the walk which are adjacent in $\mathL$, but not in the walk. This is illustrated in figure \[figure3\]; the positive elementary move creates $2$ new contacts in the walk, and so it changes the energy of the walk by $2$.

More general elementary moves (for example, the BFACF elementary moves) may contribute to the atmospheric statistics $a^+$ and $a^-$ in various ways, and may even give rise to *neutral atmospheres* which do not change the length of the walk (but which may change the energy of the walk).

Thus, in what follows, let $a^+_v(w)$ be the size of the positive atmosphere of a walk $w$ of length $\ell_n=|w|$, and which changes the energy of $w$ by $v$ units. For example, the walk on the left in figure \[figure1\] has $a^+_{0}(w)=1$ and $a^+_{1}(w)=1$, and if this was a walk in the three dimensional half-lattice $\mathL_3^+$, then $a^+_{0}(w)=3$ and $a^+_{1}(w)=1$. The walk on the left in figure \[figure3\] (with energy given by the number of contacts), has, in a similar way, $a^+_{0} (w)=2$, $a^+_{1}(w)=0$ and $a^+_{2} (w)=1$.

In exactly the same way one may define the size of the negative atmosphere of a walk $w$ of length $\ell$ which changes the energy 