\prod_{j=0}^{N-1} \beta_{\ell,u}^{\sigma(j,j{+}1)}
\prod_{j=0}^{N-1} \gamma_{\ell,u}^{\rho(j,j{+}1)}
\label{eqn9}            $$ to the sequence $\phi_N$.

The expected value of the weight over sequences of length $N$ from state $w_0$ to state $w_N$ is $$\LA W(w_0\to w_N) \RA_N = \sum_{\phi:w_0 \to w_N}  \Pr(\phi) W(\phi) .
\label{eqn10}    $$ Inserting equations and in this, and simplifying, gives $$\fl
\LA W(w_0\to w_N) \RA_N = \sum_{\phi:w_0\to w_N}
\prod_{j=1}^N 
\frac{1}{\alpha_j^{--} \plus \alpha_j^{-0} \plus \alpha_j^{+0}\thin\beta_{\ell,u} 
\plus  \alpha_j^{++}\thin \gamma_{\ell,u}}
\; {\prod}^{n} \beta_{\ell,u}
\; {\prod}^{nn} \gamma_{\ell,u}
\label{eqn11}    $$ where the product ${\prod}^{n}$ is over all the $\beta_{\ell,u}$ for transitions through *negative* elementary moves leaving the energy unchanged, and the product ${\prod}^{nn}$ is over all $\gamma_{\ell,u}$ where the transition is a *negative* elementary move decreasing the energy. As before, the $\ell
\equiv \ell(j)$ and $u\equiv u(j)$ are functions of $j$ (in other words, functions of the states $w_j$ in the sequence $\phi$).

Reverse all the sequences in equation so that the starting state is $w_N$ and the final state is $w_0$. Under this reversal all negative elementary moves become positive elementary moves and vice versa. That is, equation becomes $$\fl
\LA W(w_0\to w_N) \RA_N = \sum_{\psi:w_N\to w_0}
\prod_{j=1}^N 
\frac{1}{\alpha_j^{--} \plus \alpha_j^{-0} \plus \alpha_j^{+0}\thin\beta_{\ell,u} 
\plus \alpha_j^{++}\thin \gamma_{\ell,u}}
\; {\prod}^{n} \beta_{\ell,u}
\; {\prod}^{nn} \gamma_{\ell,u} ,
\label{eqn12}    $$ 