# The parabolic Anderson model
on a Galton-Watson tree revisited

F. den Hollander 111 Mathematical Institute, Leiden University, P.O. Box 9512,
2300 RA Leiden, The Netherlands.
Email<EMAIL_ADDRESS><EMAIL_ADDRESS>
D. Wang 11footnotemark: 1

###### Abstract

In [1] a detailed analysis was given of the large-time asymptotics of the
total mass of the solution to the parabolic Anderson model on a supercritical
Galton-Watson random tree with an i.i.d. random potential whose marginal
distribution is double-exponential. Under the assumption that the degree
distribution has bounded support, two terms in the asymptotic expansion were
identified under the quenched law, i.e., conditional on the realisation of the
random tree and the random potential. The second term contains a variational
formula indicating that the solution concentrates on a subtree with minimal
degree according to a computable profile. The present paper extends the
analysis to degree distributions with unbounded support. We identify the
weakest condition on the tail of the degree distribution under which the
arguments in [1] can be pushed through. To do so we need to control the
occurrence of large degrees uniformly in large subtrees of the Galton-Watson
tree.

MSC2010: 60H25, 82B44, 05C80.

Keywords: Parabolic Anderson model, Galton-Watson tree, double-exponential
distribution, quenched Lyapunov exponent, variational formula.

Acknowledgment: The work in this paper was supported through Gravitation-grant
NETWORKS-024.002.003 of the Netherlands Organisation for Scientific Research
(NWO). The authors thank Götz Kersting and Anton Wakolbinger for helpful
discussions on large deviation properties of the Galton-Watson process.

##  1 Introduction and main results

Section 1.1 provides a brief introduction to the parabolic Anderson model.
Section 1.2 introduces basic notation and key assumptions. Section 1.3 states
the main theorem and gives an outline of the remainder of the paper.

###  1.1 The PAM and intermittency

The _parabolic Anderson model_ (PAM) is the Cauchy problem

$\partial_{t}u(x,t)=\Delta_{\mathscr{X}}u(x,t)+\xi(x)u(x,t),\qquad
t>0,\,x\in\mathscr{X},$ (1.1)

where $\mathscr{X}$ is an ambient space, $\Delta_{\mathscr{X}}$ is a Laplace
operator acting on functions on $\mathscr{X}$, and $\xi$ is a random potential
on $\mathscr{X}$. Most of the literature considers the setting where
$\mathscr{X}$ is either $\mathbb{Z}^{d}$ or $\mathbb{R}^{d}$ with $d\geq 1$
(for mathematical surveys we refer the reader to [3], [9]). More recently,
other choices for $\mathscr{X}$ have been considered as well: the complete
graph [5], the hypercube [2], Galton-Watson trees [1], and random graphs with
prescribed degrees [1].

The main target for the PAM is a description of _intermittency_ : for large
$t$ the solution $u(\cdot,t)$ of (1.1) concentrates on well-separated regions
in $\mathscr{X}$, called _intermittent islands_. Much of the literature has
focussed on a detailed description of the size, shape and location of these
islands, and the profiles of the potential $\xi(\cdot)$ and the solution
$u(\cdot,t)$ on them. A special role is played by the case where $\xi$ is an
i.i.d. random potential with a _double-exponential_ marginal distribution

$\mathrm{P}(\xi(0)>u)=\mathrm{e}^{-\mathrm{e}^{u/\varrho}},\qquad
u\in\mathbb{R},$ (1.2)

where $\varrho\in(0,\infty)$ is a parameter. This distribution turns out to be
critical, in the sense that the intermittent islands neither grow nor shrink
with time, and therefore represents a class of its own.

The analysis of intermittency typically starts with a computation of the
large-time asymptotics of the total mass, encapsulated in what are called
_Lyapunov exponents_. There is an important distinction between the _annealed_
setting (i.e., averaged over the random potential) and the _quenched_ setting
(i.e., almost surely with respect to the random potential). Often both types
of Lyapunov exponents admit explicit descriptions in terms of _characteristic
variational formulas_ that contain information about where and how the mass
concentrates in $\mathscr{X}$. These variational formulas contain a _spatial
part_ (identifying where the concentration on islands takes place) and a
_profile part_ (identifying what the size and shape of both the potential and
the solution are on the islands).

In the present paper we focus on the case where $\mathscr{X}$ is a Galton-
Watson tree, in the quenched setting (i.e., almost surely with respect to the
random tree and the random potential). In [1] the large-time asymptotics of
the total mass was derived under the assumption that the degree distribution
has bounded support. The goal of the present paper is to relax this assumption
to unbounded degree distributions. In particular, we identify the _weakest
condition on the tail of the degree distribution_ under which the arguments in
[1] can be pushed through. To do so we need to control the occurrence of large
degrees _uniformly in large subtrees_ of the Galton-Watson tree.

###  1.2 The PAM on a graph

We begin with some basic definitions and notations (and refer the reader to
[3], [9] for more background).

Let $G=(V,E)$ be a _simple connected undirected_ graph, either finite or
countably infinite. Let $\Delta_{G}$ be the Laplacian on $G$, i.e.,

$(\Delta_{G}f)(x):=\sum_{{y\in V:}\atop{\\{x,y\\}\in E}}[f(y)-f(x)],\qquad
x\in V,\,f\colon\,V\to\mathbb{R}.$ (1.3)

Our object of interest is the non-negative solution of the Cauchy problem with
localised initial condition,

$\begin{array}[]{llll}\partial_{t}u(x,t)&=&(\Delta_{G}u)(x,t)+\xi(x)u(x,t),&x\in
V,\,t>0,\\\ u(x,0)&=&\delta_{\mathcal{O}}(x),&x\in V,\end{array}$ (1.4)

where ${\mathcal{O}}\in V$ is referred to as the _root_ of $G$. We say that
$G$ is _rooted at_ ${\mathcal{O}}$ and call $G=(V,E,{\mathcal{O}})$ a _rooted
graph_. The quantity $u(x,t)$ can be interpreted as the amount of mass present
at time $t$ at site $x$ when initially there is unit mass at ${\mathcal{O}}$.

Criteria for existence and uniqueness of the non-negative solution to (1.4)
are well-known (see [6], [7] for the case $G=\mathbb{Z}^{d}$), and rely on the
_Feynman-Kac formula_

$u(x,t)={\mathbb{E}}_{\mathcal{O}}\left[\mathrm{e}^{\int_{0}^{t}\xi(X_{s})\text{\rm
d}\mkern
0.5mus}\,{\mathchoice{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.5mu\mathrm{l}}{1\mskip-5.0mu\mathrm{l}}}\\{X_{t}=x\\}\right],$
(1.5)

where $X=(X_{t})_{t\geq 0}$ is the continuous-time random walk on the vertices
$V$ with jump rate $1$ along the edges $E$, and ${\mathbb{P}}_{\mathcal{O}}$
denotes the law of $X$ given $X_{0}={\mathcal{O}}$. We are interested in the
_total mass_ of the solution,

$U(t):=\sum_{x\in
V}u(x,t)={\mathbb{E}}_{\mathcal{O}}\left[\mathrm{e}^{\int_{0}^{t}\xi(X_{s})\text{\rm
d}\mkern 0.5mus}\right].$ (1.6)

Often we suppress the dependence on $G,\xi$ from the notation. Note that, by
time reversal and the linearity of (1.4), $U(t)=\hat{u}(0,t)$ with $\hat{u}$
the solution of (1.4) with a different initial condition, namely,
$\hat{u}(x,0)=1$ for all $x\in V$.

As in [1], throughout the paper we assume that the random potential
$\xi=(\xi(x))_{x\in V}$ consists of i.i.d. random variables with marginal
distribution satisfying:

######  Assumption 1.1.

[Asymptotic double-exponential potential]
For some $\varrho\in(0,\infty)$,

$\mathrm{P}\left(\xi(0)\geq
0\right)=1,\qquad\mathrm{P}\left(\xi(0)>u\right)=\mathrm{e}^{-\mathrm{e}^{u/\varrho}}\;\;\text{for
}u\text{ large enough.}$ (1.7)

The restrictions in (1.7) are helpful to avoid certain technicalities that
require no new ideas. In particular, (1.7) is enough to guarantee existence
and uniqueness of the non-negative solution to (1.4) on any discrete graph
with at most exponential growth (as can be inferred from the proof in [7] for
the case $G=\mathbb{Z}^{d}$). All our results remain valid under milder
restrictions (e.g. [7, Assumption (F)] plus an integrability condition on the
lower tail of $\xi(0)$).

The following _characteristic variational formula_ is important for the
description of the asymptotics of $U(t)$ when $\xi$ has a double-exponential
tail. Denote by ${\mathcal{P}}(V)$ the set of probability measures on $V$. For
$p\in{\mathcal{P}}(V)$, define

$I_{E}(p):=\sum_{\\{x,y\\}\in
E}\left(\sqrt{p(x)}-\sqrt{p(y)}\,\right)^{2},\qquad J_{V}(p):=-\sum_{x\in
V}p(x)\log p(x),$ (1.8)

and set

$\chi_{G}(\varrho):=\inf_{p\in{\mathcal{P}}(V)}[I_{E}(p)+\varrho
J_{V}(p)],\qquad\varrho\in(0,\infty).$ (1.9)

The first term in (1.9) is the quadratic form associated with the Laplacian,
describing the solution $u(\cdot,t)$ in the intermittent islands, while the
second term in (1.9) is the Legendre transform of the rate function for the
potential, describing the highest peaks of $\xi(\cdot)$ in the intermittent
islands.

###  1.3 The PAM on a Galton-Watson tree

Let $D$ be a random variable taking values in $\mathbb{N}$. Start with a root
vertex ${\mathcal{O}}$, and attach edges from ${\mathcal{O}}$ to $D$ first-
generation vertices. Proceed recursively: after having attached the $n$-th
generation of vertices, attach to each one of them independently a number of
vertices that has distribution $D$, and declare the union of these vertices to
be the $(n+1)$-th generation of vertices. Denote by
${\mathcal{G}\mathcal{W}}=(V,E)$ the graph thus obtained and by $\mathfrak{P}$
its probability law. Write $\mathcal{P}$ and $\mathcal{E}$ to denote
probability and expectation for $D$, and $\operatorname{supp}(D)$ to denote
the support of $\mathcal{P}$. The law of $D$ is the offspring distribution of
${\mathcal{G}\mathcal{W}}$, the law of $D$ is the degree distribution of
${\mathcal{G}\mathcal{W}}$.

Throughout the paper, we assume that the degree distribution satisfies:

######  Assumption 1.2.

[Exponential tails]
(1) $d_{\min}:=\min\operatorname{supp}(D)\geq 2$ and
$\mathcal{E}[D]\in(2,\infty)$.
(2) $\mathcal{E}\big{[}\mathrm{e}^{aD}\big{]}<\infty$ for all
$a\in(0,\infty)$.

Under this assumption, ${\mathcal{G}\mathcal{W}}$ is $\mathfrak{P}$-a.s. an
infinite tree. Moreover,

$\lim_{r\to\infty}\frac{\log|B_{r}({\mathcal{O}})|}{r}=\log\mathcal{E}[D]=:\vartheta\in(0,\infty)\qquad\mathfrak{P}-a.s.,$
(1.10)

where $B_{r}({\mathcal{O}})\subset V$ is the ball of radius $r$ around
${\mathcal{O}}$ in the graph distance (see e.g. [10, pp. 134–135]). Note that
this ball depends on ${\mathcal{G}\mathcal{W}}$ and therefore is random. For
our main result we need an assumption that is much stronger than Assumption
1.2(2).

######  Assumption 1.3.

[Super-double-exponential tails] There exists a function
$f\colon\,(0,\infty)\to(0,\infty)$ satisfying $\lim_{s\to\infty}f(s)=0$ and
$\lim_{s\to\infty}f^{\prime}(s)=0$ such that

$\limsup_{s\to\infty}\mathrm{e}^{-s}\log\mathcal{P}(D>s^{f(s)})<-2\vartheta.$
(1.11)

To state our main result, we define the constant

$\widetilde{\chi}(\varrho):=\inf\big{\\{}\chi_{T}(\varrho)\colon\,T\text{ is
an infinite tree with degrees in }\operatorname{supp}(D)\big{\\}},$ (1.12)

with $\chi_{G}(\varrho)$ defined in (1.9), and abbreviate

$\mathfrak{r}_{t}=\frac{\varrho t}{\log\log t}.$ (1.13)

######  Theorem 1.4.

[Quenched Lyapunov exponent] Subject to Assumptions 1.1–1.3,

$\frac{1}{t}\log
U(t)=\varrho\log(\vartheta\mathfrak{r}_{t})-\varrho-\widetilde{\chi}(\varrho)+o(1),\quad
t\to\infty,\qquad(\mathrm{P}\times\mathfrak{P})\text{-a.s.}$ (1.14)

With Theorem 1.4 we have completed our task to relax the main result in [1] to
degree distributions with unbounded support. The extension comes at the price
of having to assume a tail that decays faster than double-exponential as shown
in (1.11). This property is needed to control the occurrence of large degrees
_uniformly in large subtrees_ of ${\mathcal{G}\mathcal{W}}$. No doubt
Assumption 1.3 is stronger than is needed, but to go beyond would require a
major overhaul of the methods developed in [1], which remains a challenge.

In (1.4) the initial mass is located at the root. The asymptotics in (1.14) is
robust against different choices.

A heuristic explanation where the terms in (1.14) come from was given in [1,
Section 1.5]. The asymptotics of $U(t)$ is controlled by random walk paths in
the Feynman-Kac formula in (1.6) that run within time
$\mathfrak{r}_{t}/\varrho\log\mathfrak{r}_{t}$ to an _intermittent island_ at
distance $\mathrm{r}_{t}$ from ${\mathcal{O}}$, and afterwards stay near that
island for the rest of the time. The intermittent island turns out to consist
of a subtree with degree $d_{\min}$ where the potential has a height
$\varrho\log(\vartheta\mathfrak{r}_{t})$ and a shape that is the solution of a
variational formula restricted to that subtree. The first and third term in
(1.14) are the contribution of the path after it has reached the island, the
second term is the cost for reaching the island.

For $d\in\mathbb{N}\setminus\\{1\\}$, let ${\mathcal{T}}_{d}$ be the infinite
homogeneous tree in which every node has downward degree $d$. It was shown in
[1] that if $\varrho\geq 1/\log(d_{\rm min}+1)$, then

$\widetilde{\chi}(\varrho)=\chi_{{\mathcal{T}}_{d_{\min}}}(\varrho).$ (1.15)

Presumably ${\mathcal{T}}_{d_{\min}}$ is the _unique_ minimizer of (1.12), but
proving so would require more work.

#### Outline.

The remainder of the paper is organised as follows. Section 2 collects some
structural properties of Galton-Watson trees. Section 3 contains several
preparatory lemmas, which identify the maximum size of the islands where the
potential is suitably high, estimate the contribution to the total mass in
(1.6) by the random walk until it exits a subset of
${\mathcal{G}\mathcal{W}}$, bound the principal eigenvalue associated with the
islands, and estimate the number of locations where the potential is
intermediate. Section 4 uses these preparatory lemmas to find the contribution
to the Feynman-Kac formula in (1.6) coming from various sets of paths. Section
5 uses these contributions to prove Theorem 1.4. Appendices A–B contain some
facts about variational formulas and largest eigenvalues that are needed in
Section 3.

Assumptions 1.1–1.2 are needed throughout the paper. Only in Sections 4–5 do
we need Assumption 1.3.

##  2 Structural properties of the Galton-Watson tree

In the section we collect a few structural properties of
${\mathcal{G}\mathcal{W}}$ that play an important role throughout the paper.
None of these properties was needed in [1]. Section 2.1 looks at volumes,
Section 2.2 at degrees, Section 2.3 at tree animals.

###  2.1 Volumes

Let $Z_{k}$ be the number of offspring in generation $k$, i.e.,

$Z_{k}=|\\{x\in V\colon\,d(x,{\mathcal{O}})=k\\}|,$ (2.1)

where $d(x,{\mathcal{O}})$ is the distance from ${\mathcal{O}}$ to $x$. Let
$\mu=\mathcal{E}[D]$. Then there exists a random variable $W\in(0,\infty)$
such that

$W_{k}:=\mathrm{e}^{-k\vartheta}Z_{k}=\mu^{-k}Z_{k}\rightarrow
W\qquad\mathfrak{P}\text{-a.s. as }k\to\infty.$ (2.2)

It is shown in [13, Theorem 5] that

$\exists\,C<\infty,c>0\colon\,\quad\mathfrak{P}(|W_{k}-W|\geq\varepsilon)\leq
C\mathrm{e}^{-c\,\varepsilon^{2/3}\mu^{n/3}}\qquad\forall\,\varepsilon>0,\,k\in\mathbb{N}.$
(2.3)

In addition, it is shown in [14, Theorems 2–3] that if $D$ is bounded, then

$\displaystyle-\log\mathfrak{P}(W\geq x)$ $\displaystyle=$ $\displaystyle
x^{\gamma^{+}/(\gamma^{+}-1)}\,[L^{+}(x)+o(1)],\qquad x\to\infty,$ (2.4)
$\displaystyle-\log\mathfrak{P}(W\leq x)$ $\displaystyle=$ $\displaystyle
x^{-\gamma^{-}/(1-\gamma^{-})}\,[L^{-}(x)+o(1)],\qquad x\downarrow 0,$ (2.5)

where $\gamma^{+}\in(1,\infty)$ and $\gamma^{-}\in(0,1)$ are the unique
solutions of the equations

$\mu^{\gamma^{+}}=d_{\max},\qquad\mu^{\gamma^{-}}=d_{\min},$ (2.6)

with $L^{+},L^{-}\colon(0,\infty)\to(0,\infty)$ real-analytic functions that
are multiplicatively periodic with period $\mu^{\gamma^{+}-1}$, respectively,
$\mu^{1-\gamma^{-}}$. Note that Assumption 1.2(1) guarantees that
$\gamma^{-}\neq 1$.

