# Lower Gaussian heat kernel bounds for the Random Conductance Model in a
degenerate ergodic environment

Sebastian Andres  The University of Manchester Department of Mathematics,
Oxford Road, Manchester M13 9PL<EMAIL_ADDRESS>and  Noah
Halberstam  University of Cambridge Centre for Mathematical Sciences,
Wilberforce Road, Cambridge CB3 0WB<EMAIL_ADDRESS>
(Date: August 28, 2024)

###### Abstract.

We study the random conductance model on $\operatorname{\mathbb{Z}}^{d}$ with
ergodic, unbounded conductances. We prove a Gaussian lower bound on the heat
kernel given a polynomial moment condition and some additional assumptions on
the correlations of the conductances. The proof is based on the well-
established chaining technique. We also obtain bounds on the Green’s function.

###### Key words and phrases:

Random conductance model; heat kernel; ergodic

###### 2010 Mathematics Subject Classification:

39A12; 60J35; 60K37;82C41

##  1\. Introduction

###  1.1. The Model

We let $G=(\operatorname{\mathbb{Z}}^{d},E_{d}),$ where
$E_{d}=\\{\\{x,y\\}\in\operatorname{\mathbb{Z}}^{d}\times\operatorname{\mathbb{Z}}^{d}:|x-y|=1\\},$
be the $d$-dimensional lattice for a fixed dimension $d\geq 2$. We write
$x\sim y$ if $(x,y)\in E_{d}$. We consider the space of positive weightings on
the edges of the graph, $\Omega=(0,\infty)^{E_{d}},$ and for
$\omega\in\Omega,$ we access the weight at a particular edge $e\in E_{d}$ by
$\omega(e)$, which we will also refer to as the _conductance_ on an edge $e$.
For $x,y\in\operatorname{\mathbb{Z}}^{d}$ and $\omega\in\Omega$ we set
$\omega(x,y)=\omega(y,x)=\omega(\\{x,y\\})$ if $\\{x,y\\}\in E_{d},$ else
$\omega(x,y)=0.$ For any fixed $\omega,$ we define measures $\mu^{\omega}$ and
$\nu^{\omega}$ on $\operatorname{\mathbb{Z}}^{d}$ by

$\mu^{\omega}(x)\mathrel{\mathop{:}}=\sum_{y\sim
x}\omega(x,y)\qquad\text{and}\qquad\nu^{\omega}(x)\mathrel{\mathop{:}}=\sum_{y\sim
x}\frac{1}{\omega(x,y)}.$

For any $z\in\mathbb{Z}^{d}$ we denote by $\tau_{z}\\!:\Omega\to\Omega$ the
space shift by $z$ defined by

$\displaystyle(\tau_{z}\,\omega)(\\{x,y\\})\;\mathrel{\mathop{:}}=\;\omega(\\{x+z,y+z\\}),\qquad\forall\;\\{x,y\\}\in
E_{d}.$

We equip $\Omega$ with a $\sigma$-algebra $\mathcal{F}$. Further, we will
denote by $\operatorname{\mathbb{P}}$ a probability measure on
$(\Omega,\mathcal{F})$, and we write $\operatorname{\mathbb{E}}$ for the
expectation with respect to $\operatorname{\mathbb{P}}$. Throughout the paper
we will assume that the conductances are stationary and ergodic.

######  Assumption 1.1 (Stationarity and ergodicity).

$\operatorname{\mathbb{P}}$ is stationary and ergodic with respect to
translations of $\mathbb{Z}^{d}$, i.e.
$\operatorname{\mathbb{P}}\circ\,\tau_{x}^{-1}\\!=\operatorname{\mathbb{P}}\,$
for all $x\in\mathbb{Z}^{d}$ and $\operatorname{\mathbb{P}}[A]\in\\{0,1\\}\,$
for any $A\in\mathcal{F}$ such that $\tau_{x}(A)=A\,$ for all
$x\in\mathbb{Z}^{d}$.

We now introduce the _random conductance model (RCM)_. For a given
$\omega\in\Omega$, we consider the continuous time Markov chain
$X=\\{X_{t}:t\geq 0\\}$ on $\mathbb{Z}^{d}$ with generator

$(\mathcal{L}^{\omega}f)(x)=\frac{1}{\mu^{\omega}(x)}\sum_{y\sim
x}\omega(x,y)\,\big{(}f(y)-f(x)\big{)}.$

This stochastic process, also known as the _constant speed random walk_
(CSRW), waits at $x$ for an exponential time with mean $1$, and then chooses
the next position $y\sim x$ with probability $\omega(x,y)/\mu^{\omega}(x)$. We
also recall that the Markov chain $X$ is reversible with respect to
$\mu^{\omega}$. We denote by $\mathrm{P}_{x}^{\omega}$ the law of the walk
starting at the vertex $x\in\operatorname{\mathbb{Z}}^{d},$ and by
$\mathrm{E}_{x}^{\omega}$ the expectation with respect to this law. For
$x,y\in\operatorname{\mathbb{Z}}^{d}$ and $t>0,$ we let $p^{\omega}(t,x,y)$ be
the transition density (or the heat kernel associated with
$\mathcal{L}^{\omega}$) with respect to the measure $\mu^{\omega}$, i.e.

$p^{\omega}(t,x,y)=\frac{\mathrm{P}_{x}^{\omega}[X_{t}=y]}{\mu^{\omega}(y)}.$

###  1.2. Main Results

The random conductance model has been the subject of extensive research for
more than a decade, see [19, 38] for surveys of the model and references
therein. More recent results include the derivation of quenched functional
central limit theorems [4, 17, 25, 22] and local limit theorems [15, 5, 3, 10,
16] for the RCM with unbounded ergodic conductances under moment conditions.
In this paper we will focus on heat kernel estimates, see e.g. [24, 13, 18,
15, 14, 29, 21, 6, 7] for previous results. In particular, we will obtain
Gaussian type lower bounds on the heat kernel in the case of ergodic unbounded
conductances.

It is known that Gaussian bounds do not hold in general: for example, under
i.i.d. conductances with fat tails at zero, the heat kernel decay may be sub-
diffusive due to a trapping phenomenon – see [18, 21]. Moreover, in [5,
Theorem 5.4], it is proved that in the general ergodic setting, moment bounds
on the conductances and their reciprocals are a necessary condition for upper
and lower near-diagonal Gaussian bounds to hold. In [6], this necessary
condition is shown to be sufficient for full upper Gaussian heat kernel
bounds.

Gaussian lower bounds have been shown on i.i.d. percolation clusters in [13],
and for variable speed random walks under i.i.d. conductances bounded away
from zero in [14]. However, in the general ergodic setting, as of yet,
Gaussian lower bounds have only been proved under the stronger condition of
uniformly elliptic conductances [24], i.e. $c^{-1}\leq\omega(e)\leq c$, $e\in
E_{d}$, for some $c\geq 1$. In this paper we relax the uniform ellipticity
assumption, substituting it for the combination of a polynomial moment
condition together with an assumption concerning the correlations of the
conductances, see Assumption 1.3. It is unknown whether moment conditions by
themselves should be sufficient for the lower bound to hold. The main
available technique for proving lower bounds, the chaining method (see [28]),
fails at present in this generality (see Section 1.3 below for a more in-depth
discussion), while our assumptions are sufficient to ensure the functionality
of this method. However, it seems that other techniques would be required in
order to weaken these assumptions. One possible approach would be to use
techniques from quantitative stochastic homogenization that lead to much
stronger quantitative homogenization results for heat kernels and Green
functions, see [11, Chapters 8–9] for details. This technique has been adapted
to Bernoulli bond percolation clusters in [23], and it is expected that it
also applies to other degenerate models.

We will begin by recalling the already established Gaussian upper bound in
[6], for which we will need some more notation. For
$A\subset\operatorname{\mathbb{Z}}^{d}$ non-empty and finite, and
$p\in[1,\infty)$, we introduce space-averaged $l^{p}$ norms on functions
$\phi:A\rightarrow\operatorname{\mathbb{R}}$ by

$\left\lVert\phi\right\rVert_{p,A}\mathrel{\mathop{:}}=\bigg{(}\frac{1}{\left\lvert
A\right\rvert}\sum_{x\in
A}\big{|}\phi(x)\big{|}^{p}\bigg{)}^{\\!\frac{1}{p}}\qquad\text{and}\qquad\left\lVert\phi\right\rVert_{\infty,A}\mathrel{\mathop{:}}=\max_{x\in
A}\left\lvert\phi(x)\right\rvert,$

where $\left\lvert A\right\rvert$ denotes the cardinality of the set $A$. For
$x\in\mathbb{Z}^{d}$ we denote by
$B(x,r)\mathrel{\mathop{:}}=\\{y\in\mathbb{Z}^{d}:|x-y|<r\\}$ balls in
$\mathbb{Z}^{d}$ centered at $x$ with respect to the graph distance, where
$|x|\mathrel{\mathop{:}}=\sum_{i=1}^{d}|x_{i}|$ for
$x=(x_{1},\ldots,x_{d})\in\mathbb{Z}^{d}$. Suppose now that $\omega(e)\in
L^{p}(\operatorname{\mathbb{P}})$ and $\omega(e)^{-1}\in
L^{q}(\operatorname{\mathbb{P}})$ for any $p,q\geq 1$. Then, under Assumption
1.1, the spatial ergodic theorem gives that, $\operatorname{\mathbb{P}}$-a.s.,
for any $x\in\mathbb{Z}^{d}$,

