+ \alpha_n^{+0}\thin\beta_{\ell,u} 
+ \alpha_n^{++}\thin\gamma_{\ell,u}}, & \hbox{if $\Delta_n=-1$ and $\delta_n\leq 0$}; \\
}
\label{eqn4}   $$ The probability for any type of move can be explicitly computed for any given state $w_n$ (of length $\ell$ and energy $u$) by computing the $\alpha$’s. For example, the probability for a positive elementary move which leaves the energy unchanged is $\alpha_n^{+0}\thin\beta_{\ell,u}/(\alpha_n^{--} \plus \alpha_n^{+0}\thin\beta_{\ell,u} \plus  \alpha_n^{++}\thin\gamma_{\ell,u})$. For the state on the left in figure \[figure1\] this becomes $\beta_{16,2}/(1+\beta_{16,2}+\gamma_{16,2})$ since $\alpha_{16}^{--}=1$, $\alpha_{16}^{+0} = 1$ and $\alpha_{16}^{++}=1$.

GAS-weights
-----------

Let $w_0$ be a starting state (possibly the walk consisting of a single vertex at the origin, of length $\ell_0=0$ and energy $u_0=0$). Implement the endpoint elementary moves on $w_0$ by computing its atmosphere recursively and updating it using the transition probabilities in equation , starting at $n=0$. This generates a Markov Chain of states in a sequence $\phi_N$ (see equation ).

Assume that the parameters $\{\beta_{n,u},\gamma_{n,u}\}$ are known and fixed, so that the sampling can be implemented by simply computing the transtition probabilities and selecting positive and negative elementary moves with appropriate probabilities.

The probability of the sequence $\phi_N$ is given by $$\fl
\Pr (\phi_N) = \prod_{n=0}^{N-1}
\frac{1}{\alpha_n^{--} + \alpha_n^{-0} 
+ \alpha_n^{+0}\thin\beta_{\ell,u} 
+ \alpha_n^{++}\thin \gamma_{\ell,u}}
\; {\prod}_m^\prime \beta_{\ell,u}