The tail behaviour in (2.4) requires that $d_{\max}<\infty$. In our setting we
have $d_{\max}=\infty$, which corresponds to $\gamma^{+}=\infty$, and so we
expect exponential tail behaviour. The following lemma provides a rough bound.

######  Lemma 2.1.

[Exponential tail for generation sizes] If there exists an $a>0$ such that
$\mathcal{E}[\mathrm{e}^{aD}]<\infty$, then there exists an $a_{*}>0$ such
that $\mathfrak{E}[\mathrm{e}^{a_{*}W}]<\infty$.

###### Proof.

First note that if there exists an $a>0$ such that
$\mathcal{E}[\mathrm{e}^{aD}]<\infty$, then there exist $b>0$ large and $c>0$
small such that

$\varphi(a):=\mathcal{E}[\mathrm{e}^{aD}]\leq\mathrm{e}^{\mu
a+ba^{2}}\qquad\forall\,0<a<c.$ (2.7)

Hence

$\mathfrak{E}[\mathrm{e}^{aZ_{n+1}}]=\mathfrak{E}[\varphi(a)^{Z_{n}}]\leq\mathfrak{E}[\mathrm{e}^{(\mu
a+ba^{2})Z_{n}}]$ (2.8)

and consequently, because $\mu>1$,

$\mathfrak{E}\left[\mathrm{e}^{aW_{n+1}}\right]\leq\mathfrak{E}\left[\mathrm{e}^{(a+ba^{2}\mu^{-(n+2)})W_{n}}\right]\leq\mathfrak{E}\left[\mathrm{e}^{a\exp(bc\mu^{-(n+2)})W_{n}}\right].$
(2.9)

Put $a_{n}:=c\exp(-bc\sum_{k=0}^{n-1}\mu^{-(k+2)})$, which satisfies
$0<a_{n}\leq c$. From the last inequality in (2.9) it follows that

$\mathfrak{E}\left[\mathrm{e}^{a_{n+1}W_{n+1}}\right]\leq\mathfrak{E}\left[\mathrm{e}^{a_{n}W_{n}}\right].$
(2.10)

Since $n\mapsto a_{n}$ is decreasing with $\lim_{n\to\infty}a_{n}=a_{*}>0$,
Fatou’s lemma gives

$\mathfrak{E}\left[\mathrm{e}^{a_{*}W}\right]\leq\mathfrak{E}\left[\mathrm{e}^{a_{0}W_{0}}\right].$
(2.11)

Because $\mathcal{E}[\mathrm{e}^{a_{0}W_{0}}]=\mathrm{e}^{a_{0}}<\infty$, we
get the claim. ∎

The following lemma says that $\mathfrak{P}$-a.s. a ball of radius $R_{r}$
centred anywhere in $B_{r}({\mathcal{O}})$ has volume $\mathrm{e}^{\vartheta
R_{r}+o(R_{r})}$ as $r\to\infty$, provided $R_{r}$ is large compared to $\log
r$.

######  Lemma 2.2.

[Volumes of large balls] Subject to Assumption 1.2(1), if there exists an
$a>0$ such that $\mathcal{E}[\mathrm{e}^{aD}]<\infty$, then for any $R_{r}$
satisfying $\lim_{r\to\infty}R_{r}/\log r=\infty$,

$\liminf_{r\to\infty}\frac{1}{R_{r}}\log\Big{(}\inf_{x\in
B_{r}({\mathcal{O}})}|B_{R_{r}}(x)|\Big{)}=\limsup_{r\to\infty}\frac{1}{R_{r}}\log\Big{(}\sup_{x\in
B_{r}({\mathcal{O}})}|B_{R_{r}}(x)|\Big{)}=\vartheta\qquad\mathfrak{P}-a.s.$
(2.12)

###### Proof.

We first prove the claim for lower balls. Afterwards we use a sandwich
argument to get the claim for balls.

For $y\in{\mathcal{G}\mathcal{W}}$ that lies $k$ generations below
${\mathcal{O}}$, let $y[-i]$, $0\leq i\leq k$ be the vertex that lies $i$
generations above $y$. Define the _lower ball_ of radius around $y$ as

$B^{\downarrow}_{r}(y):=\\{x\in V\colon\,\exists\,0\leq i\leq r\ \text{with}\
x[-i]=y\\}.$ (2.13)

Note that $B^{\downarrow}_{r}({\mathcal{O}})=B_{r}({\mathcal{O}})$. Let
$\mathcal{Z}_{k}$ denote the vertices in the $k$-th generation. To get the
upper bound, pick $\delta>0$ and estimate

$\displaystyle\mathfrak{P}\Big{(}\sup_{x\in
B_{r}({\mathcal{O}})}|B_{R_{r}}^{\downarrow}(x)|\geq\mathrm{e}^{(1+\delta)\vartheta
R_{r}}\Big{)}\leq\sum_{k=0}^{r}\mathfrak{P}\Big{(}\sup_{x\in\mathcal{Z}_{k}}|B_{R_{r}}^{\downarrow}(x)|\geq\mathrm{e}^{(1+\delta)\vartheta
R_{r}}\Big{)}$ (2.14)
$\displaystyle\quad=\sum_{k=0}^{r}\sum_{l\in\mathbb{N}}\mathfrak{P}\Big{(}\sup_{x\in\mathcal{Z}_{k}}|B_{R_{r}}^{\downarrow}(x)|\geq\mathrm{e}^{(1+\delta)\vartheta
R_{r}}\leavevmode\nobreak\ \Big{|}\leavevmode\nobreak\
Z_{k}=l\Big{)}\mathfrak{P}(Z_{k}=l)$
$\displaystyle\quad\leq\sum_{k=0}^{r}\sum_{l\in\mathbb{N}}l\
\mathfrak{P}\Big{(}|B^{\downarrow}_{R_{r}}({\mathcal{O}})|\geq\mathrm{e}^{(1+\delta)\vartheta
R_{r}}\Big{)}\mathfrak{P}(Z_{k}=l)$
$\displaystyle\quad=\mathfrak{P}\Big{(}|B^{\downarrow}_{R_{r}}({\mathcal{O}})|\geq\mathrm{e}^{(1+\delta)\vartheta
R_{r}}\Big{)}\sum_{k=0}^{r}\mathfrak{E}(Z_{k}).$

By (1.10),
$\sum_{k=0}^{r}\mathfrak{E}(Z_{k})=\frac{\mathrm{e}^{\vartheta(r+1)}-1}{\mathrm{e}^{\vartheta}-1}=O(\mathrm{e}^{\vartheta
r})$, and so in order to be able to apply the Borel-Cantelli lemma, it
suffices to show that the probability in the last line decays faster than
exponentially in $r$ for any $\delta>0$. To that end, estimate

$\displaystyle\mathfrak{P}\Big{(}|B^{\downarrow}_{R_{r}}({\mathcal{O}})|\geq\mathrm{e}^{(1+\delta)\vartheta
R_{r}}\Big{)}=\mathfrak{P}\Big{(}\mathrm{e}^{-\vartheta
R_{r}}\sum_{k=0}^{R_{r}}Z_{k}\geq\mathrm{e}^{\delta\vartheta R_{r}}\Big{)}$
(2.15)
$\displaystyle\quad=\mathfrak{P}\Big{(}\sum_{k=0}^{R_{r}}W_{k}\,\mathrm{e}^{-\vartheta(R_{r}-k)}\geq\mathrm{e}^{\delta\vartheta
R_{r}}\Big{)}\leq\sum_{k=0}^{R_{r}}\mathfrak{P}\Big{(}W_{k}\,\mathrm{e}^{-\vartheta(R_{r}-k)}\geq\frac{1}{R_{r}+1}\mathrm{e}^{\delta\vartheta
R_{r}}\Big{)}$
$\displaystyle\quad=\sum_{k=0}^{R_{r}}\mathfrak{P}\Big{(}W+(W_{k}-W)\geq\frac{1}{R_{r}+1}\mathrm{e}^{\delta\vartheta
R_{r}}\mathrm{e}^{\vartheta(R_{r}-k)}\Big{)}$
$\displaystyle\quad\leq\sum_{k=0}^{R_{r}}\mathfrak{P}\Big{(}W\geq\frac{1}{2(R_{r}+1)}\mathrm{e}^{\delta\vartheta
R_{r}}\mathrm{e}^{\vartheta(R_{r}-k)}\Big{)}$
$\displaystyle\qquad\qquad+\sum_{k=0}^{R_{r}}\mathfrak{P}\Big{(}|W_{k}-W|\geq\frac{1}{2(R_{r}+1)}\mathrm{e}^{\delta\vartheta
R_{r}}\mathrm{e}^{\vartheta(R_{r}-k)}\Big{)}$
$\displaystyle\quad\leq\mathfrak{E}[\mathrm{e}^{a_{*}W}]\sum_{k=0}^{R_{r}}\exp\Big{(}-a_{*}\frac{1}{2(R_{r}+1)}\mathrm{e}^{\delta\vartheta
R_{r}}\mathrm{e}^{\vartheta(R_{r}-k)}\Big{)}$
$\displaystyle\qquad\qquad+\sum_{k=0}^{R_{r}}C\exp\Big{(}-c\Big{[}\frac{1}{2(R_{r}+1)}\,\mathrm{e}^{\delta\vartheta
R_{r}}\,\mathrm{e}^{\vartheta(R_{r}-k)}\Big{]}^{2/3}(\mathrm{e}^{\vartheta})^{k/3}\Big{)}$
$\displaystyle\quad\leq\mathfrak{E}[\mathrm{e}^{a_{*}W}](R_{r}+1)\exp\Big{(}-a_{*}\frac{1}{2(R_{r}+1)}\mathrm{e}^{\delta\vartheta
R_{r}}\Big{)}$
$\displaystyle\qquad\qquad+C(R_{r}+1)\exp\Big{(}-c\Big{[}\frac{1}{2(R_{r}+1)}\,\mathrm{e}^{\delta\vartheta
R_{r}}\Big{]}^{2/3}\Big{)},$

where we use (2.3) with $\mu=\mathrm{e}^{\vartheta}$. This produces the
desired estimate.

To get the lower bound, pick $0<\delta<1$ and estimate

$\displaystyle\mathfrak{P}\Big{(}\inf_{x\in
B_{r}({\mathcal{O}})}|B_{R_{r}}^{\downarrow}(x)|\leq\mathrm{e}^{(1-\delta)\vartheta
R_{r}}\Big{)}\leq\sum_{k=0}^{r}\mathfrak{P}\Big{(}\inf_{x\in\mathcal{Z}_{k}}|B_{R_{r}}^{\downarrow}(x)|\leq\mathrm{e}^{(1-\delta)\vartheta
R_{r}}\Big{)}$ (2.16)
$\displaystyle\quad=\sum_{k=0}^{r}\sum\limits_{l\in\mathbb{N}}\mathfrak{P}\Big{(}\inf_{x\in\mathcal{Z}_{k}}|B_{R_{r}}^{\downarrow}(x)|\leq\mathrm{e}^{(1-\delta)\vartheta
R_{r}}\leavevmode\nobreak\ \Big{|}\leavevmode\nobreak\
Z_{k}=l\Big{)}\mathfrak{P}(Z_{k}=l)$
$\displaystyle\quad\leq\sum_{k=0}^{r}\sum\limits_{l\in\mathbb{N}}l\,\mathfrak{P}\Big{(}|B^{\downarrow}_{R_{r}}({\mathcal{O}})|\leq\mathrm{e}^{(1-\delta)\vartheta
R_{r}}\Big{)}\mathfrak{P}(Z_{k}=l)$
$\displaystyle\quad=\mathfrak{P}\Big{(}|B^{\downarrow}_{R_{r}}({\mathcal{O}})|\leq\mathrm{e}^{(1-\delta)\vartheta
R_{r}}\Big{)}\sum\limits_{k=0}^{r}\mathfrak{E}(Z_{k}).$

It again suffices to show that the probability in the last line decays faster
than exponentially in $r$ for any $\delta>0$. To that end, estimate

$\displaystyle\mathfrak{P}\Big{(}|B^{\downarrow}_{R_{r}}({\mathcal{O}})|\leq\mathrm{e}^{(1-\delta)\vartheta
R_{r}}\Big{)}=\mathfrak{P}\Big{(}\mathrm{e}^{-\vartheta
R_{r}}\sum_{k=0}^{R_{r}}Z_{k}\leq\mathrm{e}^{-\delta\vartheta R_{r}}\Big{)}$
(2.17)
$\displaystyle\leq\mathfrak{P}\Big{(}W_{R_{r}}\leq\mathrm{e}^{-\delta\vartheta
R_{r}}\Big{)}\leq\mathfrak{P}(W\leq 2\,\mathrm{e}^{-\delta\vartheta
R_{r}})+\mathfrak{P}(W-W_{R_{r}}\geq\mathrm{e}^{-\delta\vartheta R_{r}})$
$\displaystyle\leq\exp\Big{(}-c^{-}(2\mathrm{e}^{\delta\vartheta
R_{r}})^{\frac{\gamma^{-}}{1-\gamma^{-}}}[1+o(1)]\Big{)}+C\exp\Big{(}-c\,[\mathrm{e}^{-\frac{2}{3}\delta\vartheta}(\mathrm{e}^{\vartheta})^{\frac{1}{3}}]^{R_{r}}\Big{)},$

where we use (2.5), (2.3) with $\mu=\mathrm{e}^{\vartheta}$, and put
$c^{-}:=\inf L^{-}\in(0,\infty)$. For $\delta$ small enough this produces the
desired estimate. This completes the proof of (2.12) for lower balls.

To get the claim for balls, we observe that

$B_{r}^{\downarrow}(x)\subseteq
B_{r}(x)\subseteq\bigcup_{k=0}^{r}B_{r}^{\downarrow}(x[-k]),$ (2.18)

and therefore

$|B_{r}^{\downarrow}(x)|\leq|B_{r}(x)|\leq\sum_{k=0}^{r}|B_{r}^{\downarrow}(x[-k])|.$
(2.19)

It follows from (2.19) that

$\inf_{x\in B_{r}({\mathcal{O}})}|B_{r}^{\downarrow}(x)|\leq\inf_{x\in
B_{r}({\mathcal{O}})}|B_{r}(x)|\leq\sup_{x\in
B_{r}({\mathcal{O}})}|B_{r}(x)|\leq(r+1)\sup_{x\in
B_{r}({\mathcal{O}})}|B_{r}^{\downarrow}(x)|.$ (2.20)

Hence we get (2.12). ∎

###  2.2 Degrees

Write $D_{x}$ to denote the degree of vertex $x$. The following lemma implies
that, $\mathfrak{P}$-a.s. and for $r\to\infty$, $D_{x}$ is bounded by a
vanishing power of $\log r$ for all $x\in B_{2r}({\mathcal{O}})$.

