%
\providecommand{\bGM}{\beta_{\mathrm{GM}}}
\providecommand{\mt}{\tilde m}
\providecommand{\MY}{M_{Y}}
\providecommand{\lPV}{\lambda_{\mathrm{PV}}}
\providecommand{\Gp}{G'}
\providecommand{\XX}{\mathbb{X}}
\providecommand{\KK}{\mathfrak{K}}
\providecommand{\RHK}{R_{\mathrm{HK}}}
\providecommand{\rcov}{r_{\mathrm{cov}}}
\providecommand{\etaw}{\eta_{w}}
\providecommand{\eps}{\varepsilon}

\section{Heat kernel, coarse-to-fine transfer and covariance bounds on slit domains}
\label{app:F}

In this appendix we give condensed proofs of Proposition~\ref{prop:6.10} and Theorem~\ref{thm:6.9} and sketch the proofs of Proposition~\ref{prop:6.11}, Theorem~\ref{thm:6.12} and Lemma~\ref{lem:F.7}. Full proofs are in the supplement: those of Proposition~\ref{prop:6.10} and Theorem~\ref{thm:6.9}, with Barlow's argument on $\Gp$, in Section~\ref{S4-S:S4.3}, which also has further details for Proposition~\ref{prop:6.11}, Theorem~\ref{thm:6.12} and Lemma~\ref{lem:F.7}, and those of the last three in Section~\ref{S:E2}. Throughout, $\Gp=(V,E)$ is an output, $d=d_{\Gp}$, $\mathcal{E}(f):=\sum_{\{x,y\}\in E}(f(x)-f(y))^{2}$, and $\partial U=\partial_{\Gp}U$ is the set of bonds of $E$ with exactly one end in $U$.

\subsection{Coarse-to-fine transfer: proof of Proposition~\ref{prop:6.10}}
\label{ssec:F.1}

A large fine set either fills many coarse cells, whose boundary (R2) controls, or is spread thinly inside cells, where box isoperimetry applies (canonical paths and Loomis\textendash Whitney). A face count joins the two scales. The coarse cells, the sites $c\in V_{c}$ of the coarse slit graph $\Gamma=(V_{c},E_{c})$ of \S\ref{ssec:6.3}, are boxes $B_{c}$ with sides in $[r_{0},2r_{0}-1]$ by Definition~\ref{def:3.4}, and by \S\ref{ssec:2.5} the box graph of $B_{c}\cup B_{c'}$, $\{c,c'\}\in E_{c}$, lies in $\Gp$. Thus a coarse bond between a \emph{heavy} cell ($|A\cap B_{c}|\ge|B_{c}|/2$) and a light one carries at least $r_{0}^{2}/4$ bonds of $\partial A$ in $B_{c}\cup B_{c'}$, and $|\partial A|\ge(r_{0}^{2}/24)|\partial_{\Gamma}H|$ for the set $H$ of heavy cells. Combined with (R2), in the form of Theorem~\ref{thm:4.7}(i), (ii), this proves (i), and the same argument with the expansion of $K$ in place of (R2) proves (iii). We deduce (ii) from (i) as in Theorem~\ref{thm:4.7}(iv), and from Theorem~\ref{thm:4.7}(ii) and $r_{0}^{3}V_{*}=2V_{*}^{f}$. Part (iv) is (iii) for the blow-up of the set $\KK^{\flat}_{c}$ of (R6) (Lemma~\ref{lem:4.16}) with Cheeger's inequality, the overlap count inherited and $|\Omega|\ge c_{1}L^{3}$ from (R5). Finally, (v) holds because balls in a connected graph of degree at most $6$ and expansion $h\le1$ grow by the factor $1+h/6$ per step up to half the vertices, so the diameter is at most $2+(24/h)\log|\KK^{\flat}|<L'$ under (L8-2). The constants are those of \S\ref{S:E1.ctf}.

