# Heat kernel fluctuations and quantitative homogenization for the one-
dimensional Bouchaud trap model

Sebastian Andres  Technische Universität Braunschweig Institut für
Mathematische Stochastik, Universitätsplatz 2, 38106 Braunschweig, Germany
<EMAIL_ADDRESS>,  David A. Croydon  Research Institute
for Mathematical Sciences Kyoto University, Kyoto 606-8502, Japan
<EMAIL_ADDRESS>and  Takashi Kumagai  Waseda University 3-4-1
Okubo, Shinjuku-ku, Tokyo 169-8555, Japan<EMAIL_ADDRESS>
###### Abstract.

We present on-diagonal heat kernel estimates and quantitative homogenization
statements for the one-dimensional Bouchaud trap model. The heat kernel
estimates are obtained using standard techniques, with key inputs coming from
a careful analysis of the volume growth of the invariant measure of the
process under study. As for the quantitative homogenization results, these
include both quenched and annealed Berry-Esseen-type theorems, as well as a
quantitative quenched local limit theorem. Whilst the model we study here is a
particularly simple example of a random walk in a random environment, we
believe the roadmap we provide for establishing the latter result in
particular will be useful for deriving quantitative local limit theorems in
other, more challenging, settings.

###### Key words and phrases:

Bouchaud trap model, heat kernel, law of iterated logarithm

###### 2000 Mathematics Subject Classification:

60K37, 60F17, 82C41, 82B43

##  1\. Introduction

A principal motivation for studying random walks in random environments is to
explore the relationships that might exist between the microscopic properties
of a disordered medium and its large scale conductivity. One particular
question of interest is whether homogenization occurs, in the sense that the
deformities in the random medium are ‘smoothed out’, so that only an averaged
effect remains; the alternative being that the local inhomogeneities persist
in the scaling limit and some kind of anomalous behaviour is seen.

In the case of the one-dimensional Bouchaud trap model (as will be introduced
precisely in Section 1.1 below), it is known that there is a phase transition
between a regime where homogenization occurs, and one where it does not. The
first goal of this article is to highlight the differences between the
behaviour of the heat kernel in these two regimes, demonstrating how one sees
a local limit theorem in the former, and almost-sure heat kernel fluctuations
that are closely linked to the inhomogeneity of the associated invariant
measure in the latter (see Section 1.2). In the case of homogenization, it is
natural to ask how quickly this occurs. Accordingly, as our second main
contribution, we derive a number of quantitative homogenization results for
the one-dimensional Bouchaud trap model (see Section 1.3 for details). These
include a Berry-Esseen-type theorem, which quantifies the rate at which the
marginals of the discrete model converge to a normal distribution. Moreover,
combining such statements with our volume estimates, we are able to establish
a quantitative quenched local limit theorem. We expect the somewhat basic
analytic approach we adopt for arriving at this conclusion is quite robust and
should be adaptable to other settings.

###  1.1. The model

The _Bouchaud trap model_ (BTM) is a random walk in a random medium given by a
landscape of traps which retain the walk for some amount of time. It has its
origins in the statistical physics literature, where it was proposed as a
simple effective model for the dynamics of spin-glasses on certain time-
scales, see [22]. The one-dimensional versions of the model considered in this
paper were first studied in [44]; for a general overview of the BTM we refer
to [18]. A distinctive feature of the BTM is that when the distribution of the
traps is sufficiently heavy-tailed, the trapping mechanism becomes relevant
even in the large scale behaviour of the system, and the model exhibits
localization [34, 35, 36, 44, 67], subdiffusivity [24, 45] and aging [17, 25,
45].

To describe the model in detail, we start by defining a trapping landscape
$\tau=(\tau_{x})_{x\in\mathbb{Z}}$, which is a collection of independent and
identically distributed (i.i.d.) random variables, built on a probability
space with probability measure $\operatorname{\mathbf{P}}$, such that

$\operatorname{\mathbf{P}}[\tau_{0}\geq u]=u^{-\alpha},\qquad\forall u\geq 1,$
(1.1)

for some $\alpha\in(0,\infty)$. Conditional on $\tau$, the dynamics of the
(symmetric) BTM are given by a continuous-time $\mathbb{Z}$-valued Markov
chain $X=(X_{t})_{t\geq 0}$, with generator

$\displaystyle\mathcal{L}f(x)=\frac{1}{2\tau_{x}}\sum_{y\sim
x}\big{(}f(y)-f(x)\big{)}.$ (1.2)

Thus the dynamics of $X$ are particularly simple. Namely, the process $X$
waits at a vertex $x$ an exponentially distributed time with mean $\tau_{x}$;
after this time, it jumps to one of the two neighbours of $x$ chosen uniformly
at random. It is easy to see that the measure
$\sum_{x\in\mathbb{Z}}\tau_{x}\delta_{x}$ is a reversible measure for the
Markov chain $X$.

In the following, we denote by $\operatorname{\mathbf{E}}$ the expectation
operator associated with the probability measure $\operatorname{\mathbf{P}}$
determining the law of $\tau$. Given a particular realisation of $\tau$, we
write $P^{\tau}_{x}$ for the law of $X$, when started from $x$; this is the
quenched law of $X$. The quenched transition density (or heat kernel) is given
by

$p^{\tau}_{t}(x,y):=\frac{P^{\tau}_{x}(X_{t}=y)}{\tau_{y}}.$ (1.3)

Moreover, we write

$\mathbb{P}(\cdot)=\int P_{0}^{\tau}(\cdot)\mathbf{P}(d\tau)$

for the annealed law of the process $X$ started from 0.

###  1.2. Scaling limits and heat kernel fluctuations

A well-known phenomenon in the study of stochastic processes in random media
is the occurrence of anomalous heat kernel behaviour when the medium is highly
inhomogeneous. Possible manifestations of such irregular behaviour are the
presence of fluctuations for the short- or long-time asymptotics of the on-
diagonal heat kernel, or scaling towards non-Gaussian processes under non-
diffusive space-time scaling. Some examples include the random conductance
model, Brownian motion on certain random fractals, and simple random walks on
the incipient infinite cluster of a critical Galton-Watson tree or on two-
dimensional uniform spanning trees, see [1] for a recent survey about the
occurrence of heat kernel fluctuations. In this paper, we add the one-
dimensional symmetric Bouchaud trap model to this class of examples. (We note
that the results of this section were announced in [1].)

We first recall the scaling limits known to hold for the process $X$. In each
case in the statement of the subsequent theorem, the topology considered is
the usual Skorohod $J_{1}$ topology on the space of càdlàg paths
$D([0,\infty))$.

######  Theorem 1.1.

  1. (i)

If $\alpha\in(0,1)$, then $(\varepsilon X_{t/\varepsilon^{1+1/\alpha}})_{t\geq
0}$ converges in distribution under the annealed law $\mathbb{P}$ to the
Fontes-Isopi-Newman diffusion of [45, Definition 1.2] with parameter $\alpha$.

  2. (ii)

If $\alpha=1$, then $(\varepsilon
X_{t\log(\varepsilon^{-1})/\varepsilon^{2}})_{t\geq 0}$ converges in
distribution under the annealed law $\mathbb{P}$ to Brownian motion.

  3. (iii)

If $\alpha>1$, then, for $\mathbf{P}$-a.s. realisation of the environment
$\tau$, $(\varepsilon X_{t\mathbf{E}(\tau_{0})/\varepsilon^{2}})_{t\geq 0}$
converges in distribution under the quenched law $P^{\tau}_{0}$ to Brownian
motion.

###### Proof.

Part (i) is contained within [16, Theorem 3.2(ii)]. (We note that all the
claims of [16, Theorem 3.2] readily extend to the case $\beta=0$.) See also
[45] for the introduction of the limiting process and a scaling limit for a
single marginal of the process, and [17] for a related result in the non-
symmetric case. Part (iii) is contained within [16, Theorem 3.2(i)]. Part (ii)
is described in [16, Remark 3.3], and can also be checked by applying the
resistance scaling techniques of [29, 31]. Indeed, via such an approach, a
similar result was checked in the more challenging case of the random walk on
the range of random walk in four dimensions in [37]. ∎

The above result shows the BTM exhibits quenched homogenization when
$\alpha>1$ and anomalous scaling behaviour for $\alpha\leq 1$. Our goal here
is to show that in the latter case, the on-diagonal part of the heat kernel
fluctuates around a leading term of polynomial order, and in the former case,
the heat kernel satisfies a quenched local limit theorem. To capture the
natural scaling of the on-diagonal heat kernel, we introduce the following
$\alpha$-dependent functions:

$\phi_{\alpha}(t):=\left\\{\begin{array}[]{ll}t^{-\frac{1}{1+\alpha}},&\hbox{if
$\alpha\in(0,1)$;}\\\ t^{-\frac{1}{2}}(\log t)^{-\frac{1}{2}},&\hbox{if
$\alpha=1$;}\\\ t^{-\frac{1}{2}},&\hbox{if $\alpha>1$.}\end{array}\right.$

Specifically, we are able to prove the following quenched heat kernel
estimates. Note that the exponents in the limsup part of (i) match those for
the limiting Fontes-Isopi-Newman diffusion, as verified in [32, Theorem 4.6].
Similarly, the liminf part of (i) essentially matches the result of [32,
Theorem 4.5], modulo a more careful computation of log exponents that is
possible in our simpler setting. (In [1], part (iii) is presented slightly
differently, and erroneously misses the factor of $\mathbf{E}(\tau_{0})$ that
arises in the scaling of the invariant measure.)

######  Theorem 1.2.

  1. (i)

If $\alpha\in(0,1)$, then it $\operatorname{\mathbf{P}}$-a.s. holds that

$\limsup_{t\rightarrow\infty}\frac{p_{t}^{\tau}(0,0)}{\phi_{\alpha}(t)(\log\log
t)^{\frac{1-\alpha}{1+\alpha}}}\in(0,\infty),$

and

$\liminf_{t\rightarrow\infty}\frac{p_{t}^{\tau}(0,0)}{\phi_{\alpha}(t)(\log
t)^{-\theta}}=\begin{cases}0,&\text{if $\theta=\frac{1}{1+\alpha}$},\\\
\infty,&\text{if $\theta>\frac{3}{\alpha}$.}\end{cases}$

  2. (ii)

If $\alpha=1$, then it $\operatorname{\mathbf{P}}$-a.s. holds that

$\limsup_{t\rightarrow\infty}\frac{p_{t}^{\tau}(0,0)}{\phi_{\alpha}(t)}\in(0,\infty),$

and

$\liminf_{t\rightarrow\infty}\frac{p_{t}^{\tau}(0,0)}{\phi_{\alpha}(t)(\log\log
t)^{-\theta}}=\begin{cases}0,&\text{if $\theta=\frac{1}{2}$},\\\
\infty,&\text{if $\theta>3$.}\end{cases}$

  3. (iii)

If $\alpha>1$, then it $\operatorname{\mathbf{P}}$-a.s. holds that, for every
$x_{0}\in[0,\infty)$ and compact interval $I\subseteq(0,\infty)$,

$\lim_{\lambda\rightarrow\infty}\sup_{|x|\leq x_{0}}\sup_{t\in
I}\left|\mathbf{E}(\tau_{0})\lambda
p^{\tau}_{\mathbf{E}(\tau_{0})\lambda^{2}t}\left(0,\lfloor\lambda
x\rfloor\right)-\frac{1}{\sqrt{2\pi t}}e^{-\frac{x^{2}}{2t}}\right|=0.$

In particular,

$\lim_{t\rightarrow\infty}\phi_{\alpha}(\mathbf{E}(\tau_{0})t)^{-1}p_{t}^{\tau}(0,0)=\frac{1}{\sqrt{2\pi}}.$

######  Remark 1.3.

Similarly to [37, Corollary 1.4], for $\alpha=1$, it would be possible to
prove a local limit theorem in probability (i.e. a statement analogous to part
(iii) above, but with the limit holding in probability, and appropriate
logarithmic factors appearing in the scaling).

We conclude the section with asymptotic statements concerning the distribution
and expectation of the heat kernel that justify the choice of $\phi_{\alpha}$
as the natural scaling function. We note that part (ii) of the result is
similar to that of [33], in that we are only able to check the appropriate
integrability of the heat kernel for a certain range of the parameter
$\alpha$.

######  Theorem 1.4.

  1. (i)

For every $\alpha>0$,

$\lim_{\lambda\rightarrow\infty}\liminf_{t\rightarrow\infty}\operatorname{\mathbf{P}}\left(\lambda^{-1}\leq\phi_{\alpha}(t)^{-1}p_{t}^{\tau}(0,0)\leq\lambda\right)=1.$

  2. (ii)

For every $\alpha>0$, there exists an $\varepsilon>0$ such that

$0<\liminf_{t\rightarrow\infty}\phi_{\alpha}(t)^{-\varepsilon}\operatorname{\mathbf{E}}\left(p_{t}^{\tau}(0,0)^{\varepsilon}\right)\leq\limsup_{t\rightarrow\infty}\phi_{\alpha}(t)^{-\varepsilon}\operatorname{\mathbf{E}}\left(p_{t}^{\tau}(0,0)^{\varepsilon}\right)<\infty.$

Moreover, if $\alpha>\frac{3}{2}$, then we may take $\varepsilon=1$.

###  1.3. Quantitative stochastic homogenization

During the last decade, starting from the works [48, 49], the area of
quantitative stochastic homogenization of elliptic and parabolic equations has
witnessed tremendous progress, see e.g. [3, 5, 7, 8, 14, 15, 38, 41, 46, 47,
50, 51], and the two monographs [4, 6]. Closely related to the topic of
homogenization in PDE theory is the problem of deriving invariance principles
for the associated random walk in random environment. For the one-dimensional
BTM with $\alpha>1$, the results stated in the last subsection include an
invariance principle in Theorem 1.1(iii) and a local limit theorem in Theorem
1.2(iii). As noted at the start of the introduction, it is natural to seek to
strengthen such statements and to provide some corresponding quantitative
stochastic homogenization results. In the case when $\alpha>2$, that is when
the variance of $\tau_{x}$ is finite, we are able give some estimates on the
speed of convergence in the invariance principle and the local limit theorem.
We first quantify the speed of convergence in the invariance principle by
means of quenched and annealed Berry-Esseen-type estimates. Throughout the
article, we use the $O$ and $o$ notation to describe asymptotic bounds (as
$t\rightarrow\infty$) in the usual way.

######  Theorem 1.5 (Berry-Esseen theorem).

Suppose that $\alpha>2$ and let
$\sigma^{2}\mathrel{\mathop{:}}=1/\operatorname{\mathbf{E}}[\tau_{0}]$.

  1. (i)

It holds that

$\sup_{x\in\mathbb{R}}\Big{|}\mathbb{P}\big{[}X_{t}\leq\sigma x\sqrt{t}\
\big{]}-\Phi(x)\Big{|}\;=\;O\left(t^{-\frac{1}{10}}(\log
t)^{\frac{1}{5}}\right),$

where $\Phi(x):=(2\pi)^{-1/2}\int_{-\infty}^{x}e^{-u^{2}/2}\,du$ denotes the
distribution function of the standard normal distribution.

