2.1 Proof of Proposition 5

The idea of the proof is to exploit that, as $\psi (\rho _-)=1$ , the process given by

\begin{equation*}W_n\,:\!=\,\sum _{|v|=n}e^{-\rho _-V(v)}\end{equation*}

is a martingale. Since $W_n$ is nonnegative it converges to some limit $W$ , which we show to be strictly positive. We then look at this martingale from the point of view of a stopping line, as discussed in [Reference Kyprianou8]. Theorem 9 in [Reference Kyprianou8] implies convergence as $t\to \infty$ of $(e^{-\rho _-t} Z_t')$ to $W$ , where

(3) \begin{equation} Z_t'\,:\!=\, \sum _{v \in \mathscr{V}} {\unicode{x1D7D9}}_{\{V(v)\lt t\}} \sum _{y \colon \boldsymbol{y}=v} e^{-\rho _-(V(y)-t)} {\unicode{x1D7D9}}_{\{V(y)\geq t\}}. \end{equation}

Observe that conditional on the $v$ with $V(v)\lt t$ the inner sums are independent with a distribution depending continuously on $V(v)-t$ . A result of Nerman [[Reference Nerman13], Theorem 3.1] therefore gives that the inner sum can be replaced by ${\unicode{x1D7D9}}_{\{t-V(v)\leq \log b\}}$ and we still get convergence to a constant multiple of $W$ .

We start the detailed proof by verifying that the limiting $W$ is strictly positive and satisfies the tail property of (1). By Biggins’ theorem for branching random walks, see e.g. [Reference Biggins2, Reference Lyons, Athreya and Jagers10], the martingale limit $W$ is strictly positive if and only if the following two conditions hold,

1. (i) $\psi (\rho _-)-\frac {\rho _-\psi '(\rho _-)}{\psi (\rho _-)}\gt 0 \, ,$

2. (ii) $\mathbb{E}_1[W_1 \log W_1]\lt \infty .$

The first one holds as $\psi (\rho _-)=1$ and $\psi '(\rho _-)\lt 0$ . For the second condition it suffices to prove the following lemma.

Proof. We define

\begin{equation*}f(x,\Pi )= e^{-\rho _- V(x)} \Big(\sum _{y\in \Pi }e^{-\rho _- V(y)}\Big)^{p-1}\, .\end{equation*}

Then $\mathbb{E}_1[W_1^p]=\mathbb{E}[\!\int f(x,\Pi ) \, \Pi (dx)]$ and by Mecke’s equation [[Reference Last and Penrose9], Theorem 4.1] we get

\begin{align*} \mathbb{E}_1[W_1^p]&= \int \mathbb{E}[f(x,\Pi +\delta _x)] \, \pi (dx) = \int e^{-\rho _-x}\mathbb{E}\Big [\big (e^{-\rho _-x}+\int e^{-\rho _- t} \, \Pi (dt) \big )^{p-1}\Big ] \pi (dx)\\ &\leq 2^{p-1} \Big (\int e^{-p \rho _-x} \pi (dx) + \mathbb{E}_1\big [ W_1^{p-1}\big ] \, \psi (\rho _-) \Big )\, . \end{align*}

The left summand is equal to $\psi (p\rho _-)$ which is finite for $1\lt p\lt \frac {1-\gamma }{\rho _-}$ . The right summand is finite if $1\lt p\leq 2$ because in this case, by Jensen’s inequality, the expectation is bounded by one. If $p\gt 2$ we iterate the argument, using the same bound but now with $1\lt p-1\lt \frac {1-\gamma }{\rho _-}$ . In each iteration the exponent is reduced by one until it is no larger than two.

Biggins’ theorem ensures not only that $W\gt 0$ on survival of $\mathscr{T}\,$ but also that $W_n\to W$ in $L^1$ . By the next lemma we can improve this to convergence in $L^p$ for $p\lt \rho _+/\rho _-$ .

Proof. By Proposition 2.1 in [Reference Iksanov, Liang and Liu6] we get that $(W_n)$ converges in $L^p$ and that $\mathbb{E}_1[W_n^p]$ is bounded if

\begin{equation*} \mathbb{E}_1 \big [W_1^p\big ] \lt \infty \text{ and } \psi (p\rho _-) \lt \psi (\rho _-)^p.\end{equation*}

The first condition is verified under the weaker condition $1\lt p\lt \frac {1-\gamma }{\rho _-}$ in Lemma 7. As $\psi (\rho _-)=1$ the second condition becomes $\psi (p\rho _-) \lt 1$ , which holds if $p\lt \rho _+/\rho _-$ .

The tail behaviour of $W$ (and later of $Y$ ) claimed in Proposition 5 now follows directly from Lemma 8 by Markov’s inequality.

As in [[Reference Mörters, Schleicher, Mailler and Wild12], Proposition 8] our next aim is to rewrite $\mathscr{T}_{0,1/u}$ started at the origin in terms of a sum over characteristics of the individuals in the population at time $t=-\log u$ of a general (Crump-Mode-Jagers) branching process. In a general branching process the location of all offspring is to the right of the parent and locations are interpreted as birth-times of offspring particles.

Figure 2.

Branching particles are marked in blue. The positions on $[0,\infty )$ of the frozen particles, which are marked in red, yield the point process $\xi$ .