$\bar{\mu}_{p}\mathrel{\mathop{:}}=\operatorname{\mathbb{E}}[\mu^{\omega}(0)^{p}]=\lim_{n\rightarrow\infty}\left\lVert\mu^{\omega}\right\rVert_{p,B(x,n)}^{p},\qquad\bar{\nu}_{q}\mathrel{\mathop{:}}=\operatorname{\mathbb{E}}[\nu^{\omega}(0)^{q}]=\lim_{n\rightarrow\infty}\left\lVert\nu^{\omega}\right\rVert_{q,B(x,n)}^{q}.$

In particular, for $\operatorname{\mathbb{P}}$-a.e. $\omega$ and each
$x\in\operatorname{\mathbb{Z}}^{d},$ there exists
$N_{1}(x)=N_{1}(\omega,x,p,q)\in\operatorname{\mathbb{N}}$ such that

$\displaystyle\sup_{n\geq
N_{1}(x)}\left\lVert\mu^{\omega}\right\rVert_{p,B(x,n)}^{p}\leq
2\bar{\mu}_{p},\qquad\sup_{n\geq
N_{1}(x)}\left\lVert\nu^{\omega}\right\rVert_{q,B(x,n)}^{q}\leq
2\bar{\nu}_{q}.$ (1.1)

We will choose $N_{1}(x)$ to be the minimal such random variable. The Gaussian
upper heat kernel bound is as follows:

######  Theorem 1.2.

Suppose that Assumption 1.1 holds and suppose there exist $p,q\in(1,\infty]$
with $1/p+1/q<2/d$ such that $\omega(e)\in L^{p}(\operatorname{\mathbb{P}})$
and $\omega(e)^{-1}\in L^{q}(\operatorname{\mathbb{P}})$ for any $e\in E_{d}$.
Then, there exist constants $c_{i}=c_{i}(d,p,q,\bar{\mu}_{p},\bar{\nu}_{q})$
such that, for $\operatorname{\mathbb{P}}$-a.e. $\omega$, for any given $t$
and $x$ with $\sqrt{t}\geq N_{1}(x)$ and all $y\in\mathbb{Z}^{d}$ the
following hold.

  1. (i)

If $|x-y|\leq c_{1}t$ then

$\displaystyle
p^{\omega}(t,x,y)\;\leq\;c_{2}\,t^{-d/2}\,\exp\\!\big{(}\\!-c_{3}\,|x-y|^{2}/t\big{)}.$

  2. (ii)

If $|x-y|\geq c_{1}t$ then

$\displaystyle
p^{\omega}(t,x,y)\;\leq\;c_{2}\,t^{-d/2}\exp\\!\big{(}\\!-c_{4}\,|x-y|(1\vee\log(|x-y|/t))\big{)}.$

###### Proof.

See [6, Theorem 1.6] and a more general version with a streamlined proof in
[7, Theorem 3.2]. ∎

We now state the additional assumptions we require, followed by our main
results. We will then discuss why these additional assumptions are needed and
how they interact with the strategy of the proof.

For $\omega,\omega^{\prime}\in\Omega$ we write $\omega\leq\omega^{\prime}$ if
$\omega(e)\leq\omega^{\prime}(e)$ for all $e\in E_{d}$. We say that a function
$f:\Omega\rightarrow\mathbb{R}$ is non-decreasing if $f(\omega)\leq
f(\omega^{\prime})$ whenever $\omega\leq\omega^{\prime}$.

######  Assumption 1.3.

At least one of the following four conditions holds.

  1. (A1)

(i) _FKG inequality._ For any finite set of edges $A\subset E_{d},$ and any
non-decreasing functions $f,g:\Omega\rightarrow\operatorname{\mathbb{R}}$
depending only on $\\{\omega(e):e\in A\\}$, we have

$\operatorname{\mathbb{C}ov}(f,g)\geq 0,$ (1.2)

whenever the covariance exists.

(ii) _Polynomial mixing._ There exist constants $\gamma>d^{2}$ and
$C_{\mathrm{mix}}\in(0,\infty)$ such that for any non-decreasing function
$f\in L^{2}(\operatorname{\mathbb{P}})$ depending only on
$\\{\omega(0,y),|y|=1\\}$, and any
$x\in\operatorname{\mathbb{Z}}^{d}\setminus\\{0\\}$,

$\displaystyle\operatorname{\mathbb{C}ov}\big{(}f,f\circ\tau_{x}\big{)}\leq
C_{\mathrm{mix}}\left\lVert
f\right\rVert_{L^{2}(\operatorname{\mathbb{P}})}^{2}\left\lvert
x\right\rvert^{-\gamma}.$

  2. (A2)

_Spectral gap._ There exists $C_{\mathrm{sg}}\in(0,\infty)$ such that

$\displaystyle\operatorname{\mathbb{E}}\big{[}(f-\operatorname{\mathbb{E}}[f])^{2}\big{]}\leq
C_{\mathrm{sg}}\sum_{e\in
E_{d}}\operatorname{\mathbb{E}}\Big{[}\big{(}\partial_{e}f\big{)}^{2}\Big{]},$
(1.3)

for any $f\in L^{2}(\operatorname{\mathbb{P}})$. Here, the ‘vertical
derivative’ $\partial_{e}f$ is defined as

$\partial_{e}f(\omega):=\limsup\limits_{h\to
0}\frac{f(\omega+h\delta_{e})-f(\omega)}{h},$

where $\delta_{e}:E_{d}\to\\{0,1\\}$ stands for the Dirac function satisfying
$\delta_{e}(e)=1$ and $\delta_{e}(e^{\prime})=0$ if $e^{\prime}\not=e$.

  3. (A3)

_Finite range dependence._ There exists a positive constant
$\mathfrak{R}\in\mathbb{N}$, such that for any
$x\in\operatorname{\mathbb{Z}}^{d}$, the collection of random variables
$\big{(}\omega(\\{x,x+e\\}):\left\lvert e\right\rvert=1\big{)}$ is independent
of $\big{(}\omega(\\{z,z+e\\}):\left\lvert
z-x\right\rvert\geq\mathfrak{R},\left\lvert e\right\rvert=1\big{)}$.

  4. (A4)

_Negative association._ For any finite set of edges $A\subset E_{d},$ and any
non-decreasing functions $f,g:\Omega\rightarrow\operatorname{\mathbb{R}}$
depending only on $\\{\omega(e):e\in A\\}$, we have

$\operatorname{\mathbb{C}ov}(f,g)\leq 0,$

whenever the covariance exists.

The FKG inequality was first (formally) investigated by Fortuin, Kastelyn and
Ginibre in [30] in connection with correlation properties of Ising spin
systems. The inequality is in fact a natural property of a very wide range of
statistical mechanics models, including the random cluster model (with $q\geq
1$) [34], Yukawa quantum field theory models [41], Gaussian free fields [20,
Proposition 5.22] and interlacement percolation [45]. We note that by [27,
Theorem 3.3], it is sufficient to check that (1.2) holds for bounded
continuous non-decreasing functions. On the other hand, the opposite
assumption of negative association (A4) also holds for some prominent models,
including the uniform spanning tree, the random cluster model (with $q\leq
1$), and simple exclusion models, we refer to [42] also for more motivation
and background for this condition. We note that in the case of Gaussian
fields, pairwise positive and negative correlation are enough to imply the FKG
inequality [36] and negative associativity [37], respectively.

The spectral gap condition in (A2) and the finite range dependence in (A3)
also appear as decorrelation assumptions in the context of quantitative
stochastic homogenization, see for instance [33] for (A2), and [11] for (A3).
In a sense, the spectral gap condition in (A2), introduced in [33], can be
interpreted as a quantified version of ergodicity, as it implies an optimal
variance decay for the semigroup associated with the “process of the
environment as seen from the particle” induced by the simple random walk on
$\mathbb{Z}^{d}$, cf. [32, Proposition 1 and Remark 5].

Throughout the paper we write $c$ to denote a positive constant which may
change on each appearance, while constants denoted $c_{i}$ will be the same
through the paper. The constants will depend only on $d,p,q$, the moments of
$\mu^{\omega}(0)$ and $\nu^{\omega}(0)$, and the parameters
$\gamma,C_{\mathrm{mix}},C_{\mathrm{sg}},\mathfrak{R}$ in Assumption 1.3 as
appropriate, unless the dependencies are specified in the particular context.

######  Theorem 1.4.

Let $d\geq 2$ and suppose that Assumptions 1.1 and 1.3 hold. Then there exist
constants $c_{5},c_{6},c_{7}\in(0,\infty)$ and $p_{0},q_{0}\in[1,\infty)$ such
that if $\omega(e)\in L^{p_{0}}(\operatorname{\mathbb{P}})$ and
$\omega(e)^{-1}\in L^{q_{0}}(\operatorname{\mathbb{P}})$ the following holds.
For $\operatorname{\mathbb{P}}$-a.e. $\omega$ and any
$x\in\operatorname{\mathbb{Z}}^{d}$, there exists a random constant
$N(x)=N(\omega,x)$ satisfying

$\operatorname{\mathbb{P}}(N(x)>r)\leq c_{5}\,r^{-\alpha},\qquad\forall r>0,$
(1.4)

for some $\alpha>d(d-1)-2$, such that for all
$y\in\operatorname{\mathbb{Z}}^{d}$ and $t\geq N(x)(1\vee|x-y|)$,

$p^{\omega}(t,x,y)\geq c_{6}\,t^{-d/2}\exp\\!\big{(}-c_{7}|x-y|^{2}/t\big{)}.$
(1.5)

######  Remark 1.5.

(i) Minimal choices for $p_{0}$ and $q_{0}$ are $p_{0}>p\kappa\chi$ and
$q_{0}>q\kappa\chi$ with $\chi:=d^{2}(1+\frac{d^{2}-2}{\gamma-d^{2}})$ under
(A1), and $p_{0}=2p\kappa d$ and $q_{0}=2q\kappa d$ under (A2), (A3) or (A4),
for any $p,q>1$ satisfying $1/p+1/q<2/d$, and with $\kappa=\kappa(p,q,d)$ as
in Proposition 3.1 below. More precisely, the quantity $\kappa$ originally
appears in the random constant of the parabolic Harnack inequality in [5],
which serves as one main ingredient in the proof of Theorem 1.4.