  2. (ii)

Fix $\theta\in(0,\frac{\alpha-2}{10(2\alpha-3)})$. For
$\operatorname{\mathbf{P}}$-a.e. $\tau$, it holds that

$\sup_{x\in\mathbb{R}}\Big{|}P_{0}^{\tau}\big{[}X_{t}\leq\sigma x\sqrt{t}\
\big{]}-\Phi(x)\Big{|}=O\left(t^{-\theta}\right).$

The annealed bound in Theorem 1.5(i) matches (up to the logarithmic
correction) the bound obtained in [65] for the random conductance model,
another model of a random walk in random environment, which has some strong
connections with the BTM. More precisely, for the random conductance model
with uniformly elliptic i.i.d. conductances, an annealed Berry-Esseen theorem
as in Theorem 1.5(i) was proven in [65] for arbitrary dimension $d\geq 1$ with
rate $t^{-1/10}$ (plus logarithmic corrections for $d=2$), and with rate
$t^{-1/5}$ for $d\geq 3$ (with some logarithmic corrections for $d=3$). In
[2], this was extended to some quenched and annealed bounds in $d\geq 3$ for
unbounded and correlated conductances satisfying strong moment conditions and
a quantified ergodicity assumption in the form of a spectral gap estimate.

Given that the random walk $(X_{t})_{t\geq 0}$ is a martingale, our approach
to proving Theorem 1.5 is based on a Berry-Esseen-type estimate for
martingales of [53] (see Theorem 4.1 below). It requires quantification of the
speed of convergence of

$E_{0}^{\tau}\left[\Big{|}\frac{\langle
X\rangle_{t}}{t}-\sigma^{2}\Big{|}^{2}\right]$

towards zero, where $\langle X\rangle$ denotes the quadratic variation process
of the martingale $X$, see Proposition 4.2 below. To this end, we exploit the
fact that $X$ is a time-change of the simple random walk on $\mathbb{Z}$. Then
it suffices to show similar quantitative estimates on the additive functional
governing the time-change, which is a continuous time version of the random
walk in random scenery studied in [39].

######  Remark 1.6.

The exponents in Theorem 1.5 are non-optimal. On the one hand, this is due to
our non-optimal (quenched) estimate for the random walk in random scenery of
Proposition 4.3. On the other hand, an additional factor of $1/5$ in the
exponent arises from the application of the general Berry-Esseen theorem for
martingales (see Theorem 4.1 below with the choice $n=2$). Recall that the
jumps of the martingale $X$ are obviously bounded. In such a situation the
results in [66] show that the decay rate given by $E_{0}^{\tau}[|\frac{\langle
X\rangle_{t}}{t}-\sigma^{2}|^{2}]^{1/5}$ is optimal (for the choice $n=2$ in
Theorem 4.1). Thus, the only possibility to improve the exponents (within this
approach) is to apply Theorem 4.1 for larger values of $n$, which requires
control on higher moments of $\big{|}\frac{\langle
X\rangle_{t}}{t}-\sigma^{2}\big{|}$ (cf. the discussions of [65, page 6] and
[2, Remark 1.12]). Note that the rate $1/2$ is the best possible one, since it
is known to be the convergence rate for the simple random walk.

By combining the Berry-Esseen estimates with suitable volume estimates we
derive the following quantitative version of the quenched local limit theorem.

######  Theorem 1.7 (Quantitative local limit theorem).

Suppose that $\alpha>2$. For any
$\theta\in(0,\frac{\alpha-2}{10(2\alpha-3)})$, for all $K>0$ and $0<T_{1}\leq
T_{2}$,

$\lim_{n\to\infty}n^{2\theta/3}\sup_{|x|\leq
K}\sup_{t\in[T_{1},T_{2}]}\bigg{|}\operatorname{\mathbf{E}}[\tau_{0}]\,n\,p^{\tau}_{\operatorname{\mathbf{E}}[\tau_{0}]n^{2}t}(0,\lfloor
nx\rfloor)-\frac{1}{\sqrt{2\pi
t}}e^{-\frac{x^{2}}{2t}}\bigg{|}=0,\qquad\mathbf{P}\text{-a.s.}$

Again it would be desirable to identify and prove optimal rates of convergence
in the local limit theorem. In the context of the random conductance model,
optimal quantitative homogenization results for heat kernels and Green
functions can be established by using techniques from quantitative stochastic
homogenization, see [6, Chapters 8–9] for details in the uniformly elliptic
case with finite range dependencies. This technique has been adapted to random
conductance models on i.i.d. percolation clusters in [38], and it is expected
that it also applies to other degenerate models.

######  Remark 1.8.

Whilst we do not pursue it here, it would be of further interest to prove an
annealed version of Theorem 1.7. Furthermore, we know that a quenched local
limit theorem holds for all $\alpha>1$, see Theorem 1.2(iii), and so it is
also relevant to ask about the speed of convergence when $\alpha\in(1,2]$.
Since we use various second moment estimates, this is beyond the techniques of
this article.

###  1.4. Organisation of the article

The remainder of the article is organised as follows. In Section 2, we
summarise classical results concerning i.i.d. sums of heavy-tailed random
variables, which will shed light on the volume growth of the invariant measure
in the BTM. (Related tail bounds are presented in Appendix A.) In Section 3,
we prove the heat kernel estimates of Theorems 1.2 and 1.4. Finally, in
Section 4, we establish the quantitative homogenization results of Theorems
1.5 and 1.7.

##  2\. Law of iterated logarithm in the $\alpha$-stable regime

The goal of this section is to summarise the asymptotic fluctuations of i.i.d.
sums with distributional tail as at (1.1). In particular, from classical
results, we will deduce the following theorem. Note that, for our heat kernel
estimates, we only need the detailed statements of parts (i) and (ii), but we
include the remaining cases for completeness. (It would potentially be
interesting to determine whether parts (iii)-(v) could be used to understand
the second order behaviour of the heat kernel in the relevant cases.) We use
the notation $x\vee y$ and $x\wedge y$ for $\max\\{x,y\\}$ and
$\min\\{x,y\\}$, respectively.

######  Theorem 2.1.

Let $X_{1},X_{2},\ldots$ be i.i.d. on some probability space
$(\Omega,\mathcal{F},\operatorname{\mathbf{P}})$ with distribution function
$F$ of the form $F(x)=(1-c_{F}x^{-\alpha})\vee 0$ with $c_{F}\in(0,\infty)$
and $\alpha\in(0,\infty)$. Write
$S_{n}\mathrel{\mathop{:}}=\sum_{i=1}^{n}X_{i}$. Then the following hold.

  1. (i)

If $\alpha\in(0,1)$, then $\operatorname{\mathbf{P}}$-a.s.

$\displaystyle\limsup_{n\to\infty}\frac{S_{n}}{n^{1/\alpha}(\log
n)^{1/\alpha}(\log\log n)^{\theta}}=\begin{cases}0,&\text{if
$\theta>1/\alpha$},\\\ \infty,&\text{if $\theta\leq 1/\alpha$,}\end{cases}$
(2.1)

and

$\displaystyle\liminf_{n\to\infty}\frac{S_{n}}{n^{1/\alpha}(\log\log
n)^{1-1/\alpha}}=1+\frac{\alpha}{1-\alpha}\Gamma(2-\alpha)^{1/\alpha}.$ (2.2)

  2. (ii)

If $\alpha=1$, then $\operatorname{\mathbf{P}}$-a.s.

$\displaystyle\limsup_{n\to\infty}\frac{S_{n}}{n(\log n)(\log\log
n)^{\theta}}=\begin{cases}0,&\text{if $\theta>1$},\\\ \infty,&\text{if
$\theta\leq 1$,}\end{cases}$ (2.3)

and

$\displaystyle\liminf_{n\to\infty}\frac{S_{n}}{n\log n}=1.$ (2.4)

  3. (iii)

If $\alpha\in(1,2)$, then $\operatorname{\mathbf{P}}$-a.s.

$\displaystyle\limsup_{n\to\infty}\frac{S_{n}-n\operatorname{\mathbf{E}}[X_{1}]}{n^{1/\alpha}(\log
n)^{1/\alpha}(\log\log n)^{\theta}}=\begin{cases}0,&\text{if
$\theta>1/\alpha$},\\\ \infty,&\text{if $\theta\leq 1/\alpha$,}\end{cases}$
(2.5)

and

$\displaystyle\liminf_{n\to\infty}\frac{S_{n}-n\operatorname{\mathbf{E}}[X_{1}]}{n^{1/\alpha}(\log\log
n)^{1-1/\alpha}}=\big{(}\Gamma(2-\alpha)^{1/\alpha}-1\big{)}\alpha(1-\alpha)^{-1}.$
(2.6)

  4. (iv)

If $\alpha=2$, then $\operatorname{\mathbf{P}}$-a.s.

$\displaystyle\limsup_{n\to\infty}\frac{S_{n}-n\operatorname{\mathbf{E}}[X_{1}]}{\sqrt{n(\log
n)(\log\log n)^{\theta}}}=\begin{cases}0,&\text{if $\theta>1$},\\\
\infty,&\text{if $\theta\leq 1$,}\end{cases}$ (2.7)

and

$\displaystyle\liminf_{n\to\infty}\frac{S_{n}-n\operatorname{\mathbf{E}}[X_{1}]}{\sqrt{n(\log
n)(\log\log n)}}=-\sqrt{2c_{F}^{-1}}.$ (2.8)

  5. (v)

If $\alpha\in(2,\infty)$, then $\operatorname{\mathbf{P}}$-a.s.

$\limsup_{n\to\infty}\frac{S_{n}-n\operatorname{\mathbf{E}}[X_{1}]}{\sqrt{n\log\log
n}}=\sqrt{2\mathrm{Var}(X_{1})},\quad\liminf_{n\to\infty}\frac{S_{n}-n\operatorname{\mathbf{E}}[X_{1}]}{\sqrt{n\log\log
n}}=-\sqrt{2\mathrm{Var}(X_{1})}.$

Towards proving Theorem 2.1, we first analyse the median of $S_{n}$ for large
$n$ in the case when $\alpha<2$.

######  Lemma 2.2.

For $n\geq 1$, let $m_{n}$ denote the median of $S_{n}$. Then, as
$n\to\infty$,

$\displaystyle
m_{n}=\begin{cases}\bar{m}_{\alpha}n^{1/\alpha}+o(n^{1/\alpha}),&\text{if
$\alpha\in(0,1)$},\\\ \bar{m}_{\alpha}n\log n+o(n\log n),&\text{if
$\alpha=1$},\\\
n\operatorname{\mathbf{E}}[X_{1}]+\bar{m}_{\alpha}n^{1/\alpha}+o(n^{1/\alpha}),&\text{if
$\alpha\in(1,2)$},\end{cases}$

with $\bar{m}_{\alpha}$ denoting the median of a totally asymmetric
$\alpha$-stable law $S_{1}^{(\alpha)}$ satisfying

$\operatorname{\mathbf{E}}\left[e^{-\lambda
S_{1}^{(\alpha)}}\right]=e^{-C\lambda^{\alpha}}$ (2.9)

for some appropriately chosen $C\in(0,\infty)$ (depending on $c_{F}$ and
$\alpha$).

###### Proof.

Recall that for $\alpha\in(1,2)$,

$\displaystyle\frac{S_{n}-n\operatorname{\mathbf{E}}[X_{1}]}{n^{1/\alpha}}\overset{d}{\longrightarrow}S_{1}^{(\alpha)}\qquad\text{as
$n\to\infty$,}$

where the law of $S_{1}^{(\alpha)}$ is characterized by (2.9), see [52,
Chapter 7] or [19, Chapter 8], for example. In particular, for any
$\varepsilon>0$,

$\displaystyle\operatorname{\mathbf{P}}\big{[}S_{n}\leq
n\operatorname{\mathbf{E}}[X_{1}]+(\bar{m}_{\alpha}-\varepsilon)n^{1/\alpha}\big{]}$
$\displaystyle=\operatorname{\mathbf{P}}\Big{[}\frac{S_{n}-n\operatorname{\mathbf{E}}[X_{1}]}{n^{1/\alpha}}\leq\bar{m}_{\alpha}-\varepsilon\Big{]}$
$\displaystyle\underset{n\to\infty}{\longrightarrow}\operatorname{\mathbf{P}}\big{[}S_{1}^{(\alpha)}\leq\bar{m}_{\alpha}-\varepsilon\big{]}<\frac{1}{2},$

and similarly

$\displaystyle\operatorname{\mathbf{P}}\big{[}S_{n}\geq
n\operatorname{\mathbf{E}}[X_{1}]+(\bar{m}_{\alpha}+\varepsilon)n^{1/\alpha}\big{]}$
$\displaystyle=\operatorname{\mathbf{P}}\Big{[}\frac{S_{n}-n\operatorname{\mathbf{E}}[X_{1}]}{n^{1/\alpha}}\geq\bar{m}_{\alpha}+\varepsilon\Big{]}$
$\displaystyle\underset{n\to\infty}{\longrightarrow}\operatorname{\mathbf{P}}\big{[}S_{1}^{(\alpha)}\geq\bar{m}_{\alpha}+\varepsilon\big{]}<\frac{1}{2}.$

This gives the result for $\alpha\in(1,2)$. In the other cases, the claim
follows by the same argument since
$S_{n}/n^{1/\alpha}\overset{d}{\longrightarrow}S_{1}^{(\alpha)}$ if
$\alpha\in(0,1)$ and $S_{n}/(n\log n)\overset{d}{\longrightarrow}S_{1}^{(1)}$
if $\alpha=1$. ∎

###### Proof of Theorem 2.1.

Case 1: $\alpha\in(0,2)$. _Upper bounds._ We first summarize some consequences
of the results in [69, Theorem 6.1]. Using the notation therein, we have in
our setting

$G_{+}(u)\mathrel{\mathop{:}}=\operatorname{\mathbf{P}}[X_{1}>u]\asymp
u^{-\alpha},\qquad G(u)=\operatorname{\mathbf{P}}[|X_{1}|>u]\asymp
u^{-\alpha},$

and

$K(u)\mathrel{\mathop{:}}=u^{-2}\int_{|y|\leq
u}y^{2}\,dF(y)=c_{F}u^{-2}\int_{1}^{u}\alpha y^{1-\alpha}\,dy\asymp
u^{-\alpha}.$

In particular,

$\liminf_{u\to\infty}\frac{G_{+}(u)}{G(u)+K(u)}>0.$

Hence, by [69, Theorem 6.1] we have $\operatorname{\mathbf{P}}$-a.s.,

$\displaystyle\limsup_{n\to\infty}\frac{S_{n}-m_{n}}{\beta_{n}}=\begin{cases}0,&\text{if
}\sum_{n}\operatorname{\mathbf{P}}[X_{1}>\beta_{n}]<\infty,\\\
\infty,&\text{if
}\sum_{n}\operatorname{\mathbf{P}}[X_{1}>\beta_{n}]=\infty,\end{cases}$

where we again write $m_{n}$ for the median of $S_{n}$. Now, choosing

$\beta_{n}=n^{1/\alpha}(\log n)^{1/\alpha}(\log\log n)^{\theta},$

so that

$\displaystyle\sum_{n}\operatorname{\mathbf{P}}[X_{1}>\beta_{n}]=c_{F}\sum_{n}n^{-1}(\log
n)^{-1}(\log\log n)^{-\alpha\theta}\begin{cases}<\infty&\text{if
$\theta>1/\alpha$},\\\ =\infty&\text{if $\theta\leq 1/\alpha$,}\end{cases}$

we obtain that, for $\alpha\in(0,2)$,

$\displaystyle\limsup_{n\to\infty}\frac{S_{n}-m_{n}}{n^{1/\alpha}(\log
n)^{1/\alpha}(\log\log n)^{\theta}}=\begin{cases}0,&\text{if
$\theta>1/\alpha$},\\\ \infty,&\text{if $\theta\leq 1/\alpha$.}\end{cases}$

Thus, from Lemma 2.2 we obtain (2.1), (2.3) and (2.5).