To this end we divide the offspring of a particle $v=(v_1,\ldots , v_n)$ at location $V(v)$ into branching particles to its left, and frozen particles to its right. The offspring of the branching particles is again divided into branching particles to the left of $V(v)$ and frozen particles to its right, until (after a finite number of steps) the offspring of all branching particles has been divided into branching and frozen particles. The frozen particles are all located to the right of $V(v)$ , they constitute the offspring process of $v$ in the general branching process. Their relative positions form a point process

\begin{equation*}\xi _v=\sum _{{w\in \mathscr{V}, |w|\gt n}\atop {(w_1,\ldots , w_n) = (v_1,\ldots , v_n)}} \delta _{V(w)-V(v)} {\unicode{x1D7D9}}_{\{V(w)\gt V(v) , V(w_1,\ldots ,w_i)\leq V(v) \forall n\leq i\lt |w|\}},\end{equation*}

and they are all copies of the point process $\xi$ depicted in Figure 2. The branching particles form a set $\mathscr{B}_v$ and their locations are all to the left of $V(v)$ .

To construct the general branching process we start with the root located at the origin, considered initially to be frozen, take the point process $\xi _\varnothing$ of frozen particles as birth times of the children of the root and apply the same procedure to every child $v$ of the root. The processes $\xi _v$ and cardinalities $|\mathscr{B}_v|$ are independent and identically distributed over all the frozen particles $v$ . The total number of particles in $\mathscr{T}_{0,1/u}$ equals

\begin{equation*}\sum _{{v \text{ frozen}}\atop {V(v) \leq t}} (1+ |\mathscr{B}_v|),\end{equation*}

where $t=-\log u$ . To obtain convergence of this quantity (properly scaled) we need to find the Malthusian parameter $\alpha \gt 0$ associated to $\xi$ , defined by

\begin{equation*}\mathbb{E} \int _0^\infty e^{-\alpha t} \xi (dt)=1. \end{equation*}

We now show that $\rho _-$ is the Malthusian parameter associated to $\xi$ . To this end we construct a martingale $(M_n)$ as follows: We start with a particle at the origin and $M_0=1$ . In every step, we replace the leftmost particle by its offspring chosen with displacements according to a Poisson process of intensity $\pi$ and leave all other particles alive. Particles in $(0,\infty )$ never branch and remain alive but frozen. If the leftmost particle is in $(0,\infty )$ the process stops and the positions of the frozen particles make up $\xi$ . The random variable $M_n$ is obtained as the sum of all particles $x$ alive after the $n$ th step weighted with $e^{-\rho _- V(x)}$ . Because $\psi (\rho _-)=1$ the process $(M_n)$ is indeed a martingale, and it clearly converges almost surely to

\begin{equation*}M_\infty =\int _0^\infty \mathrm{e}^{-\rho _- t} \xi (\mathrm{d}{t}).\end{equation*}

Now take $\alpha \gt \rho _-$ with $\psi (\alpha )\lt 1$ . Then $M_n$ is dominated by

\begin{align*} M_n \leq \sum _{u \text{ branching}} e^{- \alpha V(u)} + \sum _{u \text{ frozen}}e^{-\rho _-V(u)} \end{align*}

The right-hand side is integrable, as the sum over frozen particles born from a single particle $x$ in position $V(x)\lt 0$ has expectation at most $e^{-\alpha V(x)}$ and the expected sum over these bounds for all branching particles is itself bounded by $\frac 1{1-\psi (\alpha )}$ . By dominated convergence, we get that $\mathbb{E}[M_\infty ]=1$ and hence $\rho _-$ is the Malthusian parameter.

Theorem 3.1 in Nerman [Reference Nerman13] yields convergence of $(e^{-\rho _- t} Z_t^\phi )$ to a positive random variable $m_\phi Z$ for

\begin{align*} Z_t^\phi \,:\!=\,\sum _{v\colon V(v) \leq t}\phi _v(t-V(v)), \end{align*}

where the sum is over the particles of the general branching process born before time $t$ and the characteristics $\phi _v$ are independent, identically distributed copies of a random function $\phi \colon [0,\infty ) \to [0,\infty )$ satisfying mild technical conditions. Moreover, $Z$ is a positive random variable independent of $\phi$ and $m_\phi$ a positive constant depending on $\phi$ . The conditions of [Reference Nerman13] are satisfied for the process $(Z_t')$ in (3) by [[Reference Nerman13], Corollary 2.5], whence $Z$ is a constant multiple of $W$ , but also when the processes $(\phi (s) \colon s\ge 0)$ are bounded by an integrable random variable and $\mathbb{E}\phi \colon [0,\infty ) \to [0,\infty )$ is continuous.

We now look at the total number $I(0,b)$ of surviving particles of $\mathscr{T}_{0,1}(\log u)$ located in $(\log b,0]$ . We shift all particle positions by $t = -\log u$ . Then the killed branching random walk $ \mathscr{T}_{0,1}(\log u)$ becomes a killed branching random walk $\mathscr{T}_{0,1/u}(0)$ and $I(0,b)$ the number of surviving particles in $(t+\log b,t]$ .