(ii) Given the two-sided heat kernel bounds provided by Theorems 1.2 and 1.4,
the law of iterated logarithm (LIL) for the sample paths of the random walk
can be established, see [26, 39]. However, it is expected that the LIL can be
derived more easily under much weaker assumptions by exploiting the
decomposition of the random walk into a martingale part and a corrector
function, used in many proofs of a quenched functional central limit theorem,
together with the sublinearity of the corrector (see e.g. [2, 4, 16, 22]) and
an LIL for the martingale part.

In $d\geq 3$, we can use Theorems 1.4 and 1.2 to derive the following bound on
the Green kernel, $g^{\omega}(x,y),$ defined by

$g^{\omega}(x,y)\mathrel{\mathop{:}}=\int_{0}^{\infty}p_{t}^{\omega}(x,y)\
\textrm{d}t,\qquad x,y\in\mathbb{Z}^{d}.$

We refer to [2, Theorem 1.2] for precise estimates and asymptotics in the case
of general non-negative i.i.d. conductances, to [5, Theorem 1.14] for a local
limit theorem for $g^{\omega}$ in the case of ergodic conductances satisfying
a moment condition, and to [8] for recent results on the Green kernel in
dimension $d=2$.

######  Theorem 1.6.

Let $d\geq 3$ and suppose that Assumption 1.1 holds.

  1. (i)

Suppose there exist $p,q\in(1,\infty]$ with $1/p+1/q<2/d$ such that
$\omega(e)\in L^{p}(\operatorname{\mathbb{P}})$ and $\omega(e)^{-1}\in
L^{q}(\operatorname{\mathbb{P}})$ for any $e\in E_{d}$. For
$\operatorname{\mathbb{P}}$-a.e. $\omega$, there exist $c_{8}\in(0,\infty)$
and a random constant $N_{2}(x)=N_{2}(\omega,x)$ such that for all
$x,y\in\operatorname{\mathbb{Z}}^{d}$ with $\left\lvert x-y\right\rvert\geq
N_{2}(x)$,

$g^{\omega}(x,y)\leq c_{8}\left\lvert x-y\right\rvert^{2-d}.$ (1.6)

Additionally, suppose that Assumption 1.3 is satisfied. Then there exist
$c_{9},c_{10},c_{11}\in(0,\infty)$ and $p_{0},q_{0}\in[1,\infty)$ such that if
$\omega(e)\in L^{p_{0}}(\operatorname{\mathbb{P}})$ and $\omega(e)^{-1}\in
L^{q_{0}}(\operatorname{\mathbb{P}})$, then the following hold.

  1. (ii)

For all $x,y\in\operatorname{\mathbb{Z}}^{d}$ with $\left\lvert
x-y\right\rvert>N(x)$,

$g^{\omega}(x,y)\geq c_{9}\,|x-y|^{2-d}.$ (1.7)

  2. (iii)

For any $x,y\in\operatorname{\mathbb{Z}}^{d}$ with $x\not=y$,

$\displaystyle c_{10}\,\left\lvert
x-y\right\rvert^{2-d}\leq\operatorname{\mathbb{E}}\big{[}g^{\omega}(x,y)\big{]}\leq
c_{11}\,\left\lvert x-y\right\rvert^{2-d}.$ (1.8)

######  Example 1.7 (RCMs defined by Ginzburg-Landau $\nabla\phi$ interface
models).

One class of conductances satisfying the assumptions of Theorem 1.4 can be
constructed from the Ginzburg-Landau $\nabla\phi$-interface model (see [31]),
a well established model for an interface separating two pure thermodynamical
phases. The interface is described by a random field of height variables
$\phi=\\{\phi(x);x\in\mathbb{Z}^{d}\\}$ sampled from a Gibbs measure formally
given by $Z^{-1}\exp(-H(\varphi))\,\prod_{x\in\mathbb{Z}^{d}}d\varphi(x)$ with
formal Hamiltonian $H(\varphi)=\sum_{e\in E_{d}}V(\nabla\varphi(e))$ and
potential $V\in C^{2}(\mathbb{R};\mathbb{R}_{+})$, which we suppose to be even
and strictly convex. Note that in the special case $V(x)=\frac{1}{2}x^{2}$,
the field $\phi$ becomes a discrete Gaussian free field. In $d\geq 3$ this can
be made rigorous by taking the thermodynamical limit, while in dimension
$d\geq 1$ one considers the gradient process instead. Then, thanks to the
strict convexity we have the Brascamp-Lieb inequality, which allows one to
show that any environment with random conductances of the form
$\\{\omega(x,y)=\lambda(\nabla\phi(e)),e\in E_{d}\\}$ for any positive, even,
globally Lipschitz function $\lambda\in C^{1}(\mathbb{R})$ satisfies the
spectral gap condition in Assumption 1.3-(A2), see [9, Section 7] for details.
The Brascamp-Lieb inequality also implies that exponential moments for
gradient fields under the Gibbs measure exist (cf. [31, 40]). Thus, the
environment $\\{\omega(e),e\in E_{d}\\}$ as chosen above also satisfies the
required moment condition in Theorem 1.4. The assumption of a strictly convex
potential can be relaxed, see [10].

The $\nabla\phi$ interface model also satisfies the FKG inequality, see again
e.g. [31, 40], and for models with _massive_ Hamiltonians formally given by

$\displaystyle H(\varphi)=\sum_{e\in
E_{d}}V(\nabla\varphi(e))+\frac{m^{2}}{2}\sum_{x\in\mathbb{Z}^{d}}\phi(x)^{2},\qquad
m>0,$

we have exponential correlation decay, see [40, Theorem B]. In particular,
Assumption 1.3-(A1) holds, and Theorem 1.4 applies, for instance, to
conductances of the form $\omega(x,y)=\exp(\phi(x)+\phi(y))$, $\\{x,y\\}\in
E_{d}$.

###  1.3. The Method

It is well known that Gaussian lower and upper bounds on the heat kernel are
equivalent in many situations to a parabolic Harnack inequality (PHI), e.g. in
the case of uniformly elliptic conductances, see [24]. Indeed, the PHI implies
near-diagonal bounds which are then converted into off-diagonal bounds via the
established chaining method (see e.g. [28, 24, 13]).

In our context, a PHI has been obtained in [5]. Unfortunately, due to the
special structure of the constant in the PHI in the case of unbounded
conductances (see (3.2) below), in particular its dependence on
$\|\mu^{\omega}\|_{p,B(x,n)}$ and $\|\nu^{\omega}\|_{q,B(x,n)}$, we cannot
directly deduce off-diagonal Gaussian lower bounds from it. In order to get
effective Gaussian off-diagonal bounds using the chaining argument, one needs
to apply the Harnack inequality on a number of balls with radius $n$ over a
distance of order $n^{2}$. In general, however, the ergodic theorem does not
give the required uniform control on the convergence of space-averages of
stationary random variables over such balls (see [1]). Therefore, in order to
obtain lower Gaussian bounds we will need to make use of one of the additional
conditions on the correlations stated in Assumption 1.3. Specifically, in
Proposition 2.2 we employ any one of these conditions to derive a certain
concentration estimate. Then, in Proposition 3.2 (and Corollary 3.4), we
manipulate these to give us the desired uniform control on the space-averages
of the conductances over the aforementioned chain of balls of radius $n.$
Finally, we utilize this uniform control within the chaining argument to yield
the desired Gaussian off-diagonal lower bound.

Near-diagonal heat kernel bounds can also be deduced from a local limit
theorem, cf. [5, Lemma 5.3]. Recently, such local limit theorems have been
derived for a more general class of RCMs in [3, 10] via De Giorgi’s iteration
technique, circumventing the need for a PHI. However, the bounds obtained from
arguments in [3, 10] involve random constants which are implicit functions of
the averages $\|\mu^{\omega}\|_{p,B(x,n)}$ and $\|\nu^{\omega}\|_{q,B(x,n)}$,
while the chaining argument requires the more explicit dependence on the
averages in the PHI in [5]. Note that in [5] the PHI has only been derived for
the CSRW, so we obtain the lower heat kernel bounds in Theorem 1.4 for the
CSRW only, while the upper bounds in [7] have been established for a general
class of speed measures.

The rest of the paper is organised as follows. In Section 2, we first deduce
some concentration estimates from the correlation decay conditions in
Assumption 1.3, which are then used in Section 3 to prove the lower Gaussian
bounds in Theorem 1.4. Finally, in Section 4 we show Theorem 1.6.

##  2\. Concentration estimates under decorrelation assumptions

Recall that
$\bar{\mu}_{p}\mathrel{\mathop{:}}=\operatorname{\mathbb{E}}[\mu^{\omega}(0)^{p}]$
and
$\bar{\nu}_{q}\mathrel{\mathop{:}}=\operatorname{\mathbb{E}}[\nu^{\omega}(0)^{q}]$
for any $p,q\in[1,\infty)$. In this section we will derive some moment
estimates on the deviations of $\mu^{\omega}(x)$ and $\nu^{\omega}(x)$ from
their means under Assumption 1.3. For that purpose, we define the centred
random variables

$\Delta\mu_{p}^{\omega}(x)\mathrel{\mathop{:}}=\mu^{\omega}(x)^{p}-\bar{\mu}_{p},\qquad\Delta\nu_{q}^{\omega}(x)\mathrel{\mathop{:}}=\nu^{\omega}(x)^{q}-\bar{\nu}_{q},\qquad
x\in\mathbb{Z}^{d},$

for any $p,q\in[1,\infty)$ such that $\bar{\mu}_{p}$ and $\bar{\nu}_{q}$ are
finite. Our moment bounds on $\Delta\mu_{p}^{\omega}$ and
$\Delta\nu_{q}^{\omega}$ will take the form given in the following definition.