_Lower bounds._ Let $Q$ denote the quantile function associated with
$F(x)=(1-c_{F}x^{-\alpha})\vee 0$, that is

$Q(s)\mathrel{\mathop{:}}=\inf\big{\\{}x:F(x)\geq
s\big{\\}}=c_{F}^{-1/\alpha}\big{(}1-s\big{)}^{-1/\alpha},\qquad 0<s<1,$

set $Q(0)=Q(0+)$, and define the truncated mean

$\displaystyle\mu(s)$
$\displaystyle\mathrel{\mathop{:}}=\int_{0}^{1-s}Q(u)\,du=c_{F}^{-1/\alpha}\int_{0}^{1-s}\big{(}1-u\big{)}^{-1/\alpha}\,du=c_{F}^{-1/\alpha}\int_{s}^{1}u^{-1/\alpha}\,du$
$\displaystyle=\begin{cases}c_{F}^{-1/\alpha}\frac{\alpha}{1-\alpha}\big{[}s^{1-1/\alpha}-1\big{]},&\text{if
$\alpha\in(0,2)\backslash\\{1\\}$},\\\ -c_{F}^{-1/\alpha}\log(s),&\text{if
$\alpha=1$}.\end{cases}$

For $\alpha\in(1,2)$, note that
$\mu(s)\rightarrow\operatorname{\mathbf{E}}[X_{1}]$ as $s\to 0$. The truncated
variance is given by

$\displaystyle\sigma^{2}(s)$
$\displaystyle\mathrel{\mathop{:}}=sQ(1-s)^{2}+\int_{0}^{1-s}Q^{2}(u)\,du-\big{(}sQ(1-s)+\mu(s)\big{)}^{2}$
$\displaystyle=c_{F}^{-2/\alpha}\Big{(}s^{1-2/\alpha}+\frac{\alpha}{2-\alpha}s^{1-2/\alpha}-\big{(}s^{1-1/\alpha}+\mu(s)\big{)}^{2}\Big{)}$
$\displaystyle\sim c_{F}^{-2/\alpha}\frac{2}{2-\alpha}s^{1-2/\alpha}.$

Now setting

$\displaystyle b_{n}\mathrel{\mathop{:}}=n^{-1}\log\log n,\qquad
a_{n}\mathrel{\mathop{:}}=n^{1/2}\sigma(b_{n}),$

we have by [63, Theorem 1] that

$\displaystyle\liminf_{n\to\infty}\frac{S_{n}-n\mu(b_{n})}{a_{n}(\log\log
n)^{1/2}}=K_{\alpha},\qquad\text{$\operatorname{\mathbf{P}}$-a.s.}$ (2.10)

where the constant $K_{\alpha}\in[-\sqrt{2},0]$ is given by

$\displaystyle
K_{\alpha}=\begin{cases}\big{(}\Gamma(2-\alpha)^{1/\alpha}-1\big{)}\alpha(1-\alpha)^{-1}\big{(}(2-\alpha)/2\big{)}^{1/2},&\text{if
$\alpha\in(0,2)\backslash\\{1\\}$},\\\ -\gamma/\sqrt{2},&\text{if
$\alpha=1$,}\end{cases}$

with $\gamma$ being the Euler constant, see [63, Lemma 4].

Now for $\alpha\in(0,1)$ we have

$\displaystyle\lim_{n\to\infty}\frac{n\mu(b_{n})}{a_{n}(\log\log
n)^{1/2}}=\frac{1}{1-\alpha}\Big{(}\frac{2-\alpha}{2}\Big{)}^{1/2}=\mathrel{\mathop{:}}C_{\alpha},$

so that (2.10) can be rewritten as

$\displaystyle\liminf_{n\to\infty}\frac{S_{n}}{a_{n}(\log\log
n)^{1/2}}=C_{\alpha}+K_{\alpha},\qquad\text{$\operatorname{\mathbf{P}}$-a.s.}$

with

$\displaystyle
C_{\alpha}+K_{\alpha}=\frac{1}{1-\alpha}\Big{(}\frac{2-\alpha}{2}\Big{)}^{1/2}\Big{(}1-\alpha+\alpha\Gamma(2-\alpha)^{1/\alpha}\Big{)}>0.$

This is equivalent to (2.2).

If $\alpha=1$, then by (2.10)

$\displaystyle\liminf_{n\to\infty}\frac{S_{n}-n\log
n}{\sqrt{2}n}=\liminf_{n\to\infty}\frac{S_{n}-n\mu(b_{n})}{a_{n}(\log\log
n)^{1/2}}=-\gamma/\sqrt{2},$

and by dividing both sides by $\log n$ we derive (2.4).

Finally, if $\alpha\in(1,2)$, we have
$\mu(b_{n})\sim\operatorname{\mathbf{E}}[X_{1}]$, so that (2.10) becomes
(2.6).

Case 2: $\alpha=2$. To check the statement at (2.8), we again apply the
results of [63]. In particular, defining notation as in the previous part of
the proof, we have in this case that $\sigma^{2}(s)\sim-c_{F}^{-1}\log(s)$ as
$s\rightarrow 0$. Thus $a_{n}\sim(c_{F}^{-1}n\log(n))^{1/2}$, and so [63,
Theorem 1 and Lemma 4] yield a statement analogous to (2.10) with
$K_{2}=-\sqrt{2}$. This is equivalent to the desired result.

For (2.7), we first consider the problem for the symmetric distribution given
by $\mathbf{P}(|X_{1}|\geq x)=x^{-2}$ for $x\geq 1$. In this case, it is
explained in [43, Section 8] that, $\operatorname{\mathbf{P}}$-a.s.,

$\limsup_{n\rightarrow\infty}\frac{S_{n}}{\sqrt{n(\log n)(\log\log
n)}}=\infty.$

From this (and the conclusion at (2.8)), the corresponding result for the case
of positive $X_{1}$ readily follows. (One may alternatively deduce the result
from [69, Theorem 7.5], cf. [58, 59].) Moreover, for the case of positive
$X_{1}$, it follows from [42, Theorem 1] that, for any $\varepsilon>0$,
$\operatorname{\mathbf{P}}$-a.s.,

$\limsup_{n\rightarrow\infty}\frac{S_{n}}{\sqrt{n(\log n)(\log\log
n)^{1+\varepsilon}}}=0.$

Case 3: $\alpha>2$. Since the random variables have a finite variance in this
case, the result follows from the classical Hartman-Wintner law of the
iterated logarithm [54]. ∎

##  3\. Heat kernel estimates

In this section, we aim to establish the heat kernel estimates of Theorems 1.2
and 1.4. These are proved using what are now standard techniques in the area,
see [11, 12, 13, 28, 60, 61, 62], for example. The approaches of these papers
depend on establishing good volume growth and resistance bounds. In our
setting, the resistance is particularly simple. Indeed, if we consider the
state space $\mathbb{Z}$ to be an electrical network with unit resistors on
each edge, the resistance metric simply coincides with the Euclidean metric.
Moreover, the simple structure of the space means that, if we write
$B(x,n):=\\{y\in\mathbb{Z}:\>|x-y|<n\\}$ and the effective resistance as $R$,
then from the parallel law we obtain $R(x,B(x,n)^{c})=\frac{n}{2}$. (See [10]
for a definition of the effective resistance and background on the connections
between random walks and effective resistance.) Consequently, to establish
reasonable heat kernel estimates, it remains to study the behaviour of the
invariant measure for the Markov chain $X$. Recalling that a reversible
measure for $X$ is given by
$\mathbf{\tau}\mathrel{\mathop{:}}=\sum_{x\in\mathbb{Z}}\tau_{x}\delta_{x}$,
we define

$\displaystyle V(x,n)\mathrel{\mathop{:}}=\sum_{y=x-n}^{x+n}\tau_{y},\qquad
x\in\mathbb{Z},\,n\in\mathbb{N},$

and write $V(n)\mathrel{\mathop{:}}=V(0,n)$. Clearly, the fluctuations of
$V(n)$ for large $n$ can be immediately deduced from Theorem 2.1. In what
follows, we explain how to transfer those volume asymptotics to heat kernel
estimates. For the quenched result of Theorem 1.2, we will apply the following
bounds.

######  Proposition 3.1.

  1. (i)

Let $x\in\mathbb{Z}$. If $V(x,r)\geq V_{l}(r)$ for all $r\geq 0$, where
$V_{l}:[0,\infty)\rightarrow[0,\infty)$ is a continuous, non-decreasing
function that satisfies the doubling property (i.e. there exists a $C<\infty$
such that $V_{l}(2r)\leq CV_{l}(r)$), then there exists a constant $c<\infty$
such that

$p_{t}^{\tau}(x,x)\leq\frac{ch_{l}^{-1}(t)}{t}$

for all $t>0$, where $h_{l}(r):=rV_{l}(r)$.

  2. (ii)

For every $x\in\mathbb{Z}$ and $n\in\mathbb{N}$, it holds that

$p^{\tau}_{2nV(x,n)}(x,x)\leq\frac{2}{V(x,n-1)}.$

  3. (iii)

For every $x\in\mathbb{Z}$ and $n\in\mathbb{N}$, it holds that

$p^{\tau}_{\frac{n}{2}V(x,n)}(x,x)\geq\frac{V(x,n)^{2}}{16V(x,2n)^{3}}.$

###### Proof.

Part (i) is given by the argument of [60, Proposition 4.1], which in turn is
based on the approach of [11]. In particular, the proof of [60, Proposition
4.1] only uses a lower volume bound and the doubling property of the bounding
function. See also the proof of [28, Proposition 5].

Part (ii) can be proved by an adaptation of the argument used to establish
part (a). A version of this is stated as [28, (6.11)], but with oversight of
the fact that the volumes considered should be of the closed ball on the left-
hand side, and the open ball on the right-hand side of the inequality.

As for part (iii), we again apply a standard argument, versions of which
appear in [60, Section 4] and [62, Proposition 3.2]. This is particularly
simple in our setting due to the specific form of the Green’s function.
Indeed, by an elementary Markov chain argument, it is possible to check that

$E^{\tau}_{y}\left(T_{B(x,n)}\right)=\sum_{z}g_{B(x,n)}(y,z)\,\tau_{z},$

where $T_{B(x,n)}:=\inf\\{t\geq 0:\>X_{t}\not\in B(x,n)\\}$, $E^{\tau}_{y}$ is
the expectation under $P_{y}^{\tau}$ and

$g_{B(x,n)}(y,z)=\left\\{\begin{array}[]{ll}\frac{(n-x+y)(n+x-z)}{2n},&\hbox{if
$x-n\leq y\leq z\leq x+n$;}\\\ \frac{(n+x-y)(n-x+z)}{2n},&\hbox{if $x-n\leq
z\leq y\leq x+n$;}\\\ 0,&\hbox{otherwise.}\end{array}\right.$

In particular, it holds that $g_{B(x,n)}(y,z)\leq\frac{n}{2}$ for any $y$ and
$z$, and so

$E^{\tau}_{y}\left(T_{B(x,n)}\right)\leq\frac{n}{2}V(x,n).$

Similarly, since $g_{B(x,n)}(x,z)\geq\frac{n}{4}$ for
$z\in[x-\frac{n}{2},z+\frac{n}{2}]$, it holds that

$E^{\tau}_{x}\left(T_{B(x,n)}\right)\geq\frac{n}{4}V(x,\frac{n}{2}).$

Consequently, applying the Markov property at time $t$, we find that

$\displaystyle\frac{n}{2}V(x,n)$ $\displaystyle\leq
E^{\tau}_{x}\left(T_{B(x,2n)}\right)$ $\displaystyle\leq
t+E^{\tau}_{x}\left((T_{B(x,2n)}-t)\mathbf{1}_{\\{T_{B(x,2n)}>t\\}}\right)$
$\displaystyle\leq
t+\sup_{y}E^{\tau}_{y}\left(T_{B(x,2n)}\right)P_{x}^{\tau}\left(T_{B(x,2n)}>t\right)$
$\displaystyle\leq t+nV(x,2n)P_{x}^{\tau}\left(T_{B(x,2n)}>t\right),$

and rearranging yields

$P_{x}^{\tau}\left(T_{B(x,2n)}>t\right)\geq\frac{nV(x,n)-2t}{2nV(x,2n)}.$

To relate this inequality to the heat kernel, we apply the Cauchy-Schwarz
inequality to obtain that

$\displaystyle P_{x}^{\tau}\left(T_{B(x,2n)}>t\right)^{2}$ $\displaystyle\leq
P_{x}^{\tau}\left(X_{t}\in B(x,2n)\right)^{2}$ $\displaystyle=\left(\sum_{z\in
B(x,2n)}p^{\tau}_{t}(x,z)\tau_{z}\right)^{2}$
$\displaystyle=\sum_{z}p^{\tau}_{t}(x,z)^{2}\tau_{z}V(x,2n)$
$\displaystyle\leq p^{\tau}_{2t}(x,x)V(x,2n).$

Finally, combining the two previous inequalities and setting
$t=\frac{n}{4}V(x,n)$, we deduce the desired inequality. ∎

We are now ready to prove Theorems 1.2 and 1.4. In the following, we will
write constants depending on the particular realisation of the environment
$\tau$ as $c(\tau)$, for example; note that the value of such constants might
change from line to line.

