saving work

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@@ -456,7 +456,7 @@ The time-averaged contribution of the on-state weight can then be easily calcula
 Details of the implementation of this effective dynamics can be in found in Refs.~\onlinecite{Assaraf_1999B} and \onlinecite{Caffarel_2000}.
 
 %=======================================%
-\subsection{General domains}
+\subsection{Multi-state domains}
 \label{sec:general_domains}
 %=======================================%
 
@@ -584,21 +584,25 @@ We then have
 	\qty[ \prod_{k=0}^{p-1} \mel{ I_k }{ \qty(T_{I_k})^{n_k-1} F_{I_k} }{ I_{k+1} } ]
 	G^{(n_p-1),{\cal D}}_{I_p I_N}
 \end{multline}
+where $\delta_{i,j}$ is a Kronecker delta.
+
 This expression is the path-integral representation of the Green's matrix using only the variables $(\ket{I_k},n_k)$ of the effective dynamics defined over the set of domains. 
-The standard formula derived above, Eq.~\eqref{eq:G} may be considered as the particular case where the domain associated with each state is empty, 
+The standard formula derived above [see Eq.~\eqref{eq:G}] may be considered as the particular case where the domain associated with each state is empty, 
-In that case, $p=N$ and $n_k=1$ for $k=0$ to $N$ and we are left only with the $p$-th component of the sum, that is, $G^{(N)}_{I_0 I_N} 
+In that case, $p=N$ and $n_k=1$ for $0 \le k \le N$ and we are left only with the $p$th component of the sum, that is, $G^{(N)}_{I_0 I_N} 
 = \prod_{k=0}^{N-1} \mel{ I_k }{ F_{I_k} }{ I_{k+1} } $ where $F=T$.
 
 To express the fundamental equation for $G$ under the form of a probabilistic average, we write the importance-sampled version of the equation
-\be
+\begin{multline}
 \label{eq:Gbart}
-	{\bar G}^{(N)}_{I_0 I_N}={\bar G}^{(N),{\cal D}}_{I_0 I_N} +
+	\bar{G}^{(N)}_{I_0 I_N}=\bar{G}^{(N),\cD}_{I_0 I_N} +
 	\sum_{p=1}^{N}
-	\sum_{|I_1\rangle \notin {\cal D}_{I_0}, \hdots , |I_p\rangle \notin {\cal D}_{I_{p-1}}} 
+	\sum_{\ket{I_1} \notin \cD_{I_0}, \ldots , \ket{I_p} \notin \cD_{I_{p-1}}}
-	\sum_{n_0 \ge 1} ... \sum_{n_p \ge 1}
+	\sum_{n_0 \ge 1} \cdots \sum_{n_p \ge 1}
-	\delta(\sum_k n_k=N+1) \Big[ \prod_{k=0}^{p-1} [\frac{\PsiG_{I_{k+1}}}{\PsiG_{I_k}} \langle I_k|  T^{n_k-1}_{I_k} F_{I_k} |I_{k+1} 	\rangle \Big]
+	\\
-	{\bar G}^{(n_p-1),{\cal D}}_{I_p I_N}.
+	\times
-\ee
+	\delta_{\sum_k n_k,N+1} \qty{ \prod_{k=0}^{p-1} \qty[ \frac{\PsiG_{I_{k+1}}}{\PsiG_{I_k}} \mel{ I_k }{ \qty(T_{I_k})^{n_k-1} F_{I_k} }{ I_{k+1} 	} ] }
+	\bar{G}^{(n_p-1),\cD}_{I_p I_N}.
+\end{multline}
 Introducing the weight 
 \be
 	W_{I_k I_{k+1}} = \frac{\mel{ I_k }{ \qty(T_{I_k})^{n_k-1} F_{I_k} }{ I_{k+1} }}{\mel{ I_k }{ \qty(T^{+}_{I_k})^{n_k-1} F^+_{I_k} }{ I_{k+1} }}
@@ -606,20 +610,22 @@ Introducing the weight
 and using the effective transition probability defined in Eq.~\eqref{eq:eq3C}, we get
 \be
 \label{eq:Gbart}
-	\bar{G}^{(N)}_{I_0 I_N}=\bar{G}^{(N),\cD}_{I_0 I_N}+ \sum_{p=1}^{N}
+	\bar{G}^{(N)}_{I_0 I_N} = \bar{G}^{(N),\cD}_{I_0 I_N} + \sum_{p=1}^{N}
-	\bigg \langle
+	\expval{
-	\Big( \prod_{k=0}^{p-1} W_{I_k I_{k+1}} \Big)
+	\qty( \prod_{k=0}^{p-1} W_{I_k I_{k+1}} )
-	{\bar G}^{(n_p-1), {\cal D}}_{I_p I_N}
+	\bar{G}^{(n_p-1), {\cal D}}_{I_p I_N}
-	\bigg \rangle
+	}
 \ee
 where the average is defined as
-\be
+\begin{multline}
 	\expval{F}
-	= \sum_{|I_1\rangle \notin {\cal D}_{I_0}, \hdots , |I_p\rangle \notin {\cal D}_{I_{p-1}}} 
+	= \sum_{\ket{I_1} \notin \cD_{I_0}, \ldots , \ket{I_p} \notin \cD_{I_{p-1}}} 
 	\sum_{n_0 \ge 1} \cdots \sum_{n_p \ge 1}
 	\delta_{\sum_k n_k,N+1} 
-	\prod_{k=0}^{N-1}\cP_{I_k \to I_{k+1}}(n_k-1)  F(I_0,n_0;...;I_N,n_N)
+	\\
-\ee
+	\times
+	\prod_{k=0}^{N-1}\cP_{I_k \to I_{k+1}}(n_k-1)  F(I_0,n_0;\ldots.;I_N,n_N)
+\end{multline}
 In practice, a schematic DMC algorithm to compute the average is as follows.\\
 i) Choose some initial vector $\ket{I_0}$\\
 ii) Generate a stochastic path by running over $k$ (starting at $k=0$) as follows.\\