######  Definition 2.1.

For any $p,q\in[1,\infty)$ and $1\leq\theta<\eta<\infty$ we say that
$\operatorname{\mathbb{P}}$ satisfies a $(p,q,\eta,\theta)$-moment bound, if
there exists $c\in(0,\infty)$ such that

$\displaystyle\operatorname{\mathbb{E}}\bigg{[}\Big{|}\sum_{x\in
R}\Delta\mu_{p}^{\omega}(x)\Big{|}^{\eta}\bigg{]}\leq c\left\lvert
R\right\rvert^{\theta}\qquad\text{and}\qquad\operatorname{\mathbb{E}}\bigg{[}\Big{|}\sum_{x\in
R}\Delta\nu_{q}^{\omega}(x)\Big{|}^{\eta}\bigg{]}\leq c\left\lvert
R\right\rvert^{\theta}$ (2.1)

for all hyper-rectangles $R\subset\operatorname{\mathbb{Z}}^{d}$.

In the next proposition, which is the main result in this section, we gather
and derive the relations between Assumption 1.3 and $(p,q,\eta,\theta)$-moment
bounds.

######  Proposition 2.2.

Let $\zeta,p,q\in[1,\infty)$ and let $R\subset\operatorname{\mathbb{Z}}^{d}$
be a hyper-rectangle. Suppose that Assumptions 1.1 and 1.3 hold, and that
$\zeta<\gamma/d$ if under (A1). There exist constants
$p_{0},q_{0},\eta,\theta\in[1,\infty)$ with $\eta-\theta\geq\zeta$ such that
if $\omega(e)\in L^{p_{0}}(\operatorname{\mathbb{P}})$, $\omega(e)^{-1}\in
L^{q_{0}}(\operatorname{\mathbb{P}})$ for any $e\in E_{d}$, then the
$(p,q,\eta,\theta)$-moment bound holds.

We will prove Proposition 2.2 under each of the assumptions separately,
referencing the necessary materials before incorporating them into the proof.
The following lemma is easily implied by [12, Corollary 1].

######  Lemma 2.3.

Let $\\{Y(x):x\in\operatorname{\mathbb{Z}}^{d}\\}$ be a random field
satisfying the FKG inequality, and which is stationary with respect to
translation, and suppose that

$\sum_{|x|\geq
n}\operatorname{\mathbb{C}ov}\big{(}Y(0),Y(x)\big{)}=O(n^{-\nu})\qquad\text{and}\qquad\mathbb{E}\Big{[}\left\lvert
Y(0)\right\rvert^{\eta+\delta}\Big{]}<\infty,$

for some $\delta,\nu>0$ and $\eta>2$. Then, for any hyper-rectangle
$R\subset\operatorname{\mathbb{Z}}^{d},$

$\operatorname{\mathbb{E}}\Bigg{[}\Big{|}\sum_{x\in
R}Y(x)\Big{|}^{\eta}\bigg{]}=O(n^{\theta}),$

for
$\theta>\max\\{\eta/2,\chi(1-d^{-1}\min\\{1,\nu\delta/\chi\\})/(\eta+\delta-2)\\}$,
where $\chi=\delta+(\eta+\delta)(\eta-2).$

###### Proof.

We apply stationarity and the positivity of covariances due to the FKG
inequality to [12, Corollary 1] to give the result. Indeed, by stationarity
any hyper-rectangle can be shifted into $\operatorname{\mathbb{N}}^{d}$, and
positivity of the covariances allows us to bound the summation of covariances
over $\operatorname{\mathbb{N}}^{d}$ by the summation of covariances over
$\operatorname{\mathbb{Z}}^{d}$. ∎

###### Proof of Proposition 2.2 under (A1):.

We will deal with the moment bound on the summation of the
$\Delta\mu_{p}^{\omega}(x).$ The argument for the $\Delta\nu_{q}^{\omega}(x)$
follows identically. We will apply Lemma 2.3 to the field
$Y(x)=\mu^{\omega}(x)^{p}$, $x\in\operatorname{\mathbb{Z}}^{d}$. Then,

$\sum_{|x|\geq n}\operatorname{\mathbb{C}ov}\big{(}Y(0),Y(x)\big{)}\leq
c\sum_{|x|\geq n}\left\lvert x\right\rvert^{-\gamma}\leq cn^{-(\gamma-d)},$

where in the first inequality we used the polynomial mixing condition in (A1).
We can therefore take $\nu=\gamma-d$ in Lemma 2.3.

We now let $\eta=d\zeta$ and $p_{0}=p\alpha$ with
$\alpha>\frac{d\zeta(d\zeta-2)}{\gamma-d\zeta}+d\zeta$. Then in Lemma 2.3 we
take $\delta=\alpha-d\zeta$ and note that $\nu>d\zeta-d$, to give that (2.1)
holds with any
$\theta>\chi(1-d^{-1}\min\\{\nu\delta/\chi,1\\})/(\eta+\delta-2)$. A
computation then yields $\eta-\theta>\zeta$ for $\theta$ chosen close enough
to the lower bound above. ∎

We now turn to Proposition 2.2 under Assumption (A2). First, we recall that
under the spectral gap condition, we have the following $p$-version of the
spectral gap estimate. For $p\geq 1$ and any $f\in
L^{2p}(\Omega,\operatorname{\mathbb{P}})$ with
$\operatorname{\mathbb{E}}[f]=0$,

$\displaystyle\operatorname{\mathbb{E}}\big{[}|f|^{2p}\big{]}\leq
c(p,C_{\mathrm{sg}})$
$\displaystyle\operatorname{\mathbb{E}}\bigg{[}\Big{(}\sum_{e\in
E_{d}}\big{(}\partial_{e}f\big{)}^{2}\Big{)}^{\\!p}\bigg{]},$ (2.2)

which basically follows by applying (1.3) to the function $|u|^{p}$, see [33,
Lemma 2].

###### Proof of Proposition 2.2 under (A2):.

We will follow a similar argument given in [9, Lemma 2.10]. Again, we will
only show the moment estimate for $\Delta\mu^{\omega}_{p}$. Take $p_{0}=2\zeta
p$. Noting that $f\mathrel{\mathop{:}}=\sum_{y\in R}\Delta\mu_{p}^{\omega}(y)$
has mean zero, we use the spectral gap estimate in the form (2.2) which yields

$\operatorname{\mathbb{E}}\bigg{[}\Big{|}\sum_{y\in
R}\Delta\mu_{p}^{\omega}(y)\Big{|}^{2\zeta}\bigg{]}\leq
c\,\operatorname{\mathbb{E}}\\!\bigg{[}\Big{(}\sum_{e\in
E_{d}}\big{|}\partial_{e}u\big{|}^{2}\Big{)}^{\\!\zeta}\bigg{]}.$

Now we observe that, for any $e=\\{\bar{e},\underaccent{\bar}{e}\\}\in E_{d}$,

$\partial_{e}\big{[}\Delta\mu_{p}^{\omega}(y)\big{]}=\partial_{e}\big{[}\mu^{\omega}(y)^{p}\big{]}=p\,\mu^{\omega}(y)^{p-1}\,{\mathchoice{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.5mu\mathrm{l}}{1\mskip-5.0mu\mathrm{l}}}_{\\{\bar{e},\underaccent{\bar}{e}\\}}(y),$

so that

$\displaystyle\partial_{e}u\leq\begin{cases}p\big{(}\mu^{\omega}(\underaccent{\bar}{e})^{p-1}+\mu^{\omega}(\bar{e})^{p-1}\big{)}&\text{if
$\underaccent{\bar}{e}\in R$ or $\bar{e}\in R$,}\\\ 0&\text{else}.\end{cases}$

Hence,

$\displaystyle\operatorname{\mathbb{E}}\bigg{[}\Big{|}\sum_{y\in
R}\Delta\mu_{p}^{\omega}(y)\Big{|}^{2\zeta}\bigg{]}\leq c\left\lvert
R\right\rvert^{\zeta}\operatorname{\mathbb{E}}\big{[}\mu^{\omega}(0)^{2\zeta(p-1)}\big{]},$

and so we have obtained the requisite moment bounds with
$\eta/2=\theta=\zeta.$ ∎

######  Lemma 2.4.

Let $p\in(2,\infty)$. There exists a constant $c_{12}=c_{12}(p)$ such that if
$Y_{1},\ldots Y_{n}\in L^{p}(\operatorname{\mathbb{P}})$ are independent
random variables satisfying $\mathbb{E}\left[Y_{j}\right]=0$ for all
$j\in\\{1,\ldots,n\\}$, then

$\operatorname{\mathbb{E}}\bigg{[}\Big{|}\sum_{i=1}^{n}Y_{j}\
\Big{|}^{p}\bigg{]}^{1/p}\leq
c_{12}\,\max\Bigg{\\{}\bigg{(}\sum_{j=1}^{n}\mathbb{E}\left[\big{|}Y_{j}\big{|}^{p}\right]\bigg{)}^{\\!1/p},\bigg{(}\sum_{j=1}^{n}\mathbb{E}\left[\big{|}Y_{j}\big{|}^{2}\right]\bigg{)}^{\\!1/2}\Bigg{\\}}.$