\subsection{Nash's inequality from Faber\textendash Krahn, and the on-diagonal bound}
\label{ssec:F.2}

\begin{lemma}[Nash inequality]\label{lem:F.3}
For $f\ge0$ on $V$ with $\sum f=1$, and every $r\ge1$ with $2r^{3}\le V_{*}^{f}$, $\|f\|_{2}^{2}\le A_{N}r^{2}\mathcal{E}(f)+r^{-3}$, where $A_{N}:=24\cdot2^{2/3}/c_{F}^{2}$.
\end{lemma}

\begin{proof}
We put $s:=1/(2r^{3})$. Then $f^{2}\le2(f-s)_{+}^{2}+2\min(f,s)^{2}$ and $\sum\min(f,s)^{2}\le s$. The function $(f-s)_{+}$ vanishes off $\Omega_{s}:=\{f>s\}$, which has $|\Omega_{s}|<1/s=2r^{3}\le V_{*}^{f}$, and $\mathcal{E}((f-s)_{+})\le\mathcal{E}(f)$. By Proposition~\ref{prop:6.10}(ii), $\sum(f-s)_{+}^{2}\le(12/c_{F}^{2})(2r^{3})^{2/3}\mathcal{E}(f)$.
\qed
\end{proof}

\begin{proof}[Proof of (HK0) and (HK1)]
We obtain (HK0) from the degree bound alone, by Davies' perturbation argument. For $\psi:=-\alpha d(x,\cdot)$,
\[
\langle g,e^{\psi}\Delta_{\Gp}e^{-\psi}g\rangle=-\mathcal{E}(g)+2\sum_{\{x,y\}\in E}(\cosh(\psi(x)-\psi(y))-1)g(x)g(y)\le6(\cosh\alpha-1)\|g\|^{2},
\]
so $\|e^{\psi}P_{t}e^{-\psi}\|_{2\to2}\le e^{6(\cosh\alpha-1)t}$ and $p_{t}(x,y)=e^{\psi(y)-\psi(x)}\langle\delta_{x},e^{\psi}P_{t}e^{-\psi}\delta_{y}\rangle\le e^{-\alpha d+6(\cosh\alpha-1)t}$. At $\alpha=\frac14$, $6(\cosh\frac14-1)<\frac14-\frac1{17}$. The number $N_{t}$ of jumps up to time $t$ is dominated by a Poisson variable of mean $6t$, and $d(x,X_{t})\le N_{t}$.

For (HK1), the function $\psi(t):=\|P_{t}\delta_{x}\|^{2}=p_{2t}(x,x)$ is nonincreasing, $\psi\le1$ and $\psi'=-2\mathcal{E}(P_{t}\delta_{x})$. With $r(t)$ given by $r^{-3}=\psi/2$, the condition $2r^{3}\le V_{*}^{f}$ becomes $\psi\ge4/V_{*}^{f}$, which holds on an interval $[0,t_{*}]$. On this interval Lemma~\ref{lem:F.3} implies $-\psi'\ge\psi^{5/3}/(2^{2/3}A_{N})=c_{N}\psi^{5/3}$, hence $\psi(t)\le(\frac23c_{N}t)^{-3/2}$, while $\psi<4/V_{*}^{f}$ after $t_{*}$. Hence $p_{s}(x,x)\le\max(C_{H}s^{-3/2},4/V_{*}^{f})$, $p_{s}(x,y)\le(p_{s}(x,x)p_{s}(y,y))^{1/2}$, and $4/V_{*}^{f}=\frac{64}{27}\RHK^{-3}$ (\S\ref{S:E1.scales}). The zero modes lie below the plateau, since $p_{t}(x,x)\to1/|\KK_{x}|<1/(2V_{*}^{f})$, the last inequality by Proposition~\ref{prop:6.10}(ii).
\qed
\end{proof}