We have $I(0,b)=Z_t^\phi$ for the general branching process with offspring law $\xi$ at the time $t = -\log u$ and for the characteristic

\begin{equation*}\phi _v(s)= \sum _{w \in \mathscr{B}_v} {\unicode{x1D7D9}}_{\{s+\log b\lt V_v(w) \leq s\}},\end{equation*}

where $V_v(w)$ is the relative position of the branching particle $w$ to $v$ . Then $\phi _v(t-V(v))$ is the number of branching particles descending from $v$ (including $v$ itself) located in the interval $(t+\log b,t]$ . This process is dominated by $1+|\mathscr{B}_v|$ . To check that $|\mathscr{B}_\varnothing |$ is integrable, fix $\alpha \gt 0$ with $\psi (\alpha )\lt 1$ . Then we have for $v\in \mathscr{B}_\varnothing$ that $e^{-\alpha V(v)} \geq 1$ and

\begin{equation*}\mathbb{E} \sum _{{v\in \mathscr{B}_\varnothing }\atop {|v|=n}} e^{-\alpha V(v)} \leq \psi (\alpha )^n.\end{equation*}

Hence $\mathbb{E}[|\mathscr{B}_v|]\leq \sum _n \psi (\alpha )^n \leq \frac {1}{1-\psi (\alpha )} \lt \infty .$ As $\mathbb{E} \phi _v$ is clearly continuous the conditions of [[Reference Nerman13], Theorem 3.1] on the characteristics are satisfied.

Altogether this yields that

\begin{equation*}\lim _{u\downarrow 0} u^{\rho _-}I(0,b)= \lim _{t\uparrow \infty } e^{-\rho _- t} Z_t^\phi = Y \quad \text{ in distribution,}\end{equation*}

where the limit $Y$ is a positive, constant multiple of the positive martingale limit $W$ .

2.2 Proof of Proposition 6

Under the assumption $0\lt \beta \lt \frac {1}{4}-\frac {\gamma }{2}$ the leftmost particle of $\mathscr{T}$ drifts to the right, i.e. $\lim _{n\to \infty } \inf _{|v|=n} V(v)=\infty$ , see [[Reference Shi15], Lemma 3.1]. Hence $\inf _{v\in \mathscr{V}} V(v)$ is a finite random variable and it is easy to see that in our case its support is the entire negative half-line. Hence, given $0\lt b\lt 1$ , we can pick $\varepsilon \gt 0$ such that

\begin{equation*}\mathbb{P}_1\big( \inf _{v\in \mathscr{V}} V(v) \gt \log b\big) \geq \varepsilon .\end{equation*}

Additionally we request that, for $Y$ as in Proposition 5, $\varepsilon \gt 0$ satisfies $\mathbb{P} \left ( Y\geq \varepsilon \right ) \geq 5\varepsilon .$ This implies that

\begin{equation*}\liminf _{u \downarrow 0} \inf _{u'\in [ub,u]} \mathbb{P}_{u'} \left ( I(ub,b) \geq \varepsilon u^{-\rho _-} \right ) \geq \liminf _{u \downarrow 0} \mathbb{P}_u \left ( I(0,b) \geq \varepsilon u^{-\rho _-} \right ) - \varepsilon \geq 4 \varepsilon \end{equation*}

and, for suitably large $a\gt 1$ ,

\begin{equation*}\limsup _{u \downarrow 0} \sup _{u'\in [ub,u]} \mathbb{P}_{u'} ( I(ub,0) \geq a u^{-\rho _-}) \leq \limsup _{u \downarrow 0} \mathbb{P}_{ub} ( I(0,0) \geq a u^{-\rho _-}) \leq \frac{\varepsilon}{2}.\end{equation*}

We pick $0\lt u_0\lt b \wedge 2^{-1/\rho _-}$ such that $\inf _{u'\in [ub,u]} \mathbb{P}_{u'} ( I(ub,b) \geq \varepsilon u^{-\rho _-}) \geq 3\varepsilon$ and also $a\gt 1$ such that $\sup _{u'\in [ub,u]} \mathbb{P}_{u'} ( I(ub,0) \geq \frac {a}2 u^{-\rho _-}) \leq \varepsilon$ , for all $0\lt u\lt u_0$ . The exploration algorithm below uses $\varepsilon , a, \rho _-$ and $u_0$ as derived above from the parameter $\pi$ .

We present, for parameters $(\pi , u, m)$ with $m\in \mathbb{N}$ , an exploration algorithm with input

• • a graph $\mathscr{U}'\subset \{1,\ldots , um\}$ with at most $a m^{\rho _-}$ vertices,

• • distinct vertices $u_1 \lt \ldots \lt u_d$ in $\mathscr{U}'$ with $bum\lt u_i\leq um$ and $d\leq m^{\rho _-}$ .

The output of the algorithm are

• • a family of pairwise disjoint sets $\mathcal{Y}_1, \ldots , \mathcal{Y}_d \subset \{bm,\ldots ,m\}$ ,

• • a graph $\mathscr{U} \subset \{1,\ldots , m\}$ with at most $a (\frac {m}u)^{\rho _-}$ vertices such that $\mathscr{U}'$ is an embedded subgraph and the sets $\mathcal{Y}_i$ are contained in the connected component of $u_i$ in $\mathscr{U}$ .