###### Proof.

This can be extracted from [43, Theorem 3]. ∎

###### Proof of Proposition 2.2 under (A3):.

Take $p_{0}=2p\zeta$, again considering only the moment bound on the sum of
the $\Delta\mu_{p}^{\omega}(x)$ as the argument for
$\Delta\nu_{q}^{\omega}(x)$ is the same. Let $(e_{i})_{1\leq i\leq d}$ denote
the standard unit vectors. We call two vertices $x,y\in R$ equivalent if
$x-y=\pm\mathfrak{R}e$ for some $e\in\\{e_{1},\ldots,e_{d}\\}.$ Write the
equivalence classes as $E_{1},\dots,E_{m},$ and observe that we must have
$m\leq\mathfrak{R}^{d}$. Note that the size of each equivalence class is
trivially bounded above by $\left\lvert R\right\rvert.$ We apply the finite
range assumption to give that for each fixed $i$, the $(\mu^{\omega}(x))_{x\in
E_{i}}$ are mutually independent, and therefore

$\displaystyle\operatorname{\mathbb{E}}\bigg{[}\Big{|}\sum_{y\in
R(x)}\Delta\mu_{p}^{\omega}(y)\Big{|}^{2\zeta}\bigg{]}=\operatorname{\mathbb{E}}\bigg{[}\Big{|}\sum_{i\leq
m}\sum_{y\in E_{i}}\Delta\mu_{p}^{\omega}(y)\Big{|}^{2\zeta}\bigg{]}$
$\displaystyle\mspace{36.0mu}\leq c\sum_{i\leq
m}\operatorname{\mathbb{E}}\bigg{[}\Big{|}\sum_{y\in
E_{i}}\Delta\mu_{p}^{\omega}(y)\Big{|}^{2\zeta}\bigg{]}\leq c\,\left\lvert
R\right\rvert^{\zeta},$

where in the final step we apply Lemma 2.4 for each $i$ in the summation, with
$(Y_{j})$ an enumeration of $(\Delta\mu_{p}^{\omega}(y))_{y\in E_{i}}.$ Thus
(2.1) holds with $\eta/2=\theta=\zeta.$ ∎

######  Lemma 2.5.

Let $\left\\{Y_{i},1\leqslant i\leqslant n\right\\}$ be a negatively
associated sequence. Further, let $\left\\{Y_{i}^{*},1\leqslant i\leqslant
n\right\\}$ be a sequence of independent random variables such that
$Y_{i}^{*}$ and $Y_{i}$ have the same distribution for each $i=1,2,\ldots,n.$
Then

$\operatorname{\mathbb{E}}\bigg{[}\phi\bigg{(}\sum_{i=1}^{n}Y_{i}\bigg{)}\bigg{]}\leq\operatorname{\mathbb{E}}\bigg{[}\phi\bigg{(}\sum_{i=1}^{n}Y_{i}^{*}\bigg{)}\bigg{]}$

for any convex function $\phi$ on $\operatorname{\mathbb{R}}$, whenever the
expectation on the right hand side exists.

###### Proof.

This follows from [44, Theorem 1]. ∎

###### Proof of Proposition 2.2 under (A4):.

Let $p_{0}=2p\zeta$ and $q_{0}=2q\zeta$. Then, we apply Lemma 2.5 and Lemma
2.4, with $(Y_{i})$ an enumeration of $(\Delta\mu_{p}^{\omega}(y)^{*})_{y\in
R}$ (and $(\Delta\nu_{q}^{\omega}(y)^{*})_{y\in R}$, respectively) to give
(2.1) with $\eta/2=\theta=\zeta$. ∎

As a first consequence of the concentration estimate in Proposition 2.2 we
record the following tail estimate on the random variables $N_{1}(x)$,
$x\in\mathbb{Z}^{d}$, defined via (1.1).

######  Lemma 2.6.

Suppose that Assumption 1.1 holds and that $\operatorname{\mathbb{P}}$
satisfies a $(p,q,\eta,\theta)$-moment bound, with $\zeta:=\eta-\theta>0$.
Then there exists $c_{13}\in(0,\infty)$ such that

$\operatorname{\mathbb{P}}\big{(}N_{1}(x)>n\big{)}\leq
c_{13}\,n^{1-d\zeta},\qquad\forall n\in\mathbb{N}.$ (2.3)

###### Proof.

Note that, for any $n\in\mathbb{N}$, we have by a union bound

$\displaystyle\operatorname{\mathbb{P}}(N_{1}(x)>n)\leq\sum_{m\geq
n}\Big{(}\operatorname{\mathbb{P}}\big{[}\left\lVert\mu^{\omega}\right\rVert_{p,B(x,m)}^{p}>2\bar{\mu}_{p}\big{]}+\operatorname{\mathbb{P}}\big{[}\left\lVert\nu^{\omega}\right\rVert_{q,B(x,m)}^{q}>2\bar{\nu}_{q}\big{]}\Big{)}.$
(2.4)

For the first term we get by Proposition 2.2 and Markov’s inequality,

$\displaystyle\sum_{m\geq
n}\operatorname{\mathbb{P}}\big{[}\left\lVert\mu^{\omega}\right\rVert_{p,B(x,m)}^{p}>2\bar{\mu}_{p}\big{]}=\sum_{m\geq
n}\operatorname{\mathbb{P}}\bigg{[}\Big{|}\sum_{y\in
B(x,m)}\Delta\mu_{p}^{\omega}(y)\Big{|}^{\eta}>\bar{\mu}_{p}^{\eta}\,\big{|}B(x,m)\big{|}^{\eta}\bigg{]}$
$\displaystyle\mspace{36.0mu}\leq c\sum_{m\geq n}m^{-d\zeta}\leq
c\,n^{1-d\zeta}.$

Repeating the same argument with the second term in (2.4) gives the claim. ∎

##  3\. Heat kernel lower bounds

We first recall the near-diagonal heat kernel bound in [5, Proposition 4.7],
which will be a key ingredient in the proof of the main theorem.

######  Proposition 3.1.

Suppose that Assumption 1.1 holds, and suppose there exist $p,q\in(1,\infty]$
with $1/p+1/q<2/d$ such that $\omega(e)\in L^{p}(\operatorname{\mathbb{P}})$
and $\omega(e)^{-1}\in L^{q}(\operatorname{\mathbb{P}})$ for any $e\in E_{d}$.
Then there exists $c_{14}=c_{14}(d)$ such that for any $t\geq 1$,
$x_{1}\in\mathbb{Z}^{d}$ and $x_{2}\in B(x_{1},\frac{1}{2}\sqrt{t}),$

$p^{\omega}\left(t,x_{1},x_{2}\right)\geq\frac{c_{14}}{C_{\mathrm{PH}}}t^{-\frac{d}{2}},$
(3.1)

where
$C_{\mathrm{PH}}=C_{\mathrm{PH}}\big{(}\|\mu^{\omega}\|_{p,B(x_{1},\sqrt{t})},\|\nu^{\omega}\|_{q,B(x_{1},\sqrt{t})}\big{)}$
is the constant appearing in the parabolic Harnack inequality in [5, Theorem
1.4], more explicitly given by

$\displaystyle
C_{\mathrm{PH}}\big{(}\|\mu^{\omega}\|_{p,B},\|\nu^{\omega}\|_{q,B}\big{)}=c\exp\Big{(}c\,\big{(}1\vee\|\mu^{\omega}\|_{p,B}\big{)}^{\kappa}\big{(}1\vee\|\nu^{\omega}\|_{q,B}\big{)}^{\kappa}\Big{)}$
(3.2)

for some positive $c=c(d,p,q)$ and $\kappa=\kappa(d,p,q)\geq 1$.

Theorem 1.4 will be proven by the well-established chaining technique. More
precisely, we will apply Proposition 3.1 on a certain sequence of balls. Given
a vertex $x=(x_{1},\ldots,x_{d})\in\mathbb{Z}^{d}$ and $0<r\leq 4|x|$, we
specify a nearest-neighbour path $P[x]$ of length $D:=|x|$ from $0$ to $x$.
Setting $p_{0}(x)\mathrel{\mathop{:}}=0$ and
$p_{i}(x)\mathrel{\mathop{:}}=(x_{1},\ldots,x_{i},0,\ldots,0)\in\mathbb{Z}^{d}$,
$1\leq i\leq d$, we define $P[x]$ to be the path that consists of $d$
consecutive straight line segments connecting
$p_{0}(x),p_{1}(x),\ldots,p_{d}(x)$. Next, for any $k\in\mathbb{N}$ with
$\frac{12D}{r}\leq k\leq\frac{16D}{r}$, we choose a subset $\\{z_{0},\ldots
z_{k}\\}\subset P[x]$ such that $z_{0}=0$, $z_{k}=x$,
$d(z_{j},z_{j-1})\leq\frac{r}{12}$ for $1\leq j\leq k$ and such that, for each
$j\leq k$, $\big{|}B(z_{j},r)\cap\\{z_{0},\ldots z_{k}\\}\big{|}\leq c$ for
some $c=c(d)$. Set $B_{j}\mathrel{\mathop{:}}=B(z_{j},r/48)$. Finally, we let
$s\mathrel{\mathop{:}}=Dr/k,$ then $\frac{1}{16}r^{2}\leq
s\leq\frac{1}{12}r^{2}.$

######  Proposition 3.2.

Suppose that Assumption 1.1 holds. Further, for fixed $p,q\in[1,\infty)$
assume that $\operatorname{\mathbb{P}}$ satisfies a $(p,q,\eta,\theta)$-moment
bound with $\zeta:=\eta-\theta>d$. Then there exist constants
$c_{15},c_{16}\in(0,\infty)$ and a random variable $N_{3}=N_{3}(\omega)$
satisfying