###### Proof of Theorem 1.2(i).

Recall that this part of the theorem applies to $\alpha\in(0,1)$. We start by
checking the limsup part of the result. Firstly, by applying (2.2), it is
possible to deduce that, $\operatorname{\mathbf{P}}$-a.s., the assumptions of
Proposition 3.1(a) hold with $V_{l}(r):=c(\tau)r^{1/\alpha}(\log\log
r)^{1-1/\alpha}$. (Strictly speaking, to handle small $r$, we should replace
$r^{1/\alpha}(\log\log r)^{1-1/\alpha}$ by a function such as
$\max\\{1,r^{1/\alpha}(\log\log r)^{1-1/\alpha}\\}$, but this does not affect
the substance of what follows, and we choose to omit the truncation from the
notation for brevity.) Noting that

$h_{l}^{-1}(t)\sim C(\tau)t^{\frac{\alpha}{1+\alpha}}(\log\log
t)^{\frac{1-\alpha}{1+\alpha}}$

as $t\rightarrow\infty$, we thus obtain from Proposition 3.1(a) that the
limsup of the statement of Theorem 1.2(i) is finite. Secondly, observe that
(2.2) further implies that there exists a subsequence $(n_{i})_{i\geq 1}$ such
that

$c_{1}(\tau)\,n_{i}^{1/\alpha}\,(\log\log n_{i})^{1-1/\alpha}\leq
V(0,n_{i})\leq V(0,2n_{i})\leq c_{2}(\tau)\,n_{i}^{1/\alpha}\,(\log\log
n_{i})^{1-1/\alpha}.$

Hence Proposition 3.1(iii) yields that, along the subsequence
$t_{i}=\frac{n_{i}}{2}V(0,n_{i})$,

$p_{t_{i}}^{\tau}(0,0)\geq\frac{c(\tau)}{n_{i}^{1/\alpha}(\log\log
n_{i})^{1-1/\alpha}}.$

Since $t_{i}$ bounded above and below by constant multiples of
$n_{i}^{1+1/\alpha}(\log\log n_{i})^{1-1/\alpha}$, it follows that

$p_{t_{i}}^{\tau}(0,0)\geq c(\tau)\,t_{i}^{-\frac{1}{1+\alpha}}\,(\log\log
t_{i})^{\frac{1-\alpha}{1+\alpha}},$

which establishes that the limsup of the statement of Theorem 1.2(a) is
strictly positive.

We next consider the liminf, starting with the upper bound. By (2.1), for any
$\varepsilon>0$, it $\operatorname{\mathbf{P}}$-a.s. holds that there exists a
subsequence $(n_{i})_{i\geq 1}$ such that

$\displaystyle V(0,n_{i}-1)$ $\displaystyle\geq
c(\tau)\,n_{i}^{1/\alpha}\,(\log n_{i})^{1/\alpha}(\log\log n_{i})^{1/\alpha}$
$\displaystyle V(0,n_{i})$ $\displaystyle\leq c(\tau)\,n_{i}^{1/\alpha}\,(\log
n_{i})^{1/\alpha}(\log\log n_{i})^{1/\alpha+\varepsilon}.$

Consequently, taking $t_{i}=2n_{i}V(0,n_{i})$, we deduce from Proposition
3.1(b) that

$p_{t_{i}}^{\tau}(0,0)\leq\frac{c(\tau)}{n_{i}^{1/\alpha}(\log
n_{i})^{1/\alpha}(\log\log n_{i})^{1/\alpha}}.$

Since $t_{i}\leq c(\tau)n_{i}^{1+1/\alpha}(\log n_{i})^{1/\alpha}(\log\log
n_{i})^{1/\alpha+\varepsilon}$, it follows that

$p_{t_{i}}^{\tau}(0,0)\leq c(\tau)\,t_{i}^{-\frac{1}{1+\alpha}}(\log
t_{i})^{-\frac{1}{1+\alpha}}\,(\log\log
t_{i})^{-\frac{1-\varepsilon}{1+\alpha}},$

as desired. As for the lower bound on the liminf in the statement of Theorem
1.2(a), we apply both (2.1) and (2.2) to deduce that, for any $\varepsilon>0$,
$\operatorname{\mathbf{P}}$-a.s., for all $n\in\mathbb{N}$,

$c_{1}(\tau)\,n^{1/\alpha}\,(\log\log n)^{1-1/\alpha}\leq V(0,n-1)\leq
V(0,2n)\leq c_{2}(\tau)\,n^{1/\alpha}\,(\log n)^{1/\alpha+\varepsilon}.$

Consequently, if $t$ satisfies

$\frac{n-1}{2}V(0,n-1)\leq t\leq\frac{n}{2}V(0,n),$

then the monotonicity of the on-diagonal heat kernel (see [10, Lemma 5.11],
for example) and Proposition 3.1(iii) imply that

$p_{t}^{\tau}(0,0)\geq
p_{\frac{n}{2}V(0,n)}^{\tau}(0,0)\geq\frac{V(0,n)^{2}}{16V(0,2n)^{3}}\geq\frac{c(\tau)}{n^{1/\alpha}(\log
n)^{3/\alpha+4\varepsilon}}\geq\frac{c(\tau)}{t^{\frac{1}{1+\alpha}}(\log
t)^{\frac{3}{\alpha}+5\varepsilon}},$

from which the result follows. ∎

###### Proof of Theorem 1.2(ii).

The proof of Theorem 1.2(ii) is essentially identical to that of Theorem
1.2(i), only replacing the estimates (2.1) and (2.2) by (2.3) and (2.4),
respectively. ∎

###### Proof of Theorem 1.2(iii).

Let $\alpha>1$. Given the scaling limit of Theorem 1.1(iii), together with the
fact that, $\operatorname{\mathbf{P}}$-a.s.,

$n^{-1}\sum_{z}\tau_{z}\delta_{n^{-1}z}$

converges vaguely to the Lebesgue measure multiplied by
$\mathbf{E}(\tau_{0})$, it is straightforward to check that a continuous-time
version of [30, Assumption 1] with scaling factors
$\alpha(n)=n,\beta(n)=n\mathbf{E}(\tau_{0})$ and
$\gamma(n)=n^{2}\mathbf{E}(\tau_{0})$ holds when we consider the graphs
$G^{n}$ of [30] to be given by $n^{-1}\mathbb{Z}$ (equipped with nearest
neighbour edges), $d_{G^{n}}$ to be the usual shortest path metric on $G^{n}$,
$\nu^{n}$ to be the measure on $n^{-1}\mathbb{Z}$ placing mass $\tau_{x}$ at
$x$, and $X^{n}$ to be the continuous-time random walk of the associated
symmetric Bouchaud trap model. Thus, as commented at the end of the
introduction of [30], it is possible to deduce Theorem 1.2(iii) from a
continuous-time version of [30, Theorem 1] if one can further establish a
continuous-time version of [30, Assumption 2], i.e.,
$\operatorname{\mathbf{P}}$-a.s., for every $x_{0}\in[0,\infty)$ and compact
interval $I\subseteq(0,\infty)$,

$\lim_{\delta\rightarrow
0}\limsup_{\lambda\rightarrow\infty}\sup_{\begin{subarray}{c}x,y:\>|x|,|y|\leq
x_{0}\\\ |x-y|\leq\delta\end{subarray}}\sup_{t\in
I}\mathbf{E}(\tau_{0})\lambda\left|\,p^{\tau}_{\mathbf{E}(\tau_{0})\lambda^{2}t}\left(0,\lfloor\lambda
x\rfloor\right)-\,p^{\tau}_{\mathbf{E}(\tau_{0})\lambda^{2}t}\left(0,\lfloor\lambda
y\rfloor\right)\right|=0.$ (3.1)

Towards checking this, we first observe that, $\mathbf{P}$-a.s., $\tau_{x}\geq
1$ for all $x$ and so $V(0,n)\geq n$. Therefore we have from Proposition
3.1(i) that, $\mathbf{P}$-a.s.

$p^{\tau}_{t}(0,0)\leq C\,t^{-1/2}.$ (3.2)

Next, it is a simple application of the Cauchy-Schwarz inequality to deduce
that, for all $x,y\in\mathbb{Z}$,

$\left|p^{\tau}_{t}(0,x)-p^{\tau}_{t}(0,y)\right|^{2}\leq|y-x|\sum_{z}\left(p^{\tau}_{t}(0,z)-p^{\tau}_{t}(0,z+1)\right)^{2}$

The sum in the right-hand side here is the Dirichlet energy of the function
$p^{\tau}_{t}(0,\cdot)$, and the argument of [9, Proposition 4.16] gives that
this is bounded above by $t^{-1}p^{\tau}_{t}(0,0)$. Consequently, recalling
the bound at (3.2), we find that, $\operatorname{\mathbf{P}}$-a.s.,

$\left|p^{\tau}_{t}(0,x)-p^{\tau}_{t}(0,y)\right|^{2}\leq
C|y-x|t^{-3/2},\qquad\forall x,y\in\mathbb{Z}.$ (3.3)

From this estimate, the equicontinuity condition at (3.1) readily follows, and
thus the proof is complete. ∎

###### Proof of Theorem 1.4(i).

The result can be checked by applying a continuous-time version of [62,
Proposition 1.3]. The required input is given by the limit

$\lim_{\lambda\rightarrow\infty}\inf_{n\in\mathbb{N}}\operatorname{\mathbf{P}}\left(\lambda^{-1}v_{\alpha}(n)\leq
V(0,n)\leq\lambda v_{\alpha}(n)\right)=1,$ (3.4)

where

$v_{\alpha}(r):=\left\\{\begin{array}[]{ll}r^{1/\alpha},&\hbox{if
$\alpha\in(0,1)$;}\\\ r\max\\{1,\log r\\},&\hbox{if $\alpha=1$;}\\\
r,&\hbox{if $\alpha>1$.}\end{array}\right.$ (3.5)

Since (3.4) is a straightforward consequence of the convergence in
distribution of $V(0,n)/v_{\alpha}(n)$ to a random variable $S_{\alpha}$ that
takes values in $(0,\infty)$ (cf. the convergence results described in the
proof of Lemma 2.2), we are done. ∎

###### Proof of Theorem 1.4(ii).

For any $\varepsilon>0$, the lower bound is an immediate consequence of
Theorem 1.4(i). As for the upper bound, we will require a quantitative version
of (3.4). In particular, writing $h_{\alpha}(r)=rv_{\alpha}(r)$, suppose that
the following conditions hold for some $n\geq 2$, $t>0$ and $\lambda\geq 1$:

$h^{-1}_{\alpha}(t/\lambda)\in[n,n+1),$ $\lambda^{-1}v_{\alpha}(n)\leq
V(0,n-1)\leq 2V(0,n)\leq\lambda v_{\alpha}(n).$

We then have from Proposition 3.1(ii) that

$p_{t}^{\tau}(0,0)\leq
p_{2nV(0,n)}^{\tau}(0,0)\leq\frac{2}{V(0,n-1)}\leq\frac{2\lambda}{v_{\alpha}(n)}=\frac{2\lambda
n}{h_{\alpha}(n)}\leq\frac{c\lambda^{2}n}{t}\leq\frac{c\lambda^{2}h_{\alpha}^{-1}(t/\lambda)}{t},$

where $c$ is deterministic. So,

$p_{t}^{\tau}(0,0)\leq\left\\{\begin{array}[]{ll}c\lambda^{\frac{1+2\alpha}{1+\alpha}}\phi_{\alpha}(t),&\hbox{if
$\alpha\in(0,1)$;}\\\ c\lambda^{2}\phi_{\alpha}(t),&\hbox{if $\alpha=1$;}\\\
c\lambda^{\frac{3}{2}}\phi_{\alpha}(t),&\hbox{if
$\alpha>1$,}\end{array}\right.$

where for $\alpha=1$, we simply applied the bound
$h_{\alpha}^{-1}(t/\lambda)\leq h_{\alpha}^{-1}(t)$, and otherwise, we applied
the exact expression for $h_{\alpha}^{-1}$. Hence, setting
$\theta=\frac{1+2\alpha}{1+\alpha}$ if $\alpha\in(0,1)$, $\theta=2$ if
$\alpha=1$, and $\theta=\frac{3}{2}$ if $\alpha>1$, and reparametrising, we
can conclude that: if $h^{-1}_{\alpha}(t/\lambda^{1/\theta})\in[n,n+1)$ for
some $n\geq 2$, then

$\displaystyle\operatorname{\mathbf{P}}\left(p_{t}^{\tau}(0,0)\geq
c\lambda\phi_{\alpha}(t)\right)$
$\displaystyle\leq\operatorname{\mathbf{P}}\left(V(0,n-1)<\lambda^{-1/\theta}v_{\alpha}(n)\right)+\operatorname{\mathbf{P}}\left(2V(0,n)\geq\lambda^{1/\theta}v_{\alpha}(n)\right)$
$\displaystyle\leq C\lambda^{-c/\theta},$ (3.6)

where we have applied Proposition A.1 to obtain the second inequality.
Furthermore, if it holds that $h^{-1}_{\alpha}(t/\lambda^{1/\theta})<2$, then
$t\leq\lambda^{1/\theta}h_{\alpha}(2)$, which implies

$\lambda\phi_{\alpha}(t)\geq\lambda\phi_{\alpha}(c\lambda^{1/\theta})\geq\lambda^{\varepsilon}$

for suitably small $\varepsilon$. Consequently, applying the fact that
$\tau_{x}\geq 1$ and Markov’s inequality,

$\operatorname{\mathbf{P}}\left(p_{t}^{\tau}(0,0)\geq
c\lambda\phi_{\alpha}(t)\right)\leq\operatorname{\mathbf{P}}\left(\tau_{0}^{-1}\geq
c\lambda^{\varepsilon}\right)\leq C\lambda^{-c},$

where $c$ can be taken arbitrarily large. Hence, we have shown that: for all
$t>0$ and $\lambda\geq 1$,

$\operatorname{\mathbf{P}}\left(p_{t}^{\tau}(0,0)\geq
c\lambda\phi_{\alpha}(t)\right)\leq C\lambda^{-c}.$

This readily implies the upper bound of Theorem 1.4(ii). To check the final
claim of Theorem 1.4, it will suffice to check that $c$ can be taken to be
strictly greater than 1 in the previous inequality when $\alpha>3/2$. This is
straightforward since, when $\alpha>3/2$, we have $\theta=3/2$, and
Proposition A.1 allows us to take $c>3/2$ in (3.6). ∎

##  4\. Quantitative homogenization for $\alpha>2$

###  4.1. Quantitative CLT

In this subsection, we will prove Theorem 1.5. Our argument will be based on
the following general quantitative central limit theorem for martingales.

######  Theorem 4.1.

Let $(N_{t})_{t\geq 0}$ be a locally square-integrable martingale (with
respect to some probability measure $P$) and denote by $\Delta
N_{t}:=N_{t}-N_{t-}$ its jump process and by $\langle N\rangle_{t}$ its
quadratic variation process. Then, for any $n>1$, there exists a constant
$c>0$ depending only on $n$ such that

$\sup_{x\in\mathbb{R}}\left|P\big{[}N_{1}\leq x\
\big{]}-\Phi(x)\right|\;\leq\;c\,\Big{(}E\big{[}\big{|}\langle
N\rangle_{1}-1\big{|}^{n}\big{]}+E\Big{[}\sum_{0\leq t\leq 1}\big{|}\Delta
N_{t}\big{|}^{2n}\Big{]}\Big{)}^{1/(2n+1)}.$

###### Proof.

See [53, Theorem 2] (cf. [56]). ∎

To apply this, we will establish the following quenched and annealed estimates
for the BTM.

