of $w$, and the number of edges in the positive atmosphere is denoted by $a^+(w)$. For example, for the walk on the left in figure \[figure1\], $a^+(w)=2$.

Similarly, the set of edges which may be removed from the endpoint of a walk to decrease its length by one, is the *negative atmosphere* of the walk. For endpoint elementary moves, the last step is always the sole negative atmospheric edge, so that the size of the negative atmosphere for endpoint elementary moves is always $a^-(w)=1$, if $w$ is not the trivial walk of length $0$.

An elementary move may change the energy of a walk. For example, in a model of adsorbing walks in $\mathL^+_2$, the energy is the number of *visits* of the walk to the adsorbing line $\partial\mathL^+_2$ (the boundary of the half-lattice $\mathL^+_2$). The positive move in figure \[figure1\] does not change the number of visits, but the move in figure \[figure2\] increases the number of visits (and so the energy) by $1$. The negative move in figure \[figure2\] similarly reduces the number of visits by $1$.

figure4-.tex

A similar situation arises if a model of collapsing walks with energy given by nearest neighbour *contacts* between vertices in 