$\displaystyle\operatorname{\mathbb{P}}(N_{3}>\rho)\leq
c_{15}\rho^{-d(\zeta-1)+2},\qquad\forall\rho>0,$ (3.3)

such that, $\operatorname{\mathbb{P}}$-a.s., for all $r\geq N_{3}$,
$x\in\mathbb{Z}^{d}$ and $(B_{j})_{1\leq j\leq k}$ defined as right above, the
following holds. If $r\leq 4|x|$, for any collection of vertices
$y_{0},\ldots,y_{k}$ with $y_{0}=0$, $y_{j}\in B_{j}$ for $1\leq j\leq k-1$
and $y_{k}=x$ we have

$\sum_{j=0}^{k-1}\left(1\vee\left\|\mu^{\omega}\right\|_{p,B(y_{j},\sqrt{s})}\right)\left(1\vee\left\|\nu^{\omega}\right\|_{q,B(y_{j},\sqrt{s})}\right)\leq
c_{16}\,k.$ (3.4)

###### Proof.

Set $B_{y_{j}}\mathrel{\mathop{:}}=B(y_{j},\sqrt{s})$ for abbreviation. Then
note that there exists $c=c(d)\in(0,\infty)$ such that

$\displaystyle\big{|}\big{\\{}j\in\\{1,\ldots k\\}:z\in
B_{y_{j}}\big{\\}}\big{|}\leq c,\qquad\forall z\in\bigcup_{j\leq k}B_{y_{j}}.$
(3.5)

We divide the rest of the proof into several steps.

_Step 1._ For $x\in\mathbb{Z}^{d}$ and $r$ as in the statement, we will define
a collection $(\mathcal{R}^{x,r}_{i})_{0\leq i\leq d}$ of $d+1$ hyper-
rectangles in $\mathbb{Z}^{d}$ which covers $\bigcup_{j\leq k}B_{y_{j}}$ for
any selection $y_{j}\in B_{j}$. For simplicity, we will only give the
definition for $x\in\mathbb{Z}^{d}\cap[0,\infty)^{d}$ – it can be easily
adjusted to the other regions of $\mathbb{Z}^{d}$. For any
$m,l\in\operatorname{\mathbb{N}}$, $u\in\mathbb{Z}^{d}$ and $i=1,\ldots d$ we
write

$\displaystyle
R_{i}(u,m,l)\mathrel{\mathop{:}}=u+\big{\\{}v\in\operatorname{\mathbb{Z}}^{d}:0\leq
v_{i}\leq l,\,\left\lvert v_{j}\right\rvert\leq m,\text{ for all }j\neq
i\big{\\}}$

for the $d$-dimensional hyper-rectangle with base point $u$ and dimension $l$
along the $e_{i}$ axis and $m$ along the remaining coordinate axes. Now define

$\displaystyle\mathcal{R}_{0}^{x,r}\mathrel{\mathop{:}}=\big{(}[0,r]\times[-r,0]^{d-1}\big{)}\cap\mathbb{Z}^{d}\quad\text{and}\quad\mathcal{R}_{i}^{x,r}\mathrel{\mathop{:}}=R_{i}\big{(}p_{i-1}(x),r,x_{i}+r\big{)},\quad
1\leq i\leq d.$

Then note that $\bigcup_{0\leq i\leq
d}\mathcal{R}_{i}^{x,r}\supseteq\bigcup_{j\leq k}B_{y_{j}}$.

_Step 2._ In this step we will show that there exists a random $N_{3}$
satisfying (3.3) such that for all $x\in\mathbb{Z}^{d}$ and all $r\geq N_{3}$,

$\displaystyle\sum_{j\leq
k}\left\lVert\mu^{\omega}\right\rVert_{p,B_{y_{j}}}^{p}\leq
c\,k,\qquad\text{and}\qquad\sum_{j\leq
k}\left\lVert\nu^{\omega}\right\rVert_{q,B_{y_{j}}}^{q}\leq c\,k.$ (3.6)

We will only discuss the first inequality as the arguments for the second are
identical. By (3.5) and the fact that the hyper-rectangles
$(\mathcal{R}^{x,r}_{i})_{0\leq i\leq d}$ cover $\bigcup_{j\leq k}B_{y_{j}}$,
we have that

$\displaystyle\sum_{j\leq
k}\left\lVert\mu^{\omega}\right\rVert_{p,B_{y_{j}}}^{p}$ $\displaystyle\leq
c\,r^{-d}\sum_{0\leq i\leq
d}{\sum_{y\in\mathcal{R}_{i}^{x,r}}}\mu^{\omega}(y)^{p}$ $\displaystyle\leq
c\,r^{-d}\bigg{(}\big{(}|x|+r\big{)}r^{d-1}\bar{\mu}_{p}+\sum_{0\leq i\leq
d}\sum_{y\in\mathcal{R}_{i}^{x,r}}\Delta\mu_{p}^{\omega}(y)\bigg{)}$
$\displaystyle\leq c\,k+c\,r^{-d}\sum_{0\leq i\leq
d}\sum_{y\in\mathcal{R}_{i}^{x,r}}\Delta\mu_{p}^{\omega}(y),$ (3.7)

where we have used that $\left\lvert x\right\rvert\leq ckr$. Now we apply the
moment-bound hypothesis with Proposition 2.2 and Markov’s inequality to give

$\operatorname{\mathbb{P}}\bigg{(}\sum_{0\leq i\leq
d}\sum_{y\in\mathcal{R}_{i}^{x,r}}\Delta\mu_{p}^{\omega}(y)>kr^{d}\bigg{)}\leq
c\,\big{(}\left\lvert x\right\rvert r^{d-1}\big{)}^{-\zeta},$

where we have used that $kr^{d}\geq c\left\lvert x\right\rvert r^{d-1}$. Now
fix $\rho,l\in\operatorname{\mathbb{N}}$ with $l\geq\rho$. By applying a union
bound, and summing over $\partial B(l):=\\{x\in\mathbb{Z}^{d}:\left\lvert
x\right\rvert=l\\}$ and $r\geq\rho$, we get

$\displaystyle\operatorname{\mathbb{P}}\bigg{(}\exists x\in\partial
B(l),r\in[\rho,4l]\cap\mathbb{N}:\sum_{0\leq i\leq
d}\sum_{y\in\mathcal{R}_{i}^{x,r}}\Delta\mu_{p}^{\omega}(y)>kr^{d}\bigg{)}\leq
c\,l^{d-1-\zeta}\rho^{-\zeta(d-1)+1}.$

Set

$A_{\rho}:=\bigg{\\{}\exists
x\in\mathbb{Z}^{d},r\in\mathbb{N}:|x|\geq\rho,\,r\in\big{[}\rho,4|x|\big{]},\sum_{0\leq
i\leq
d}\sum_{y\in\mathcal{R}_{i}^{x,r}}\Delta\mu_{p}^{\omega}(y)>kr^{d}\bigg{\\}}.$

Since $\zeta>d$, we can apply another union bound over $l\geq\rho$ to obtain

$\operatorname{\mathbb{P}}(A_{\rho})\leq
c\,\rho^{d-\zeta}\rho^{-\zeta(d-1)+1}=c\,\rho^{d(1-\zeta)+1}.$ (3.8)

However, $d(1-\zeta)+1<-1$, so by Borel-Cantelli, for
$\operatorname{\mathbb{P}}$-a.e. $\omega,$ there exists $N_{3}=N_{3}(\omega)$
such that $A_{\rho}$ does not occur for $\rho\geq N_{3}$. Substituting into
(3) completes the proof of (3.6). Moreover, via a union bound, (3.8) implies
that $N_{3}$ can be constructed such that (3.3) holds.

_Step 3._ By Hölder’s inequality,

$\displaystyle\sum_{j\leq
k}\left(1\vee\left\|\mu^{\omega}\right\|_{p,B_{y_{j}}}\right)\left(1\vee\left\|\nu^{\omega}\right\|_{q,B_{y_{j}}}\right)$
$\displaystyle\mspace{36.0mu}\leq
k^{1-\frac{1}{p}-\frac{1}{q}}\,\bigg{(}\sum_{j\leq
k}\Big{(}1\vee{\left\lVert\mu^{\omega}\right\rVert_{p,B_{y_{j}}}^{p}}\Big{)}\bigg{)}^{\\!1/p}\bigg{(}\sum_{j\leq
k}\Big{(}1\vee{\left\lVert\nu^{\omega}\right\rVert_{q,B_{y_{j}}}^{q}}\Big{)}\bigg{)}^{\\!1/q},$

so that the statement follows from Step 2. ∎

######  Remark 3.3.

With a more convoluted covering argument, replacing union bounds with bounds
on maxima, the requirement of $\zeta>d$ in Proposition 3.2 can be decreased to
$\zeta>d-1,$ and thus we only need $\gamma>d(d-1)$, and the minimal moment
conditions of Remark 1.5 can be reduced to $p_{0}>p\kappa\chi$ and
$q_{0}>q\kappa\chi$ with
$\chi=d(d-1)\big{[}1+\frac{d(d-1)-2}{\gamma-d(d-1)}\big{]}$ under (A1), and
$p_{0}=2p\kappa(d-1)$ and $q_{0}=2q\kappa(d-1)$ under (A2), (A3) or (A4). We
do not include this argument as it brings greatly increased complication for
very limited improvement.

