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Proof of Lemma 2

$Z$ becomes in that case

$\displaystyle Z$ $\textstyle =$ $\displaystyle W_+ e^{- \frac{1}{2} \Delta} + W_- e^{\frac{1}{2} \Delta} \:\:,$ (32)

where $\Delta=\vec{v}[1] - \vec{v}[0]$. There are five different values for $\Delta$, giving rise to nine different $\vec{v}$:

\begin{eqnarray*}
\Delta = & +2 & \Rightarrow \vec{v} = (-1, +1) \:\:,\\
\Delta...
...0) \:\:,\\
\Delta = & -2 & \Rightarrow \vec{v} = (+1, -1) \:\:.
\end{eqnarray*}



Fix $\Delta = k$ where $k \in \{-2, -1, 0, 1, 2\}$. $\forall k \in \{-1, 0, 1, 2\}$, the value $\Delta = k$ should be preferred to the value $\Delta = k-1$ iff the corresponding $Z$ is smaller, that is :
$\displaystyle W_+\times e^{-\frac{k}{2}} + W_- \times e^{\frac{k}{2}}$ $\textstyle <$ $\displaystyle W_+ \times e^{-\frac{k-1}{2}} + W_- \times e^{\frac{k-1}{2}} \:\:.$ (33)

Rearranging terms gives $W_- < W_+ \times \frac{1}{e^{k-\frac{1}{2}}}$. This leads to the rule of the lemma.



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