to account for such transitions in models where this is not the case. This assumption leaves the following atmospheric statistics: $\{\alpha_n^{--},\alpha_n^{-0},\alpha_n^{+0},\alpha_n^{++}\}$. That is, distinguish between negative moves which decrease the energy, or negative moves which leave the energy unchanged, or positive moves which leave the energy unchanged, and positive moves which increase the energy.

Introduce parameters $\{\beta_{\ell,u}\}$ to control positive elementary moves on walks of length $\ell$ and energy $u$, and which leave the energy unchanged (that is, the elementary moves contributing to $\alpha_n^{+0}$). Similarly, introduce the parameters $\{\gamma_{\ell,u}\}$ to control positive elementary moves which increase the energy on walks of length $\ell$ and energy $u$.

The transition probabilities of positive elementary moves which leave the energy unchanged will be proportional to $\beta_{\ell,u}$; if the state has length $\ell$ and energy $u$. For example, since $\ell=16$ and $u=2$ in the walk in figure \[figure1\], the transition probability of the positive move in that figure is proporsional to $\beta_{16,2}$. Similarly, the positive elementary move in figure \[figure2\] increases the energy; so here the transition probability is proportional to $\gamma_{16,2}$, instead. In figure \[figure3\] the positive elementary move also increases the energy, and so its transition probability is proportional to $\gamma_{16,4}$.

Thus, 