######  Lemma 2.3.

[Maximal degree in a ball around the root]
(a) Subject to Assumption 1.2(2), for every $\delta>0$,

$\sum_{r\in\mathbb{N}}\mathfrak{P}\big{(}\exists\,x\in
B_{2r}({\mathcal{O}})\colon\,D_{x}>\delta r\big{)}<\infty.$ (2.21)

(b) Subject to Assumption 1.3, there exists a function
$\delta_{r}\colon\,(0,\infty)\to(0,\infty)$ satisfying
$\lim_{r\to\infty}\delta_{r}$ $=0$ and
$\lim_{r\to\infty}r\frac{\mathrm{d}}{\mathrm{d}r}=0$ such that

$\sum_{r\in\mathbb{N}}\mathfrak{P}\big{(}\exists\,x\in
B_{2r}({\mathcal{O}})\colon\,D_{x}>(\log r)^{\delta_{r}}\big{)}<\infty.$
(2.22)

###### Proof.

(a) Estimate

$\displaystyle\mathfrak{P}\big{(}\exists\,x\in
B^{\downarrow}_{2r}({\mathcal{O}})\colon\,D_{x}>\delta
r\big{)}\leq\sum_{k=0}^{2r}\mathfrak{P}\big{(}\exists\,x\in\mathcal{Z}_{k}\colon\,D_{x}>\delta
r\big{)}$ (2.23)
$\displaystyle=\sum_{k=0}^{2r}\sum_{l\in\mathbb{N}}\mathfrak{P}\big{(}\exists\,x\in\mathcal{Z}_{k}\colon\,D_{x}>\delta
r\mid Z_{k}=l\big{)}\,\mathfrak{P}(Z_{k}=l)$
$\displaystyle\leq\mathcal{P}(D>\delta
r)\sum_{k=0}^{2r}\sum_{l\in\mathbb{N}}l\,\mathfrak{P}\big{(}Z_{k}=l)=\mathcal{P}(D>\delta
r)\sum_{k=0}^{2r}\mathfrak{E}(Z_{k}).$

Since
$\sum_{k=0}^{2r}\mathfrak{E}(Z_{k})=\frac{\mathrm{e}^{(2r+1)\vartheta}-1}{\mathrm{e}^{\vartheta}-1}=O(\mathrm{e}^{2r\vartheta})$,
it suffices to show that $\mathcal{P}(D>\delta r)=O(\mathrm{e}^{-cr})$ for
some $c>2\vartheta$. Since $\mathcal{P}(D>\delta r)\leq\mathrm{e}^{-a\delta
r}\mathcal{E}(\mathrm{e}^{aD})$, the latter is immediate from Assumption
1.2(2) when we choose $a>2\vartheta/\delta$.
(b) The only change is that in the last line $\mathcal{P}(D>\delta r)$ must be
replaced by $\mathcal{P}(D>(\log r)^{\delta_{r}})$. To see that the latter is
$O(\mathrm{e}^{-cr})$ for some $c>2\vartheta$, we use the tail condition in
(1.11) with $\delta_{r}=f(s)$ and $s=\log r$. ∎

###  2.3 Tree animals

For $n\in\mathbb{N}_{0}$ and $x\in B_{r}({\mathcal{O}})$, let

$\mathcal{A}_{n}(x)=\\{\Lambda\subset B_{n}(x)\colon\,\Lambda\text{ is
connected},\Lambda\ni x,|\Lambda|=n+1\\}$ (2.24)

be the set of _tree animals_ of size $n+1$ that contain $x$. Put
$a_{n}(x)=|\mathcal{A}_{n}(x)|$.

######  Lemma 2.4.

[Number of tree animals] Subject to Assumption 1.2(2), $\mathfrak{P}$-a.s.
there exists an $r_{0}\in\mathbb{N}$ such that $a_{n}(x)\leq r^{n}$ for all
$r\geq r_{0}$, $x\in B_{r}({\mathcal{O}})$ and $0\leq n\leq r$.

###### Proof.

We first prove the claim for lower tree animals. Afterwards we us a sandwich
argument to get the claim for tree animals.

For $n\in\mathbb{N}_{0}$ and $x\in B^{\downarrow}_{r}({\mathcal{O}})$, let

$\mathcal{A}^{\downarrow}_{n}(x)=\\{\Lambda\subset
B^{\downarrow}_{n}(x)\colon\,\Lambda\text{ is connected},\Lambda\ni
x,|\Lambda|=n+1\\}$ (2.25)

be the set of _lower tree animals_ of size $n+1$ that contain $x$. Put
$a^{\downarrow}_{n}(x)=|\mathcal{A}^{\downarrow}_{n}(x)|$. Fix $\delta>0$. By
Lemma 2.3(a) and the Borel-Cantelli lemma, $\mathfrak{P}$-a.s. there exists an
$r_{0}=r_{0}(\delta)\in\mathbb{N}$ such that $D_{x}\leq\delta r$ for all $x\in
B^{\downarrow}_{2r}({\mathcal{O}})$. Any lower tree animal of size $n+1$
containing a vertex in $B^{\downarrow}_{r}({\mathcal{O}})$ is contained in
$B^{\downarrow}_{r+n}({\mathcal{O}})$. Any lower tree animal of size $n+1$ can
be created by adding a vertex to the outer boundary of a lower tree animal of
size $n$. This leads to the recursive inequality

$a^{\downarrow}_{n}(x)\leq(\delta r)a^{\downarrow}_{n-1}(x)\qquad\forall\,x\in
B^{\downarrow}_{r}({\mathcal{O}}),\quad 1\leq n\leq r.$ (2.26)

Since $a^{\downarrow}_{0}(x)=1$, it follows that

$a^{\downarrow}_{n}(x)\leq(\delta r)^{n}\qquad\forall\,x\in
B^{\downarrow}_{r}({\mathcal{O}}),\,\,0\leq n\leq r.$ (2.27)

Pick $\delta=1$ to get the claim for lower tree animals.

To get the claim for tree animals, note that
$a_{n}(x)\leq\sum_{k=0}^{n}a^{\downarrow}_{n}(x[-k])$ (compare with (2.19)),
and so $a_{n}(x)\leq(n+1)r^{n}$ for all $x\in B_{r}({\mathcal{O}})$ and all
$0\leq n\leq r$. ∎

##  3 Preliminaries

In this section we extend the lemmas in [1, Section 2]. Section 3.1 identifies
the maximum size of the islands where the potential is suitably high. Section
3.2 estimates the contribution to the total mass in (1.6) by the random walk
until it exits a subset of ${\mathcal{G}\mathcal{W}}$. Section 3.3 gives a
bound on the principal eigenvalue associated with the islands. Section 3.5
estimates the number of locations where the potential is intermediate.

Abbreviate $L_{r}=L_{r}({\mathcal{G}\mathcal{W}})=|B_{r}(\mathcal{O})|$ and
put

$S_{r}:=(\log r)^{\alpha},\qquad\alpha\in(0,1).$ (3.1)

###  3.1 Maximum size of the islands

For every $r\in\mathbb{N}$ there is a unique $a_{r}$ such that

$\mathrm{P}(\xi(0)>a_{r})=\frac{1}{r}.$ (3.2)

By Assumption 1.1, for $r$ large enough

$a_{r}=\varrho\log\log r$ (3.3)

For $r\in\mathbb{N}$ and $A>0$, let

$\Pi_{r,A}=\Pi_{r,A}(\xi):=\\{z\in
B_{r}({\mathcal{O}})\colon\,\xi(z)>a_{L_{r}}-2A\\}$ (3.4)

be the set of vertices in $B_{r}({\mathcal{O}})$ where the potential is close
to maximal,

$D_{r,A}=D_{r,A}(\xi):=\\{z\in
B_{r}({\mathcal{O}})\colon\,\mathrm{dist}(z,\Pi_{r,A})\leq S_{r}\\}$ (3.5)

be the $S_{r}$-neighbourhood of $\Pi_{r,A}$, and $\mathfrak{C}_{r,A}$ be the
set of connected components of $D_{r,A}$ in ${\mathcal{G}\mathcal{W}}$, which
we think of as _islands_. For $M_{A}\in\mathbb{N}$, define the event

$\mathcal{B}_{r,A}:=\big{\\{}\exists\,\mathcal{C}\in\mathfrak{C}_{r,A}\colon\,|\mathcal{C}\cap\Pi_{r,A}|>M_{A}\big{\\}}.$
(3.6)

Note that $\Pi_{r,A},D_{r,A},\mathcal{B}_{r,A}$ depend on
${\mathcal{G}\mathcal{W}}$ and therefore are random.

######  Lemma 3.1.

[Maximum size of the islands] Subject to Assumptions 1.1–1.2, for every $A>0$
there exists an $M_{A}\in\mathbb{N}$ such that

$\sum_{r\in\mathbb{N}}\mathrm{P}(\mathcal{B}_{r,A})<\infty\qquad\mathfrak{P}-a.s.$
(3.7)

###### Proof.

We follow [4, Lemma 6.6]. By Assumption 1.1, for every $x\in V$ and $r$ large
enough,

$\mathrm{P}(x\in\Pi_{r,A})=\mathrm{P}(\xi(x)>a_{L_{r}}-2A)=L_{r}^{-c_{A}}$
(3.8)

with $c_{A}=e^{-2A/\varrho}$. By Lemma 2.2, $\mathfrak{P}$-a.s. for every
$y\in B_{r}({\mathcal{O}})$ and $r$ large enough,

$|B_{S_{r}}(y)|\leq|B_{o(r)}({\mathcal{O}})|=L_{o(r)}=L_{r}^{o(1)},$ (3.9)

where we use that $S_{r}=o(\log r)=o(r)$, and hence for every
$m\in\mathbb{N}$,

$\mathrm{P}(|B_{S_{r}}(y)\cap\Pi_{r,A}|\geq
m)\leq\binom{|B_{S_{r}}(y)|}{m}L_{r}^{-c_{A}m}\leq(|B_{S_{r}}(y)|L_{r}^{-c_{A}})^{m}\leq
L_{r}^{-c_{A}m[1+o(1)]}.$ (3.10)

Consequently, $\mathfrak{P}$-a.s.

$\displaystyle\mathrm{P}(\exists\,\mathcal{C}\in\mathfrak{C}_{r,A}\colon\,|\mathcal{C}\cap\Pi_{r,A}|\geq
m)$ $\displaystyle\leq\mathrm{P}(\exists\,y\in
B_{r}({\mathcal{O}})\colon\,|B_{S_{r}}(y)\cap\Pi_{r,A}|\geq m)$ (3.11)
$\displaystyle\leq|B_{r}({\mathcal{O}})|L_{r}=L_{r}^{(1-c_{A}m)[1+o(1)]}.$

By choosing $m>1/c_{A}$, we see that the above probability becomes summable in
$r$, and so we have proved the claim with $M_{A}=\lceil 1/c_{A}\rceil$. ∎

Lemma 3.1 implies that $(\mathrm{P}\times\mathfrak{P})$-a.s.
$\mathcal{B}_{r,A}$ does not occur eventually as $r\to\infty$. Note that
$\mathfrak{P}$-a.s. on the event $[\mathcal{B}_{r,A}]^{c}$,

$\forall\,\mathcal{C}\in\mathfrak{C}_{r,A}\colon\,|\mathcal{C}\cap\Pi_{r,A}|\leq
M_{A},\,\operatorname{diam}_{\mathcal{G}\mathcal{W}}(\mathcal{C})\leq
2M_{A}S_{r},\,|\mathcal{C}|\leq\mathrm{e}^{2\vartheta M_{A}S_{r}},$ (3.12)

where the last inequality follows from Lemma 2.2.

###  3.2 Mass up to an exit time

######  Lemma 3.2.

[Mass up to an exit time] Subject to Assumption 1.2(2), $\mathfrak{P}$-a.s.
for any $\delta>0$, $r\geq r_{0}$, $y\in\Lambda\subset B_{r}({\mathcal{O}})$,
$\xi\in[0,\infty)^{V}$ and
$\gamma>\lambda_{\Lambda}=\lambda_{\Lambda}(\xi,{\mathcal{G}\mathcal{W}})$,

$\mathbb{E}_{y}\left[\mathrm{e}^{\int_{0}^{\tau_{\Lambda^{\text{\rm
c}}}}(\xi(X_{s})-\gamma)\,\text{\rm d}\mkern 0.5mus}\right]\leq
1+\frac{(\delta r)\,|\Lambda|}{\gamma-\lambda_{\Lambda}}.$ (3.13)

###### Proof.

We follow the proof of [7, Lemma 2.18] and [12, Lemma 4.2]. Define

$u(x):=\mathbb{E}_{x}\left[\mathrm{e}^{\int_{0}^{\tau_{\Lambda^{\text{\rm
c}}}}(\xi(X_{s})-\gamma)\,\text{\rm d}\mkern 0.5mus}\right].$ (3.14)

This is the solution to the boundary value problem

$\displaystyle\begin{split}(\Delta+\xi-\gamma)u&=0\quad\text{on}\ \Lambda\\\
u&=1\quad\text{on}\ \Lambda^{\text{\rm c}}.\end{split}$ (3.15)

Via the substitution $u=:1+v$, this turns into

$\displaystyle\begin{split}(\Delta+\xi-\gamma)v&=\gamma-\xi\quad\text{on}\
\Lambda\\\ v&=0\qquad\quad\text{on}\ \Lambda^{\text{\rm c}}.\end{split}$
(3.16)

It is readily checked that for $\gamma>\lambda_{\Lambda}$ the solution exists
and is given by

$v=\mathcal{R}_{\gamma}(\xi-\gamma),$ (3.17)

where $\mathcal{R}_{\gamma}$ denotes the resolvent of $\Delta+\xi$ in
$\ell^{2}(\Lambda)$ with Dirichlet boundary condition. Hence

$v(x)\leq(\delta r)\,(\mathcal{R}_{\gamma}\mathds{1})(x)\leq(\delta
r)\,\langle\mathcal{R}_{\gamma}\mathds{1},\mathds{1}\rangle_{\Lambda}\leq\frac{(\delta
r)\,|\Lambda|}{\gamma-\lambda_{\Lambda}},\quad x\in\Lambda,$ (3.18)

where $\mathds{1}$ denotes the constant function equal to $1$, and
$\langle\cdot,\cdot\rangle_{\Lambda}$ denotes the inner product in
$\ell^{2}(\Lambda)$. To get the first inequality, we combine Lemma 2.3(a) with
the lower bound in (B.2) from Lemma B.1, to get
$\xi-\gamma\leq\lambda_{\Lambda}+\delta r-\gamma\leq\delta r$ on $\Lambda$.
The positivity of the resolvent gives

$0\leq[\mathcal{R}_{\gamma}(\delta\log r-(\xi-\gamma))](x)=(\delta
r)\,[\mathcal{R}_{\gamma}\mathds{1}](x)-[\mathcal{R}_{\gamma}(\xi-\gamma)](x).$
(3.19)

To get the second inequality, we write

$(\delta r)\,(\mathcal{R}_{\gamma}\mathds{1})(x)\leq(\delta
r)\sum_{x\in\Lambda}(\mathcal{R}_{\gamma}\mathds{1})(x)=(\delta
r)\sum_{x\in\Lambda}(\mathcal{R}_{\gamma}\mathds{1})(x)\mathds{1}(x)=(\delta
r)\,\langle\mathcal{R}_{\gamma}\mathds{1},\mathds{1}\rangle_{\Lambda}.$ (3.20)

To get the third inequality, we use the Fourier expansion of the resolvent
with respect to the orthonormal basis of eigenfunctions of $\Delta+\xi$ in
$\ell^{2}(\Lambda)$. ∎

###  3.3 Principal eigenvalue of the islands

The following lemma provides a spectral bound.