By construction the output sets $\mathcal{Y}_1, \ldots , \mathcal{Y}_d$ are pairwise disjoint and $u_i$ is connected to $\mathcal{Y}_i$ by edges in $\mathscr{U}$ . Also, for every $i\in \{1,\ldots ,d\}$ the algorithm adds at most $\frac {a}2u^{-\rho _-}+1$ vertices to the graph $\mathscr{U}$ , so that its output $\mathscr{U}$ satisfies

\begin{align*} |\mathscr{U}| & \leq |\mathscr{U}'|+ d\bigg(\frac{a}{2}u^{-\rho _-}+1\bigg) \leq a (m/u)^{\rho _-} \bigg ( u^{\rho _-} +\frac{1}{2} + \frac{1}{a} u^{\rho _-}\bigg ) \leq a (m/u)^{\rho _-}, \end{align*}

for all $0\lt u\lt u_0$ by choice of $u_0$ . In the following we show how the algorithm can be used to construct a suitably large subgraph of $\mathscr{G}_m$ .

We run the algorithm with parameter $(\tilde \pi ,u,m)$ for an intensity measure with a slightly decreased density parameter $0\lt \tilde \beta \lt \beta$ , $0\lt u\lt u_0$ and some large $m$ . This leads to a slightly smaller value of $\rho _-$ which is referred to as $\rho$ in the statement of Proposition 6. The next lemma shows that the probability of edges inserted by the algorithm is bounded from above by the edge probabilities in $\mathscr{G}_m$ .

Lemma 9. There exists $m(u)\in \mathbb{N}$ such that, for all $m\ge m(u)$ , for all $m \ge s,r \geq bum$ with $s \not =r$ the probability that a particle $v$ in location $V(v)$ with $\pi _m(V(v))=r$ has an offspring $y$ with location $V(y)$ satisfying $\pi _m(V(y))=s$ is at most

\begin{equation*}\beta (r\wedge s)^{-\gamma } (r\vee s)^{\gamma -1}.\end{equation*}

Proof. For a fixed particle $v$ in location $V(v)$ with $\pi _m(V(v))=r$ the probability that it has an offspring $y$ with location $V(y)$ satisfying $\pi _m(V(y))=s$ equals

(4) \begin{equation} 1-\exp \Bigg(\!-\tilde \pi \Bigg( -\sum _{k=s}^m \frac 1k -V(v), -\sum _{k=s+1}^m \frac 1k-V(v)\Bigg]\Bigg). \end{equation}

As $\pi _m(V(v))=r$ we have

\begin{equation*}-\sum _{k=r}^m \frac 1k \lt V(v) \leq -\sum _{k=r+1}^m \frac 1k.\end{equation*}

The probability in (4) is therefore largest when $V(v)=-\sum _{k=r}^m \frac 1k$ . It therefore remains to show that, for $bum\leq s\lt r$ , we have

(5) \begin{equation} 1-\exp \Bigg (-\tilde \pi \Bigg( -\sum _{k=s}^{r-1} \frac 1k, -\sum _{k=s+1}^{r-1}\frac 1k\Bigg]\Bigg ) \leq \beta s^{-\gamma }r^{\gamma -1}, \end{equation}

and, for $bum\leq r\lt s$ , we have

(6) \begin{equation} 1-\exp \Bigg (-\tilde \pi \Bigg(\sum _{k=r}^{s-1} \frac 1k, \sum _{k=r}^{s} \frac 1k\Bigg]\Bigg) \leq \beta s^{\gamma -1}r^{-\gamma }. \end{equation}

For (5) we find that, for some constant $C\gt 0$ , if $m\geq m(u)$ for a suitable $m(u)\in \mathbb{N}$ ,

\begin{align*} \tilde \pi \Bigg( -\sum _{k=s}^{r-1} \frac 1k, -\sum _{k=s+1}^{r-1}\frac 1k\Bigg] & = \frac {\tilde \beta }{1-\gamma }\, \exp \Bigg({-(1-\gamma )\sum _{k=s}^{r-1} \frac 1k} \Bigg)(e^{\frac {1-\gamma }{s+1}}-1)\\ & \leq \bigg (\frac {\tilde \beta }{s+1}+ \frac {C}{(bum)^2}\bigg )\, \exp \big (-(1-\gamma ) (\log (\tfrac {r-1}{s-1}) - \tfrac {C}{bum}) \big )\\[2mm] & \leq \beta s^{-\gamma }r^{\gamma -1}. \end{align*}

Hence, using that $1-e^{-x}\le x$ , we get (5).

For (6) we find that, for some constant $C\gt 0$ , if $m\geq m(u)$ for a suitable $m(u)\in \mathbb{N}$ ,

\begin{align*} \tilde \pi \Bigg( \sum _{k=r}^{s-1} \frac 1k, \sum _{k=r}^{s}\frac 1k\Bigg] & = \frac {\tilde \beta }{\gamma }\exp \bigg({\gamma \sum _{k=r}^{s-1} \frac 1k} \bigg)(e^{\frac {\gamma }{s}-1})\\ & \leq \Bigg(\frac {\tilde \beta }{s}+\frac {C}{(bum)^2}\Bigg)\, \exp \Big(\gamma (\log \Big(\frac {s-1}{r-1}\Big) - \tfrac {C}{bum}) \Big)\\[2mm] & \leq \beta s^{\gamma -1}r^{-\gamma }. \end{align*}

Hence, using that $1-e^{-x}\le x$ , we get (6).