######  Corollary 3.4.

In the setting of Proposition 3.2, assume that $\operatorname{\mathbb{P}}$
satisfies a $(\kappa p,\kappa q,\eta,\theta)$-moment bound. Then there exists
a constant $c_{17}\in(0,\infty)$ and a random variable $N_{4}=N_{4}(\omega)$
satisfying (3.3) such that, $\operatorname{\mathbb{P}}$-a.s., for all $r\geq
N_{4}$, $x\in\mathbb{Z}^{d}$ with $r\leq 4|x|$,

$\sum_{j=0}^{k-1}\left(1\vee\left\|\mu^{\omega}\right\|_{p,B(y_{j},\sqrt{s})}\right)^{\kappa}\left(1\vee\left\|\nu^{\omega}\right\|_{q,B(y_{j},\sqrt{s})}\right)^{\kappa}\leq
c_{17}k.$ (3.9)

###### Proof.

This follows exactly as Proposition 3.2 after applying Jensen’s inequality to
$\left\|\mu^{\omega}\right\|_{p,B_{y_{j}}}$ and
$\left\|\nu^{\omega}\right\|_{q,B_{y_{j}}}$, and then replacing $\mu^{\omega}$
by $(\mu^{\omega})^{\kappa}$ and $\nu^{\omega}$ by $(\nu^{\omega})^{\kappa}$.
∎

###### Proof of Theorem 1.4.

By translation invariance of the measure it suffices to show a lower bound on
$p^{\omega}(t,0,x)$. To begin with, we must establish the necessary moment
conditions to deploy the tools developed in the previous section.

We assume that there exist some $p,q\in(1,\infty)$ with $1/p+1/q<2/d$ such
that $\omega(e)\in L^{p}(\operatorname{\mathbb{P}})$ and $\omega(e)^{-1}\in
L^{q}(\operatorname{\mathbb{P}})$. This will allow us to apply Proposition 3.1
involving the constant $\kappa=\kappa(p,q,d)$. If working under assumption
(A1), recall that $\gamma>d^{2}$ and fix $d<\zeta<\gamma/d$; otherwise, just
fix $\zeta>d$. Then Proposition 2.2 provides us with $p_{0},q_{0}\geq 1$ such
that if $\omega(e)\in L^{p_{0}}(\operatorname{\mathbb{P}})$ and
$\omega(e)^{-1}\in L^{q_{0}}(\operatorname{\mathbb{P}})$, then
$(1,1,\eta,\theta)$ and $(\kappa p,\kappa q,\eta,\theta)$-moment bounds hold
with $\eta-\theta\geq\zeta$. This will allow us to apply Lemma 2.6,
Proposition 3.2, and Corollary 3.4 as required. We then set
$N\mathrel{\mathop{:}}=N_{1}(0)\vee N_{3}\vee N_{4}$ and combine the tail
bounds in Lemma 2.6 and (3.3) to obtain that $N$ satisfies the tail bound in
(1.4).

Set again $D\mathrel{\mathop{:}}=\left\lvert x\right\rvert$, and assume now
that $t\geq N(D\vee 1)$. We will split the proof into two cases, $D^{2}/t\leq
1/4$ and $D^{2}/t>1/4$.

_Case 1:_ $D^{2}/t\leq 1/4$. Then $x\in B(0,\frac{1}{2}\sqrt{t})$, so by
Proposition 3.1,

$p^{\omega}(t,0,x)\geq\frac{c_{14}}{C_{\mathrm{PH}}}t^{-\frac{d}{2}}$

with
$C_{\mathrm{PH}}=C_{\mathrm{PH}}\big{(}\|\mu^{\omega}\|_{p,B(0,\sqrt{t})},\|\nu^{\omega}\|_{q,B(0,\sqrt{t})}\big{)}$.
Since
$C_{\mathrm{PH}}\big{(}\|\mu^{\omega}\|_{p,B},\|\nu^{\omega}\|_{q,B}\big{)}$
is increasing in $\left\lVert\mu^{\omega}\right\rVert_{p,B}$ and
$\left\lVert\nu^{\omega}\right\rVert_{q,B}$ (cf. (3.2) above) and $t\geq
N_{1}(0)$,

$\displaystyle
C_{\mathrm{PH}}\big{(}\|\mu^{\omega}\|_{p,B(0,\sqrt{t})},\|\nu^{\omega}\|_{q,B(0,\sqrt{t})}\big{)}\leq
C_{\mathrm{PH}}\Big{(}\big{(}2\bar{\mu}_{p}\big{)}^{1/p},\big{(}2\bar{\nu}_{q}\big{)}^{1/q}\Big{)},$

and therefore $p^{\omega}(t,0,x)\geq ct^{-d/2}$.

_Case 2:_ $D^{2}/t>1/4$. Set $r\mathrel{\mathop{:}}=t/D\geq 1\vee N_{3}\vee
N_{4}$. We deploy the chaining setup as introduced right below Proposition
3.1. Recall that $s\mathrel{\mathop{:}}=Dr/k=t/k$ with $\frac{12D}{r}\leq
k\leq\frac{16D}{r}$ so that $1\leq\frac{1}{16}r^{2}\leq
s\leq\frac{1}{12}r^{2}$, and note that $k\geq 3$. Then, for any collection of
vertices $y_{0},\ldots y_{k}$ with $y_{0}=0$, $y_{j}\in B_{j}$ for $1\leq
j\leq k-1$ and $y_{k}=x$, we have $d(y_{i},y_{i+1})\leq r/8\leq\sqrt{s}/2$ so
that by Proposition 3.1,

$\displaystyle p^{\omega}(s,y_{i},y_{i+1})$
$\displaystyle\geq\frac{c_{14}}{C_{\mathrm{PH}}\big{(}\|\mu^{\omega}\|_{p,B_{y_{j}}},\|\nu^{\omega}\|_{q,B_{y_{j}}}\big{)}}\,s^{-\frac{d}{2}}\geq\frac{c}{C_{\mathrm{PH}}\big{(}\|\mu^{\omega}\|_{p,B_{y_{j}}},\|\nu^{\omega}\|_{q,B_{y_{j}}}\big{)}}\,r^{-d},$

with $B_{y_{j}}\mathrel{\mathop{:}}=B(y_{j},\sqrt{s})$. Further, recall the
representation of $C_{\mathrm{PH}}$ in (3.2) and that
$\mathrm{P}_{y_{j}}^{\omega}[X_{s}=y_{j+1}]=p^{\omega}(s,y_{j},y_{j+1})\mu^{\omega}(y_{j+1})$.
Hence, by the Markov property,

$\displaystyle\mathrm{P}_{0}^{\omega}\left[X_{t}=x\right]=\mathrm{P}_{0}^{\omega}\left[X_{ks}=x\right]\geq\mathrm{P}_{0}^{\omega}\left[X_{js}\in
B_{j},1\leq j\leq k-1,X_{ks}=x\right]$
$\displaystyle\mspace{36.0mu}\geq\sum_{y_{1}\in B_{1},\dots,y_{k-1}\in
B_{k-1}}\frac{c^{k}\left(\prod_{j=1}^{k-1}r^{-d}\mu^{\omega}(y_{j})\right)s^{-d/2}\mu^{\omega}(x)}{\exp\Big{(}c\sum_{j=0}^{k-1}\big{(}1\vee\|\mu^{\omega}\|_{p,B_{y_{j}}}\big{)}^{\kappa}\big{(}1\vee\|\nu^{\omega}\|_{q,B_{y_{j}}}\big{)}^{\kappa}\Big{)}}$
$\displaystyle\mspace{36.0mu}\geq
c^{k}\bigg{(}\prod_{j=1}^{k-1}\|\mu^{\omega}\|_{1,B_{j}}\bigg{)}\,s^{-d/2}\mu^{\omega}(x),$

where we used Corollary 3.4 in the last step. In particular,

$p^{\omega}(t,0,x)\geq
c^{k}\,\bigg{(}\prod_{j=1}^{k-1}\|\mu^{\omega}\|_{1,B_{j}}\bigg{)}\,t^{-d/2}.$

Now, by the harmonic-geometric mean inequality and Jensen’s inequality, we
have

$\displaystyle\bigg{(}\prod_{j=1}^{k-1}\|\mu^{\omega}\|_{1,B_{j}}\bigg{)}^{\frac{1}{k-1}}\geq\frac{k-1}{\sum_{j=1}^{k-1}\left\lVert\mu^{\omega}\right\rVert_{1,B_{j}}^{-1}}\geq\frac{c\,(k-1)}{\sum_{j=1}^{k-1}\left\lVert\nu^{\omega}\right\rVert_{1,B_{j}}}.$

We use Proposition 3.2 (setting $p=q=1$, $y_{j}=z_{j}$, and replacing
$B_{y_{j}}$ with $B_{j}$) to obtain

$\displaystyle\sum_{j=1}^{k-1}\left\lVert\nu^{\omega}\right\rVert_{1,B_{j}}\leq
c\,(k-1),$

so that

$\displaystyle\bigg{(}\prod_{j=1}^{k-1}\|\mu^{\omega}\|_{1,B_{j}}\bigg{)}\geq
c^{k-1}.$

Combining this with (3) yields $p^{\omega}(t,0,x)\geq c\,c_{18}^{k}t^{-d/2}$
for some $c_{18}\in(0,1)$, which gives the bound (1.5) by the choice of $k$. ∎

##  4\. Green kernel estimates

In this final section we utilize Theorems 1.2 and 1.4 to establish Theorem
1.6. We refer to [15, Section 6] for similar arguments.