######  Lemma 3.3.

[Principal eigenvalues of the islands] Subject to Assumptions 1.1 and 1.2(2),
for any $\varepsilon>0$, $(\mathrm{P}\times\mathfrak{P})$-a.s. eventually as
$r\to\infty$,

$\text{all}\ \mathcal{C}\in\mathfrak{C}_{r,A}\
\text{satisfy}\colon\,\lambda_{\mathcal{C}}(\xi;{\mathcal{G}\mathcal{W}})\leq
a_{L_{r}}-\widehat{\chi}_{\mathcal{C}}({\mathcal{G}\mathcal{W}})+\varepsilon.$
(3.21)

###### Proof.

We follow the proof of [1, Lemma 2.3]. For $\varepsilon>0$ and $A>0$, define
the event

$\bar{\mathcal{B}}_{r,A}:=\left\\{\begin{subarray}{c}\text{there exists a
connected subset }\Lambda\subset V\text{ with }\Lambda\cap
B_{r}({\mathcal{O}})\neq\emptyset,\\\ |\Lambda|\leq\mathrm{e}^{2\vartheta
M_{A}S_{r}},\,\lambda_{\Lambda}(\xi;{\mathcal{G}\mathcal{W}})>a_{L_{r}}-\widehat{\chi}_{\Lambda}({\mathcal{G}\mathcal{W}})+\varepsilon\end{subarray}\right\\}$
(3.22)

with $M_{A}$ as in Lemma 3.1. Note that, by (1.1),
$\mathrm{e}^{\xi(x)/\varrho}$ is stochastically dominated by $Z\vee N$, where
$Z$ is an $\mathrm{Exp}(1)$ random variable and $N>0$ is a constant. Thus, for
any $\Lambda\subset V$, using [1, Eq. (2.17)], putting
$\gamma=\sqrt{\mathrm{e}^{\varepsilon/\varrho}}>1$ and applying Markov’s
inequality, we may estimate

$\displaystyle\mathrm{P}\left(\lambda_{\Lambda}(\xi;{\mathcal{G}\mathcal{W}})>a_{L_{r}}-\widehat{\chi}_{\Lambda}({\mathcal{G}\mathcal{W}})+\varepsilon\right)\leq\mathrm{P}\left({\mathcal{L}}_{\Lambda}(\xi-
a_{L_{r}}-\varepsilon)>1\right)$ (3.23)
$\displaystyle=\mathrm{P}\left(\gamma^{-1}{\mathcal{L}}_{\Lambda}(\xi)>\gamma\log
L_{r}\right)\leq\mathrm{e}^{-\gamma\log
L_{r}}\mathrm{E}[\mathrm{e}^{\gamma^{-1}{\mathcal{L}}_{\Lambda}(\xi)}]\leq\mathrm{e}^{-\gamma\log
L_{r}}K_{\gamma}^{|\Lambda|}$

with $K_{\gamma}=\mathrm{E}[\mathrm{e}^{\gamma^{-1}(Z\vee N)}]\in(1,\infty)$.
Next, by Lemma 2.4, for any $x\in B_{r}({\mathcal{O}})$ and $1\leq n\leq r$,
the number of connected subsets $\Lambda\subset V$ with $x\in\Lambda$ and
$|\Lambda|=n+1$ is $\mathfrak{P}$-a.s. at most
$(n+1)r^{n}\leq\mathrm{e}^{2n\log r}$ for $r\geq r_{0}$. Noting that
$\mathrm{e}^{S_{r}}\leq r$, we use a union bound and that by Lemma 2.2 $\log
L_{r}=\vartheta r+o(r)$ as $r\to\infty\ \mathfrak{P}$-a.s., to estimate for
$r$ large enough,

$\displaystyle\mathrm{P}(\bar{\mathcal{B}}_{r,A})$
$\displaystyle\leq\mathrm{e}^{-(\gamma-1)\log
L_{r}}\sum_{n=1}^{\lfloor\mathrm{e}^{2\vartheta
M_{A}S_{r}}\rfloor}\mathrm{e}^{2n\log r}K_{\gamma}^{n}$ (3.24)
$\displaystyle\leq\mathrm{e}^{2\vartheta
M_{A}S_{r}}\exp\left\\{-\vartheta(\gamma-1)r+o(r)+(2\log r+\log
K_{\gamma})\,\mathrm{e}^{2\vartheta M_{A}S_{r}}\right\\}$
$\displaystyle=r^{o(1)}\exp\left\\{-\vartheta(\gamma-1)r+o(r)+(\log
r)\,r^{o(1)}\right\\}\leq\mathrm{e}^{-\tfrac{1}{2}\vartheta(\gamma-1)r}.$

Via the Borel-Cantelli lemma this implies that
$(\mathrm{P}\times\mathfrak{P})$-a.s. $\bar{\mathcal{B}}_{r,A}$ does not occur
eventually as $r\to\infty$. The proof is completed by invoking Lemma 3.1. ∎

######  Corollary 3.4.

[Uniform bound on principal eigenvalue of the islands] Subject to Assumptions
1.1–1.2, for $\vartheta$ as in (1.10), and any $\varepsilon>0$,
$(\mathrm{P}\times\mathfrak{P})$-a.s. eventually as $r\to\infty$,

$\max_{\mathcal{C}\in\mathfrak{C}_{r,A}}\lambda^{{{\scriptscriptstyle{({1}})}}}_{\mathcal{C}}(\xi;G)\leq
a_{L_{r}}-\widetilde{\chi}(\varrho)+\varepsilon.$ (3.25)

###### Proof.

See [1, Corollary 2.8]. The proof carries over verbatim because the degrees
play no role. ∎

###  3.4 Maximum of the potential

The next lemma shows that $a_{L_{r}}$ is the leading order of the maximum of
$\xi$ in $B_{r}({\mathcal{O}})$.

######  Lemma 3.5.

[Maximum of the potential] Subject to Assumptions 1.1–1.2, for any
$\vartheta>0$, $(\mathrm{P}\times\mathfrak{P})$-a.s. eventually as
$r\to\infty$,

$\left|\max_{x\in
B_{r}({\mathcal{O}})}\xi(x)-a_{L_{r}}\right|\leq\frac{2\varrho\log
r}{\vartheta r}.$ (3.26)

###### Proof.

See [1, Lemma 2.5]. The proof carries over verbatim and uses Lemma 2.2. ∎

###  3.5 Number of intermediate peaks of the potential

We recall the following Chernoff bound for a binomial random variable with
parameters $n$ and $p$ (see e.g. [11, Lemma 5.9]):

$P\left(\textnormal{Bin}(n,p)\geq
u\right)\leq\mathrm{e}^{-u[\log(\frac{u}{np})-1]},\qquad u>0.$ (3.27)

######  Lemma 3.6.

[Number of intermediate peaks of the potential] Subject to Assumptions 1.1 and
1.2(2), for any $\beta\in(0,1)$ and $\varepsilon\in(0,\tfrac{1}{2}\beta)$ the
following holds. For a self-avoiding path $\pi$ in ${\mathcal{G}\mathcal{W}}$,
set

$N_{\pi}=N_{\pi}(\xi):=|\\{z\in\operatorname{supp}(\pi)\colon\,\xi(z)>(1-\varepsilon)a_{L_{r}}\\}|.$
(3.28)

Define the event

$\mathcal{B}_{r}:=\left\\{\begin{subarray}{c}\text{there exists a self-
avoiding path }\pi\text{ in ${\mathcal{G}\mathcal{W}}$ with }\\\
\operatorname{supp}(\pi)\cap
B_{r}\neq\emptyset,\,|\operatorname{supp}(\pi)|\geq(\log L_{r})^{\beta}\text{
and
}N_{\pi}>\frac{|\operatorname{supp}(\pi)|}{(\log{L_{r}})^{\varepsilon}}\end{subarray}\right\\}.$
(3.29)

Then

$\sum_{r\in\mathbb{N}_{0}}\mathrm{P}(\mathcal{B}_{r})<\infty\qquad\mathfrak{P}-a.s.$
(3.30)

###### Proof.

We follow the proof of [1, Lemma 2.9]. Fix $\beta\in(0,1)$ and
$\varepsilon\in(0,\frac{1}{2}\beta)$. (1.7) implies

$p_{r}:=\mathrm{P}(\xi(0)>(1-\varepsilon)a_{L_{r}})=\exp\left\\{-(\log
L_{r})^{1-\varepsilon}\right\\}.$ (3.31)

Fix $x\in B_{r}({\mathcal{O}})$ and $k\in\mathbb{N}$. The number of self-
avoiding paths $\pi$ in $B_{r}({\mathcal{O}})$ with
$|\operatorname{supp}(\pi)|=k$ and $\pi_{0}=x$ is at most $\mathrm{e}^{k\log
r}$ by Lemma 2.4 for $r$ sufficiently large. For such a $\pi$, the random
variable $N_{\pi}$ has a Bin($k$, $p_{r}$)-distribution. Using (3.27), we
obtain

$\mathrm{P}\Bigl{(}\exists\,\text{ self-avoiding }\pi\text{ with
}|\operatorname{supp}(\pi)|=k,\pi_{0}=x\text{ and }N_{\pi}>k/(\log
L_{r})^{\varepsilon}\Bigr{)}\\\ \leq\exp\Big{\\{}-k\Big{(}(\log
L_{r})^{1-2\varepsilon}-\log r-\frac{1+\varepsilon\log\log L_{r}}{(\log
L_{r})^{\varepsilon}}\Big{)}\Big{\\}}.$ (3.32)

By the definition of $\varepsilon$, together with the fact that $L_{r}>r$ and
$x\mapsto(\log\log x)/(\log x)^{\varepsilon}$ is eventually decreasing, the
expression in parentheses above is at least $\frac{1}{2}(\log
L_{r})^{1-2\varepsilon}$. Summing over $k\geq(\log L_{r})^{\beta}$ and $x\in
B_{r}({\mathcal{O}})$, we get $\mathfrak{P}-a.s.$

$\displaystyle\mathrm{P}\left(\mathcal{B}_{r}\right)\leq
2L_{r}\exp\Big{\\{}-\tfrac{1}{2}(\log
L_{r})^{1+\beta-2\varepsilon}\Big{\\}}\leq c_{1}\exp\Big{\\{}-c_{2}(\log
L_{r})^{1+\delta}\Big{\\}}$ (3.33)

for some $c_{1},c_{2},\delta>0$. Since $L_{r}>r$, (3.33) is summable in $r$. ∎

Lemma 3.6 implies that $(\mathrm{P}\times\mathfrak{P})$-a.s. for $r$ large
enough, all self-avoiding paths $\pi$ in ${\mathcal{G}\mathcal{W}}$ with
$\operatorname{supp}(\pi)\cap B_{r}\neq\emptyset$ and
$|\operatorname{supp}(\pi)|\geq(\log L_{r})^{\beta}$ satisfy
$N_{\pi}\leq\frac{|\operatorname{supp}(\pi)|}{(\log L_{r})^{\varepsilon}}$.

######  Lemma 3.7.

[Number of high exceedances of the potential] Subject to Assumptions 1.1 and
1.2(2), for any $A>0$ there is a $C\geq 1$ such that, for all
$\delta\in(0,1)$, the following holds. For a self-avoiding path $\pi$ in
${\mathcal{G}\mathcal{W}}$, let

$N_{\pi}:=|\\{x\in\operatorname{supp}(\pi)\colon\,\xi(x)>a_{L_{r}}-2A\\}|.$
(3.34)

Define the event

$\mathcal{B}_{r}:=\left\\{\begin{subarray}{c}\text{there exists a self-
avoiding path }\pi\text{ in $G$ with }\\\ \operatorname{supp}(\pi)\cap
B_{r}\neq\emptyset,\,|\operatorname{supp}(\pi)|\geq C(\log
L_{r})^{\delta}\text{ and
}N_{\pi}>\frac{|\operatorname{supp}(\pi)|}{(\log{L_{r}})^{\delta}}\end{subarray}\right\\}.$
(3.35)

Then
$\sum_{r\in\mathbb{N}_{0}}\sup_{G\in\mathfrak{G}_{r}}\mathrm{P}(\mathcal{B}_{r})<\infty$.
In particular, $(\mathrm{P}\times\mathfrak{P})$-a.s. for $r$ large enough, all
self-avoiding paths $\pi$ in ${\mathcal{G}\mathcal{W}}$ with
$\operatorname{supp}(\pi)\cap B_{r}\neq\emptyset$ and
$|\operatorname{supp}(\pi)|\geq C(\log L_{r})^{\delta}$ satisfy

$N_{\pi}=|\\{x\in\operatorname{supp}(\pi)\colon\,\xi(x)>a_{L_{r}}-2A\\}|\leq\frac{|\operatorname{supp}(\pi)|}{(\log
L_{r})^{\delta}}.$ (3.36)

###### Proof.

Proceed as for Lemma 3.6, noting that this time

$p_{r}:=\mathrm{P}\big{(}\xi(0)>a_{L_{r}}-2A\big{)}=L_{r}^{-\epsilon}$ (3.37)

where $\epsilon=\mathrm{e}^{-2A/\varrho}$, and taking $C>2/\epsilon$. ∎

##  4 Path expansions

In this section we extend [1, Section 3]. Section 4.1 proves three lemmas that
concern the contribution the total mass in (1.6) coming from various sets of
paths. Section 4.2 proves a key proposition that controls the entropy
associated with a key set of paths. The proof is based on the three lemmas in
Section 4.1.

######  Lemma 4.1.

[Mass up to an exit time] Subject to Assumption 1.3, $\mathfrak{P}$-a.s. for
any $r\geq r_{0}$, $y\in\Lambda\subset B_{r}({\mathcal{O}})$,
$\xi\in[0,\infty)^{V}$ and
$\gamma>\lambda_{\Lambda}=\lambda_{\Lambda}(\xi,{\mathcal{G}\mathcal{W}})$,

$\mathbb{E}_{y}\left[\mathrm{e}^{\int_{0}^{\tau_{\Lambda^{\text{\rm
c}}}}(\xi(X_{s})-\gamma)\,\text{\rm d}\mkern 0.5mus}\right]\leq 1+\frac{(\log
r)^{\delta r}\,|\Lambda|}{\gamma-\lambda_{\Lambda}}.$ (4.1)

###### Proof.

The proof is identical to that of Lemma 3.2, with $\delta r$ replaced by
$(\log r)^{\delta r}$ (recall Lemma 2.3). ∎

We need various sets of nearest-neighbour paths in
${\mathcal{G}\mathcal{W}}=(V,E,{\mathcal{O}})$, defined in [1]. For
$\ell\in\mathbb{N}_{0}$ and subsets $\Lambda,\Lambda^{\prime}\subset V$, put

$\displaystyle\mathscr{P}_{\ell}(\Lambda,\Lambda^{\prime}):=\left\\{(\pi_{0},\ldots,\pi_{\ell})\in
V^{\ell+1}\colon\,\begin{array}[]{ll}&\pi_{0}\in\Lambda,\pi_{\ell}\in\Lambda^{\prime},\\\
&\\{\pi_{i},\pi_{i-1}\\}\in E\;\forall\,1\leq i\leq\ell\end{array}\right\\},$
(4.2)
$\displaystyle\mathscr{P}(\Lambda,\Lambda^{\prime}):=\bigcup_{\ell\in\mathbb{N}_{0}}\mathscr{P}_{\ell}(\Lambda,\Lambda^{\prime}),$

and set

$\mathscr{P}_{\ell}:=\mathscr{P}_{\ell}(V,V),\qquad\mathscr{P}:=\mathscr{P}(V,V).$
(4.3)

When $\Lambda$ or $\Lambda^{\prime}$ consists of a single point, write $x$
instead of $\\{x\\}$. For $\pi\in\mathscr{P}_{\ell}$, set $|\pi|:=\ell$. Write
$\operatorname{supp}(\pi):=\\{\pi_{0},\ldots,\pi_{|\pi|}\\}$ to denote the set
of points visited by $\pi$.

Let $X=(X_{t})_{t\geq 0}$ be the continuous-time random walk on $G$ that jumps
from $x\in V$ to any neighbour $y\sim x$ at rate $1$. Denote by
$(T_{k})_{k\in\mathbb{N}_{0}}$ the sequence of jump times (with $T_{0}:=0$).
For $\ell\in\mathbb{N}_{0}$, let

$\pi^{{{\scriptscriptstyle{({\ell}})}}}(X):=(X_{0},\ldots,X_{T_{\ell}})$ (4.4)

be the path in $\mathscr{P}_{\ell}$ consisting of the first $\ell$ steps of
$X$. For $t\geq 0$, let

$\pi(X_{[0,t]})=\pi^{{{\scriptscriptstyle{({\ell_{t}}})}}}(X),\quad\text{ with
}\ell_{t}\in\mathbb{N}_{0}\,\text{ satisfying }\,T_{\ell_{t}}\leq
t<T_{\ell_{t}+1},$ (4.5)

denote the path in $\mathscr{P}$ consisting of all the steps taken by $X$
between times $0$ and $t$.

Recall the definitions from Section 3.1. For $\pi\in\mathscr{P}$ and $A>0$,
define

$\lambda_{r,A}(\pi):=\sup\big{\\{}\lambda^{{{\scriptscriptstyle{({1}})}}}_{\mathcal{C}}(\xi;G)\colon\,\mathcal{C}\in\mathfrak{C}_{r,A},\,\operatorname{supp}(\pi)\cap\mathcal{C}\cap\Pi_{r,A}\neq\emptyset\big{\\}},$
(4.6)

with the convention $\sup\emptyset=-\infty$. This is the largest principal
eigenvalue among the components of $\mathfrak{C}_{r,A}$ in
${\mathcal{G}\mathcal{W}}$ that have a point of high exceedance visited by the
path $\pi$.

###  4.1 Mass of the solution along excursions

######  Lemma 4.2.

[Path evaluation] For $\ell\in\mathbb{N}_{0}$, $\pi\in\mathscr{P}_{\ell}$ and
$\gamma>\max_{0\leq i<|\pi|}\\{\xi(\pi_{i})-D_{\pi_{i}}\\}$,

${\mathbb{E}}_{\pi_{0}}\left[\mathrm{e}^{\int_{0}^{T_{\ell}}(\xi(X_{s})-\gamma)\,\text{\rm
d}\mkern 0.5mus}\leavevmode\nobreak\ \Big{|}\leavevmode\nobreak\
\pi^{{{\scriptscriptstyle{({\ell}})}}}(X)=\pi\right]=\prod_{i=0}^{\ell-1}\frac{D_{\pi_{i}}}{\gamma-[\xi(\pi_{i})-D_{\pi_{i}}]}.$
(4.7)

###### Proof.

The proof is identical to that of [1, Lemma 3.2]. The left-hand side of (4.7)
can be evaluated by using the fact that $T_{\ell}$ is the sum of $\ell$
independent Exp($\deg(\pi_{i})$) random variables that are independent of
$\pi^{{{\scriptscriptstyle{({\ell}})}}}(X)$. The condition on $\gamma$ ensures
that all $\ell$ integrals are finite. ∎

For a path $\pi\in\mathscr{P}$ and $\varepsilon\in(0,1)$, we write

$M^{r,\varepsilon}_{\pi}:=\big{|}\bigl{\\{}0\leq
i<|\pi|\colon\,\xi(\pi_{i})\leq(1-\varepsilon)a_{L_{r}}\bigr{\\}}\big{|},$
(4.8)

with the interpretation that $M^{r,\varepsilon}_{\pi}=0$ if $|\pi|=0$.