Let $E_i$ be the event that the exploration of $u_i$ was successful. This is the case if $\mathcal{Y}_i\not =\emptyset$ or, equivalently, $|\mathcal{Y}_i|\geq \varepsilon u^{-\rho _-}$ . Let $\mathscr{U}_i$ be the graph in Algorithm1 at the time when the exploration of $u_i$ is completed and $(\mathscr{F}_{\mathscr{U}_i} \colon i=0,\ldots ,d)$ the natural filtration associated with this process. Note that

\begin{equation*}\mathscr{U}'\,=:\,\mathscr{U}_0 \subset \mathscr{U}_1 \subset \ldots \subset \mathscr{U}_d=\mathscr{U},\end{equation*}

and that $E_i\in \mathscr{F}_{\mathscr{U}_i}$ for all $i\in \{1,\ldots ,d\}$ .

Lemma 10. For $0\lt u \lt u_0$ there exists $m(u)\in \mathbb{N}$ such that, for all $m \geq m(u)$ , almost surely,

\begin{equation*}\mathbb{P}(E_{i+1} \mid \mathscr{F}_{\mathscr{U}_i}) \ge \varepsilon .\end{equation*}

Algorithm 1

Branching Random Walk Exploration (π,u,m)

To complete the construction we have to remove the possible dependence of the size of the sets $\mathcal{Y}_1, \ldots , \mathcal{Y}_{d}$ by means of the following decoupling lemma.

Lemma 11. Let $\mathcal{Y}_1, \ldots , \mathcal{Y}_{d}$ be random sets such that, almost surely,

\begin{equation*}\mathbb{P}(|\mathcal{Y}_{i+1}|\ge k \mid \mathscr{F}_{\mathscr{U}_i}) \ge \epsilon ,\end{equation*}

then there exist random sets ${\mathcal{X}_i} \subset \mathcal{Y}_i$ with $X_i$ elements such that $X_1,\ldots X_d$ are independent and $\mathbb{P}(X_{i}= k) = \epsilon \text{ and } \mathbb{P}(X_{i}=0) = 1- \epsilon .$

Proof. Let $U_1,\ldots , U_d$ be independent and uniformly distributed on $(0,1)$ . Let $\mathscr{E}_1$ be the event that $\mathcal{Y}_1$ has at least $k$ elements and $U_1\leq \frac \epsilon {\mathbb{P}(|\mathcal{Y}_1|\ge k)}$ so that $\mathbb{P}(\mathscr{E}_1)=\epsilon$ . On the event $\mathscr{E}_1$ we draw $k$ elements from $\mathcal{Y}_1$ without replacement and put $\mathcal{X}_1$ to be the set of elements thus drawn. On $\mathscr{E}_0=\mathscr{E}_1^c$ we set ${\mathcal{X}_1}=\emptyset$ . Then $X_1$ has the desired distribution.

Now let $\mathscr{E}_{i1}$ be the event $\mathscr{E}_i$ intersected with the event that $\mathcal{Y}_2$ has at least $k$ elements and $U_2\leq \frac \epsilon {\mathbb{P}(|\mathcal{Y}_2|\ge k \mid \mathscr{E}_i)}$ . Then

\begin{equation*}\mathbb{P}(\mathscr{E}_{i1})= \mathbb{P}(\mathscr{E}_i) \mathbb{P}\big ( |\mathcal{Y}_2|\ge k, U_2\leq \tfrac \epsilon {\mathbb{P}(|\mathcal{Y}_2|\ge k \mid \mathscr{E}_i)}\mid \mathscr{E}_i\big ) = \epsilon \mathbb{P}(\mathscr{E}_i).\end{equation*}

On the event $\mathscr{E}_{i1}$ we draw $k$ elements from $\mathcal{Y}_2$ without replacement and put $\mathcal{X}_2$ to be the set of elements thus drawn. On $\mathscr{E}_{i0}=\mathscr{E}_i \setminus \mathscr{E}_{i1}$ we set ${\mathcal{X}_2}=\emptyset$ . Then

\begin{align*} \mathbb{P}(X_1=k, X_2=k) = \mathbb{P}( \mathscr{E}_{11}) = \epsilon ^2, \qquad & \mathbb{P}(X_1=0, X_2=k) = \mathbb{P}( \mathscr{E}_{01}) = (1-\epsilon )\epsilon , \\ \mathbb{P}(X_1=k, X_2=0) = \mathbb{P}( \mathscr{E}_{10}) = \epsilon (1-\epsilon ), \qquad & \mathbb{P}(X_1=0, X_2=0) = \mathbb{P}( \mathscr{E}_{00}) = (1-\epsilon )^2. \end{align*}

This implies that $X_2$ has the desired distribution and $X_1$ and $X_2$ are independent. We continue with this method until $\mathcal{X}_1 , \ldots , \mathcal{X}_{d}$ are constructed.

