saving work

g.tex

@@ -243,7 +243,7 @@ $e^{-N\tau H}$ which is usually referred to as the imaginary-time dependent Gree
 \titou{Introducing the set of $N-1$ intermediate states, $\{ \ket{i_k} \}_{1 \le k \le N-1}$, in the $N$th product of $T$,} $G^{(N)}$ can be written in the following expanded form
 \be
 \label{eq:cn}
-	G^{(N)}_{i_0 i_N} = \sum_{i_1} \sum_{i_2} ... \sum_{i_{N-1}} \prod_{k=0}^{N-1} T_{i_{k} i_{k+1}},
+	G^{(N)}_{i_0 i_N} = \sum_{i_1} \sum_{i_2} \cdots \sum_{i_{N-1}} \prod_{k=0}^{N-1} T_{i_{k} i_{k+1}},
 \ee
 where $T_{ij} =\mel{i}{T}{j}$.
 Here, each index $i_k$ runs over all basis vectors.
@@ -460,25 +460,27 @@ Details of the implementation of this effective dynamics can be in found in Refs
 \label{sec:general_domains}
 %=======================================%
 
-Let us now extend the results of the preceding section to a general domain. For that, 
-let us associate to each state $\ket{i}$ a set of states, called the domain of $\ket{i}$ and
-denoted $\cD_i$, consisting of the state $\ket{i}$ plus a certain number of states. No particular constraints on the type of domains 
-are imposed, for example domains associated with different states can be identical, or may have or not common states. The only important condition is 
-that the set of all domains ensures the ergodicity property of the effective stochastic dynamics (that is, starting from any state there is a 
-non-zero-probability to reach any other state in a finite number of steps). In practice, it is not difficult to impose such a condition.
+Let us now extend the results of Sec.~\ref{sec:single_domains} to a general domain. 
+To do so, let us associate to each state $\ket{i}$ a set of states, called the domain of $\ket{i}$ denoted $\cD_i$, consisting of the state $\ket{i}$ plus a certain number of states. 
+No particular constraints on the type of domains are imposed. 
+For example, domains associated with different states can be identical, and they may or may not have common states. 
+The only important condition is that the set of all domains ensures the ergodicity property of the effective stochastic dynamics (that is, starting from any state there is a non-zero-probability to reach any other state in a finite number of steps). 
+In practice, it is not difficult to impose such a condition.
 
 Let us write an arbitrary path of length $N$ as 
 \be
 	\ket{i_0} \to \ket{i_1} \to \cdots \to \ket{i_N} 
 \ee
-where the successive states are drawn using the transition probability matrix, $p_{i \to j}$. This series can be rewritten 
+where the successive states are drawn using the transition probability matrix, $p_{i \to j}$. 
+This series can be rewritten 
 \be
 \label{eq:eff_series}
 	(\ket*{I_0},n_0) \to  (\ket*{I_1},n_1) \to \cdots \to (\ket*{I_p},n_p) 
 \ee
-where $\ket{I_0}=\ket{i_0}$ is the initial state, $n_0$ the number of times the walker remains within the domain of $\ket{i_0}$ ($n_0=1$ to $N+1$), $\ket{I_1}$ is the first exit state, that is not belonging to $\cD_{i_0}$, $n_1$ is the number of times the walker remains within $\cD_{i_1}$ ($n_1=1$ to $N+1-n_0$), $\ket{I_2}$ the second exit state, and so on.
-Here, the integer $p$ goes from 0 to $N$ and indicates the number of exit events occurring along the path. The two extreme cases, $p=0$ and $p=N$, correspond to the cases where the walker remains for ever within the initial domain, and to the case where the walker leaves the current domain at each step, respectively. 
-In what follows, we shall systematically write the integers representing the exit states in capital letter.
+where $\ket{I_0}=\ket{i_0}$ is the initial state, $n_0$ is the number of times the walker remains within the domain of $\ket{i_0}$ (with $1 \le n_0 \le N+1$), $\ket{I_1}$ is the first exit state that does not belong to $\cD_{i_0}$, $n_1$ is the number of times the walker remains in $\cD_{i_1}$ (with $1 \le n_1 \le N+1-n_0$), $\ket{I_2}$ is the second exit state, and so on.
+Here, the integer $p$ goes from 0 to $N$ and indicates the number of exit events occurring along the path. 
+The two extreme cases, $p=0$ and $p=N$, correspond to the cases where the walker remains in the initial domain during the entire path, and to the case where the walker exits a domain at each step, respectively. 
+\titou{In what follows, we shall systematically write the integers representing the exit states in capital letter.}
 