######  Lemma 4.3.

[Mass of excursions] Subject to Assumptions 1.1–1.3, for every
$A,\varepsilon>0$, $(\mathrm{P}\times\mathfrak{P})$-a.s. there exists an
$r_{0}\in\mathbb{N}$ such that, for all $r\geq r_{0}$, all
$\gamma>a_{L_{r}}-A$ and all
$\pi\in\mathscr{P}(B_{r}({\mathcal{O}}),B_{r}({\mathcal{O}}))$ satisfying
$\pi_{i}\notin\Pi_{r,A}$ for all $0\leq i<\ell:=|\pi|$,

${\mathbb{E}}_{\pi_{0}}\left[\mathrm{e}^{\int_{0}^{T_{\ell}}(\xi(X_{s})-\gamma)\,\text{\rm
d}\mkern 0.5mus}\leavevmode\nobreak\ \Big{|}\leavevmode\nobreak\
\pi^{{{\scriptscriptstyle{({\ell}})}}}(X)=\pi\right]\leq
q_{r,A}^{\ell}\mathrm{e}^{M^{r,\varepsilon}_{\pi}\log[(\log
r)^{\delta_{r}}/a_{L_{r},A,\varepsilon}q_{r,A}]},$ (4.9)

where

$a_{L_{r},A,\varepsilon}:=\varepsilon a_{L_{r}}-A,\qquad
q_{r,A}:=\left(1+\frac{A}{(\log r)^{\delta_{r}}}\right)^{-1}.$ (4.10)

Note that $\pi_{\ell}\in\Pi_{r,A}$ is allowed.

###### Proof.

The proof is identical to that of [1, Lemma 3.3], with $d_{\max}$ replaced by
$(\log r)^{\delta_{r}}$ (recall Lemma 2.3). ∎

We follow [1, Definition 3.4] and [11, Section 6.2]. Note that the distance
between $\Pi_{r,A}$ and $D_{r,A}^{\text{\rm c}}$ in ${\mathcal{G}\mathcal{W}}$
is at least $S_{r}=(\log L_{r})^{\alpha}$ (recall (3.4)–(3.5)).

######  Definition 4.4.

[Concatenation of paths] (a) When $\pi$ and $\pi^{\prime}$ are two paths in
$\mathscr{P}$ with $\pi_{|\pi|}=\pi^{\prime}_{0}$, we define their
_concatenation_ as

$\pi\circ\pi^{\prime}:=(\pi_{0},\ldots,\pi_{|\pi|},\pi^{\prime}_{1},\ldots,\pi^{\prime}_{|\pi^{\prime}|})\in\mathscr{P}.$
(4.11)

Note that $|\pi\circ\pi^{\prime}|=|\pi|+|\pi^{\prime}|$.

(b) When $\pi_{|\pi|}\neq\pi^{\prime}_{0}$, we can still define the _shifted
concatenation_ of $\pi$ and $\pi^{\prime}$ as $\pi\circ\hat{\pi}^{\prime}$,
where
$\hat{\pi}^{\prime}:=(\pi_{|\pi|},\pi_{|\pi|}+\pi^{\prime}_{1}-\pi^{\prime}_{0},\ldots,\pi_{|\pi|}+\pi^{\prime}_{|\pi^{\prime}|}-\pi^{\prime}_{0})$.
The shifted concatenation of multiple paths is defined inductively via
associativity.

Now, if a path $\pi\in\mathscr{P}$ intersects $\Pi_{r,A}$, then it can be
decomposed into an initial path, a sequence of excursions between $\Pi_{r,A}$
and $D_{r,A}^{\text{\rm c}}$, and a terminal path. More precisely, there
exists $m_{\pi}\in\mathbb{N}$ such that

$\pi=\check{\pi}^{{{\scriptscriptstyle{({1}})}}}\circ\hat{\pi}^{{{\scriptscriptstyle{({1}})}}}\circ\cdots\circ\check{\pi}^{{{\scriptscriptstyle{({m_{\pi}}})}}}\circ\hat{\pi}^{{{\scriptscriptstyle{({m_{\pi}}})}}}\circ\bar{\pi},$
(4.12)

where the paths in (4.12) satisfy

$\displaystyle\check{\pi}^{{{\scriptscriptstyle{({1}})}}}$
$\displaystyle\in\mathscr{P}(V,\Pi_{r,A})$ with
$\displaystyle\check{\pi}^{{{\scriptscriptstyle{({1}})}}}_{i}$
$\displaystyle\notin\Pi_{r,A},$ $\displaystyle\quad\,0\leq
i<|\check{\pi}^{{{\scriptscriptstyle{({1}})}}}|,$ (4.13)
$\displaystyle\hat{\pi}^{{{\scriptscriptstyle{({k}})}}}$
$\displaystyle\in\mathscr{P}(\Pi_{r,A},D_{r,A}^{\text{\rm c}})$ with
$\displaystyle\hat{\pi}^{{{\scriptscriptstyle{({k}})}}}_{i}$ $\displaystyle\in
D_{r,A},$ $\displaystyle\quad\,0\leq
i<|\hat{\pi}^{{{\scriptscriptstyle{({k}})}}}|,\;1\leq k\leq m_{\pi}-1,$
$\displaystyle\check{\pi}^{{{\scriptscriptstyle{({k}})}}}$
$\displaystyle\in\mathscr{P}(D_{r,A}^{\text{\rm c}},\Pi_{r,A})$ with
$\displaystyle\check{\pi}^{{{\scriptscriptstyle{({k}})}}}_{i}$
$\displaystyle\notin\Pi_{r,A},$ $\displaystyle\quad\,0\leq
i<|\check{\pi}^{{{\scriptscriptstyle{({k}})}}}|,\;2\leq k\leq m_{\pi},$
$\displaystyle\hat{\pi}^{{{\scriptscriptstyle{({m_{\pi}}})}}}$
$\displaystyle\in\mathscr{P}(\Pi_{r,A},V)$ with
$\displaystyle\hat{\pi}^{{{\scriptscriptstyle{({m_{\pi}}})}}}_{i}$
$\displaystyle\in D_{r,A},$ $\displaystyle\quad\,0\leq
i<|\hat{\pi}^{{{\scriptscriptstyle{({m_{\pi}}})}}}|,$

while

$\begin{array}[]{ll}\bar{\pi}\in\mathscr{P}(D_{r,A}^{\text{\rm c}},V)\text{
and }\bar{\pi}_{i}\notin\Pi_{r,A}\;\forall\,i\geq 0&\text{ if
}\hat{\pi}^{{{\scriptscriptstyle{({m_{\pi}}})}}}\in\mathscr{P}(\Pi_{r,A},D^{\text{\rm
c}}_{r,A}),\\\ \bar{\pi}_{0}\in D_{r,A},|\bar{\pi}|=0&\text{
otherwise.}\end{array}$ (4.14)

Note that the decomposition in (4.12)–(4.14) is unique, and that the paths
$\check{\pi}^{{{\scriptscriptstyle{({1}})}}}$,
$\hat{\pi}^{{{\scriptscriptstyle{({m_{\pi}}})}}}$ and $\bar{\pi}$ can have
zero length. If $\pi$ is contained in $B_{r}({\mathcal{O}})$, then so are all
the paths in the decomposition.

Whenever $\operatorname{supp}(\pi)\cap\Pi_{r,A}\neq\emptyset$ and
$\varepsilon>0$, we define

$\displaystyle
s_{\pi}:=\sum_{i=1}^{m_{\pi}}|\check{\pi}^{{{\scriptscriptstyle{({i}})}}}|+|\bar{\pi}|,\qquad
k^{r,\varepsilon}_{\pi}:=\sum_{i=1}^{m_{\pi}}M^{r,\varepsilon}_{\check{\pi}^{{{\scriptscriptstyle{({i}})}}}}+M^{r,\varepsilon}_{\bar{\pi}}$
(4.15)

to be the total time spent in exterior excursions, respectively, on moderately
low points of the potential visited by exterior excursions (without their last
point).

In case $\operatorname{supp}(\pi)\cap\Pi_{r,A}=\emptyset$, we set
$m_{\pi}:=0$, $s_{\pi}:=|\pi|$ and
$k^{r,\varepsilon}_{\pi}:=M^{r,\varepsilon}_{\pi}$. Recall from (4.6) that, in
this case, $\lambda_{r,A}(\pi)=-\infty$.

We say that $\pi,\pi^{\prime}\in\mathscr{P}$ are _equivalent_ , written
$\pi^{\prime}\sim\pi$, if $m_{\pi}=m_{\pi^{\prime}}$,
$\check{\pi}^{\prime{{\scriptscriptstyle{({i}})}}}=\check{\pi}^{{{\scriptscriptstyle{({i}})}}}$
for all $i=1,\ldots,m_{\pi}$, and $\bar{\pi}^{\prime}=\bar{\pi}$. If
$\pi^{\prime}\sim\pi$, then $s_{\pi^{\prime}}$,
$k^{r,\varepsilon}_{\pi^{\prime}}$ and $\lambda_{r,A}(\pi^{\prime})$ are all
equal to the counterparts for $\pi$.

To state our key lemma, we define, for $m,s\in\mathbb{N}_{0}$,

$\mathscr{P}^{(m,s)}=\left\\{\pi\in\mathscr{P}\colon\,m_{\pi}=m,s_{\pi}=s\right\\},$
(4.16)

and denote by

$C_{r,A}:=\max\\{|\mathcal{C}|\colon\,\mathcal{C}\in\mathfrak{C}_{r,A}\\}$
(4.17)

the maximal size of the islands in $\mathfrak{C}_{r,A}$.

######  Lemma 4.5.

[Mass of an equivalence class] Subject to Assumptions 1.1 and 1.3, for every
$A,\varepsilon>0$, $(\mathrm{P}\times\mathfrak{P})$-a.s. there exists an
$r_{0}\in\mathbb{N}$ such that, for all $r\geq r_{0}$, all
$m,s\in\mathbb{N}_{0}$, all $\pi\in\mathscr{P}^{(m,s)}$ with
$\operatorname{supp}(\pi)\subset B_{r}({\mathcal{O}})$, all
$\gamma>\lambda_{r,A}(\pi)\vee(a_{L_{r}}-A)$ and all $t\geq 0$,

$\qquad{\mathbb{E}}_{\pi_{0}}\left[\text{\rm
e}^{\int_{0}^{t}(\xi(X_{u})-\gamma)\,\text{\rm d}\mkern
0.5muu}\,{\mathchoice{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.5mu\mathrm{l}}{1\mskip-5.0mu\mathrm{l}}}_{\\{\pi(X_{[0,t]})\sim\pi\\}}\right]\\\
\leq\left(C_{r,A}^{1/2}\right)^{{\mathchoice{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.5mu\mathrm{l}}{1\mskip-5.0mu\mathrm{l}}}_{\\{m>0\\}}}\left(1+\frac{(\log
r)^{\delta_{r}}\,C_{r,A}}{\gamma-\lambda_{r,A}(\pi)}\right)^{m}\left(\frac{q_{r,A}}{(\log
r)^{\delta_{r}}}\right)^{s}\mathrm{e}^{k^{r,\varepsilon}_{\pi}\log[(\log
r)^{\delta_{r}}/a_{L_{r},A,\varepsilon}q_{r,A}]}.$ (4.18)

###### Proof.

The proof is identical to that of [1, Lemma 3.5], with $d_{\max}$ is replaced
by $(\log r)^{\delta_{r}}$ (recall Lemma 2.3). ∎

###  4.2 Key proposition

The main result of this section is the following proposition.

######  Proposition 4.6.

[Entropy reduction] Let $\alpha\in(0,1)$ and $\kappa\in(\alpha,1)$. Subject to
Assumption 1.3, there exists an $A_{0}(r)$ such that, for all $A\geq
A_{0}(r)$, with $\mathfrak{P}$-probability tending to one as $r\to\infty$, the
following statement is true. For each $x\in B_{r}({\mathcal{O}})$, each
$\mathcal{N}\subset\mathscr{P}(x,B_{r}({\mathcal{O}}))$ satisfying
$\operatorname{supp}(\pi)\subset B_{r}({\mathcal{O}})$ and
$\max_{1\leq\ell\leq|\pi|}\operatorname{dist}_{G}(\pi_{\ell},x)\geq(\log
L_{r})^{\kappa}$ for all $\pi\in\mathcal{N}$, and each assignment
$\pi\mapsto(\gamma_{\pi},z_{\pi})\in\mathbb{R}\times V$ satisfying

$\gamma_{\pi}\geq\left(\lambda_{r,A}(\pi)+\text{\rm
e}^{-S_{r}}\right)\vee(a_{L_{r}}-A)\qquad\forall\,\,\pi\in\mathcal{N}$ (4.19)

and

$z_{\pi}\in\operatorname{supp}(\pi)\cup\bigcup_{\begin{subarray}{c}\mathcal{C}\in\mathfrak{C}_{r,A}\colon\\\
\operatorname{supp}(\pi)\cap\mathcal{C}\cap\Pi_{r,A}\neq\emptyset\end{subarray}}\mathcal{C}\qquad\forall\,\,\pi\in\mathcal{N},$
(4.20)

the following inequality holds for all $t\geq 0$:

$\log{\mathbb{E}}_{x}\left[\text{\rm e}^{\int_{0}^{t}\xi(X_{s})\text{\rm
d}\mkern
0.5mus}{\mathchoice{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.5mu\mathrm{l}}{1\mskip-5.0mu\mathrm{l}}}_{\\{\pi(X_{[0,t]})\in\mathcal{N}\\}}\right]\leq\sup_{\pi\in\mathcal{N}}\Big{\\{}t\gamma_{\pi}+\operatorname{dist}_{G}(x,z_{\pi})\log[(\log
r)^{\delta_{r}}/a_{L_{r},A,\varepsilon}q_{r,A}]\Big{\\}}.$ (4.21)

###### Proof.

The proof is based on [1, Section 3.4]. First fix $c_{0}>2$ and define

$A_{0}(r)=(\log r)^{\delta_{r}}\left(\mathrm{e}^{3c_{0}(\log
r)^{1-\alpha}}-1\right).$ (4.22)

Fix $A\geq A_{0}(r)$, $\beta\in(0,\alpha)$ and
$\varepsilon\in(0,\frac{1}{2}\beta)$ as in Lemma 3.6. Let $r_{0}\in\mathbb{N}$
be as given in Lemma 4.5, and take $r\geq r_{0}$ so large that the conclusions
of Lemmas 2.3, 3.1, 3.3 and 3.6 hold, i.e., assume that the events
$\mathcal{B}_{r}$ and $\mathcal{B}_{r,A}$ in these lemmas do not occur. Fix
$x\in B_{r}({\mathcal{O}})$. Recall the definitions of $C_{r,A}$ and
$\mathscr{P}^{(m,s)}$. Note that the relation $\sim$ is an equivalence
relation in $\mathscr{P}^{(m,s)}$, and define

$\widetilde{\mathscr{P}}^{(m,s)}_{x}:=\big{\\{}\text{equivalence classes of
the paths in }\mathscr{P}(x,V)\cap\mathscr{P}^{(m,s)}\big{\\}}.$ (4.23)

The following bounded on the cardinality of this set is needed.

######  Lemma 4.7.

[Bound equivalence classes] Subject to Assumption 1.3,
$\mathfrak{P}$-a.s.,$|\widetilde{\mathscr{P}}^{(m,s)}_{x}|$
$\leq(2C_{r,A})^{m}(\log r)^{\delta_{r}(m+s)}$ for all $m,s\in\mathbb{N}_{0}$.

###### Proof.

We can copy the proof of [1, Lemma 3.6], replacing $d_{\max}$ by $(\log
r)^{\delta_{r}}$.