Proof of Proposition 6. To prove Proposition 6 we run Algorithm 1 with parameters $(\tilde \pi ,u,m)$ for the intensity measure

\begin{equation*}\tilde \pi (dx)=\tilde \beta (e^{\gamma x} {\unicode{x1D7D9}}_{x\gt 0} + e^{(1-\gamma ) x} {\unicode{x1D7D9}}_{x\lt 0}) \, dx,\end{equation*}

with a slightly decreased density parameter $0\lt \tilde \beta \lt \beta$ , $0\lt u\lt u_0$ and $m\ge m(u)$ . Input are the vertices $u_1,\ldots , u_d\in \mathcal{U}'$ and a graph $\mathscr{U}'$ distributed like the restriction of $\mathscr{G}_m$ to its vertex set $\mathcal{U}'$ . Lemma 9 ensures that the algorithm inserts an edge into $\mathscr{U}$ with a probability no larger than the edge probabilities in $\mathscr{G}_m$ . Hence the output graph $\mathscr{U}$ is stochastically dominated by the restriction of $\mathscr{G}_m$ to its vertex set, denoted $\mathcal{U}$ . We can therefore add vertices to $\mathscr{U}$ so that is distributed like the restriction of $\mathscr{G}_m$ to $\mathcal{U}$ .

By Lemma 10 the set $\mathcal{U}$ contains disjoint subsets $\mathcal{Y}_1, \ldots , \mathcal{Y}_d$ with

\begin{equation*}\mathbb{P}(|\mathcal{Y}_{i+1}|\geq \varepsilon u^{-\rho _-} \mid \mathscr{F}_{\mathscr{U}_i}) \ge \varepsilon ,\end{equation*}

and Lemma 11 gives the existence of random sets $\mathcal{X}_i \subset \mathcal{Y}_i$ with size $X_i=|\mathcal{X}_i|$ such that

\begin{equation*} X_i = {\begin{cases} \lceil \varepsilon u^{-\rho _-} \rceil , & \text{with probability } \varepsilon \gt 0, \\ 0, & \text{with probability } 1 - \varepsilon , \end{cases}} \end{equation*}

and $X_1,\ldots , X_d$ independent. By construction the connected component of $u_i$ in $\mathscr{U}$ contains $\mathcal{X}_i$ with $\mathcal{X}_i \cap \mathcal{X}_j = \emptyset$ for all $i \neq j$ . The construction can be completed by embedding $\mathscr{U}$ into $\mathscr{G}_m$ by adding vertices and independent edges.

3.1 Proof of Proposition 12

For the coupling we sample a killed branching random walk $\mathscr{T}_{0,1}(-X)$ started in $-X$ with intensity measure $\tilde \pi$ which has a slightly increased (but still subcritical) density parameter $\tilde \beta \gt \beta$ . All particles on the positive halfline and their descendants are killed. We use the projection $\pi _n$ defined in (2) to project all particle locations on the negative half-line onto vertices in $\{1,\ldots ,n\}$ and retain all edges as in the genealogical tree of $\mathscr{T}_{0,1}(-X)$ . We denote the resulting multigraph by $\mathscr{K}_n$ . To prove that this coupling has the properties claimed in Proposition 12 we use the following lemma.

When the killed branching random walk is sampled we first check whether it has a particle $y$ with location $V(y)$ satisfying $\pi _n(V(y))\leq n_0$ . If this is the case then, for sufficiently large $n$ , we have $V(y) \leq -(1-\epsilon )\log n$ . If this is not the case, then all particle locations of $\mathscr{T}_{0,1}(-X)$ are projected onto vertices with index at least $n_0$ . Using Lemma 14 to compare the edge probabilities, we see that in this case $\mathscr{G}_n$ is dominated by $\mathscr{K}_n$ , which proves Proposition 12.

Proof of Lemma 14 (i). The probability of the event that a fixed particle $x$ in location $V(x)$ with $\pi _n(V(x))=r$ has an offspring $y$ with location $V(y)$ satisfying $\pi _n(V(y))=m$ equals

(8) \begin{equation} 1-\exp \Bigg (-\tilde \pi \Bigg( -\sum _{k=m}^n \frac 1k -V(x), -\sum _{k=m+1}^n \frac 1k-V(x)\Bigg]\Bigg). \end{equation}

As $\pi _n(V(x))=r$ we have

\begin{equation*}-\sum _{k=r}^n \frac 1k \lt V(x) \leq -\sum _{k=r+1}^n \frac 1k.\end{equation*}

The probability in (8) is therefore smallest when $V(x)=-\sum _{k=r+1}^n \frac 1k$ . It therefore remains to show that there exists $n_0\in \mathbb{N}$ such that, for $n_0\leq m\lt r$ , we have

(9) \begin{equation} 1-\exp \Bigg(-\tilde \pi \Bigg( -\sum _{k=m}^{r} \frac 1k, -\sum _{k=m+1}^{r}\frac 1k\Bigg]\Bigg) \geq \beta m^{-\gamma }r^{\gamma -1}, \end{equation}

and, for $n_0\leq r\lt m$ , we have

(10) \begin{equation} 1-\exp \Bigg(-\tilde \pi \Bigg(\sum _{k=r+1}^{m-1} \frac 1k, \sum _{k=r+1}^{m} \frac 1k\Bigg]\Bigg) \geq \beta m^{\gamma -1}r^{-\gamma }. \end{equation}