######  Proposition 4.2.

Suppose that $\alpha>2$ and let
$\sigma^{2}\mathrel{\mathop{:}}=1/\operatorname{\mathbf{E}}[\tau_{0}]$.

  1. (i)

It holds that

$\displaystyle\operatorname{\mathbf{E}}\Bigg{[}E_{0}^{\tau}\bigg{[}\Big{|}\frac{\langle
X\rangle_{t}}{t}-\sigma^{2}\Big{|}^{2}\bigg{]}\Bigg{]}=O\left(t^{-1/2}\log
t\right).$

  2. (ii)

For any $\eta>0$ and $\operatorname{\mathbf{P}}$-a.e. $\tau$, it holds that

$\displaystyle E_{0}^{\tau}\bigg{[}\Big{|}\frac{\langle
X\rangle_{t}}{t}-\sigma^{2}\Big{|}^{2}\bigg{]}=O\left(t^{-\frac{1}{4}+\frac{1}{4(2\alpha-3)}+\eta}\right).$

In order to show Proposition 4.2, we consider a continuous time simple random
walk $(Y_{t})_{t\geq 0}$ on $\mathbb{Z}$ with unit jump rates and associated
heat kernel denoted by $p^{Y}_{t}(x,y)$, and set

$\displaystyle A_{t}\mathrel{\mathop{:}}=\int_{0}^{t}\tau_{Y_{s}}\,ds,\qquad
t\geq 0.$

The latter process is a continuous time version of a random walk in random
scenery, as studied in [39, 40]. (The original model in discrete time was
first introduced by Borodin [20, 21] and Kesten-Spitzer [57] independently.)
For any configuration $\tau=(\tau_{x})_{x\in\mathbb{Z}}$, in a slight abuse of
notation, we denote by $E_{x}^{\tau}$ the quenched law of
$(\tau_{Y_{t}})_{t\geq 0}$ when the random walk $(Y_{t})_{t\geq 0}$ starts at
$x\in\mathbb{Z}$. Importantly, we note that

$X_{t}=Y_{A_{t}^{-1}},$

where $A^{-1}$ denotes the right-continuous inverse of $A$. We will show that
Proposition 4.2 is a consequence of the following statement on the time-change
functional $A$.

######  Proposition 4.3.

Suppose that $\alpha>2$.

  1. (i)

It holds that

$\displaystyle\operatorname{\mathbf{E}}\bigg{[}E_{0}^{\tau}\Big{[}\big{|}A_{t}-\operatorname{\mathbf{E}}[\tau_{0}]t\big{|}^{2}\Big{]}\bigg{]}=O\left(t^{3/2}\right).$
(4.1)

  2. (ii)

For any $\eta>0$ and $\operatorname{\mathbf{P}}$-a.e. $\tau$, it holds that

$E_{0}^{\tau}\Big{[}\big{|}A_{t}-\operatorname{\mathbf{E}}[\tau_{0}]t\big{|}^{2}\Big{]}=O\left(t^{\frac{7}{4}+\frac{1}{4(2\alpha-3)}+\eta}\right).$

###### Proof.

(i) For any $t\geq 0$, by the on-diagonal heat kernel bound
$p^{Y}_{t}(x,x)\leq c(1\wedge t^{-1/2})$ of [10, Theorem 6.28],

$\displaystyle\operatorname{\mathbf{E}}\bigg{[}E_{0}^{\tau}\Big{[}\big{|}A_{t}-\operatorname{\mathbf{E}}[\tau_{0}]t\big{|}^{2}\Big{]}\bigg{]}$
$\displaystyle=\int_{0}^{t}\int_{0}^{t}\Big{(}\operatorname{\mathbf{E}}[E^{\tau}_{0}[\tau_{Y_{u}}\tau_{Y_{v}}]]-\operatorname{\mathbf{E}}[\tau_{0}]^{2}\Big{)}\,du\,dv$
$\displaystyle=2\,\mathrm{Var}(\tau_{0})\int_{0}^{t}\int_{v}^{t}\,p^{Y}_{u-v}(0,0)\,du\,dv$
$\displaystyle\leq
c\int_{0}^{t}\int_{v}^{t}\,\big{(}1\wedge(u-v)^{-1/2}\big{)}\,du\,dv$
$\displaystyle\leq c\int_{0}^{t}\big{(}1+(t-v)^{1/2}\big{)}\,dv$
$\displaystyle\leq c\,t^{3/2},$

which shows (i).

(ii) For this part of the proof, we will use similar arguments to those of
[39]. Define
$\bar{A}_{t}\mathrel{\mathop{:}}=\int_{0}^{t}\bar{\tau}_{Y_{s}}\,ds$, $t\geq
0$, where
$\bar{\tau}_{x}\mathrel{\mathop{:}}=\tau_{x}-\operatorname{\mathbf{E}}[\tau_{0}]$,
$x\in\mathbb{Z}$. Further, for any $\varepsilon>0$, set
$\bar{A}_{t}^{\varepsilon}\mathrel{\mathop{:}}=\int_{0}^{t}\bar{\tau}_{Y_{s}}\wedge
t^{\varepsilon}\,ds$, $t\geq 0$, so that we have the obvious bound
$|\bar{A}_{t}^{\varepsilon}|\leq t^{1+\varepsilon}$.

_Step 1._ In our first step, we show that for $\operatorname{\mathbf{P}}$-a.e.
$\tau$,

$\displaystyle\Big{|}E_{0}^{\tau}\big{[}\big{|}\bar{A}_{t}^{\varepsilon}\big{|}^{2}\big{]}-\operatorname{\mathbf{E}}\big{[}E_{0}^{\tau}\big{[}\big{|}\bar{A}_{t}^{\varepsilon}\big{|}^{2}\big{]}\big{]}\Big{|}=O\left(t^{7/4+3\varepsilon}\right).$
(4.2)

For two configurations $\tau^{1}$ and $\tau^{2}$, by using the Cauchy-Schwarz
inequality,

$\displaystyle
E_{0}^{\tau^{1}}\big{[}\big{|}\bar{A}_{t}^{\varepsilon}\big{|}^{2}\big{]}-E_{0}^{\tau^{2}}\big{[}\big{|}\bar{A}_{t}^{\varepsilon}\big{|}^{2}\big{]}$
$\displaystyle=E_{0}^{\tau^{1},\tau^{2}}\bigg{[}\int_{0}^{t}\big{(}\bar{\tau}^{1}_{Y_{s}}\wedge
t^{\varepsilon}-\bar{\tau}^{2}_{Y_{s}}\wedge
t^{\varepsilon}\big{)}\,ds\,\big{(}\bar{A}_{t}^{\varepsilon}(\tau^{1})+\bar{A}_{t}^{\varepsilon}(\tau^{2})\big{)}\bigg{]}$
$\displaystyle\leq
2t^{1+\varepsilon}E_{0}^{\tau^{1},\tau^{2}}\bigg{[}\int_{0}^{t}\big{|}\bar{\tau}^{1}_{Y_{s}}\wedge
t^{\varepsilon}-\bar{\tau}^{2}_{Y_{s}}\wedge
t^{\varepsilon}\big{|}\big{(}\mathbbm{1}_{\\{|Y_{s}|\leq
2t\\}}+\mathbbm{1}_{\\{|Y_{s}|>2t\\}}\big{)}\,ds\,\bigg{]}$ $\displaystyle\leq
2t^{1+\varepsilon}\int_{0}^{t}\sum_{y\in[-2t,2t]}p^{Y}_{s}(0,y)\big{|}\bar{\tau}^{1}_{y}\wedge
t^{\varepsilon}-\bar{\tau}^{2}_{y}\wedge
t^{\varepsilon}\big{|}\,ds+2t^{2(1+\varepsilon)}\sup_{s\leq
t}P_{0}(|Y_{s}|>2t)$ $\displaystyle\leq
2t^{1+\varepsilon}\int_{0}^{t}\sum_{y\in[-2t,2t]}p^{Y}_{s}(0,y)\big{|}\bar{\tau}^{1}_{y}\wedge
t^{\varepsilon}-\bar{\tau}^{2}_{y}\wedge t^{\varepsilon}\big{|}\,ds+ce^{-ct}$
$\displaystyle\leq
2t^{1+\varepsilon}\int_{0}^{t}\bigg{(}\sum_{y\in[-2t,2t]}p^{Y}_{s}(0,y)^{2}\bigg{)}^{1/2}\bigg{(}\sum_{y\in[-2t,2t]}\big{|}\bar{\tau}^{1}_{y}\wedge
t^{\varepsilon}-\bar{\tau}^{2}_{y}\wedge
t^{\varepsilon}\big{|}^{2}\bigg{)}^{1/2}\,ds+c\,e^{-ct}$ $\displaystyle\leq
2t^{1+\varepsilon}\,\|\bar{\tau}^{1}\wedge
t^{\varepsilon}-\bar{\tau}^{2}\wedge
t^{\varepsilon}\|_{2}\int_{0}^{t}p^{Y}_{2s}(0,0)^{1/2}\,ds+c\,e^{-ct}$
$\displaystyle\leq 2t^{7/4+\varepsilon}\,\|\bar{\tau}^{1}\wedge
t^{\varepsilon}-\bar{\tau}^{2}\wedge t^{\varepsilon}\|_{2}+c\,e^{-ct},$

where we write
$E_{0}^{\tau^{1},\tau^{2}}\mathrel{\mathop{:}}=E_{0}^{\tau^{1}}\times
E_{0}^{\tau^{2}}$ and $\|\tau\|_{2}:=(\sum_{y\in[-2t,2t]}\tau_{y}^{2})^{1/2}$,
and the exponential bound on the probability $P_{0}(|Y_{s}|>2t)$ follows from
[10, Lemmas 5.21, 5.22 and Theorem 6.28]. This allows us to apply Talagrand’s
concentration inequality (see [70, Theorem 6.6] and the argument below [70,
Equation (4.2)] to replace the median by the mean) to obtain

$\displaystyle\operatorname{\mathbf{P}}\bigg{[}\Big{|}E_{0}^{\tau}\big{[}\big{|}\bar{A}_{t}^{\varepsilon}\big{|}^{2}\big{]}-\operatorname{\mathbf{E}}\big{[}E_{0}^{\tau}\big{[}\big{|}\bar{A}_{t}^{\varepsilon}\big{|}^{2}\big{]}\big{]}\Big{|}>t^{7/4+3\varepsilon}\bigg{]}$
$\displaystyle\;\leq\;4\exp\big{(}-t^{7/2+6\varepsilon}/(16t^{7/2+4\varepsilon})\big{)}$
$\displaystyle\;\leq\;c\exp\big{(}-ct^{2\varepsilon}\big{)}.$

(We use [70, Theorem 6.6] with the choice
$f(\tau)=E_{0}^{\tau}\big{[}\big{|}\bar{A}_{t}^{\varepsilon}\big{|}^{2}\big{]}$,
$\sigma=t^{7/4+2\varepsilon}$, $X_{x}=t^{-\varepsilon}(\tau_{x}\wedge
t^{\varepsilon})\in[0,1]$, $N=|[-2t,2t]\cap\mathbb{Z}|$ and $b=0$.) Hence, by
the Borel-Cantelli lemma we obtain (4.2), as required.

_Step 2._ We next show that, for an arbitrary $\delta>0$ and
$\operatorname{\mathbf{P}}$-a.e. $\tau$,

$E_{0}^{\tau}\left[\left(\int_{0}^{t}\bar{\tau}_{Y_{s}}\mathbbm{1}_{\\{\bar{\tau}_{Y_{s}}>t^{\varepsilon}\\}}\right)^{\\!2}\right]=O\left(t^{2+2\delta-2(\alpha-2)\varepsilon}\right).$
(4.3)

To this end, we first note that if $(p_{t}^{Y}(x,y))_{x,y\in\mathbb{Z},t>0}$
is the transition density of $Y$, then the standard Gaussian bounds that hold
for this (see [10, Theorem 6.28]) imply that

$\displaystyle\int_{0}^{t}p^{Y}_{s}(x,y)ds$
$\displaystyle\leq\int_{0}^{t}Cs^{-1/2}e^{-\frac{|x-y|^{2}}{Cs}\wedge\frac{|x-y|}{C}}ds$
$\displaystyle\leq
Ce^{-\frac{|x-y|^{2}}{Ct}\wedge\frac{|x-y|}{C}}\int_{0}^{t}s^{-1/2}ds$
$\displaystyle=2Ct^{1/2}e^{-\frac{|x-y|^{2}}{Ct}\wedge\frac{|x-y|}{C}}.$

Hence

$\displaystyle
E_{0}^{\tau}\left[\left(\int_{0}^{t}\bar{\tau}_{Y_{s}}\mathbbm{1}_{\\{\bar{\tau}_{Y_{s}}>t^{\varepsilon}\\}}\right)^{\\!2}\right]$
(4.4) $\displaystyle=$ $\displaystyle
2\sum_{x,y\in\mathbb{Z}}\bar{\tau}_{x}\mathbbm{1}_{\\{\bar{\tau}_{x}>t^{\varepsilon}\\}}\bar{\tau}_{y}\mathbbm{1}_{\\{\bar{\tau}_{y}>t^{\varepsilon}\\}}\int_{0}^{t}\int_{0}^{s}p^{Y}_{r}(0,y)\,p^{Y}_{s-r}(y,x)\,dr\,ds$
$\displaystyle\leq$ $\displaystyle
Ct\sum_{x,y\in\mathbb{Z}}\bar{\tau}_{x}\mathbbm{1}_{\\{\bar{\tau}_{x}>t^{\varepsilon}\\}}\bar{\tau}_{y}\mathbbm{1}_{\\{\bar{\tau}_{y}>t^{\varepsilon}\\}}e^{-\frac{|x|^{2}}{Ct}\wedge\frac{|x|}{C}}e^{-\frac{|x-y|^{2}}{Ct}\wedge\frac{|x-y|}{C}}.$

To bound the sum here, we will use the following basic almost-sure bounds on
the i.i.d. random variables in the collection $(\tau_{x})_{x\in\mathbb{Z}}$.
First, for each $\delta>0$, it $\mathbf{P}$-a.s. holds that there exists a
(possibly random) constant $C$ such that

$\sup_{|x|\leq n}\tau_{x}\leq Cn^{\frac{1}{\alpha}+\delta},\qquad\forall
n\in\mathbb{N}.$ (4.5)

Second, writing $N(n,m):=\\#\\{x:\>|x|\leq n,\>\bar{\tau}_{x}\geq m\\}$, if
$\beta>0$, then it $\mathbf{P}$-a.s. holds that there exists a (possibly
random) constant $C$ such that

$N(n,n^{\beta})\leq Cn^{1-\alpha\beta},\qquad\forall n\in\mathbb{N}.$ (4.6)

