leave unchanged, the energy of the walk. This is done by defining atmospheric statistics as follows: Let the states $w_n$ be walks of length $\ell_n = |w_n|$. Define $$\begin{aligned}
\alpha_n^{--} = \sum_{v< 0} a_{v}^- (w_n),\qq & 
\alpha_n^{-0} = \sum_{v=0} a_{v}^-(w_n), \qq 
\alpha_n^{-+} = \sum_{v>0} a_{v}^-(w_n); \\
\alpha_n^{0-} = \sum_{v< 0} a_{v}^0 (w_n),\qq & 
\alpha_n^{00} = \sum_{v=0} a_{v}^0(w_n), \qq 
\alpha_n^{0+} = \sum_{v>0} a_{v}^0(w_n); \\
\alpha_n^{+-} = \sum_{v< 0} a_{v}^+ (w_n),\qq & 
\alpha_n^{+0} = \sum_{v=0} a_{v}^+(w_n), \qq 
\alpha_n^{++} = \sum_{v>0} a_{v}^+(w_n).  \end{aligned}$$ For example, $\alpha_n^{--}$ is the number of negative elementary moves which also *decreases* the energy of the walk, and $\alpha_n^{-+}$ is the number of negative elementary moves which also *increases* the energy of the walk. The rest of the $\alpha$’s are similarly defined.

With these atmospheric statistics defined, a rule needs to be constructed in order to realise the sequence $\phi_N$ in equation .

The endpoint elementary moves in figures \[figure2\] and \[figure3\] have the property that no positive elementary move can decrease the energy, and no negative elementary move can increase the energy. Moreover, there are no neutral moves amongst the elementary moves. Thus, assume that $$\alpha_n^{-+} = 0,\q\alpha_n^{+-}=0,\q\alpha_n^{0-}=0,\q\alpha_n^{00}=0 
\q\hbox{and}\; \alpha_n^{0+} = 0.$$ The algorithm can be modified appropriately 