For (9) let $\beta \lt \beta '\lt \tilde \beta$ and hence

\begin{align*} \tilde \pi \Bigg(\!-\sum _{k=m}^{r} \frac 1k, -\sum _{k=m+1}^{r}\frac 1k\Bigg] & = \frac {\tilde \beta }{1-\gamma }\exp \Bigg({-(1-\gamma )\sum _{k=m+1}^{r} \frac 1k} \Bigg)( e^{\frac {1-\gamma }{m+1}}-1)\\ & \geq \frac {\tilde \beta }{m+1} \exp \Big (-(1-\gamma ) \left(\log \left(\tfrac {r}m\right) + \tfrac {C}{n_0}\right) \Big )\\[2mm] & \geq \beta ' m^{-\gamma }r^{\gamma -1}, \end{align*}

for some constant $C\gt 0$ , if $n_0\leq m\lt r$ for a suitable $n_0\in \mathbb{N}$ . Hence, using that $1-e^{-x}\ge (\beta /\beta ')x$ for sufficiently small $x$ , we get

\begin{align*} 1-\exp \left(\!-\tilde \pi \left(-\sum _{k=m}^{r} \frac 1k, -\sum _{k=m+1}^{r}\frac 1k\right]\right) & \geq \beta m^{-\gamma }r^{\gamma -1}. \end{align*}

The calculation giving (10) is analogous.

Proof of Lemma 14 (ii). For $\beta \lt \beta '\lt \beta ''\lt \tilde \beta$ we have

\begin{align*} \tilde \pi \left ( -\infty , -\sum _{k=n_0+1}^{r}\frac 1k\right] & = \frac {\tilde \beta }{1-\gamma }\exp \left ({-(1-\gamma )\sum _{k=n_0+1}^{r} \frac 1k} \right)\\ & \geq \frac {\tilde \beta }{1-\gamma } \exp \left (-(1-\gamma ) \left(\log \left(\tfrac {r}{n_0}\right) + \tfrac {C}{n_0}\right) \right)\\[2mm] & \geq \frac {\beta ''}{1-\gamma } n_0^{1-\gamma }r^{\gamma -1} \geq \beta ' \sum _{m=1}^{n_0} m^{-\gamma }r^{\gamma -1}, \end{align*}

for some constant $C\gt 0$ and $n_0\in \mathbb{N}$ sufficiently large such that $\tilde \beta e^{-(1-\gamma )C/n_0}\ge \beta ''$ . Hence

\begin{equation*}1-\exp \left(-\tilde \pi \left( -\infty , -\sum _{k=n_0+1}^{r}\frac 1k\right]\right) \geq 1- \prod _{m=1}^{n_0} e^{-\beta ' m^{-\gamma }r^{\gamma -1}}.\end{equation*}

Using that $e^{-x}\le 1-(\beta /\beta ')x$ for sufficiently small $x$ the result follows.

3.2 Proof of Proposition 13

This section is concerned with the large deviations results for the killed branching random walk. The results here are rough versions of the very fine asymptotic results presented in [Reference Aïdékon, Hu and Zindy1]. The key difference is that our branching random walk has an infinite offspring distribution so that the moment requirements that are crucially used in [Reference Aïdékon, Hu and Zindy1] are not satisfied here. Instead we use the moment bound on

(11) \begin{equation} W_n=\sum _{|x|=n} e^{-\rho _- V(x)}, \end{equation}

which we received in Lemmas 7 and 8 exploiting the Poisson property of the offspring distribution. Recall that $\mathbb{P}_u, \mathbb{E}_u$ refer to initial particles in position $\log u$ . Shifting $\log u$ to the origin in Lemma 8 we get, for $1\lt p\lt \rho _+/\rho _-$ that

(12) \begin{equation} \mathbb{E}_u \big [W_n^p\big ] \leq C u^{-p\rho _-}, \end{equation}

for some constant $C\gt 0$ . We now look at several generations and allow the starting point to be a uniform random variable, note that the expectation $\mathbb{E}$ refers to exactly this situation.

Lemma 15. For $1\lt p\lt \rho _+/\rho _-$ we have, for all $N\in \mathbb{N}$ , that

\begin{equation*}\mathbb{E} \left[\left(\sum _{n=1}^N W_n \right)^p\right] \leq C N^{p+1}.\end{equation*}

Proof. We first estimate

\begin{equation*}\left( \sum _{n=1}^N W_n \right)^p \leq N^p \max _{n=1}^N W_n^p \leq N^p \sum _{n=1}^N W_n^p.\end{equation*}

By (12) we infer from this that

\begin{equation*}\mathbb{E}_u\left[\left( \sum _{n=1}^N W_n \right)^p \right] \leq N^p \sum _{n=1}^N \mathbb{E}_u\big [W_n^p\big ] \leq C N^{p+1} u^{-p\rho _-}.\end{equation*}

Now take $u$ uniformly random, split according to its value and apply the above, to get

\begin{align*} \mathbb{E} \left[\left( \sum _{n=1}^N W_n \right)^p\right] & \leq \sum _{j=0}^\infty \mathbb{P}\left( j\leq -\log U \lt j+1\right) \mathbb{E}_{e^{-j-1}}\left[ \left( \sum _{n=1}^N W_n \right)^p \right]\\ & \leq C N^{p+1} \sum _{j=0}^\infty e^{-j+p\rho _-(j+1)} \\ & \leq C N^{p+1} \frac {e^{-p\rho _-} }{1-e^{p\rho _-{-}1}}, \end{align*}

as $p\rho _-\lt 1$ . This completes the proof.