In particular, to apply these bounds, it will be convenient to break the sum
$\sum_{x,y\in\mathbb{Z}}$ at (4.4) into four pieces:

$\sum_{\begin{subarray}{c}x\in\mathbb{Z}:\\\ |x|\leq
t^{\frac{1}{2}+\delta}\end{subarray}}\sum_{\begin{subarray}{c}y\in\mathbb{Z}:\\\
|y-x|\leq
t^{\frac{1}{2}+\delta}\end{subarray}}+\sum_{\begin{subarray}{c}x\in\mathbb{Z}:\\\
|x|>t^{\frac{1}{2}+\delta}\end{subarray}}\sum_{\begin{subarray}{c}y\in\mathbb{Z}:\\\
|y-x|\leq
t^{\frac{1}{2}+\delta}\end{subarray}}+\sum_{\begin{subarray}{c}x\in\mathbb{Z}:\\\
|x|\leq
t^{\frac{1}{2}+\delta}\end{subarray}}\sum_{\begin{subarray}{c}y\in\mathbb{Z}:\\\
|y-x|>t^{\frac{1}{2}+\delta}\end{subarray}}+\sum_{\begin{subarray}{c}x\in\mathbb{Z}:\\\
|x|>t^{\frac{1}{2}+\delta}\end{subarray}}\sum_{\begin{subarray}{c}y\in\mathbb{Z}:\\\
|y-x|>t^{\frac{1}{2}+\delta}\end{subarray}};$

we will call these pieces $\Sigma_{1}$, $\Sigma_{2}$, $\Sigma_{3}$,
$\Sigma_{4}$, respectively. Applying (4.5) and assuming $\delta\in(0,1/2)$,
for the fourth piece we have that

$\displaystyle\Sigma_{4}$ $\displaystyle\leq$ $\displaystyle
C\sum_{\begin{subarray}{c}x\in\mathbb{Z}:\\\
|x|>t^{\frac{1}{2}+\delta}\end{subarray}}|x|^{\frac{1}{\alpha}+\delta}e^{-C^{-1}|x|^{4\delta/(1+2\delta)}}\sum_{\begin{subarray}{c}y\in\mathbb{Z}:\\\
|y-x|>t^{\frac{1}{2}+\delta}\end{subarray}}|y|^{\frac{1}{\alpha}+\delta}e^{-C^{-1}|y-x|^{4\delta/(1+2\delta)}}$
$\displaystyle\leq$ $\displaystyle
C\sum_{\begin{subarray}{c}x\in\mathbb{Z}:\\\
|x|>t^{\frac{1}{2}+\delta}\end{subarray}}|x|^{\frac{1}{\alpha}+\delta}e^{-C^{-1}|x|^{4\delta/(1+2\delta)}}\sum_{\begin{subarray}{c}y\in\mathbb{Z}:\\\
|y|>t^{\frac{1}{2}+\delta}\end{subarray}}\left(|y|^{\frac{1}{\alpha}+\delta}+|x|^{\frac{1}{\alpha}+\delta}\right)e^{-C^{-1}|y|^{4\delta/(1+2\delta)}}$
$\displaystyle\leq$ $\displaystyle
C\left(\sum_{\begin{subarray}{c}x\in\mathbb{Z}:\\\
|x|>t^{\frac{1}{2}+\delta}\end{subarray}}|x|^{\frac{2}{\alpha}+2\delta}e^{-C^{-1}|x|^{4\delta/(1+2\delta)}}\right)^{2},$

and it is a consequence of the monotone convergence theorem that this
expression converges to 0 as $t\rightarrow\infty$. For the third piece, we
proceed similarly, applying (4.5) to deduce that

$\displaystyle\Sigma_{3}$ $\displaystyle\leq$ $\displaystyle
C\sum_{\begin{subarray}{c}x\in\mathbb{Z}:\\\ |x|\leq
t^{\frac{1}{2}+\delta}\end{subarray}}|x|^{\frac{1}{\alpha}+\delta}e^{-C^{-1}|x|^{4\delta/(1+2\delta)}}\sum_{\begin{subarray}{c}y\in\mathbb{Z}:\\\
|y-x|>t^{\frac{1}{2}+\delta}\end{subarray}}|y|^{\frac{1}{\alpha}+\delta}e^{-C^{-1}|y-x|^{4\delta/(1+2\delta)}}$
$\displaystyle\leq$ $\displaystyle
Ct^{\left(\frac{1}{2}+\delta\right)\left(\frac{1}{\alpha}+\delta\right)}\sum_{\begin{subarray}{c}y\in\mathbb{Z}:\\\
|y|>t^{\frac{1}{2}+\delta}\end{subarray}}|y|^{\frac{1}{\alpha}+\delta}e^{-C^{-1}|y|^{4\delta/(1+2\delta)}}$
$\displaystyle\leq$ $\displaystyle
C\sum_{\begin{subarray}{c}y\in\mathbb{Z}:\\\
|y|>t^{\frac{1}{2}+\delta}\end{subarray}}|y|^{\frac{2}{\alpha}+2\delta}e^{-C^{-1}|y|^{4\delta/(1+2\delta)}},$

and again one can use the monotone convergence theorem to check that this
expression converges to 0 as $t\rightarrow\infty$. Also, for the second piece,
one has from a similar argument that

$\Sigma_{2}\leq C\sum_{\begin{subarray}{c}x\in\mathbb{Z}:\\\
|x|>t^{\frac{1}{2}+\delta}\end{subarray}}|x|^{\frac{2}{\alpha}+2\delta}e^{-C^{-1}|x|^{4\delta/(1+2\delta)}},$

which obviously also converges to 0 as $t\rightarrow\infty$. Thus it remains
to bound the first term, for which we straightforwardly have that

$\Sigma_{1}\leq\sum_{\begin{subarray}{c}x\in\mathbb{Z}:\\\ |x|\leq
t^{\frac{1}{2}+\delta}\end{subarray}}\sum_{\begin{subarray}{c}y\in\mathbb{Z}:\\\
|y-x|\leq
t^{\frac{1}{2}+\delta}\end{subarray}}\bar{\tau}_{x}\mathbbm{1}_{\\{\bar{\tau}_{x}>t^{\varepsilon}\\}}\bar{\tau}_{y}\mathbbm{1}_{\\{\bar{\tau}_{y}>t^{\varepsilon}\\}}\leq\left(\sum_{\begin{subarray}{c}x\in\mathbb{Z}:\\\
|x|\leq
t^{\frac{1}{2}+\delta}\end{subarray}}\bar{\tau}_{x}\mathbbm{1}_{\\{\bar{\tau}_{x}>t^{\varepsilon}\\}}\right)^{2}.$

Now, it readily holds that

$\sum_{\begin{subarray}{c}x\in\mathbb{Z}:\\\ |x|\leq
t^{\frac{1}{2}+\delta}\end{subarray}}\bar{\tau}_{x}\mathbbm{1}_{\\{\bar{\tau}_{x}>t^{\varepsilon}\\}}\leq\sum_{k=1}^{\infty}N\left(t^{\frac{1}{2}+\delta},t^{\varepsilon
k}\right)t^{(k+1)\varepsilon}.$

Moreover, from (4.5), when $t$ is sufficiently large, the terms in this
expression are equal to 0 for $k\geq k_{0}$, where
$k_{0}:=1+\lceil\varepsilon^{-1}(\frac{1}{2}+\delta)(\frac{1}{\alpha}+\delta)\rceil$.
Therefore, recalling the bound at (4.6), we have that

$\sum_{\begin{subarray}{c}x\in\mathbb{Z}:\\\ |x|\leq
t^{\frac{1}{2}+\delta}\end{subarray}}\bar{\tau}_{x}\mathbbm{1}_{\\{\bar{\tau}_{x}>t^{\varepsilon}\\}}\leq
C\sum_{k=1}^{k_{0}}t^{\frac{1}{2}+\delta-\alpha\varepsilon
k}t^{(k+1)\varepsilon}=Ct^{\frac{1}{2}+\delta+\varepsilon}\sum_{k=1}^{k_{0}}t^{-(\alpha-1)k\varepsilon}.$

Since $\alpha\geq 1$, the sum is bounded by $k_{0}t^{-(\alpha-1)\varepsilon}$,
and so we obtain (for sufficiently large $t$)

$\sum_{\begin{subarray}{c}x\in\mathbb{Z}:\\\ |x|\leq
t^{\frac{1}{2}+\delta}\end{subarray}}\bar{\tau}_{x}\mathbbm{1}_{\\{\bar{\tau}_{x}>t^{\varepsilon}\\}}\leq
Ct^{\frac{1}{2}+\delta-(\alpha-2)\varepsilon}.$

This implies that $\Sigma_{1}=O(t^{1+2\delta-2(\alpha-2)\varepsilon})$, and
combining this with the decay of the other terms, we deduce from (4.4) that
(4.3) holds.

_Step 3._ This is the annealed version of Step 2. In particular, we aim to
check that, for $\operatorname{\mathbf{P}}$-a.e. $\tau$,

$\mathbf{E}\left[E_{0}^{\tau}\left[\left(\int_{0}^{t}\bar{\tau}_{Y_{s}}\mathbbm{1}_{\\{\bar{\tau}_{Y_{s}}>t^{\varepsilon}\\}}\right)^{\\!2}\right]\right]=O\left(t^{\frac{3}{2}-(\alpha-2)\varepsilon}\right)+O\left(t^{2-2(\alpha-1)\varepsilon}\right).$
(4.7)

To check this, we first take expectations at (4.4) to yield

$\displaystyle\mathbf{E}\left[E_{0}^{\tau}\left[\left(\int_{0}^{t}\bar{\tau}_{Y_{s}}\mathbbm{1}_{\\{\bar{\tau}_{Y_{s}}>t^{\varepsilon}\\}}\right)^{\\!2}\right]\right]$
$\displaystyle\leq
Ct\sum_{x,y\in\mathbb{Z}}\mathbf{E}\left[\bar{\tau}_{x}\mathbbm{1}_{\\{\bar{\tau}_{x}>t^{\varepsilon}\\}}\bar{\tau}_{y}\mathbbm{1}_{\\{\bar{\tau}_{y}>t^{\varepsilon}\\}}\right]e^{-\frac{|x|^{2}}{Ct}\wedge\frac{|x|}{C}}e^{-\frac{|x-y|^{2}}{Ct}\wedge\frac{|x-y|}{C}}.$
(4.8)

Now, an elementary calculation gives

$\mathbf{E}\left[\bar{\tau}_{x}\mathbbm{1}_{\\{\bar{\tau}_{x}>t^{\varepsilon}\\}}\bar{\tau}_{y}\mathbbm{1}_{\\{\bar{\tau}_{y}>t^{\varepsilon}\\}}\right]\leq\left\\{\begin{array}[]{ll}Ct^{-(\alpha-2)\varepsilon},&\hbox{if
$x=y$;}\\\ Ct^{-2(\alpha-1)\varepsilon},&\hbox{otherwise.}\end{array}\right.$

Using these estimates and the fact that

$\sum_{x\in\mathbb{Z}}e^{-\frac{|x|^{2}}{Ct}\wedge\frac{|x|}{C}}\leq
Ct^{1/2},$

we obtain (4.7) from (4.8), where the two terms correspond to the sums over
$x=y$ and that over $x\neq y$, respectively.

_Step 4._ Finally, putting together the estimates of (4.1), (4.2), (4.3) and
(4.7), we find that, for any $\delta>0$ and $\operatorname{\mathbf{P}}$-a.e.
$\tau$, it holds that

$E_{0}^{\tau}\Big{[}\big{|}A_{t}-\operatorname{\mathbf{E}}[\tau_{0}]t\big{|}^{2}\Big{]}=O\left(t^{7/4+3\varepsilon}\right)+O\left(t^{2+2\delta-2(\alpha-2)\varepsilon}\right).$

As $\delta$ can be chosen arbitrarily small and $\alpha>2$, optimising over
$\varepsilon$ gives an upper bound of the desired form, i.e.

$O\left(t^{\frac{7}{4}+\frac{1}{4(2\alpha-3)}+\eta}\right),$

for arbitrarily small $\eta$. ∎

######  Remark 4.4.

Whilst the exponent improves from 2 to $7/4$ as $\alpha$ is increased from 2
to $\infty$, we do not obtain the bound of $t^{3/2}$ that holds in the
deterministic case. It would be of interest to determine the optimal exponent
for the Bouchaud trap model.

###### Proof of Proposition 4.2.

Set $\sigma^{2}\mathrel{\mathop{:}}=1/\operatorname{\mathbf{E}}[\tau_{0}]$.
First notice that, for $\operatorname{\mathbf{P}}$-a.e. $\tau$,
$A^{-1}_{t}\leq t$ for all $t\geq 0$. In particular,

$P_{0}^{\tau}\Big{[}\big{|}A_{t}^{-1}-\sigma^{2}t\big{|}\geq x\Big{]}=0$

for all $x\geq(1-\sigma^{2})t$. Moreover, by Proposition 4.3(ii) we have for
$x\in(0,\sigma^{2}t]$,

$\displaystyle P_{0}^{\tau}\Big{[}\big{|}A_{t}^{-1}-\sigma^{2}t\big{|}\geq
x\Big{]}$ $\displaystyle=P_{0}^{\tau}\Big{[}A_{t}^{-1}-\sigma^{2}t\geq
x\Big{]}+P_{0}^{\tau}\Big{[}A_{t}^{-1}-\sigma^{2}t\leq-x\Big{]}$
$\displaystyle=P_{0}^{\tau}\Big{[}A_{\sigma^{2}t+x}\leq
t\Big{]}+P_{0}^{\tau}\Big{[}A_{\sigma^{2}t-x}\geq t\Big{]}$ $\displaystyle\leq
P_{0}^{\tau}\Big{[}\big{|}A_{\sigma^{2}t+x}-\sigma^{-2}(\sigma^{2}t+x))\big{|}\geq
x/\sigma^{2}\Big{]}+P_{0}^{\tau}\Big{[}\big{|}A_{\sigma^{2}t-x}-\sigma^{-2}(\sigma^{2}t-x))\big{|}\geq
x/\sigma^{2}\Big{]}$
$\displaystyle\leq\frac{\sigma^{4}}{x^{2}}\bigg{(}E_{0}^{\tau}\Big{[}\big{|}A_{\sigma^{2}t+x}-\sigma^{-2}(\sigma^{2}t+x))\big{|}^{2}\Big{]}+E_{0}^{\tau}\Big{[}\big{|}A_{\sigma^{2}t-x}-\sigma^{-2}(\sigma^{2}t-x))\big{|}^{2}\Big{]}\bigg{)}$
$\displaystyle\leq\frac{c\sigma^{4}}{x^{2}}\big{(}\sigma^{2}t+x\big{)}^{\frac{7}{4}+\frac{1}{4(2\alpha-3)}+\eta},$

and similarly, for $x\geq\sigma^{2}t$,

$P_{0}^{\tau}\Big{[}\big{|}A_{t}^{-1}-\sigma^{2}t\big{|}\geq
x\Big{]}=P_{0}^{\tau}\Big{[}A_{\sigma^{2}t+x}\leq
t\Big{]}\leq\frac{c\sigma^{4}}{x^{2}}\big{(}\sigma^{2}t+x\big{)}^{\frac{7}{4}+\frac{1}{4(2\alpha-3)}+\eta}.$