The estimate is clear when $m=0$. To prove that it holds for $m\geq 1$, write
$\partial\Lambda:=\\{z\notin\Lambda\colon\,\operatorname{dist}_{G}(z,\Lambda)=1\\}$
for $\Lambda\subset V$. Then $|\partial\mathcal{C}\cup\mathcal{C}|\leq((\log
r)^{\delta_{r}}+1)|\mathcal{C}|\leq 2(\log r)^{\delta_{r}}C_{r,A}$ by Lemma
2.3. Define the map
$\Phi\colon\widetilde{\mathscr{P}}^{(m,s)}_{x}\to\mathscr{P}_{s}(x,V)\times\\{1,\ldots,2(\log
r)^{\delta_{r}}C_{r,A}\\}^{m}$ as follows. For each $\Lambda\subset V$ with
$1\leq|\Lambda|\leq 2(\log r)^{\delta_{r}}C_{r,A}$, fix an injection
$f_{\Lambda}\colon\Lambda\to\\{1,\ldots,2(\log r)^{\delta_{r}}C_{r,A}\\}$.
Given a path $\pi\in\mathscr{P}^{(m,s)}\cap\mathscr{P}(x,V)$, decompose $\pi$,
and denote by $\widetilde{\pi}\in\mathscr{P}_{s}(x,V)$ the shifted
concatenation of
$\check{\pi}^{{{\scriptscriptstyle{({1}})}}},\ldots,\check{\pi}^{{{\scriptscriptstyle{({m}})}}}$,
$\bar{\pi}$. Note that, for $2\leq k\leq m$, the point
$\check{\pi}^{{{\scriptscriptstyle{({k}})}}}_{0}$ lies in
$\partial\mathcal{C}_{k}$ for some $\mathcal{C}_{k}\in\mathfrak{C}_{r,A}$,
while
$\bar{\pi}_{0}\in\partial\overline{\mathcal{C}}\cup\overline{\mathcal{C}}$ for
some $\overline{\mathcal{C}}\in\mathfrak{C}_{r,A}$. Thus, it is possible to
set

$\Phi(\pi):=\bigl{(}\widetilde{\pi},f_{\partial\mathcal{C}_{2}}(\check{\pi}^{{{\scriptscriptstyle{({2}})}}}_{0}),\dots,f_{\partial\mathcal{C}_{m}}(\check{\pi}^{{{\scriptscriptstyle{({m}})}}}_{0}),f_{\partial\bar{\mathcal{C}}\cup\bar{\mathcal{C}}}(\bar{\pi}_{0})\bigr{)}.$
(4.24)

It is readily checked that $\Phi(\pi)$ depends only on the equivalence class
of $\pi$ and, when restricted to equivalence classes, $\Phi$ is injective.
Hence the claim follows. ∎

Now take $\mathcal{N}\subset\mathscr{P}(x,V)$ as in the statement, and set

$\widetilde{\mathcal{N}}^{(m,s)}:=\big{\\{}\text{equivalence classes of paths
in
}\mathcal{N}\cap\mathscr{P}^{(m,s)}\big{\\}}\subset\widetilde{\mathscr{P}}^{(m,s)}_{x}.$
(4.25)

For each $\mathcal{M}\in\widetilde{\mathcal{N}}^{(m,s)}$, choose a
representative $\pi_{\mathcal{M}}\in\mathcal{M}$, and use Lemma (4.7) to write

$\displaystyle{\mathbb{E}}_{x}\left[\text{\rm
e}^{\int_{0}^{t}\xi(X_{u})\text{\rm d}\mkern
0.5muu}{\mathchoice{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.5mu\mathrm{l}}{1\mskip-5.0mu\mathrm{l}}}_{\\{\pi(X_{[0,t]})\in\mathcal{N}\\}}\right]=\sum_{m,s\in\mathbb{N}_{0}}\sum_{\mathcal{M}\in\widetilde{\mathcal{N}}^{(m,s)}}{\mathbb{E}}_{x}\left[\text{\rm
e}^{\int_{0}^{t}\xi(X_{u})\text{\rm d}\mkern
0.5muu}{\mathchoice{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.5mu\mathrm{l}}{1\mskip-5.0mu\mathrm{l}}}_{\\{\pi(X_{[0,t]})\sim\pi_{\mathcal{M}}\\}}\right]$
$\displaystyle\quad\qquad\leq\sum_{m,s\in\mathbb{N}_{0}}(2(\log
r)^{\delta_{r}}C_{r,A})^{m}((\log
r)^{\delta_{r}})^{s}\sup_{\pi\in\mathcal{N}^{(m,s)}}{\mathbb{E}}_{x}\left[\text{\rm
e}^{\int_{0}^{t}\xi(X_{u})\text{\rm d}\mkern
0.5muu}{\mathchoice{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.5mu\mathrm{l}}{1\mskip-5.0mu\mathrm{l}}}_{\\{\pi(X_{[0,t]})\sim\pi\\}}\right]$
(4.26)

with the convention $\sup\emptyset=0$. For fixed $\pi\in\mathcal{N}^{(m,s)}$,
by (4.19), apply (4.18) and Lemma 3.1 to obtain, for all $r$ large enough and
with $c_{0}>2$ ,

$\displaystyle(2(c\log r)^{\delta_{r}})^{m}(\log
r)^{\delta_{r}s}\,{\mathbb{E}}_{x}\left[\text{\rm
e}^{\int_{0}^{t}\xi(X_{u})\text{\rm d}\mkern
0.5muu}{\mathchoice{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.5mu\mathrm{l}}{1\mskip-5.0mu\mathrm{l}}}_{\\{\pi(X_{[0,t]})\sim\pi\\}}\right]$
(4.27) $\displaystyle\qquad\leq\text{\rm e}^{t\gamma_{\pi}}\text{\rm
e}^{c_{0}m\log r}q_{r,A}^{s}\,\mathrm{e}^{k^{r,\varepsilon}_{\pi}\log[(\log
r)^{\delta_{r}}/a_{L_{r},A,\varepsilon}q_{r,A}]}.$

We next claim that, for $r$ large enough and $\pi\in\mathcal{N}^{(m,s)}$,

$s\geq\left[(m-1)\vee 1\right]S_{r}.$ (4.28)

Indeed, when $m\geq 2$,
$|\operatorname{supp}(\check{\pi}^{{{\scriptscriptstyle{({i}})}}})|\geq S_{r}$
for all $2\leq i\leq m$. When $m=0$,
$|\operatorname{supp}(\pi)|\geq\max_{1\leq\ell\leq|\pi|}|\pi_{\ell}-x|\geq(\log
L_{r})^{\kappa}\gg S_{r}$ by assumption. When $m=1$, the latter assumption and
Lemma 3.1 together imply that $\operatorname{supp}(\pi)\cap D^{\text{\rm
c}}_{r,A}\neq\emptyset$, and so either
$|\operatorname{supp}(\check{\pi}^{{{\scriptscriptstyle{({1}})}}})|\geq S_{r}$
or $|\operatorname{supp}(\bar{\pi})|\geq S_{r}$. Thus, (4.28) holds by the
definition of $S_{r}$ and $s$.

Note that $q_{r,A}^{S_{r}}<\mathrm{e}^{-3c_{0}\log r}$, so

$\sum_{m\geq 0}\sum_{s\geq[(m-1)\vee 1]S_{r}}\mathrm{e}^{c_{0}m\log
r}q_{r,A}^{s}=\frac{q_{r,A,}^{S_{r}}+\mathrm{e}^{c_{0}\log
r}q_{r,A}^{S_{r}}+\sum_{m\geq 2}\mathrm{e}^{mc_{0}\log
r}q_{r,A}^{(m-1)S_{r}}}{1-q_{r,A}}\leq\frac{4\mathrm{e}^{-c_{0}\log
r}}{1-q_{r,A}}<1$ (4.29)

for $r$ large enough. Inserting this back into (4.2), we obtain

$\log{\mathbb{E}}_{x}\left[\text{\rm e}^{\int_{0}^{t}\xi(X_{s})\text{\rm
d}\mkern
0.5mus}{\mathchoice{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.5mu\mathrm{l}}{1\mskip-5.0mu\mathrm{l}}}_{\\{\pi(X_{0,t})\in\mathcal{N}\\}}\right]\leq\sup_{\pi\in\mathcal{N}}\Big{\\{}t\gamma_{\pi}+k^{r,\varepsilon}_{\pi}\log[(\log
r)^{\delta_{r}}/a_{L_{r},A,\varepsilon}q_{r,A}]\Big{\\}}.$ (4.30)

Thus the proof will be finished once we show that, for some
$\varepsilon^{\prime}>0$ and whp, respectively, a.s. eventually as
$r\to\infty$,

$k^{r,\varepsilon}_{\pi}\geq\operatorname{dist}_{G}(x,z_{\pi})(1-2(\log
L_{r})^{-\varepsilon^{\prime}})\qquad\forall\,\pi\in\mathcal{N}.$ (4.31)

We can copy the argument at the end of [1, Section 3.4]. For each
$\pi\in\mathcal{N}$ define an auxiliary path $\pi_{\star}$ as follows. First
note that by using our assumptions we can find points
$z^{\prime},z^{\prime\prime}\in\operatorname{supp}(\pi)$ (not necessarily
distinct) such that

$\operatorname{dist}_{G}(x,z^{\prime})\geq(\log
L_{r})^{\kappa},\qquad\operatorname{dist}_{G}(z^{\prime\prime},z_{\pi})\leq
2M_{A}S_{r},$ (4.32)

where the latter holds by (3.12). Write
$\\{z_{1},z_{2}\\}=\\{z^{\prime},z^{\prime\prime}\\}$ with $z_{1}$, $z_{2}$
ordered according to their hitting times by $\pi$, i.e.,
$\inf\\{\ell\colon\pi_{\ell}=z_{1}\\}\leq\inf\\{\ell\colon\pi_{\ell}=z_{2}\\}$.
Define $\pi_{e}$ as the concatenation of the loop erasure of $\pi$ between $x$
and $z_{1}$ and the loop erasure of $\pi$ between $z_{1}$ and $z_{2}$. Since
$\pi_{e}$ is the concatenation of two self-avoiding paths, it visits each
point at most twice. Finally, define $\pi_{\star}\sim\pi_{e}$ by replacing the
excursions of $\pi_{e}$ from $\Pi_{r,A}$ to $D_{r,A}^{\text{\rm c}}$ by direct
paths between the corresponding endpoints, i.e., replace each
$\hat{\pi}_{e}^{{{\scriptscriptstyle{({i}})}}}$ by
$|\hat{\pi}_{e}^{{{\scriptscriptstyle{({i}})}}}|=\ell_{i}$,
$(\hat{\pi}_{e}^{{{\scriptscriptstyle{({i}})}}})_{0}=x_{i}\in\Pi_{r,A}$, and
$(\hat{\pi}_{e}^{{{\scriptscriptstyle{({i}})}}})_{\ell_{i}}=y_{i}\in
D_{r,A}^{\text{\rm c}}$ by a shortest-distance path
$\widetilde{\pi}_{\star}^{{{\scriptscriptstyle{({i}})}}}$ with the same
endpoints and
$|\widetilde{\pi}_{\star}^{{{\scriptscriptstyle{({i}})}}}|=\operatorname{dist}_{G}(x_{i},y_{i})$.
Since $\pi_{\star}$ visits each $x\in\Pi_{r,A}$ at most $2$ times,

$\displaystyle k^{r,\varepsilon}_{\pi}\geq k^{r,\varepsilon}_{\pi_{\star}}\geq
M^{r,\varepsilon}_{\pi_{\star}}-2|\operatorname{supp}(\pi_{\star})\cap\Pi_{r,A}|(S_{r}+1)\geq
M^{r,\varepsilon}_{\pi_{\star}}-4|\operatorname{supp}(\pi_{\star})\cap\Pi_{r,A}|S_{r}.$
(4.33)

Note that
$M_{\pi_{\star}}^{r,\varepsilon}\geq\left|\\{x\in\operatorname{supp}(\pi_{\star})\colon\,\xi(x)\leq(1-\varepsilon)a_{L_{r}}\\}\right|-1$
and, by (4.32),
$|\operatorname{supp}(\pi_{\star})|\geq\operatorname{dist}_{G}(x,z^{\prime})\geq(\log
L_{r})^{\kappa}\gg(\log L_{r})^{\alpha+2\varepsilon^{\prime}}$ for some
$0<\varepsilon^{\prime}<\varepsilon$. Applying Lemmas 3.6–3.7 and using (3.1)
and $L_{r}>r$, we obtain, for $r$ large enough,

$\displaystyle k^{r,\varepsilon}_{\pi}$
$\displaystyle\geq|\operatorname{supp}(\pi_{\star})|\left(1-\frac{2}{(\log
L_{r})^{\varepsilon}}-\frac{4S_{r}}{(\log
L_{r})^{\alpha+2\varepsilon^{\prime}}}\right)\geq|\operatorname{supp}(\pi_{\star})|\left(1-\frac{1}{(\log
L_{r})^{\varepsilon^{\prime}}}\right).$ (4.34)

On the other hand, since $|\operatorname{supp}(\pi_{\star})|\geq(\log
L_{r})^{\kappa}$, by (4.32) we have

$\displaystyle\left|\operatorname{supp}(\pi_{\star})\right|$
$\displaystyle=\big{(}\left|\operatorname{supp}(\pi_{\star})\right|+2M_{A}S_{r}\big{)}-2M_{A}S_{r}$
(4.35)
$\displaystyle=\big{(}\left|\operatorname{supp}(\pi_{\star})\right|+2M_{A}S_{r}\big{)}\left(1-\frac{2M_{A}S_{r}}{\left|\operatorname{supp}(\pi_{\star})\right|+2M_{A}S_{r}}\right)$
$\displaystyle\geq\left(\operatorname{dist}_{G}(x,z^{\prime\prime})+2M_{A}S_{r}\right)\left(1-\frac{2M_{A}S_{r}}{(\log
L_{r})^{\kappa}}\right)$
$\displaystyle\geq\operatorname{dist}_{G}(x,z_{\pi})\left(1-\frac{1}{(\log
L_{r})^{\varepsilon^{\prime}}}\right),$

where the first inequality uses that the distance between two points on
$\pi_{\star}$ is less than the total length of $\pi_{\star}$. Now (4.31)
follows from (4.34)–(4.35). ∎

##  5 Proof of the main theorem

Define

$U^{*}(t):=\mathrm{e}^{t[\varrho\log(\vartheta\mathfrak{r}_{t})-\varrho-\widetilde{\chi}(\varrho)]},$
(5.1)

where we recall (1.13). To prove Theorem 1.4 we show that

$\frac{1}{t}\log U(t)-\frac{1}{t}\log U^{*}(t)=o(1),\quad
t\to\infty,\qquad(\mathrm{P}\times\mathfrak{P})\text{-a.s.}$ (5.2)

The proof proceeds via upper and lower bound, proved in Sections 5.1 and 5.2,
respectively. Throughout this section, Assumptions 1.1, 1.2(1) and 1.3 are in
force.

###  5.1 Upper bound

We follow [1, Section 4.2]. The proof of the upper bound in (5.2) relies on
two lemmas showing that paths staying inside a ball of radius $\lceil
t^{\gamma}\rceil$ for some $\gamma\in(0,1)$ or leaving a ball of radius $t\log
t$ have a negligible contribution to (1.6), the total mass of the solution.

######  Lemma 5.1.

[No long paths] For any $\ell_{t}\geq t\log t$,

$\lim_{t\to\infty}\frac{1}{U^{*}(t)}\,{\mathbb{E}}_{{\mathcal{O}}}\left[\mathrm{e}^{\int_{0}^{t}\xi(X_{s})\mathrm{d}s}{\mathchoice{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.5mu\mathrm{l}}{1\mskip-5.0mu\mathrm{l}}}_{\\{\tau_{[B_{\ell_{t}}]^{\text{\rm
c}}}<t\\}}\right]=0\quad(\mathrm{P}\times\mathfrak{P})-a.s.$ (5.3)

###### Proof.

We follow [1, Lemma 4.2]. For $r\geq\ell_{t}$, let

$\mathcal{B}_{r}:=\left\\{\max_{x\in B_{r}({\mathcal{O}})}\xi(x)\geq
a_{L_{r}}+2\varrho\right\\}.$ (5.4)

Since $\lim_{t\to\infty}\ell_{t}=\infty$, Lemma 3.5 gives that
$\mathrm{P}$-a.s.