\subsection{Barlow's argument: proof of Theorem~\ref{thm:6.9}}
\label{ssec:F.3}

From now on we assume $\bGM\ge\bGMo$, so that $\mt\le K_{0}(2C_{H})^{-1/3}$, $\RHK\ge K_{0}/\mt\ge(2C_{H})^{1/3}$, and $p_{t}\le C_{H}t^{-3/2}$ on $[1,\RHK^{2}]$ by (HK1). We obtain (HK1$'$) as in \cite[Lemma~3.3(b), Prop.~3.4]{Ba04}, (HK2) as in \cite[Lemmas~3.5, 3.6, Prop.~3.7]{Ba04}, and (HK3) as the fixed-time form of \cite[Thm~3.8]{Ba04}, with the following adjustments, each forced by $\Gp$:
\par(a) \cite[Prop.~3.1, Cor.~3.2]{Ba04}, the on-diagonal decay in a very good ball for $t\le cR^{2}/\log R$, are replaced by (HK1), which holds at all centres. Every ball is a very good ball with $N_{B}=1$ \cite[Def.~1.7]{Ba04}, there is no localization and no $\log R$ loss, and the window is $t\le\eps_{1}^{\mathrm{HK}}\RHK^{2}$. The exit-time bound $\frac12+4C_{\mathrm{mo}}^{2}t/r^{2}$ of \cite[Lemma~3.5]{Ba04} holds at all centres for $t\le\RHK^{2}/2$, and exceeds $1$ for larger $t$ when $r\le2C_{\mathrm{mo}}\RHK$.
\par(b) The walk is the variable-speed walk with counting measure on a finite, possibly disconnected graph, and \cite[(3.8) to (3.10)]{Ba04} hold without change, with $a_{xy}=1$, $\mu\equiv1$, as finite sums.
\par(c) The volume bound \cite[(1.1)]{Ba04} enters only \cite[Lemma~3.3(b)]{Ba04}, through $\sum_{y}e^{-a\,d(x,y)}\le c_{4}a^{-3}$ for $0<a\le2$, which follows from $d_{\Gp}\ge\|\cdot\|_{1}$: $\sum_{y}e^{-a\,d(x,y)}\le\sum_{z\in\Zthree}e^{-a\|z\|_{1}}=\coth(a/2)^{3}\le(1+2/a)^{3}\le64a^{-3}$.
\par(d) \cite[Lemma~1.1]{Ba04} is replaced by (HK0), (HK1) and the Poisson domination of $N_{t}$. Let $M(t):=\mathbb{E}_{x}d(x,X_{t})$. By (HK1), $M(t)\ge\mathbb{P}_{x}(X_{t}\ne x)\ge1-C_{H}t^{-3/2}\ge\frac12$ for $t\ge(2C_{H})^{2/3}$, and the bound by $N_{t}$ gives $M(t)\le6t$ at earlier times. For $\rho>A_{1}t/4$ ($A_{1}$ of \S\ref{S:E1.HK}), the event $\tau(x,\rho)<t$ forces $N_{t}>\rho$.
\par(e) (HK4) is not in \cite{Ba04}, but it follows from (HK2). With $D:=d(w,X_{s})$, we split the sum in $\mathbb{E}e^{\theta D}-1\le\theta\sum_{k\ge1}e^{\theta k}\mathbb{P}(D\ge k)$ at $k=s$ into a Gaussian part, summed with $\theta\le s^{-1/2}$, and a Poisson part, summed with $\theta\le c_{\mathrm{ex}}/2$. For $s<1$, $D\le N_{s}$.
\par For (HK3), Chapman\textendash Kolmogorov gives $p_{t}(x,y)\le2\sup_{z,w}p_{t/2}(z,w)\sup_{w}\mathbb{P}_{w}(d(w,X_{t/2})\ge d/2)$, and (HK1), (HK2) bound the two factors. The constants $C_{\mathrm{mo}},c_{\mathrm{ex}},A_{1},C_{\mathrm{gs}},c_{\mathrm{gs}},C_{M}$ of \S\ref{S:E1.HK} are functions of $C_{H}$.