Hence,

$\displaystyle
E_{0}^{\tau}\Big{[}\big{|}A^{-1}_{t}-\sigma^{2}t\big{|}^{2}\Big{]}$
$\displaystyle=\int_{0}^{\infty}P_{0}^{\tau}\Big{[}\big{|}A_{t}^{-1}-\sigma^{2}t\big{|}^{2}\geq
x\Big{]}\,dx$
$\displaystyle\leq\int_{0}^{(1-\sigma^{2})^{2}t^{2}}P_{0}^{\tau}\Big{[}\big{|}A_{t}^{-1}-\sigma^{2}t\big{|}\geq\sqrt{x}\Big{]}\,dx$
$\displaystyle\leq\int_{0}^{(1-\sigma^{2})^{2}t^{2}}\bigg{(}1\wedge\frac{c\sigma^{4}}{x}\big{(}\sigma^{2}t+\sqrt{x}\big{)}^{\frac{7}{4}+\frac{1}{4(2\alpha-3)}+\eta}\bigg{)}\,dx$
$\displaystyle=O\left(t^{\frac{7}{4}+\frac{1}{4(2\alpha-3)}+\eta}\log
t\right).$ (4.9)

In order to show the statement, recall that $\langle X\rangle_{t}=\langle
Y\rangle_{A^{-1}_{t}}$, where $\langle Y\rangle_{t}$ equals the number of
jumps of $Y$ up to time $t$. In particular, $(\langle Y\rangle_{t})_{t\geq 0}$
is a Poisson process with rate $1$ and $(\langle Y\rangle_{t}-t)_{t\geq 0}$ is
martingale. Thus, by decomposing $\langle X\rangle_{t}$ as $\langle
X\rangle_{t}=A_{t}^{-1}+\langle Y\rangle_{A^{-1}_{t}}-A_{t}^{-1}$, we get

$\displaystyle E_{0}^{\tau}\bigg{[}\Big{|}\frac{\langle
X\rangle_{t}}{t}-\sigma^{2}\Big{|}^{2}\bigg{]}$ $\displaystyle\leq
2\bigg{(}E_{0}^{\tau}\bigg{[}\Big{|}\frac{A_{t}^{-1}}{t}-\sigma^{2}\Big{|}^{2}\bigg{]}+t^{-2}E_{0}^{\tau}\Big{[}\big{(}\langle
Y\rangle_{A_{t}^{-1}}-A^{-1}_{t}\big{)}^{2}\Big{]}\bigg{)}$ $\displaystyle\leq
O\left(t^{-\frac{1}{4}+\frac{1}{4(2\alpha-3)}+\eta}\log
t\right)+t^{-2}E_{0}^{\tau}\Big{[}\sup_{s\leq t}\big{(}\langle
Y\rangle_{s}-s\big{)}^{2}\Big{]},$

where we used (4.1) and again that $A^{-1}_{t}\leq t$. Using Doob’s inequality
and observing that $E_{0}^{\tau}\big{[}\big{(}\langle
Y\rangle_{t}-t\big{)}^{2}\big{]}$ equals the variance of a Poisson distributed
random variable with parameter $t$, we can bound the second term from above as

$\displaystyle t^{-2}E_{0}^{\tau}\Big{[}\sup_{s\leq t}\big{(}\langle
Y\rangle_{s}-s\big{)}^{2}\Big{]}\bigg{)}\leq
ct^{-2}E_{0}^{\tau}\Big{[}\big{(}\langle
Y\rangle_{t}-t\big{)}^{2}\Big{]}\bigg{)}=O\left(t^{-1}\right).$

Since $\eta>0$ is arbitrary, this shows (ii). Part (i) follows from
Proposition 4.3(i) by the same arguments. ∎

We next control the jumps of $X$.

######  Proposition 4.5.

For any $n\in\mathbb{N}$ and $\mathbb{P}$-a.e. $\tau$,

$E_{0}^{\tau}\Big{[}\sum_{0\leq s\leq
t}\big{|}X_{s}-X_{s-}\big{|}^{n}\Big{]}\;\leq\;t,\qquad\forall t>0.$

###### Proof.

According to the Lévy system (see e.g. [26, Appendix A]), for any continuous
bounded non-negative function
$f:\mathbb{Z}^{d}\times\mathbb{Z}^{d}\to\mathbb{R}$ that vanishes on the
diagonal, for $\mathbb{P}$-a.e. $\tau$,

$E_{0}^{\tau}\bigg{[}\sum_{0\leq s\leq
t}f(X_{s-},X_{s})\bigg{]}=E_{0}^{\tau}\bigg{[}\int_{(0,t]}\frac{1}{2\tau_{X_{s-}}}\sum_{y\sim
X_{s-}}f(X_{s-},y)\,ds\bigg{]}.$

Then choosing $f(x,y)=1\wedge\big{|}x-y\big{|}^{n}$ and recalling that
$\tau_{x}\geq 1$ for all $x$, we obtain

$E_{0}^{\tau}\Big{[}\frac{1}{t}\sum_{0\leq s\leq
t}\big{|}X_{s}-X_{s-}\big{|}^{n}\Big{]}=\frac{1}{t}\int_{0}^{t}ds\,E^{\tau}_{0}\Big{[}\frac{1}{2\tau_{X_{s-}}}\sum_{y\sim
X_{s-}}\big{|}y-X_{s-}\big{|}^{n}\Big{]}\leq 1,$

which finishes the proof. ∎

###### Proof of Theorem 1.5.

We shall apply the general result of Theorem 4.1 with the choice $n=2$ on the
martingale

$N_{s}\;\mathrel{\mathop{:}}=\;\frac{X_{st}}{\sqrt{t}\,\sigma},\qquad 0\leq
s\leq 1.$

Since

$\langle N\rangle_{1}-1\;=\;\frac{\langle
X\rangle_{t}}{t\,\sigma^{2}}-1\;=\;\frac{1}{\sigma^{2}}\bigg{(}\frac{\langle
X\rangle_{t}}{t}-\sigma^{2}\bigg{)},$

we get by Proposition 4.2 that

$\displaystyle\operatorname{\mathbf{E}}\Big{[}E_{0}^{\tau}\big{[}|\langle
N\rangle_{1}-1|^{2}\big{]}\Big{]}=O\left(t^{-1/2}\log
t\right),\>\>E_{0}^{\tau}\big{[}|\langle
N\rangle_{1}-1|^{2}\big{]}=O\left(t^{-\frac{1}{4}+\frac{1}{4(2\alpha-3)}+\eta}\right).$
(4.10)

Furthermore,

$\displaystyle\sum_{s\leq 1}\big{|}\Delta
N_{s}\big{|}^{4}\;=\;(t\sigma^{2})^{-2}\sum_{s\leq
t}\big{|}X_{s}-X_{s-}\big{|}^{4},$

and we obtain from Proposition 4.5 that

$\displaystyle E_{0}^{\tau}\Big{[}\sum_{0\leq s\leq 1}\big{|}\Delta
N_{s}\big{|}^{4}\Big{]}\;\leq t^{-1}.$ (4.11)

Now we apply Theorem 4.1, under the annealed measure $\mathbb{P}$, which gives

$\displaystyle\sup_{x\in\mathbb{R}}\left|\mathbb{P}\big{[}X_{t}\leq x\sqrt{t}\
\big{]}-\Phi(\tfrac{x}{\sigma})\right|\;=\;\sup_{x\in\mathbb{R}}\left|\mathbb{P}\big{[}N_{1}\leq
x\ \big{]}-\Phi(x)\right|$
$\displaystyle\mspace{36.0mu}\leq\;c\Bigg{(}\operatorname{\mathbf{E}}\bigg{[}E_{0}^{\tau}\Big{[}\big{|}\langle
N\rangle_{1}-1\big{|}^{2}\Big{]}\bigg{]}+\operatorname{\mathbf{E}}\bigg{[}E_{0}^{\tau}\Big{[}\sum_{0\leq
s\leq 1}\big{|}\Delta N_{s}\big{|}^{4}\Big{]}\bigg{]}\Bigg{)}^{\frac{1}{5}},$

and an analogous estimate holds under the quenched measure $P_{0}^{\tau}$.
Hence, the claim follows from (4.10) and (4.11). ∎

###  4.2. Quantitative local CLT

In this section, we derive the quantitative quenched local limit result of
Theorem 1.7. We first give a general statement of a quantitative local CLT for
BTM on a class of metric measure spaces and then prove it applies in our case.
The approach is quite generic and will apply to other stochastic processes and
settings, but we do not attempt to give the broadest presentation of the
argument here, so as to not obscure the main ideas.

Let $(M,d,\mu)$ be a metric measure space, where $M$ is a locally compact
metric space that is also a vector space, $d$ be a metric on $M$ and $\mu$ be
a Radon measure on $M$ of full support. Let $0\in M$ be the origin of $M$ as
the vector space. Let $G\subset M$ be the vertices of locally finite,
connected graph and for $n\in\mathbb{Z}$, let $G_{n}:=n^{-1}G$. Suppose $G$
consists of ‘cells’ $\\{V^{(i)}\\}_{i\in\Lambda}$ such that $|V^{(i)}|=L$ for
some $L\geq 2$ and $\cup_{i\in\Lambda}V^{(i)}=G$, and moreover, $M$ can be
decomposed into ‘complexes’ $\\{K^{(i)}\\}_{i\in\Lambda}$, where
$K^{(i)}:=\mbox{Conv}(V^{(i)})$, with $\mbox{Conv}\,(H)$ being the convex hull
of $H$. We assume that $\sup_{i\in\Lambda}\,\mbox{diam}\,K^{(i)}<\infty$,
$\cup_{i\in\Lambda}K^{(i)}=M$, and there exists a $d_{f}\geq 1$ such that
$\mu(n^{-1}K^{(i)})=n^{-d_{f}}\mu(K^{(i)})$ for all $i\in\Lambda$ and
$n\in\mathbb{Z}$. In the following, we write $V^{(i)}_{n}:=n^{-1}V^{(i)}$ and
$K^{(i)}_{n}:=n^{-1}K^{(i)}$. Concerning the dynamics on $M$, we assume that
there exists a diffusion process $\\{B_{t}\\}_{t}$ on $M$ that has transition
density $p_{\mathrm{BM}}(t,x,y)$ (with respect to $\mu$) enjoying the
following estimates for all $t\in(0,\infty)$, $x,y\in M$

$\displaystyle
c_{3}t^{-d_{f}/d_{w}}\exp\Big{(}-c_{4}\Big{(}\frac{d(x,y)^{d_{w}}}{t}\Big{)}^{1/(d_{w}-1)}\Big{)}\leq
p_{\mathrm{BM}}(t,x,y)$ $\displaystyle~{}~{}~{}~{}~{}~{}\leq
c_{5}t^{-d_{f}/d_{w}}\exp\Big{(}-c_{6}\Big{(}\frac{d(x,y)^{d_{w}}}{t}\Big{)}^{1/(d_{w}-1)}\Big{)},$
(4.12)

where $d_{w}\geq 2$. We write the Markov process on $G$ corresponding to (1.2)
as $\\{X_{t}\\}_{t}$ (where $x\sim y$ means $x$ and $y$ are connected by a
bond), and write its transition density, defined as at (1.3), as
$p_{t}^{\tau}(x,y)$. Fix $h\in(0,\infty)$, $a>1$ and $\frac{1}{2}<\eta$. For
$N\in\mathbb{N}$, set

$\mathcal{I}_{N}:=\left\\{K^{(i)}_{a^{-\lfloor{\eta
N}\rfloor}}:i\in\Lambda,\>K^{(i)}_{a^{-\lfloor{\eta N}\rfloor}}\cap
B(0,ha^{N+1})\neq\emptyset\right\\},$

where $B(0,r)$ is a ball of centered $0$ and radius $r$ with respect to the
distance $d$. For $x\in B(0,h)$ and $n\in[a^{N},a^{N+1})$, let $I_{x}^{n}$ be
an element in $\mathcal{I}_{N}$ containing $nx$. For $I\in\mathcal{I}_{N}$,
define $V(I):=\sum_{y\in I}\tau_{y}$. The following assumption captures the
key estimates we need to deduce a quantitative local limit theorem.

######  Assumption 4.6.

There exists $\theta\in(0,d_{f}/(2d_{w}))$ such that for each
$\frac{d_{f}}{2}<\kappa<\eta d_{f}<d_{f}$, $h>0$ and $0<T_{1}<T_{2}<\infty$,
the following hold $\operatorname{\mathbf{P}}\mbox{-a.s.}$:

$\displaystyle\sup_{I\in\mathcal{I}_{N}}\left|V(I)-\mu(I)\right|$
$\displaystyle\leq C(\tau)a^{\kappa N},$ (4.13)
$\displaystyle\sup_{\begin{subarray}{c}|x|<h,\\\ t\in[T_{1},T_{2}],\\\
n\in[a^{N},a^{N+1})\end{subarray}}\left|P^{\tau}_{0}\big{[}X_{n^{d_{w}}t}\in
I_{x}^{n}\big{]}-\int_{n^{-1}I_{x}^{n}}p_{\mathrm{BM}}(t,0,y)\,dy\right|$
$\displaystyle\leq C(\tau)a^{-d_{w}\theta N},$ (4.14)
$\displaystyle\sup_{\begin{subarray}{c}|x|<h:\>nx\in G,\\\
t\in[T_{1},T_{2}],\\\ n\in[a^{N},a^{N+1})\end{subarray}}n^{d_{f}}\sup_{y\in
I_{x}^{n}}\left|p^{\tau}_{n^{d_{w}}t}(0,y)-p^{\tau}_{n^{d_{w}}t}(0,nx)\right|$
$\displaystyle\leq C(\tau)a^{\frac{1}{2}N(\eta-1)},$ (4.15)
$\displaystyle\sup_{t\in[T_{1},T_{2}]}\left|p_{\mathrm{BM}}(t,0,y)-p_{\mathrm{BM}}(t,0,x)\right|$
$\displaystyle\leq Cd(x,y)^{\frac{1}{2}}.$ (4.16)

######  Remark 4.7.

Note that, since $\mu$ has full support, it follows from (4.13) that for every
$x\in M$, there exists a sequence $x_{n}\in G_{n}$, $n\geq 1$, such that
$d(x_{n},x)\rightarrow 0$.