 %Generalizing what has been done for domains consisting of only one single state, the general idea here is to integrate out exactly the stochastic dynamics over the 
 %set of all paths having the same representation, Eq.(\ref{eff_series}). As a consequence, an effective Monte Carlo dynamics including only exit states 
@@ -488,56 +490,57 @@ Let us define the probability of being $n$ times within the domain of $\ket{I_0}
 It is given by
 \be
 \label{eq:eq1C}
-	\cP_{I_0 \to I}(n) = \sum_{|i_1\rangle \in {\cal D}_{I_0}} ... \sum_{|i_{n-1}\rangle \in {\cal D}_{I_0}} 
-p_{I_0 \to i_1} \ldots p_{i_{n-2} \to i_{n-1}} p_{i_{n-1} \to I}
+	\cP_{I_0 \to I}(n) 
+	= \sum_{\ket{i_1} \in \cD_{I_0}} \cdots \sum_{\ket{i_{n-1}} \in \cD_{I_0}} 
+	p_{I_0 \to i_1} \ldots p_{i_{n-2} \to i_{n-1}} p_{i_{n-1} \to I}
 \ee
 To proceed we need to introduce the projector associated with each domain
 \be
-	P_I=  \sum_{\ket{k} \in \cD_I} \dyad{k}{k}
-\label{pi}
+\label{eq:pi}
+	P_I = \sum_{\ket{k} \in \cD_I} \dyad{k}{k}
 \ee
 and to define the restriction of the operator $T^+$ to the domain
 \be
 	T^+_I= P_I T^+ P_I.
 \ee
-$T^+_I$ is the operator governing the dynamics of the walkers moving within ${\cal D}_{I}$.
-Using Eqs.(\ref{eq1C}) and (\ref{pij}), the probability can be rewritten as
+$T^+_I$ is the operator governing the dynamics of the walkers moving within $\cD_{I}$.
+Using Eqs.~\eqref{eq:pij} and \eqref{eq:eq1C}, the probability can be rewritten as
 \be
-	\cP+{I_0 \to I}(n) = \frac{1}{\PsiG_{I_0}} \mel{I_0}{\qty(T^+_{I_0})^{n-1} F^+_{I_0}}{I} \PsiG_{I}
-\label{eq3C}
+\label{eq:eq3C}
+	\cP+{I_0 \to I}(n) = \frac{1}{\PsiG_{I_0}} \mel{I_0}{\qty(T^+_{I_0})^{n-1} F^+_{I_0}}{I} \PsiG_{I},
 \ee
 where the operator $F$, corresponding to the last move connecting the inside and outside regions of the 
 domain, is given by
 \be
-F^+_I =  P_I T^+ (1-P_I),
-\label{Fi}
+\label{eq:Fi}
+	F^+_I =  P_I T^+ (1-P_I),
 \ee
-that is, $(F^+_I)_{ij}= T^+_{ij}$ when $(|i\rangle \in {\cal D}_{I}, |j\rangle \notin {\cal D}_{I})$, and zero
+that is, $(F^+_I)_{ij}= T^+_{ij}$ when $(\ket{i} \in \cD_{I}, \ket{j} \notin \cD_{I})$, and zero
 otherwise.
 Physically, $F$ may be seen as a flux operator through the boundary of ${\cal D}_{I}$.
 