\subsection{Resolvent decay and covariance bounds}
\label{ssec:F.4}\label{ssec:F.5}

For Proposition~\ref{prop:6.11}, (HK4) gives $\mathbb{E}_{w}e^{\theta_{M}d(w,X_{s})}\le\frac32$ for $s\le T_{1}:=\min(M^{-2},\eps_{1}^{\mathrm{HK}}\RHK^{2})$. Using blocks of length $T_{1}$, we obtain $\mathbb{E}_{x}e^{\theta_{M}d(x,X_{t})}\le\frac32e^{M^{2}t/2}$, and the mass term outweighs this growth in $\sum_{y}v^{(M)}_{\Gp}(x,y)e^{\theta_{M}d(x,y)}=\int_{0}^{\infty}e^{-M^{2}t}\mathbb{E}_{x}e^{\theta_{M}d(x,X_{t})}dt$. For Theorem~\ref{thm:6.12} we write $C_{0}=\int_{0}^{\infty}k(t)P_{t}\,dt$ with $0\le k(t)\le\min(\MY^{2}t,\varpi)e^{-r_{-}t}$ (\S\ref{S:E1.cov}). With this representation, (H1), (H1$'$), (H2) follow from (HK0), (HK1), (HK3), \eqref{eq:6.LW} and the blocking bound by splitting the time integral, and (H$_{C}$) follows from the Schur test and a Neumann series in $\|\cdot\|_{\rcov}$.

\subsection{Bounds on the lattice Yukawa kernel}
\label{ssec:F.6}

\begin{lemma}[Lattice Yukawa kernel]\label{lem:F.7}
Let $0<M\le\frac15$ and $v^{(M)}:=(-\Delta_{\Zthree}+M^{2})^{-1}$. There is an absolute constant $C_{L}$ such that for all $x\in\Zthree$ and all unit vectors $e$,
\begin{equation}\label{eq:F.L1L2}
v^{(M)}(x)\le C_{L}(1+|x|)^{-1}e^{-3M|x|/4},\qquad|v^{(M)}(x+e)-v^{(M)}(x)|\le C_{L}(1+|x|)^{-2}e^{-3M|x|/4}.
\end{equation}
If $\Lambda'=\prod_{\mu}[1,n_{\mu}]$ is a box with $\min_{\mu}n_{\mu}\ge M^{-1}$ and $u_{x}:=v^{(M)}_{\Lambda'}(\cdot,x)$ is the kernel of the box with all its bonds, then for $x,y,y+e\in\Lambda'$, with $C_{L}'$ absolute,
\begin{gather*}
\mathrm{(L1)}\ \ u_{x}(y)\le C_{L}'(1+|y-x|)^{-1}e^{-M|y-x|/2},\\
\mathrm{(L2)}\ \ |u_{x}(y+e)-u_{x}(y)|\le C_{L}'(1+|y-x|)^{-2}e^{-M|y-x|/2}.
\end{gather*}
\end{lemma}

\noindent\emph{Sketch.} Exponential tilts of the three one-dimensional heat kernels and the unimodality of the Poisson law give \eqref{eq:F.L1L2} after the time integral, the rate $3M/4$ coming from $M^{2}t+\frac{9}{64}|x|^{2}/t\ge\frac34M|x|$. The box kernel is the image sum $v^{(M)}_{\Lambda'}(y,x)=\sum_{g\in\mathcal{G}}v^{(M)}(y-gx)$ over the group $\mathcal{G}$ generated by the reflections in the face planes. Since $\min_{\mu}n_{\mu}\ge M^{-1}$, the images cost a factor $(1+M|y-x|)^{3}$, which the loss of rate from $3M/4$ to $M/2$ absorbs.