The next auxiliary lemma we need for the proof of Proposition 13 is an easy large deviations bound for the position of the leftmost particle in the branching random walk. Recall that $t^*\in (\rho _-, \rho _+)$ is uniquely defined as the solution of

\begin{equation*} \frac 1{t^*} \log \psi (t^*) = \frac {\psi '(t^*)}{\psi (t^*)}.\end{equation*}

Lemma 16. For every $0\lt \delta \lt -\frac {\psi '(t^*)}{\psi (t^*)}$ there exists $J(\delta )\gt 0$ such that

\begin{equation*}\mathbb{P}_1 \left( \min _{|x|=N} V(x) \leq \delta N \right) \leq e^{-N {J(\delta )}}.\end{equation*}

Proof. We pick $\rho _-\lt t\lt t^*$ and by strict convexity of $\log \psi$ we get

\begin{equation*} \frac 1{t} \log \psi (t) \lt \frac {\psi '(t^*)}{\psi (t^*)}.\end{equation*}

Now we use the exponential Chebyshev inequality to get

\begin{align*} \mathbb{P}_1 \left( \min _{|x|=N} V(x) \leq \delta N \right) & \leq \mathbb{P}_1 \left( e^{-t \min \limits _{|x|=N} V(x)} \geq e^{-t\delta N} \right)\\ & \leq e^{t\delta N} \mathbb{E}_1 \left[ \sum _{|x|=N} e^{-t V(x)} \right] =\exp \left ( tN(\delta + \frac 1t \log \psi (t) \right )\\ & \leq \exp \left( tN\left (\delta + \frac {\psi '(t^*)}{\psi (t^*)} \right )\right). \end{align*}

Now we let ${J(\delta )}=-\delta -\frac {\psi '(t^*)}{\psi (t^*)} \gt 0$ to get the desired result.

Combining the last two lemmas gives us the main step in the proof of Proposition 13. Let $\tilde {W}_n$ be as in (11) but with the sum restricted to the particles of the killed branching random walk.

Lemma 17. We have, for every $\epsilon \gt 0$ and all sufficiently large $n$ ,

\begin{equation*}\mathbb{P} \left( \sum _{k=0}^{\infty } {\tilde {W}_k} \ge n^{\rho _-+\epsilon } \right) \leq n^{-\rho _++\epsilon }.\end{equation*}

Proof. We fix $N\in \mathbb{N}$ arbitrarily and determine the value we require later. Then we split the left-hand side according to survival up to generation $N$ . This yields

\begin{align*} {\mathbb{P} \left( \sum _{k=0}^{\infty } \tilde {W}_k \ge n^{\rho _-+\epsilon } \right) } & \leq \mathbb{P} \left( \sum _{k=0}^{N-1} W_k \ge n^{\rho _-+\epsilon } \right) + \mathbb{P} \left ( {\tilde {W}_N}\gt 0 \right ) \\ & \leq n^{-p(\rho _-+\epsilon )} \mathbb{E} \left[ \left( \sum _{k=0}^{N-1} W_k \right)^p \right] + \mathbb{P} \left( \min _{|x|=N} V(x) \leq 0 \right) \\ & \leq CN^{p+1} \, n^{-p(\rho _-+\epsilon )} + \mathbb{P}_1 \left( \min _{|x|=N} V(x) \leq X \right), \end{align*}

using Lemma 15 in the last step. We pick $0\lt \delta \lt -\frac {\psi '(t^*)}{\psi (t^*)}$ and get, by Lemma 16,

\begin{align*} \mathbb{P}_1 \bigg( \min _{|x|=N} V(x) \leq X \bigg) & \leq \mathbb{P}_1 \bigg( \min _{|x|=N} V(x) \leq \delta N \bigg) + \mathbb{P}_1 \big ( X \geq \delta N\big ) \leq e^{-N ({J(\delta )}+\delta )}. \end{align*}

Setting $N =\lceil (\log n) \frac {\rho _+}{{J(\delta )}+\delta } \rceil$ completes the proof.

Proof of Proposition 13 (a). Let $\mathscr{Z}_k$ be the set of locations of the $k$ th generation in the killed branching random walk. Then if there is $x \in \mathscr{Z}_k$ with $V(x) \leq -(1-\epsilon )\log n$ we have ${\tilde {W}_k} \ge e^{\rho _- (1-\epsilon )\log n}=n^{\rho _- (1-\epsilon )}$ . Hence

\begin{align*} \mathbb{P}\left ( \exists x \in \mathscr{T}\,(\!-X) \text{ with } V(x) \leq -(1-\epsilon )\log n \right ) \leq \mathbb{P}\left( \sum _{k=0}^\infty {\tilde {W}_k} \geq n^{\rho _- (1-\epsilon )} \right), \end{align*}

so that the required bound holds by Lemma 17.

Proof of Proposition 13 (b). We first replace the total population size of the killed branching random by the sum of weighted particles with the same starting point,

\begin{equation*}|\mathscr{T}_{0,1}(-X)| \leq \sum _{n=0}^\infty {\tilde {W}_n},\end{equation*}

using that in the sum defining $\tilde {W}_n$ all particles located to the left of the origin get weight at least one. Hence

\begin{align*} \mathbb{P} \left ( |\mathscr{T}_{0,1}(-X)| \ge n^{\rho _-+\epsilon } \right) & \leq \mathbb{P} \left( \sum _{n=0}^{\infty } {\tilde {W}_n} \ge n^{\rho _-+\epsilon } \right), \end{align*}

and again the required bound holds by Lemma 17.