\mu_n$ and ${\mathbb E} \left[(D_1^{n})^2\right]=\left(2+\lambda n ^{-\tfrac{\alpha-1}{\alpha+1}}\right) \mu_n$ , we get, by integrating three times, that
  (29) \begin{align}\begin{split}{\mathcal L}_{D_1^n}(s)&=1-\mu_ns+\left(1+\frac{\lambda}{2} n^{-\tfrac{\alpha-1}{\alpha+1}}\right)\mu_ns^2-\frac{C^{(n)}}{\alpha+1}s^{\alpha+1}+o(s^{\alpha+1})\\&=\exp\!\left({-}\,\mu_n s + \frac{\mu_n(2+\lambda n ^{-\tfrac{\alpha-1}{\alpha+1}}-\mu_n)}{2}s^2-\frac{C^{(n)}}{\alpha+1}s^{\alpha+1}+o(s^{\alpha+1})\right)\end{split}\end{align}
  as $s\rightarrow 0$ .

• For $\lambda=0$ and a degree distribution that does not depend on n, for $m=\lfloor tn^{\alpha/(\alpha+1)}\rfloor$ , by [Reference Conchon-Kerjan and Goldschmidt17, Lemma 4.6] it holds that, as $n\to\infty$ ,

  \begin{equation*}\exp \left(m- \frac{ 2+\mu_n}{2\mu_n}\frac{m^2}{n}\right)\left[{\mathcal L}\left(\frac{m}{n\mu_n}\right)\right] ^{n-m} \rightarrow \exp\!\left({-}\, \frac{C_{\alpha}}{(\alpha+1 )\mu^{\alpha+1}}t^{\alpha+1}\right).\end{equation*}
  By (29) we straightforwardly obtain that, in our set-up, as $n\rightarrow \infty$ ,
  (30) \begin{equation}\exp \left(m- \frac{ 2+\mu_n}{2\mu_n}\frac{m^2}{n}\right)\left[{\mathcal L}\left(\frac{m}{n\mu_n}\right)\right] ^{n-m} \rightarrow \exp\!\left({-}\, \frac{C_{\alpha}}{(\alpha+1 )\mu^{\alpha+1}}t^{\alpha+1}+\frac{\lambda t^2}{2\mu}\right).\end{equation}

• For $\lambda=0$ and a degree distribution that does not depend on n, for $s(0)=0$ and $s(i)=\sum_{j=1}^i (k_j-2)$ , for $i\geq 1$ , by [Reference Conchon-Kerjan and Goldschmidt17, Lemma 4.7] it holds that

  $$\phi_m^{n}(k_1,\dots,k_m)\geq \exp\!\left(\frac{1}{n\mu_n}\sum\limits_{i=0}^{m}(s(i)-s(m))-\frac{C_\alpha}{(\alpha+1 )\mu^{\alpha+1}}t^{\alpha+1} \right)(1+o(1)) .$$
  Adapting the proof to our set-up, using (30), we find that for $s(0)=0$ and $s(i)=\sum_{j=1}^i (k_j-2)$ , for $i\geq 1$ ,
  $$\phi_m^{n}(k_1,\dots,k_m)\geq (1+o(1))\exp\!\left(\frac{1}{n\mu_n}\sum\limits_{i=0}^{m}(s(i)-s(m))-\frac{C_\alpha t^{\alpha+1}}{(\alpha+1 )\mu^{\alpha+1}}+\frac{\lambda t^2}{2\mu}\right) .$$
  The proof of this can be found in the appendix.

Now we have that

$$\phi(n,t)\geq \underline{\phi}(n,t)\,{:\!=}\, \exp\!\left(\frac{1}{n\mu_n}\sum\limits_{i=0}^{m}(S^n(i)-S^n(m))-\frac{C_\alpha}{(\alpha+1) \mu^{\alpha+1}}t^{\alpha+1}+\frac{\lambda t^2}{2\mu}\right).$$

But then, by (27) and (28), since

$$\frac{1}{n}\sum\limits_{k=0}^{\lfloor tn^{{\alpha/(\alpha+1)}}\rfloor -1} \left(S^n(k)-S^n(\lfloor tn^{{\alpha/(\alpha+1)}}\rfloor -1)\right) \xrightarrow{d} \int_0^t(L_s-L_t) ds-\frac{\lambda t^2}{2},$$

we get that

$$ \underline{\phi}(n,t)\xrightarrow{d}\exp \left(\frac{1}{\mu}\int_0^t (L_s-L_t)ds-\frac{C_{\alpha}}{(\alpha+1) \mu^{\alpha+1}}t^{\alpha+1}\right).$$

We then finish the proof in the same way as the proof of [Reference Conchon-Kerjan and Goldschmidt17, Proposition 3.3] to conclude that

$$\phi(n,t)\xrightarrow{d}\phi(t)$$

as $n\rightarrow \infty$ , and that the sequence $\{\phi(n,t), n\in{\mathbb N}\}$ is uniformly integrable. The details can be found in the appendix.