$\bigcup_{r\geq\ell_{t}}\mathcal{B}_{r}\text{ does not occur eventually as
}t\to\infty.$ (5.5)

Therefore we can work on the event
$\bigcap_{r\geq\ell_{t}}[\mathcal{B}_{r}]^{\text{\rm c}}$. On this event, we
write

$\displaystyle{\mathbb{E}}_{{\mathcal{O}}}\left[\mathrm{e}^{\int_{0}^{t}\xi(X_{s})\mathrm{d}s}{\mathchoice{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.5mu\mathrm{l}}{1\mskip-5.0mu\mathrm{l}}}_{\\{\tau_{[B_{\ell_{t}}]^{\text{\rm
c}}}<t\\}}\right]$
$\displaystyle=\sum_{r\geq\ell_{t}}{\mathbb{E}}_{{\mathcal{O}}}\left[\mathrm{e}^{\int_{0}^{t}\xi(X_{s})\mathrm{d}s}{\mathchoice{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.5mu\mathrm{l}}{1\mskip-5.0mu\mathrm{l}}}_{\\{\sup_{s\in[0,t]}|X_{s}|=r\\}}\right]$
$\displaystyle\leq\mathrm{e}^{2\varrho
t}\sum_{r\geq\ell_{t}}\,\mathrm{e}^{\varrho t\log r+\log(\delta_{r}\log\log
r)}\,{\mathbb{P}}_{{\mathcal{O}}}\left(J_{t}\geq r\right),$ (5.6)

where $J_{t}$ is the number of jumps of $X$ up to time $t$, and we use that
$|B_{r}({\mathcal{O}})|\leq(\log r)^{\delta_{r}r}$. Next, $J_{t}$ is
stochastically dominated by a Poisson random variable with parameter $t(\log
r)^{\delta_{r}}$. Hence

${\mathbb{P}}_{{\mathcal{O}}}\left(J_{t}\geq
r\right)\leq\frac{[\mathrm{e}t\,(\log
r)^{\delta_{r}}]^{r}}{r^{r}}\leq\exp\left\\{-r\log\left(\frac{r}{\mathrm{e}t\,(\log
r)^{\delta_{r}}}\right)\right\\}$ (5.7)

for large $r$. Using that $\ell_{t}\geq t\log t$, we can easily check that,
for $r\geq\ell_{t}$ and $t$ large enough,

$\varrho t\log r-r\log\left(\frac{r}{\mathrm{e}t\,(\log
r)^{\delta_{r}}}\right)<-3r,\qquad r\geq\ell_{t}.$ (5.8)

Thus (5.1) is at most

$\mathrm{e}^{2\varrho
t}\sum_{r\geq\ell_{t}}\,\mathrm{e}^{-3r+\log(\delta_{r}\log\log
r)}\,\leq\mathrm{e}^{2\varrho t}\sum_{r\geq\ell_{t}}\,\mathrm{e}^{-2r}\leq
2\,\mathrm{e}^{2\varrho
t}\,\mathrm{e}^{-2\ell_{t}}\leq\mathrm{e}^{-\ell_{t}}.$ (5.9)

Since $\lim_{t\to\infty}\ell_{t}=\infty$ and
$\lim_{t\to\infty}U^{*}(t)=\infty$, this settles the claim. ∎

######  Lemma 5.2.

[No short paths] For any $\gamma\in(0,1)$,

$\lim_{t\to\infty}\frac{1}{U^{*}(t)}\,{\mathbb{E}}_{{\mathcal{O}}}\left[\mathrm{e}^{\int_{0}^{t}\xi(X_{s})\mathrm{d}s}{\mathchoice{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.5mu\mathrm{l}}{1\mskip-5.0mu\mathrm{l}}}_{\\{\tau_{[B_{\lceil
t^{\gamma}\rceil}]^{\text{\rm
c}}}>t\\}}\right]=0\quad(\mathrm{P}\times\mathfrak{P})-a.s.$ (5.10)

###### Proof.

We follow [1, Lemma 4.3]. By Lemma 3.5 with $r=\lceil t^{\gamma}\rceil$, we
may assume that

$\max_{x\in B_{\lceil t^{\gamma}\rceil}}\xi(x)\leq\varrho\log\log L_{\lceil
t^{\gamma}\rceil}+\frac{2\varrho\log\lceil t^{\gamma}\rceil}{\vartheta\lceil
t^{\gamma}\rceil}\leq\gamma\varrho\log t+O(1),\quad t\to\infty,$ (5.11)

where the second inequality uses that $\log L_{\lceil
t^{\gamma}\rceil}\sim\log|B_{\lceil
t^{\gamma}\rceil}({\mathcal{O}})|\sim\vartheta\lceil t^{\gamma}\rceil$. Hence

$\frac{1}{U^{*}(t)}\,{\mathbb{E}}_{{\mathcal{O}}}\left[\mathrm{e}^{\int_{0}^{t}\xi(X_{s})\mathrm{d}s}{\mathchoice{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.5mu\mathrm{l}}{1\mskip-5.0mu\mathrm{l}}}_{\\{\tau_{[B_{\lceil
t^{\gamma}\rceil}]^{\text{\rm
c}}}>t\\}}\right]\leq\frac{1}{U^{*}(t)}\,\mathrm{e}^{\gamma\varrho t\log
t+O(1)}\leq\mathrm{e}^{(1-\gamma)\varrho t\log t+C\log\log\log t},\quad
t\to\infty,$ (5.12)

for any constant $C>1$. ∎

The proof of the upper bound in (5.2) also relies on a third lemma estimating
the contribution of paths leaving a ball of radius $\lceil t^{\gamma}\rceil$
for some $\gamma\in(0,1)$ but staying inside a ball of radius $t\log t$. We
slice to annulus between these two balls into layers, and derive an estimate
for paths that reach a given layer but do not reach the next layer. To that
end, fix $\gamma\in(\alpha,1)$ with $\alpha$ as in (3.1), and let

$K_{t}:=\lceil t^{1-\gamma}\log t\rceil,\qquad r^{(k)}_{t}:=k\lceil
t^{\gamma}\rceil,\quad 1\leq k\leq K_{t},\qquad\ell_{t}:=K_{t}\lceil
t^{\gamma}\rceil\geq t\log t.$ (5.13)

For $1\leq k\leq K_{t}$, define (recall (4.2))

${\mathcal{N}}^{{{\scriptscriptstyle{({k}})}}}_{t}:=\left\\{\pi\in\mathscr{P}({\mathcal{O}},V)\colon\,\operatorname{supp}(\pi)\subset
B_{r^{{{\scriptscriptstyle{({k+1}})}}}_{t}}({\mathcal{O}}),\,\operatorname{supp}(\pi)\cap
B^{\text{\rm
c}}_{r^{{{\scriptscriptstyle{({k}})}}}_{t}}({\mathcal{O}})\neq\emptyset\right\\}$
(5.14)

and set

$U^{{{\scriptscriptstyle{({k}})}}}(t):={\mathbb{E}}_{\mathcal{O}}\left[\mathrm{e}^{\int_{0}^{t}\xi(X_{s})\mathrm{d}s}{\mathchoice{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.5mu\mathrm{l}}{1\mskip-5.0mu\mathrm{l}}}_{\\{\pi_{[0,t]}(X)\in{\mathcal{N}}^{{{\scriptscriptstyle{({k}})}}}_{t}\\}}\right].$
(5.15)

######  Lemma 5.3.

[Upper bound on $U^{{{\scriptscriptstyle{({k}})}}}(t)$] For any
$\varepsilon>0$, $(\mathrm{P}\times\mathfrak{P})$-a.s. eventually as
$t\to\infty$,

$\sup_{1\leq k\leq K_{t}}\frac{1}{t}\log
U^{{{\scriptscriptstyle{({k}})}}}_{t}\leq\frac{1}{t}\log
U^{*}(t)+\varepsilon.$ (5.16)

###### Proof.

We follow [1, Lemma 4.4] Fix $k\in\\{1,\ldots,K_{t}\\}$. For
$\pi\in{\mathcal{N}}^{{{\scriptscriptstyle{({k}})}}}_{t}$, let

$\gamma_{\pi}:=\lambda_{r^{{{\scriptscriptstyle{({k+1}})}}}_{t},A}(\pi)+\mathrm{e}^{-S_{\lceil
t^{\gamma}\rceil}},\qquad
z_{\pi}\in\operatorname{supp}(\pi),|z_{\pi}|>r^{{{\scriptscriptstyle{({k}})}}}_{t},$
(5.17)

chosen such that (4.19)–(4.20) are satisfied. By Proposition 4.6 and (4.10),
$(\mathrm{P}\times\mathfrak{P})$-a.s. eventually as $t\to\infty$,

$\displaystyle\frac{1}{t}\log
U^{{{\scriptscriptstyle{({k}})}}}_{t}\leq\gamma_{\pi}-\frac{|z_{\pi}|}{t}\left(\log[\varepsilon\varrho\log(\vartheta
r^{(k+1)}_{t})]-\delta_{r}\log[\log(r^{(k+1)}_{t})]+o(1)\right).$ (5.18)

Using Corollary 3.4 and $\log L_{r}\sim\vartheta r$, we bound

$\displaystyle\gamma_{\pi}\leq\varrho\log(\vartheta
r^{(k+1)}_{t})-\widetilde{\chi}(\varrho)+\tfrac{1}{2}\varepsilon+o(1).$ (5.19)

Moreover, $|z_{\pi}|>r^{{{\scriptscriptstyle{({k+1}})}}}_{t}-\lceil
t^{\gamma}\rceil$ and

$\displaystyle\frac{\lceil
t^{\gamma}\rceil}{t}\left(\log[\varepsilon\varrho\log(\vartheta
r^{(k+1)}_{t})]-\delta_{r}\log[\log(r^{(k+1)}_{t})]\right)$ (5.20)
$\displaystyle\qquad\leq\frac{1}{t^{1-\gamma}}\log\log(2t\log t)=o(1).$

Hence

$\gamma_{\pi}\leq
F_{t}(r^{(k+1)}_{t})-\widetilde{\chi}(\varrho)+\tfrac{1}{2}\varepsilon+o(1)$
(5.21)

with

$F_{t}(r):=\varrho\log(\vartheta
r)-\frac{r}{t}\big{[}\log(\varepsilon\varrho\log(\vartheta
r))-\delta_{r}\log(\log r)\big{]},\qquad r>0.$ (5.22)

The function $F_{t}$ is maximized at any point $r_{t}$ satisfying

$\varrho t=r_{t}\left[\log(\varepsilon\varrho\log(\vartheta
r_{t}))-(\delta_{r}+r\tfrac{\mathrm{d}}{\mathrm{d}r}\delta_{r})\log\log
r+\frac{1}{\log(\vartheta r_{t})}-\frac{\delta_{r}}{\log r_{t}}\right].$
(5.23)

In particular, $r_{t}=\mathfrak{r}_{t}[1+o(1)]$, which implies that

$\sup_{r>0}F_{t}(r)\leq\varrho\log(\vartheta\mathfrak{r}_{t})-\varrho+o(1),\qquad
t\to\infty.$ (5.24)

Inserting (5.24) into (5.21), we obtain $\displaystyle\frac{1}{t}\log
U^{{{\scriptscriptstyle{({k}})}}}_{t}<\varrho\log(\vartheta\mathfrak{r}_{t})-\varrho-\widetilde{\chi}(\varrho)+\varepsilon$,
which is the desired upper bound because $\varepsilon>0$ is arbitrary. ∎

###### Proof of the upper bound in (5.2).

To avoid repetition, all statements hold $(\mathfrak{P}\times\mathrm{P})$-a.s.
eventually as $t\to\infty$. Set

$U^{{{\scriptscriptstyle{({0}})}}}(t):={\mathbb{E}}_{\mathcal{O}}\left[\mathrm{e}^{\int_{0}^{t}\xi(X_{s})\mathrm{d}s}{\mathchoice{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.5mu\mathrm{l}}{1\mskip-5.0mu\mathrm{l}}}_{\\{\tau_{[B_{\lceil
t^{\gamma}\rceil}]^{\text{\rm c}}}>t\\}}\right],\quad
U^{{{\scriptscriptstyle{({\infty}})}}}(t):={\mathbb{E}}_{\mathcal{O}}\left[\mathrm{e}^{\int_{0}^{t}\xi(X_{s})\mathrm{d}s}{\mathchoice{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.5mu\mathrm{l}}{1\mskip-5.0mu\mathrm{l}}}_{\\{\tau_{[B_{\lceil
t\log t\rceil}]^{\text{\rm c}}}\leq t\\}}\right].$ (5.25)

Then

$U(t)\leq
U^{{{\scriptscriptstyle{({0}})}}}(t)+U^{{{\scriptscriptstyle{({\infty}})}}}(t)+K_{t}\max_{1\leq
k\leq K_{t}}U^{{{\scriptscriptstyle{({k}})}}}(t).$ (5.26)

From Lemmas 5.1–5.3 and the fact that $K_{t}=o(t)$, we get

$\limsup_{t\to\infty}\left\\{\frac{1}{t}\log U(t)-\frac{1}{t}\log
U^{*}(t)\right\\}\leq\varepsilon.$ (5.27)

Since $\varepsilon>0$ is arbitrary, this completes the proof of the upper
bound in (1.14). ∎

###  5.2 Lower bound

We follow [1, Section 4.1]. Fix $\varepsilon>0$. By the definition of
$\widetilde{\chi}$, there exists an infinite rooted tree
$T=(V^{\prime},E^{\prime},\mathcal{Y})$ with degrees in
$\operatorname{supp}(D_{g})$ such that
$\chi_{T}(\varrho)<\widetilde{\chi}(\varrho)+\tfrac{1}{4}\varepsilon$. Let
$Q_{r}=B^{T}_{r}(\mathcal{Y})$ be the ball of radius $r$ around $\mathcal{Y}$
in $T$. By Proposition A.1 and (A.2), there exist a radius $R\in\mathbb{N}$
and a potential profile $q\colon B^{T}_{R}\to\mathbb{R}$ with
${\mathcal{L}}_{Q_{R}}(q;\varrho)<1$ (in particular, $q\leq 0$) such that

$\lambda_{Q_{R}}(q;T)\geq-\widehat{\chi}_{Q_{R}}(\varrho;T)-\tfrac{1}{2}\varepsilon>-\widetilde{\chi}(\varrho)-\varepsilon.$
(5.28)

For $\ell\in\mathbb{N}$, let $B_{\ell}=B_{\ell}({\mathcal{O}})$ denote the
ball of radius $\ell$ around ${\mathcal{O}}$ in ${\mathcal{G}\mathcal{W}}$. We
will show next that, $(\mathfrak{P}\times\mathrm{P})$-a.s. eventually as
$\ell\to\infty$, $B_{\ell}$ contains a copy of the ball $Q_{R}$ where the
potentail $\xi$ is bounded from below by $\varrho\log\log|B_{\ell}|+q$.

######  Proposition 5.4.

[Balls with high exceedances] $(\mathfrak{P}\times\mathrm{P})$-almost surely
eventually as $\ell\to\infty$, there exists a vertex $z\in B_{\ell}$ with
$B_{R+1}(z)\subset B_{\ell}$ and an isomorphism $\varphi:B_{R+1}(z)\to
Q_{R+1}$ such that $\xi\geq\varrho\log\log|B_{\ell}|+q\circ\varphi$ in
$B_{R}(z)$. In particular,

$\lambda_{B_{R}(z)}(\xi;{\mathcal{G}\mathcal{W}})>\varrho\log\log|B_{\ell}|-\widetilde{\chi}(\varrho)-\varepsilon.$
(5.29)

Any such $z$ necessarily satisfies $|z|\geq c\ell$
$(\mathfrak{P}\times\mathrm{P})$-a.s. eventually as $\ell\to\infty$ for some
constant $c=c(\varrho,\vartheta,\widetilde{\chi}(\varrho),\varepsilon)>0$.