 Now, the probability of being trapped $n$ times within ${\cal D}_{I}$ is given by
 \be
-P_{I}(n)=
-\frac{1}{\PsiG_{I}} \langle I | {T^+_{I}}^{n-1} F^+_{I}|\PsiG \rangle.
-\label{PiN}
+\label{eq:PiN}
+	P_{I}(n) = \frac{1}{\PsiG_{I}} \mel{ I }{ {T^+_{I}}^{n-1} F^+_{I} }{ \PsiG }.
 \ee
 Using the fact that 
 \be
-{T^+_I}^{n-1} F^+_I= {T^+_I}^{n-1} T^+ - {T^+_I}^n 
-\label{relation}
+\label{eq:relation}
+	{T^+_I}^{n-1} F^+_I = {T^+_I}^{n-1} T^+ - {T^+_I}^n,
 \ee
 we have
 \be
-\sum_{n=0}^{\infty} P_{I}(n) = \frac{1}{\PsiG_{I}}  \sum_{n=1}^{\infty} \Big( \langle I | {T^+_{I}}^{n-1} |\PsiG\rangle 
--  \langle I | {T^+_{I}}^{n} |\PsiG\rangle \Big) = 1
+	\sum_{n=0}^{\infty} P_{I}(n) 
+	= \frac{1}{\PsiG_{I}}  \sum_{n=1}^{\infty} \qty( \mel{ I }{ {T^+_{I}}^{n-1} }{ \PsiG } 
+	-  \mel{ I }{ {T^+_{I}}^{n} }{ \PsiG } ) = 1
 \ee
 and the average trapping time
 \be 
-t_{I}={\bar n}_{I} \tau=  \frac{1}{\PsiG_{I}}  \langle I | P_{I} \frac{1}{H^+ -E_L^+}  P_{I} | \PsiG\rangle 
+	t_{I}={\bar n}_{I} \tau = \frac{1}{\PsiG_{I}}  \mel{ I }{ P_{I} \frac{1}{H^+ - \EL^+}  P_{I} }{ \PsiG }. 
 \ee
-In practice, the various quantities restricted to the domain are computed by diagonalizing the matrix $(H^+-E_L^+)$ in ${\cal D}_{I}$. Note that 
-it is possible only if the dimension of the domains is not too large (say, less than a few thousands).
+In practice, the various quantities restricted to the domain are computed by diagonalizing the matrix $(H^+-\EL^+)$ in $\cD_{I}$. 
+Note that it is possible only if the dimension of the domains is not too large (say, less than a few thousands).
 
 %=======================================%
 \subsection{Expressing the Green's matrix using domains}
@@ -548,12 +551,12 @@ it is possible only if the dimension of the domains is not too large (say, less
 \subsubsection{Time-dependent Green's matrix}
 \label{sec:time}
 %--------------------------------------------%
-In this section we generalize the path-integral expression of the Green's matrix, Eqs.(\ref{G}) and (\ref{cn_stoch}), to the case where domains are used.
+In this section we generalize the path-integral expression of the Green's matrix, Eqs.~\eqref{eq:G} and \eqref{eq:cn_stoch}, to the case where domains are used.
 For that we introduce the Green's matrix associated with each domain 
 \be
-G^{(N),{\cal D}}_{IJ}= \langle J| T_I^N| I\rangle.
+	G^{(N),\cD}_{IJ}= \mel{ J }{ T_I^N }{ I }.
 \ee
-Starting from Eq.(\ref{cn})
+Starting from Eq.~\eqref{eq:cn}
 \be
 G^{(N)}_{i_0 i_N}= \sum_{i_1,...,i_{N-1}} \prod_{k=0}^{N-1} \langle i_k|  T |i_{k+1} \rangle.
 \ee
@@ -575,12 +578,12 @@ $$
 \sum_{n_0 \ge 1} ... \sum_{n_p \ge 1}
 \ee
 \be
+\label{eq:Gt}
 \delta(\sum_{k=0}^p n_k=N+1) \Big[ \prod_{k=0}^{p-1} \langle I_k|T^{n_k-1}_{I_k} F_{I_k} |I_{k+1} \rangle \Big]
 G^{(n_p-1),{\cal D}}_{I_p I_N}
-\label{Gt}
 \ee
 This expression is the path-integral representation of the Green's matrix using only the variables $(|I_k\rangle,n_k)$ of the effective dynamics defined over the set
-of domains. The standard formula derived above, Eq.(\ref{G}) may be considered as the particular case where the domain associated with each state is empty, 
+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, 
 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} 
 = \prod_{k=0}^{N-1} \langle I_k|F_{I_k}|I_{k+1} \rangle $
 where $F=T$.