Remember that $n^{-1/(\alpha+1)}\left( S^n\left(\lfloor tn^{{\alpha/(\alpha+1)}}\rfloor\right)\!, t\geq 0\right)$ converges in law to $(L_t+\lambda t, t\geq 0)$ as $n\to \infty$ . We can get from the process $\left(S^n\left(\lfloor sn^{{\alpha/(\alpha+1)}}\rfloor\right)\!,0\leq s \leq t\right)$ to the process $\left(\tilde{S}^{n}\left(\lfloor sn^{{\alpha/(\alpha+1)}}\rfloor\right)\!,0\leq s \leq t \right)$ via the measure change $\phi(n,t)$ . The random variable $\phi(n,t)$ converges in law to $\phi(t)$ as $n\to\infty$ . Therefore, we will define the process $(\tilde{K}^{\lambda},\tilde{H}^\lambda)$ via the following measure change. For $t\geq 0$ , for every non-negative integrable functional $F\,{:}\,{\mathbb D}([0,t],{\mathbb R}^2)\to{\mathbb R}$ ,

(31) \begin{align}\begin{split} {\mathbb E}[F(\tilde{K}^{\lambda}_s,\tilde{H}^{\lambda}_s, 0\leq s \leq t)]&={\mathbb E}\left[\phi(t)F(L_s+\lambda s,H^\lambda_s,0\leq s \leq t)\right].\end{split}\end{align}

Proposition 6. We have

\begin{align*} &\left(n^{-1/(\alpha+1)} \tilde{S}^{n}\left(\lfloor sn^{{\alpha/(\alpha+1)}}\rfloor\right)\!, n^{-\alpha/(\alpha+1)} \tilde{H}^{n}\left(\lfloor sn^{{\alpha/(\alpha+1)}}\rfloor\right)\!, 0\leq s \leq t\right)\\&\quad \xrightarrow{d} (\tilde{K}^\lambda_s, \tilde{H}^\lambda_s ,0\leq s \leq t)\end{align*}

in ${\mathbb D}({\mathbb R}_+,{\mathbb R}^2)$ as $n\to\infty$ .

Proof. We want to show that for any $t\geq 0$ and any bounded continuous test function $f\,{:}\,{\mathbb D}([0,t],{\mathbb R}^2) \rightarrow {\mathbb R}$ , for $\tilde{H}^n$ the height process corresponding to $\tilde{S}^n$ ,

\begin{align*}&{\mathbb E}\left[f\left( n^{-1/(\alpha+1)} \tilde{S}^{n}\left(\lfloor sn^{{\alpha/(\alpha+1)}}\rfloor\right)\!, n^{-\alpha/(\alpha+1)} \tilde{H}^{n}\left(\lfloor sn^{{\alpha/(\alpha+1)}}\rfloor\right)\!, 0\leq s \leq t\right)\right] \\&\rightarrow {\mathbb E} \left[f(\tilde{K}^\lambda_s, \tilde{H}^\lambda_s ,0\leq s \leq t)\right] \end{align*}

as $n\rightarrow \infty$ . Using our measure change, this is equivalent to showing that for any $t\geq 0$ and any bounded continuous test function $f\,{:}\,{\mathbb D}([0,t],{\mathbb R}^2)\rightarrow {\mathbb R}$ ,

\begin{align*}&{\mathbb E} \left[\phi(n,t) f\left( n^{-1/(\alpha+1)} {S}^{n}\left(\lfloor sn^{{\alpha/(\alpha+1)}}\rfloor\right)\!, n^{-\alpha/(\alpha+1)}{H}^{n}\left(\lfloor sn^{{\alpha/(\alpha+1)}}\rfloor\right)\!, 0\leq s \leq t\right) \right] \\&\rightarrow {\mathbb E} [\phi(t)f(L_s+\lambda s,H^\lambda_s, 0\leq s\leq t)].\end{align*}

We now finish as in the proof of [Reference Conchon-Kerjan and Goldschmidt17, Theorem 4.1] to obtain the result.

The following proposition characterizes the law of $\tilde{K}^\lambda$ .

Proposition 7. For $\tilde{K}^\lambda(t)$ as defined in (31), for $\theta>0$ ,

\begin{align*}&{\mathbb E}\left[\,\!\exp\!\left({-}\,\theta \tilde{K}^\lambda(t) \right)\right]\\&\quad=\exp\!\left( -\theta \lambda t + \theta C_\alpha \frac{t^\alpha}{\mu^\alpha}+\int_0^t ds \int_0 ^\infty \frac{c}{\mu}x^{-(\alpha+1)}e^{-xs/\mu}(e^{-\theta x}-1+\theta x)dx\right) .\end{align*}

The proof can be found in the appendix.