if $w_n$ is the current state (of length $\ell_n$ and energy $u_n$), and $w_{n+1}$ is the next state (of length $\ell_{n+1}$ and energy $u_{n+1}$), then define the change in length by $\Delta_n = \ell_{n+1}\minus \ell_n$, and the change in energy by $\delta_n = u_{n+1} \minus u_n$. Notice that $\Delta_n=\pm 1$ for endpoint elementary moves, and that $\delta_n = \pm 1$, or $\delta_n = 0$, for the model of adsorbing walks in figures \[figure1\] and \[figure2\] (but these quantities may take on other values in the model in figure \[figure3\]). The transition probabilities are chosen such that $$\Pr(w_n\to w_{n+1}) \propto
\cases{
\beta_{\ell,u}, & \hbox{if $\Delta_n = +1$ and $\delta_n=0$;} \\
1 & \hbox{if $\Delta_n = -1$ and $\delta_n=0$},
}$$ where $\ell=\ell_n$ and $u=u_n$ are functions of $n$. In the case that the energy is changed, then the transition probability is constructed such that $$\Pr(w_n\to w_{n+1}) \propto
\cases{
\gamma_{\ell,u}, & \hbox{if $\Delta_n=+1$ and $\delta_n>0$;} \\
1 & \hbox{if $\Delta_n=-1$ and $\delta_n < 0$}.
}$$ Normalising the transition probabilities gives $$\fl
\Pr(w_n \to w_{n+1})
= \cases{
\frac{\beta_{\ell,u}}{\alpha_n^{--} + \alpha_n^{-0} + \alpha_n^{+0}\thin\beta_{\ell,u} 
+ \alpha_n^{++}\thin\gamma_{\ell,u}}, & \hbox{if $\Delta_n=1$ and $\delta_n=0$}; \\
\frac{\gamma_{\ell,u}}{\alpha_n^{--} +\alpha_n^{-0} + \alpha_n^{+0}\thin\beta_{\ell,u} 
+ \alpha_n^{++}\thin\gamma_{\ell,u}}, & \hbox{if $\Delta_n=1$ and $\delta_n>0$}; \\
\frac{1}{\alpha_n^{--} +\alpha_n^{-0} 