######  Theorem 4.8.

Under Assumption 4.6, it holds that for all $h>0$ and $0<T_{1}\leq T_{2}$ and
any $\varepsilon>0$,

$\displaystyle\lim_{n\to\infty}n^{\frac{d_{w}\theta}{1+2d_{f}}-\varepsilon}\sup_{|x|\leq
h:\>nx\in
G}\sup_{t\in[T_{1},T_{2}]}\big{|}n^{d_{f}}p^{\tau}_{n^{d_{w}}t}(0,nx)-p_{\mathrm{BM}}(t,0,x)\big{|}=0,\qquad\operatorname{\mathbf{P}}\mbox{-a.s.}$

######  Remark 4.9.

If $G_{n}:=b^{-n}G$ for $b>1$ and Assumption 4.6 holds with $b^{n}$ instead of
$n$, then Theorem 4.8 holds with $b^{n}$ instead of $n$.

###### Proof.

Suppose $n\in[a^{N},a^{N+1})$ and $x\in B(0,h)\cap n^{-1}G$. It then holds
that

$\displaystyle\big{|}n^{d_{f}}p^{\tau}_{n^{d_{w}}t}(0,nx)-p_{\mathrm{BM}}(t,0,x)\big{|}$
$\displaystyle=\Big{|}n^{d_{f}}p^{\tau}_{n^{d_{w}}t}(0,nx)-\frac{n^{d_{f}}P^{\tau}_{0}[X_{n^{d_{w}}t}\in
I_{x}^{n}]}{V(I_{x}^{n})}\Big{|}$
$\displaystyle\qquad+\Big{|}\frac{n^{d_{f}}}{V(I_{x}^{n})}\Big{(}P^{\tau}_{0}\big{[}X_{n^{d_{w}}t}\in
I_{x}^{n}\big{]}-\int_{n^{-1}I_{x}^{n}}p_{\mathrm{BM}}(t,0,y)\,dy\Big{)}\Big{|}$
$\displaystyle\qquad+\frac{1}{n^{-d_{f}}\mu(I_{x}^{n})}\int_{n^{-1}I_{x}^{n}}p_{\mathrm{BM}}(t,0,y)\,\mu(dy)\cdot\Big{|}1-\frac{\mu(I_{x}^{n})}{V(I_{x}^{n})}\Big{|}$
$\displaystyle\qquad+\Big{|}\frac{1}{n^{-d_{f}}\mu(I_{x}^{n})}\int_{n^{-1}I_{x}^{n}}p_{\mathrm{BM}}(t,0,y)\,\mu(dy)-p_{\mathrm{BM}}(t,0,x)\Big{|}$
$\displaystyle=:\Upsilon_{1}+\Upsilon_{2}+\Upsilon_{3}+\Upsilon_{4}.$

By (4.16),

$\displaystyle\sup_{t\in[T_{1},T_{2}]}\sup_{|x|<h}\Upsilon_{4}$
$\displaystyle\leq$
$\displaystyle\sup_{t\in[T_{1},T_{2}]}\sup_{|x|<h}\sup_{y\in
n^{-1}I_{x}^{n}}\left|p_{\mathrm{BM}}(t,0,y)-p_{\mathrm{BM}}(t,0,x)\right|$
$\displaystyle\leq$ $\displaystyle Cn^{-1/2}a^{\lfloor\eta N\rfloor/2}$
$\displaystyle\leq$ $\displaystyle Cn^{\frac{1}{2}(\eta-1)}.$

Similarly, by (4.15), we have that

$\displaystyle\sup_{\begin{subarray}{c}|x|<h:\>nx\in G,\\\
t\in[T_{1},T_{2}]\end{subarray}}\Upsilon_{1}$ $\displaystyle\leq$
$\displaystyle\sup_{\begin{subarray}{c}|x|<h:\>nx\in G,\\\
t\in[T_{1},T_{2}]\end{subarray}}n^{d_{f}}\sup_{y\in
I_{x}^{n}}\left|p^{\tau}_{n^{d_{w}}t}(0,y)-p^{\tau}_{n^{d_{w}}t}(0,nx)\right|\leq
Cn^{\frac{1}{2}(\eta-1)}.$

As for $\Upsilon_{2}$, from (4.14), we deduce that

$\sup_{\begin{subarray}{c}|x|<h,\\\
t\in[T_{1},T_{2}]\end{subarray}}\left|P_{0}\big{[}X_{n^{d_{w}}t}\in
I_{x}^{n}\big{]}-\int_{n^{-1}I_{x}^{n}}p_{\mathrm{BM}}(t,0,y)\,\mu(dy)\right|\leq
Cn^{-{d_{w}}\theta}.$

From the volume bound (4.13), we also have that

$\sup_{|x|<h}\frac{n^{d_{f}}}{V(I_{x}^{n})}\leq\frac{n^{d_{f}}}{a^{\lfloor\eta
N\rfloor d_{f}}-a^{\kappa N}}\leq Cn^{d_{f}(1-\eta)}.$

Combining the two previous bounds yields

$\sup_{\begin{subarray}{c}|x|<h,\\\
t\in[T_{1},T_{2}]\end{subarray}}\Upsilon_{2}\leq
Cn^{d_{f}(1-\eta)-d_{w}\theta}.$

Moreover, in the third term, $\Upsilon_{3}$, we have that

$\sup_{\begin{subarray}{c}|x|<h,\\\
t\in[T_{1},T_{2}]\end{subarray}}\frac{1}{n^{-d_{f}}\mu(I_{x}^{n})}\int_{n^{-1}I_{x}^{n}}p_{\mathrm{BM}}(t,0,y)\,dy\leq\sup_{x\in
M,\>t\in[T_{1},T_{2}]}p_{\mathrm{BM}}(t,x,x)\leq C,$

and also, again from (4.13),

$\sup_{|x|<h}\Big{|}1-\frac{\mu(I_{x}^{n})}{V(I_{x}^{n})}\Big{|}\leq
Cn^{\kappa-\eta d_{f}},$

hence

$\Upsilon_{3}\leq Cn^{\kappa-\eta d_{f}}.$

Putting these pieces together, we have established that

$\displaystyle\sup_{|x|\leq h:\>nx\in
G}\sup_{t\in[T_{1},T_{2}]}\big{|}n^{d_{f}}p^{\omega}_{n^{d_{w}}t}(0,nx)-p_{\mathrm{BM}}(t,0,x)\big{|}$
$\displaystyle\leq
C\left(n^{\frac{1}{2}(\eta-1)}+n^{d_{f}(1-\eta)-d_{w}\theta}+n^{\kappa-\eta
d_{f}}\right).$

Using that $\theta<d_{f}/(2d_{w})$, it is an elementary exercise to check that
this bound is optimised over the $\kappa$ and $\eta$ satisfying the
constraints of Assumption 4.6 by taking $\kappa$ suitably close to $d_{f}/2$
and $\eta=\frac{1+2d_{f}-2d_{w}\theta}{1+2d_{f}}$. With this choice, we obtain
the desired result. ∎

We now apply the above result to our case. In particular, we let $M$ be given
by $\mathbb{R}$ equipped with the Euclidean metric, $\mu$ be Lebesgue measure
multiplied by $\mathbf{E}(\tau_{0})$, $G=\mathbb{Z}$, $K^{(i)}$ be the unit
interval $[i,i+1]$ and $V^{(i)}$ is its boundary, and $B$ be Brownian motion
on $\mathbb{R}$, slowed down by a factor $\mathbf{E}(\tau_{0})$. Clearly, in
this setting, (4.12) is satisfied with $d_{f}=1$, $d_{w}=2$. Moreover, taking
$a=2$ and $x_{i}:=i\lfloor 2^{\eta N}\rfloor$, $\mathcal{I}_{N}$ is given by

$\mathcal{I}_{N}:=\left\\{[x_{i-1},x_{i}]:\>x_{i}\geq-h2^{N+1},\>x_{i-1}\leq
h2^{N+1}\right\\}.$

###### Proof of Theorem 1.7.

It is enough to check Assumption 4.6. We first prove (4.13). By a union bound,
we have (for sufficiently large $N$) that

$\displaystyle\mathbf{P}\left(\sup_{I\in\mathcal{I}_{N}}\left|V(I)-\mu(I)-\mathbf{E}(\tau_{0})\right|>a^{\kappa
N}\right)$ $\displaystyle\leq$
$\displaystyle\left|\mathcal{I}_{N}\right|\frac{\mathrm{Var}\left(V(I)\right)}{a^{2\kappa
N}}$ $\displaystyle\leq$ $\displaystyle\frac{Ca^{N}}{a^{\lfloor{\eta
N}\rfloor}}\times\frac{a^{\lfloor{\eta
N}\rfloor}\mathrm{Var}\left(\tau_{0}\right)}{a^{2\kappa N}}$ $\displaystyle=$
$\displaystyle Ca^{(1-2\kappa)N}.$

Thus the result is a simple application of the Borel-Cantelli lemma. The
assumption at (4.14) is a consequence of Theorem 1.5 (ii), where we may take
$\theta$ as in that result. The bound at (4.15) follows from (3.3), and (4.16)
is a standard estimate for $1$-dimensional Brownian motion (which can be
proved similarly to (3.3)). Hence, with some adjustment of the constants, we
obtain the theorem. ∎

##  Appendix A Tail bounds for the volume of balls in the BTM

In this section, we present tail bounds for $S_{n}$, which imply corresponding
bounds on the volume of balls in the Bouchaud trap model, as will be needed in
the proof of Theorem 1.4. The results are similar to ones established in [23,
27, 55, 64, 68], for example, but for the convenience of the reader we present
self-contained proofs. The proof of part (ii) in particular is modelled on the
argument of [23, Proposition 2.1]. The definition of $v_{\alpha}$ should be
recalled from (3.5).

######  Proposition A.1.

  1. (i)

For any $\alpha>0$, it holds that: for $n\in\mathbb{N}$ and $\lambda\geq 1$,

$\operatorname{\mathbf{P}}\left(S_{n}\leq\lambda^{-1}v_{\alpha}(n)\right)\leq
Ce^{-c\lambda^{\gamma}},$

where we can take $\gamma=\min\\{1,\alpha\\}$.

  2. (ii)

For any $\alpha>0$, it holds that: for $n\in\mathbb{N}$ and $\lambda\geq 1$,

$\operatorname{\mathbf{P}}\left(S_{n}\geq\lambda v_{\alpha}(n)\right)\leq
C\lambda^{-\rho},$

where we can take $\rho$ to be any number strictly smaller that $\alpha$.

###### Proof.

Towards checking part (i), we first note that, for $\theta<1$,

$\operatorname{\mathbf{E}}\left(e^{-\theta\tau_{1}}\right)\leq\left\\{\begin{array}[]{ll}1-c\theta^{\alpha},&\hbox{if
$\alpha<1$;}\\\ 1-c\theta\log(\theta^{-1}),&\hbox{if $\alpha=1$;}\\\
1-c\theta,&\hbox{if $\alpha>1$.}\end{array}\right.$

Hence, for $n\lambda\leq v_{\alpha}(n)$,

$\displaystyle\operatorname{\mathbf{P}}\left(S_{n}\leq\lambda^{-1}v_{\alpha}(n)\right)$
$\displaystyle\leq C\operatorname{\mathbf{E}}\left(e^{-\lambda
S_{n}/v_{\alpha}(n)}\right)$
$\displaystyle=C\operatorname{\mathbf{E}}\left(e^{-\lambda\tau_{1}/v_{\alpha}(n)}\right)^{n}$
$\displaystyle\leq\left\\{\begin{array}[]{ll}C\left(1-\frac{c\lambda^{\alpha}}{n}\right)^{n},&\hbox{if
$\alpha<1$;}\\\ C\left(1-\frac{c\lambda}{n}\right)^{n},&\hbox{if $\alpha\geq
1$.}\end{array}\right.$

Since $1-x\leq e^{-x}$, the desired conclusion thus holds in the range
$n\lambda<v_{\alpha}(n)$. If $n\lambda>v_{\alpha}(n)$, then it trivially holds
that
$\operatorname{\mathbf{P}}\left(S_{n}\leq\lambda^{-1}v_{\alpha}(n)\right)\leq\operatorname{\mathbf{P}}\left(S_{n}<n\right)=0$,
since each individual summand is bounded below by 1.

As for part (ii), elementary calculations yield that, for $\theta<1$,

$\operatorname{\mathbf{E}}\left(e^{\theta\tau_{1}}\mathbf{1}_{\\{\tau_{1}\leq\theta^{-1}\\}}\right)\leq\left\\{\begin{array}[]{ll}1+c\theta^{\alpha},&\hbox{if
$\alpha<1$;}\\\ 1+c\theta\log(\theta^{-1}),&\hbox{if $\alpha=1$;}\\\
1+c\theta,&\hbox{if $\alpha>1$.}\end{array}\right.$

Hence, for $\alpha<1$, if $m=\lambda^{\eta}n^{1/\alpha}$ for some
$\eta\in(0,1)$, then

$\displaystyle\operatorname{\mathbf{P}}\left(S_{n}\geq\lambda
v_{\alpha}(n)\right)$ $\displaystyle\leq
n\operatorname{\mathbf{P}}\left(\tau_{1}\geq
m\right)+\operatorname{\mathbf{E}}\left(e^{S_{n}/m}\mathbf{1}_{\\{\tau_{1},\dots,\tau_{n}\leq
m\\}}\right)e^{-\lambda v_{\alpha}(n)/m}$ $\displaystyle\leq
nm^{-\alpha}+\left(1+cm^{-\alpha}\right)^{n}e^{-\lambda v_{\alpha}(n)/m}$
$\displaystyle\leq\lambda^{-\eta\alpha}+e^{c\lambda^{-\eta\alpha}-\lambda^{1-\eta}}$
$\displaystyle\leq C\lambda^{-\eta\alpha}.$

This completes the proof in this case. Taking $m=\lambda^{\eta}n$ for some
$\eta\in(0,1)$, the argument for $\alpha=1$ is similar, and so we omit it. For
$\alpha>1$, set $m=\lambda^{\eta}n^{1/\alpha}$ for some $\eta\in(0,1)$, then

$\displaystyle\operatorname{\mathbf{P}}\left(S_{n}\geq\lambda
v_{\alpha}(n)\right)$ $\displaystyle\leq
nm^{-\alpha}+\left(1+cm^{-1}\right)^{n}e^{-\lambda v_{\alpha}(n)/m}$
$\displaystyle\leq\lambda^{-\eta\alpha}+e^{-(\lambda-c)\lambda^{-\eta}n^{1-1/\alpha}}$
$\displaystyle\leq C\lambda^{-\eta\alpha}+Ce^{-c\lambda^{1-\eta}}$
$\displaystyle\leq C\lambda^{-\eta\alpha},$

which completes the proof. ∎

Acknowledgements: This research was supported by JSPS Grant-in-Aid for
Scientific Research (A) 22H00099, 23KK0050, (C) 19K03540, and the Research
Institute for Mathematical Sciences, an International Joint Usage/Research
Center located in Kyoto University. The authors thank Ryoki Fukushima for
useful comments concerning the proof of Proposition 4.3 (ii).

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