ox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
 {\overset{}{\longleftrightarrow}}}y}\leq P_{\Lambda;y}^{\prime{{\scriptscriptstyle}(0)}}(o,x),\end{aligned}$$ which will be used in Section \[ss:pijbd\] to obtain the bounds on $\pi_\Lambda^{{\scriptscriptstyle}(j)}(x)$ for $j\ge1$.

Since the inequality is trivial if $x=o$, we restrict our attention to the case of $x\ne o$.

First we note that, for each current configuration ${{\bf n}}$ with ${\partial}{{\bf n}}=\{o,x\}$ and ${\mathbbm{1}{\scriptstyle\{o{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
 {\overset{}{\Longleftrightarrow}}}x\}}}=1$, there are at least *three edge-disjoint* paths on ${{\mathbb G}}_{{\bf n}}$ between $o$ and $x$. See, for example, the first term on the right-hand side in Figure \[fig:1stpiv\]. Suppose that the thick line in that picture, referred to as $\zeta_1$ and split into $\zeta_{11}{\:\Dot{\cup}\:}\zeta_{12}{\:\Dot{\cup}\:}\zeta_{13}$ from $o$ to $x$, consists of bonds $b$ with $n_b=1$, and that the thin lines, referred to as $\zeta_2$ and $\zeta_3$ that terminate at $o$ and $x$ respectively, consist of bonds $b'$ with $n_{b'}=2$. Let $\zeta'_i$, for $i=2,3$, be the duplication of $\zeta_i$. Then, the three paths $\zeta_2{\:\Dot{\cup}\:}\zeta_{13}$, $\zeta'_2{\:\Dot{\cup}\:}\zeta_{12}{\:\Dot{\cup}\:}\zeta_3$ and $\zeta_{11}{\:\Dot{\cup}\:}\zeta'_3$ are edge-disjoint.

Then, by multiplying $\pi_\Lambda^{{\scriptscriptstyle}(0)}(x)$ by *two* dummies $(Z_\Lambda/Z_\Lambda)^2\,(\equiv1)$, we obtain $$\begin{aligned}
{\label{eq:pi0*Z2}}
\pi_\Lambda^{{\scriptscriptstyle}(0)}(x)&=\sum_{\substack{{\partial}{{\bf n}}=\{o,x\}\\ {\partial}{{\bf m}}'={\partial}{{\bf m}}''
 ={\varnothing}}}\frac{w_\Lambda({{\bf n}})}{Z_\Lambda}\,\frac{w_\Lambda({{\bf m}}')}{Z_\Lambda}
 \,\frac{w_\Lambda({{\bf m}}'')}{Z_\Lambda}\,{\mathbbm{1}{\raisebox{-2pt}{$\scriptstyle \{o{\underset{\raisebox{5pt}{${\scriptscriptstyle}{{\bf n}}$}}
 {\overset{}{\Longleftrightarrow}}}x\}$}}}{\nonumber}\\
&=\sum_{{\partial}{{\bf N}}=\{o,x\}}\frac{w_\Lambda({{\bf N}})}{Z_