###### Proof.

See [1, Proposition 4.1]. The proof carries over verbatim because the degrees
play no role. ∎

###### Proof of the lower bound in (1.14).

Let $z$ be as in Proposition 5.4. Write $\tau_{z}$ for the hitting time of $z$
by the random walk $X$. For $s\in(0,t)$, we estimate

$\displaystyle U(t)$
$\displaystyle\geq{\mathbb{E}}_{\mathcal{O}}\Big{[}\mathrm{e}^{\int_{0}^{t}\xi(X_{u})\,\mathrm{d}u}\,{\mathchoice{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.5mu\mathrm{l}}{1\mskip-5.0mu\mathrm{l}}}_{{\\{\tau_{z}\leq
s\\}}}\,{\mathchoice{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.5mu\mathrm{l}}{1\mskip-5.0mu\mathrm{l}}}_{{\\{X_{u}\in
B_{R}(z)\,\forall u\in[\tau_{z},t]\\}}}\Big{]}$ (5.30)
$\displaystyle={\mathbb{E}}_{\mathcal{O}}\Big{[}\mathrm{e}^{\int_{0}^{\tau_{z}}\xi(X_{u})\,\mathrm{d}u}\,{\mathchoice{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.5mu\mathrm{l}}{1\mskip-5.0mu\mathrm{l}}}_{{\\{\tau_{z}\leq
s\\}}}\,{\mathbb{E}}_{z}\Big{[}\mathrm{e}^{\int_{0}^{v}\xi(X_{u})\,\mathrm{d}u}\,{\mathchoice{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.5mu\mathrm{l}}{1\mskip-5.0mu\mathrm{l}}}_{{\\{X_{u}\in
B_{R}(z)\,\forall u\in[0,v]\\}}}\Big{]}\Big{|}_{v=t-\tau_{z}}\Big{]},$

where we use the strong Markov property at time $\tau_{z}$. We first bound the
last term in the integrand in (5.30). Since
$\xi\geq\varrho\log\log|B_{\ell}|+q$ in $B_{R}(z)$,

$\displaystyle{\mathbb{E}}_{z}\Big{[}\mathrm{e}^{\int_{0}^{v}\xi(X_{u})\,\mathrm{d}u}{\mathchoice{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.5mu\mathrm{l}}{1\mskip-5.0mu\mathrm{l}}}_{\\{X_{u}\in
B_{R}(z)\,\forall u\in[0,v]\\}}\Big{]}$
$\displaystyle\geq\mathrm{e}^{v\varrho\log\log|B_{\ell}|}{\mathbb{E}}_{\mathcal{Y}}\Big{[}\mathrm{e}^{\int_{0}^{v}q(X_{u})\,\mathrm{d}u}{\mathchoice{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.5mu\mathrm{l}}{1\mskip-5.0mu\mathrm{l}}}_{\\{X_{u}\in
Q_{R}\,\forall u\in[0,v]\\}}\Big{]}$ (5.31)
$\displaystyle\geq\mathrm{e}^{v\varrho\log\log|B_{\ell}|}\mathrm{e}^{v\lambda_{Q_{R}}(q;T)}\phi^{{{\scriptscriptstyle{({1}})}}}_{Q_{R}}(\mathcal{Y})^{2}$
$\displaystyle>\exp\big{\\{}v\left(\varrho\log\log|B_{\ell}|-\widetilde{\chi}(\varrho)-\varepsilon\right)\big{\\}}$

for large $v$, where we used that $B_{R+1}(z)$ is isomorphic to $Q_{R+1}$ for
the indicators in the first inequality, and applied Lemma B.2 and (5.28) to
obtain the second and third inequalities, respectively. On the other hand,
since $\xi\geq 0$,

${\mathbb{E}}_{\mathcal{O}}\Big{[}\mathrm{e}^{\int_{0}^{\tau_{z}}\xi(X_{u})\,\mathrm{d}u}{\mathchoice{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.5mu\mathrm{l}}{1\mskip-5.0mu\mathrm{l}}}{\\{\tau_{z}\leq
s\\}}\Big{]}\geq{\mathbb{P}}_{\mathcal{O}}(\tau_{z}\leq s),$ (5.32)

and we can bound the latter probability from below by the probability that the
random walk runs along a shortest path from the root ${\mathcal{O}}$ to $z$
within a time at most $s$. Such a path $(y_{i})_{i=0}^{|z|}$ has
$y_{0}={\mathcal{O}}$, $y_{|z|}=z$, $y_{i}\sim y_{i-1}$ for $i=1,\ldots,|z|$,
has at each step from $y_{i}$ precisely $\deg(y_{i})$ choices for the next
step with equal probability, and the step is carried out after an exponential
time $E_{i}$ with parameter $\deg(y_{i})$. This gives

$\displaystyle\mathrm{P}_{\mathcal{O}}(\tau_{z}\leq s)$
$\displaystyle\geq\Big{(}\prod_{i=1}^{|z|}\frac{1}{\deg(y_{i})}\Big{)}P\Big{(}\sum_{i=1}^{|z|}E_{i}\leq
s\Big{)}\geq((\log|z|)^{\delta_{\ell}})^{-|z|}{\rm Poi}_{d_{\rm
min}s}([|z|,\infty)),$ (5.33)

where ${\rm Poi}_{\gamma}$ is the Poisson distribution with parameter
$\gamma$, and $P$ is the generic symbol for probability. Summarising, we
obtain

$\displaystyle U(t)$
$\displaystyle\geq((\log|z|)^{\delta_{l}})^{-|z|}\mathrm{e}^{-d_{\rm
min}s}\frac{(d_{\rm
min}s)^{|z|}}{|z|!}\mathrm{e}^{(t-s)\left[\varrho\log\log|B_{\ell}|-\widetilde{\chi}(\varrho)-\varepsilon\right]}$
(5.34)
$\displaystyle\geq\exp\left\\{-d_{\min}s+(t-s)\left[\varrho\log\log|B_{\ell}|-\widetilde{\chi}(\varrho)-\varepsilon\right]-|z|\log\left(\frac{(\log|z|)^{\delta_{\ell}}}{d_{\min}}\frac{|z|}{s}\right)\right\\}$
$\displaystyle\geq\exp\left\\{-d_{\min}s+(t-s)\left[\varrho\log\log|B_{\ell}|-\widetilde{\chi}(\varrho)-\varepsilon\right]-\ell\log\left(\frac{(\log\ell)^{\delta_{\ell}}}{d_{\min}}\frac{\ell}{s}\right)\right\\},$

where in the last inequality we use that $s\leq|z|$ and $\ell\geq|z|$. Further
assuming that $\ell=o(t)$, we see that the optimum over $s$ is obtained at

$s=\frac{\ell}{d_{\min}+\varrho\log\log|B_{\ell}|-\widetilde{\chi}(\varrho)-\varepsilon}=o(t).$
(5.35)

Note that, by Proposition 5.4, this $s$ indeed satisfies $s\leq|z|$. Applying
(1.10) we get, after a straightforward computation,
$(\mathfrak{P}\times\mathrm{P})$-a.s. eventually as $t\to\infty$,

$\frac{1}{t}\log
U(t)\geq\varrho\log\log|B_{\ell}|-\frac{\ell}{t}\log\log\ell-\frac{\ell}{t}\delta_{\ell}\log\log\ell-\widetilde{\chi}(\varrho)-\varepsilon+O\left(\frac{\ell}{t}\right).$
(5.36)

Inserting $\log|B_{\ell}|\sim\vartheta\ell$, we get

$\frac{1}{t}\log U(t)\geq
F_{\ell}-\widetilde{\chi}(\varrho)-\varepsilon+o(1)+O\left(\frac{\ell}{t}\right)$
(5.37)

with

$F_{\ell}=\varrho\log(\vartheta\ell)-\frac{\ell}{t}\log\log\ell-\frac{\ell}{t}\delta_{\ell}\log\log\ell.$
(5.38)

The optimal $\ell$ for $F_{\ell}$ satisfies

$\varrho
t=\ell\big{[}1+(\delta_{\ell}+\ell\tfrac{\mathrm{d}}{\mathrm{d}\ell}\delta_{\ell})]\log\log\ell+\frac{\ell\delta_{\ell}}{\log\ell}+\frac{\ell}{\log\ell},$
(5.39)

i.e., $\ell=\mathfrak{r}_{t}[1+o(1)]$. For this choice we obtain

$\frac{1}{t}\log
U(t)\geq\varrho\log(\vartheta\mathfrak{r}_{t})-\varrho-\widetilde{\chi}(\varrho)-\varepsilon+o(1).$
(5.40)

Hence $(\mathfrak{P}\times\mathrm{P})$-a.s.

$\liminf_{t\to\infty}\left\\{\frac{1}{t}\log U(t)-\frac{1}{t}\log
U^{*}(t)\right\\}\geq-\varepsilon.$ (5.41)

Since $\varepsilon>0$ is arbitrary, this completes the proof of the lower
bound in (1.14). ∎

#### REMARK:

It is clear from (5.23) and (5.39) that, in order to get the correct
asymptotics, it is crucial that both $\delta_{r}$ and
$r\frac{\mathrm{d}}{\mathrm{d}r}\delta_{r}$ tend to zero as $r\to\infty$. This
is why Assumption 1.3 is the weakest condition on the tail of the degree
distribution under which the arguments in [1] can be pushed through.

##  Appendix A Dual variational formula

We introduce alternative representations for $\chi$ in (1.9) in terms of a
‘dual’ variational formula. Fix $\varrho\in(0,\infty)$ and a graph $G=(V,E)$.
The functional

${\mathcal{L}}(q;G):=\sum_{x\in
V}\mathrm{e}^{q(x)/\varrho}\in[0,\infty],\qquad
q\colon\,V\to[-\infty,\infty),$ (A.1)

plays the role of a large deviation rate function for the potential $\xi$ in
$V$ (compare with (1.7)). For $\Lambda\subset V$, define

$\widehat{\chi}_{\Lambda}(G):=-\sup_{\begin{subarray}{c}q\colon
V\to[-\infty,\infty),\\\ {\mathcal{L}}(q;G)\leq
1\end{subarray}}\lambda_{\Lambda}(q;G)\in[0,\infty).$ (A.2)

The condition ${\mathcal{L}}(q;G)\leq 1$ under the supremum ensures that the
potentials $q$ have a fair probability under the i.i.d. double-exponential
distribution. Write $\widehat{\chi}(G)=\widehat{\chi}_{V}(G)$.

######  Proposition A.1.

[Alternative representations for $\chi$] For any graph $G=(V,E)$ and any
$\Lambda\subset V$,

$\widehat{\chi}_{\Lambda}(\varrho;G)\geq\widehat{\chi}_{V}(\varrho;G)=\widehat{\chi}_{G}(\varrho)=\chi_{G}(\varrho).$
(A.3)

###### Proof.

See [1, Section A.1] ∎

##  Appendix B Largest eigenvalue

We recall the Rayleigh-Ritz formula for the principal eigenvalue of the
Anderson Hamiltonian. For $\Lambda\subset V$ and
$q\colon\,V\to[-\infty,\infty)$, let $\lambda_{\Lambda}(q;G)$ denote the
largest eigenvalue of the operator $\Delta_{G}+q$ in $\Lambda$ with Dirichlet
boundary conditions on $V\setminus\Lambda$, i.e.,

$\displaystyle\lambda_{\Lambda}(q;G):=\sup\big{\\{}\langle(\Delta_{G}+q)\phi,\phi\rangle_{\ell^{2}(V)}\colon\,\phi\in\mathbb{R}^{V},\,\operatorname{supp}\phi\subset\Lambda,\,\|\phi\|_{\ell^{2}(V)}=1\big{\\}}.$
(B.1)

######  Lemma B.1.

[Spectral bounds]

  1. (1)

For any $\Gamma\subset\Lambda\subset V$,

$\max_{z\in\Gamma}q(z)-D_{\bar{z}}\leq\lambda_{\Gamma}(q;G)\leq\lambda_{\Lambda}(q;G)\leq\max_{z\in\Lambda}q(z)$
(B.2)

with $\bar{z}=\mathrm{arg}\max_{z\in\Gamma}q(z)$ and $D_{\bar{z}}$ the degree
of $\bar{z}$.

  2. (2)

The eigenfunction corresponding to $\lambda_{\Lambda}(q;G)$ can be taken to be
non-negative.

  3. (3)

If $q$ is real-valued and $\Gamma\subsetneq\Lambda$ is finite and connected in
$G$, then the second inequality in (B.2) is strict and the eigenfunction
corresponding to $\lambda_{\Lambda}(q;G)$ is strictly positive.

###### Proof.

Write

$\displaystyle\langle(\Delta_{G}+q)\phi,\phi\rangle_{\ell^{2}(V)}$
$\displaystyle=\sum_{x\in\Lambda}\left[(\Delta_{G}\phi)(x)+q(x)\phi(x)\right]\phi(x)$
(B.3) $\displaystyle=\sum_{x\in\Lambda}\sum_{{y\in\Lambda:}\atop{\\{x,y\\}\in
E_{\Lambda}}}[\phi(y)-\phi(x)]\phi(x)+\sum_{x\in\Lambda}q(x)\phi(x)^{2}$
$\displaystyle=-\tfrac{1}{2}\sum_{{x,y\in\Lambda:}\atop{\\{x,y\\}\in
E_{\Lambda}}}[\phi(x)-\phi(y)]^{2}+\sum_{x\in\Lambda}q(x)\phi(x)^{2},$

where the first sum in the last line runs over all ordered pairs $(x,y)$ with
$(x,y)\neq(y,x)$, which gives rise to the factor $\tfrac{1}{2}$. The upper
bound in (B.2) follows from the estimate

$\langle(\Delta_{G}+q)\phi,\phi\rangle\leq\sum_{x\in\Lambda}q(x)\phi(x)^{2}\leq\max_{z\in\Lambda}q(z)\sum_{x\in\Lambda}\phi(x)^{2}=\max_{z\in\Lambda}q(z).$
(B.4)

To get the lower bound in (B.2), we use the fact that $\lambda_{\Lambda}$ is
non-decreasing in $q$. Hence, replacing $q(z)$ by $-\infty$ for every
$z\neq\bar{z}$ and taking as test function $\phi=\bar{\phi}=\delta_{\bar{z}}$,
we get from (B.3) that

$\displaystyle\lambda_{\Lambda}(q;G)$
$\displaystyle\geq-\tfrac{1}{2}\sum_{{x,y\in\Lambda:}\atop{\\{x,y\\}\in
E_{\Lambda}}}\left[\bar{\phi}(x)-\bar{\phi}(y)\right]^{2}+\sum_{x\in\Lambda}q(x)\bar{\phi}(x)^{2}$
(B.5) $\displaystyle=-\tfrac{1}{2}\sum_{{y\in\Lambda:}\atop{\\{\bar{z},y\\}\in
E_{\Lambda}}}1+q(\bar{z})=-D_{\bar{z}}+\max_{z\in\Lambda}q(z),$

which settles the claim in (1). The claims in (2) and (3) are standard. ∎

Inside ${\mathcal{G}\mathcal{W}}$, fix a finite connected subset
$\Lambda\subset V$, and let $H_{\Lambda}$ denote the Anderson Hamiltonian in
$\Lambda$ with zero Dirichlet boundary conditions on
$\Lambda^{c}=V\backslash\Lambda$ (i.e., the restriction of the operator
$H_{G}=\Delta_{G}+\xi$ to the class of functions supported on $\Lambda$). For
$y\in\Lambda$, let $u^{y}_{\Lambda}$ be the solution of

$\begin{array}[]{llll}\partial_{t}u(x,t)&=&(H_{\Lambda}u)(x,t),&x\in\Lambda,\,t>0,\\\
u(x,0)&=&\delta_{y}(x),&x\in\Lambda,\end{array}$ (B.6)

and set $U^{y}_{\Lambda}(t):=\sum_{x\in\Lambda}u^{y}_{\Lambda}(x,t)$. The
solution admits the Feynman-Kac representation

$u^{y}_{\Lambda}(x,t)={\mathbb{E}}_{y}\left[\exp\left\\{\int_{0}^{t}\xi(X_{s})\text{\rm
d}\mkern
0.5mus\right\\}{\mathchoice{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.5mu\mathrm{l}}{1\mskip-5.0mu\mathrm{l}}}\\{\tau_{\Lambda^{{\text{\rm
c}}}}>t,X_{t}=x\\}\right],$ (B.7)

where $\tau_{\Lambda^{\text{\rm c}}}$ is the hitting time of
$\Lambda^{\text{\rm c}}$. It also admits the spectral representation

$u^{y}_{\Lambda}(x,t)=\sum_{k=1}^{|\Lambda|}\text{\rm
e}^{t\lambda^{{{\scriptscriptstyle{({k}})}}}_{\Lambda}}\phi_{\Lambda}^{{{\scriptscriptstyle{({k}})}}}(y)\phi^{{{\scriptscriptstyle{({k}})}}}_{\Lambda}(x),$
(B.8)

where
$\lambda^{{{\scriptscriptstyle{({1}})}}}_{\Lambda}\geq\lambda^{{{\scriptscriptstyle{({2}})}}}_{\Lambda}\geq\cdots\geq\lambda^{{{\scriptscriptstyle{({|\Lambda|}})}}}_{\Lambda}$
and
$\phi^{{{\scriptscriptstyle{({1}})}}}_{\Lambda},\phi^{{{\scriptscriptstyle{({2}})}}}_{\Lambda},\ldots,\phi^{{{\scriptscriptstyle{({|\Lambda|}})}}}_{\Lambda}$
are, respectively, the eigenvalues and the corresponding orthonormal
eigenfunctions of $H_{\Lambda}$. These two representations may be exploited to
obtain bounds for one in terms of the other, as shown by the following lemma.

######  Lemma B.2.

[Bounds on the solution] For any $y\in\Lambda$ and any $t>0$,

$\qquad\text{\rm
e}^{t\lambda^{{{\scriptscriptstyle{({1}})}}}_{\Lambda}}\phi^{{{\scriptscriptstyle{({1}})}}}_{\Lambda}(y)^{2}\leq{\mathbb{E}}_{y}\left[\text{\rm
e}^{\int_{0}^{t}\xi(X_{s})\text{\rm d}\mkern
0.5mus}{\mathchoice{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.5mu\mathrm{l}}{1\mskip-5.0mu\mathrm{l}}}_{\\{\tau_{\Lambda^{\text{\rm
c}}}>t,X_{t}=y\\}}\right]\\\ \leq{\mathbb{E}}_{y}\left[\text{\rm
e}^{\int_{0}^{t}\xi(X_{s})\text{\rm d}\mkern
0.5mus}{\mathchoice{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.5mu\mathrm{l}}{1\mskip-5.0mu\mathrm{l}}}_{\\{\tau_{\Lambda^{\text{\rm
c}}}>t\\}}\right]\leq\text{\rm
e}^{t\lambda^{{{\scriptscriptstyle{({1}})}}}_{\Lambda}}|\Lambda|^{1/2}.\qquad$
(B.9)

###### Proof.

The first and third inequalities follow from (B.7)–(B.8) after a suitable
application of Parseval’s identity. The second inequality is elementary. ∎

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