###### Proof of Theorem 1.6.

(i) First we deduce the upper bound (1.6) on the Green kernel. For any
distinct $x,y\in\operatorname{\mathbb{Z}}^{d}$, we decompose the integral as

$g^{\omega}(x,y)=\frac{1}{\mu^{\omega}(x)}\int_{0}^{N_{1}(x)^{2}}\textrm{P}_{y}^{\omega}(X_{t}=x)\,\textrm{d}t+\int_{N_{1}(x)^{2}}^{N_{x,y}}p_{t}^{\omega}(x,y)\
\textrm{d}t+\int_{N_{x,y}}^{\infty}p_{t}^{\omega}(x,y)\ \textrm{d}t$ (4.1)

with $N_{x,y}\mathrel{\mathop{:}}=N_{1}(x)^{2}\vee(\left\lvert
x-y\right\rvert/{c_{1}})$, where we used that
$p_{t}^{\omega}(x,y)=p_{t}^{\omega}(y,x)$ by the symmetry of the heat kernel.
Using Theorem 1.2 we can bound the last two terms of (4.1) by

$\displaystyle\int_{N_{1}(x)^{2}}^{N_{x,y}}p_{t}^{\omega}(x,y)\
\textrm{d}t\leq c_{2}e^{-c_{4}\left\lvert
x-y\right\rvert}\int_{1}^{\infty}t^{-d/2}\ \textrm{d}t\leq c\left\lvert
x-y\right\rvert^{2-d},$

and

$\displaystyle\int_{N_{x,y}}^{\infty}p_{t}^{\omega}(x,y)\ \textrm{d}t\leq
c_{2}\int_{0}^{\infty}t^{-d/2}\,e^{-c_{3}|x-y|^{2}/t}\ \textrm{d}t\leq
c\left\lvert x-y\right\rvert^{2-d}.$

It is left to bound the first term in the right hand side of (4.1). Recall
that the random walk $X$ spends i.i.d. $\mathrm{Exp}(1)$-distributed waiting
times between its jumps. Set $\lambda\mathrel{\mathop{:}}=N_{1}(x)^{2}$ and
$r:=\left\lvert x-y\right\rvert\geq 1$. In particular, the random walk
starting at $y$ needs to perform at least $r$ jumps to get to $x$. Thus,

$\displaystyle\frac{1}{\mu^{\omega}(x)}\int_{0}^{\lambda}\textrm{P}_{y}^{\omega}(X_{t}=x)\,\textrm{d}t\leq\frac{\lambda}{\mu^{\omega}(x)}\textrm{P}_{y}^{\omega}\big{(}X_{t}=x\text{
for any }t\in[0,\lambda]\big{)}$
$\displaystyle\mspace{36.0mu}\leq\frac{\lambda}{\mu^{\omega}(x)}\mathrm{Pois}(\lambda)\big{(}[r,\infty)\big{)}.$
(4.2)

Here $\textrm{Pois}(\lambda)$ denotes the Poisson distribution with parameter
$\lambda$, which we recall to have exponential tails (see e.g. [35, Remark
2.6]). So there exists ${N_{2}=N_{2}(\omega,x)}$ such that for each
$y\in\operatorname{\mathbb{Z}}^{d}$ with $\left\lvert x-y\right\rvert\geq
N_{2}(\omega,x)$ the first term in (4.1) is bounded from above by
$c\left\lvert x-y\right\rvert^{2-d}$, which completes the proof of (1.6).

(ii) This follows directly from Theorem 1.4, which gives for
$x,y\in\operatorname{\mathbb{Z}}^{d}$ with $\left\lvert x-y\right\rvert>N(x)$,

$\displaystyle\int_{0}^{\infty}p_{t}^{\omega}(x,y)\ \textrm{d}t$
$\displaystyle\geq\int_{N(x)|x-y|}^{\infty}c_{6}\,t^{-d/2}e^{-c_{7}|x-y|^{2}/t}\
\textrm{d}t\geq\int_{|x-y|^{2}}^{\infty}c_{6}\,t^{-d/2}e^{c_{7}|x-y|^{2}/t}\
\textrm{d}t$
$\displaystyle=|x-y|^{2-d}\int_{1}^{\infty}c_{6}\,t^{-d/2}\,e^{-c_{7}/t}\,\textrm{d}t=c\,|x-y|^{2-d}.$

(iii) First, we carry out some preparation for the proof of the upper bound.
In particular, we show the Green kernel has finite second moments. By the
symmetry of the heat kernel and the on-diagonal part of the upper bound in
Theorem 1.2, note that

$\displaystyle g^{\omega}(x,y)=\int_{0}^{N_{1}(x)^{2}}p^{\omega}_{t}(y,x)\
\mathrm{d}t+\int_{N_{1}(x)^{2}}^{\infty}p^{\omega}_{t}(x,y)\
\mathrm{d}t\leq\frac{N_{1}(x)^{2}}{\mu^{\omega}(x)}+cN_{1}(x)^{2-d}$
$\displaystyle\mspace{36.0mu}\leq cN_{1}(x)^{2}\nu^{\omega}(x),$

where we used Jensen’s inequality in the last step. Assuming that
$\omega(e)\in L^{p_{0}}(\operatorname{\mathbb{P}})$ and $\omega(e)^{-1}\in
L^{q_{0}}(\operatorname{\mathbb{P}})$ for suitable $p_{0},q_{0}\in(1,\infty)$,
we apply Proposition 2.2 together with Lemma 2.6 (with $\zeta=d$) to obtain
that $N_{1}\in L^{\beta}(\operatorname{\mathbb{P}})$ for any $\beta<d^{2}-1$.
Thus, by Hölder’s inequality,

$\displaystyle\operatorname{\mathbb{E}}\big{[}g^{\omega}(x,y)^{\beta}\big{]}<\infty,$
(4.3)

for any $\beta<(d^{2}-1)(q_{0}-1)/(2q_{0})$. Then, as $d\geq 3$, assuming
$q_{0}>2$ ensures that the second moment of the Green kernel exists.

We can now prove the upper bound of (1.8). To do so, we show that the random
variable $N_{2}$ introduced in (i) satisfies the tail bound

$\displaystyle\operatorname{\mathbb{P}}\big{[}N_{2}>u]\leq
c\,u^{2-d},\qquad\forall u\geq 1.$ (4.4)

Indeed, if (4.4) holds true, then we obtain

$\displaystyle\operatorname{\mathbb{E}}\big{[}g^{\omega}(x,y)\big{]}\leq
c\left\lvert
x-y\right\rvert^{2-d}+\operatorname{\mathbb{E}}\big{[}g^{\omega}(x,y)^{2}\big{]}^{1/2}\,\operatorname{\mathbb{P}}\big{[}N_{2}>|x-y|\big{]}\leq
c\,\left\lvert x-y\right\rvert^{2-d},$

where we used (1.6) and the Cauchy-Schwarz inequality in the first step, and
(4.4) and (4.3) in the second step.

In order to show (4.4), recall that $N_{2}$ has been chosen as a value of $r$
such that
$\frac{\lambda}{\mu^{\omega}(x)}\mathrm{Pois}(\lambda)\big{(}[r,\infty)\big{)}\leq
cr^{2-d}$ with $\lambda\mathrel{\mathop{:}}=N_{1}(x)^{2}$. By Chernoff’s
inequality (cf. e.g. [35, Corollary 2.4 and Remark 2.6]),
$\mathrm{Pois}(\lambda)\big{(}[r,\infty)\big{)}\leq e^{-r+7\lambda}$ for
$r>7\lambda$. Hence, $N_{2}$ can be chosen as a constant times
$\lambda=N_{1}^{2}$, so the tail bound on $N_{2}$ can be dominated by a tail
bound on $N_{1}^{2}$, which is provided by Proposition 2.2 and Lemma 2.6 (with
the choice $\zeta=d$) under suitable moment conditions on $\omega(e)$ and
$\omega(e)^{-1}$. More precisely,

$\operatorname{\mathbb{P}}(N_{2}>u)\leq\operatorname{\mathbb{P}}(N_{1}^{2}>cu)\leq
c\,u^{\frac{1-d^{2}}{2}}\leq c\,u^{2-d},\qquad u\geq 1,$

since $(1-d^{2})/2<2-d$ for $d\geq 3$, which completes the proof of (4.4).

Finally, we prove the lower bound of (1.8), which follows again from Theorem
1.4. Choose $K\in(0,\infty)$ such that $\operatorname{\mathbb{P}}(N(x)\leq
K)=\operatorname{\mathbb{P}}(N(0)\leq K)\geq 1/2$, then

$\operatorname{\mathbb{E}}\big{[}g^{\omega}(x,y)\big{]}\geq\operatorname{\mathbb{E}}\Big{[}\int_{N(x)}^{\infty}p_{t}^{\omega}(x,y)\
\textrm{d}t\Big{]}\geq\frac{1}{2}\int_{K}^{\infty}c_{6}\,t^{-d/2}\,e^{-c_{7}|x-y|^{2}/t}\
\textrm{d}t.$

If $\left\lvert x-y\right\rvert^{2}\geq K$ then we can bound the integral on
the right hand side as in the proof of (1.7) to give (1.8). On the other hand,
there are only finitely many vertices $z\in B(0,\sqrt{K})$, and for each such
$z$ we have $\mathbb{E}\left[g^{\omega}(0,z)\right]>0$. Therefore, $\inf_{y\in
B(x,\sqrt{K})}\mathbb{E}\left[g^{\omega}(x,y)\right]=\inf_{z\in
B(0,\sqrt{K})}\mathbb{E}\left[g^{\omega}(0,z)\right]>0$. Thus, we can adjust
the constant $c_{10}$ such that (1.8) also holds for $\left\lvert
x-y\right\rvert^{2}\leq K$. ∎

#### Acknowledgement

We thank Scott Armstrong for some valuable comments on an earlier version of
this paper. N.H. has been supported by the doctoral training centre, Cambridge
Mathematics of Information (CMI).

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