saving work

g.tex

@@ -189,7 +189,7 @@ In their pioneering work, \cite{Kalos_1974} Kalos and collaborators introduced t
 The domain used was the Cartesian product of small spheres around each particle, the Hamiltonian being approximated by the kinetic part only within the domain.
 Some time later, Kalos proposed to extend these ideas to more general domains such as rectangular and/or cylindrical domains. \cite{Kalos_2000} In both works, the size of the domains is infinitely small in the limit of a vanishing time step. 
 Here, the domains are of arbitrary size, thus greatly increasing the efficiency of the approach.
-Note also that some general equations for arbitrary domains in continuous space have also been proposed by \titou{some} of us in Ref.~\onlinecite{Assaraf_1999B}.
+Note also that some general equations for arbitrary domains in continuous space have also been proposed by some of us in Ref.~\onlinecite{Assaraf_1999B}.
 
 Finally, from a general perspective, it is interesting to mention that the method proposed here is an illustration of how valuable and efficient can be the combination of stochastic and deterministic techniques.
 In recent years, a number of works have exploited this idea and proposed hybrid stochastic/deterministic schemes.
@@ -273,14 +273,12 @@ This is the central theme of the present work.
 %%% FIG 1 %%%
 \begin{figure*}
 \includegraphics[width=0.7\textwidth]{fig1}
-\label{fig:paths}
 \caption{
-\titou{$\Psi_0$ or $\Phi_0$?}
-Path integral representation of the exact coefficient $c_i=\braket{i}{\Psi_0}$ of the ground-state wave function $\ket{\Psi_0}$ obtained as an infinite sum of paths starting from $\ket{i_0}$ and ending at $\ket{i}$ [see Eq.\eqref{eq:G}]. 
+Path integral representation of the exact coefficient $c_i=\braket{i}{\Phi_0}$ of the ground-state wave function $\ket{\Phi_0}$ obtained as an infinite sum of paths starting from $\ket{i_0}$ and ending at $\ket{i}$ [see Eq.\eqref{eq:G}]. 
 Each path carries a weight $\prod_k T_{i_{k} i_{k+1}}$ computed along it. 
-The result is independent of the choice of the initial state  $\ket{i_0}$, provided that $\braket{i_0}{\Psi_0} \neq 0$.
-Here, only four paths of infinite length have been represented. 
-}
+The result is independent of the choice of the initial state  $\ket{i_0}$, provided that $\braket{i_0}{\Phi_0} \neq 0$.
+Here, only four paths of infinite length have been represented.}
+\label{fig:paths}
 \end{figure*}
 
 
@@ -422,7 +420,7 @@ To calculate the probabilistic averages, an artificial (mathematical) ``particle
 During the Monte Carlo simulation, the walker moves in configuration space by drawing new states with 
 probability $p_{i_k \to i_{k+1}}$, thus realizing the path of probability  $\text{Prob}_{i_0}$. 
 Note that, instead of using a single walker, it is common to introduce a population of independent walkers and to calculate the averages over this population.
-In addition, thanks to the ergodic property of the stochastic matrix (see, for example, Ref.~\onlinecite{Caffarel_1988}), a unique path of infinite length from which sub-paths of length $N$ can be extracted may also be used. 
+In addition, thanks to the ergodicity property of the stochastic matrix (see, for example, Ref.~\onlinecite{Caffarel_1988}), a unique path of infinite length from which sub-paths of length $N$ can be extracted may also be used. 
 We shall not insist here on these practical details that are discussed, for example, in Refs.~\onlinecite{Foulkes_2001,Kolorenc_2011}.
 
 %{\it Spawner representation} In this representation, we no longer consider moving particles but occupied or non-occupied states $|i\rangle$.
@@ -501,7 +499,7 @@ 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, $p_{i \to j}$. 
+where the successive states are drawn using the transition probability matrix $p_{i \to j}$. 
 This series can be recast
 \be
 \label{eq:eff_series}
@@ -510,19 +508,17 @@ This series can be recast
 where $\ket{I_0}=\ket{i_0}$ is the initial state, $n_0$ is the number of times the walker remains in 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 $p$ (with $0 \le p \le N$) indicates the number of exit events occurring along the path. 
 The two extreme values, $p=0$ and $p=N$, correspond to the cases where the walker remains in the initial domain during the entire path, and where the walker exits a domain at each step, respectively. 
-In what follows, we shall systematically write the integers representing the exit states in capital letter, while small letters will be used for 
-denoting the elementary states $\ket{i_k}$ generated with $p_{i \to j}$. Making this distinction is important since the effective 
-stochastic dynamics used in practical Monte Carlo calculations will only involve exit states, the contribution of the elementary states, $\ket{i_k}$, being 
-exactly integrated out.
-Figure \ref{fig2} exemplifies how a path can be decomposed as proposed in Eq.(\ref{eq:eff_series}).
-To make things as clear as possible, let us explicit in detail how the path drawn in Figure \ref{fig2} evolves in time.
-The walker realizing the path starts at $\ket{i_0}$ within the domain $\cD_{i_0}$. It then makes two steps to arrive at $\ket{i_1}$, then $\ket{i_2}$ and, finally, leaves the domain 
-at $\ket{i_3}$. The state $\ket{i_3}$ is the first exit state and is denoted as $\ket{I_1}(=\ket{i_3})$ following our convention of denoting exit states 
-with capital letters. The trapping time in $\cD_{i_0}$ is $n_0=3$ since three states of the domain have been visited 
-(namely, $\ket{i_0}$,$\ket{i_1}$,and $\ket{i_2}$).
+In what follows, we shall systematically label exit states with upper-case letters, while lower-case letters denote elementary states $\ket{i_k}$.
+Making this distinction is important since the effective stochastic dynamics used in practical Monte Carlo calculations only involve exit states $\ket{I_k}$, the contribution from the elementary states $\ket{i_k}$ being exactly integrated out.
+
+\titou{Figure \ref{fig:domains} exemplifies how a path can be decomposed as proposed in Eq.~\eqref{eq:eff_series}.
+To make things as clear as possible, let us explicit in detail how the path drawn in Fig.~\ref{fig:domains} evolves in time.
+The walker realizing the path starts at $\ket{i_0}$ within the domain $\cD_{i_0}$. It then makes two steps to arrive at $\ket{i_1}$, then $\ket{i_2}$ and, finally, leaves the domain at $\ket{i_3}$. 
+The state $\ket{i_3}$ is the first exit state and is denoted as $\ket{I_1}(=\ket{i_3})$ following our convention of denoting exit states with capital letters. 
+The trapping time in $\cD_{i_0}$ is $n_0=3$ since three states of the domain have been visited (namely, $\ket{i_0}$,$\ket{i_1}$,and $\ket{i_2}$).
 During the next steps the domains $\cD_{I_1}$, $\cD_{I_2}$, and $\cD_{I_3}$ are successively visited with $n_1=2$, $n_2=3$, and $n_3=1$, respectively. 
-The corresponding exit states are $\ket{I_2}=\ket{i_5}$, $\ket{I_3}=\ket{i_8}$, and $\ket{I_4}=\ket{i_9}$, respectively. This work takes advantage of 
-the fact that each possible path can be decomposed in this way.
+The corresponding exit states are $\ket{I_2}=\ket{i_5}$, $\ket{I_3}=\ket{i_8}$, and $\ket{I_4}=\ket{i_9}$, respectively. 
+This work takes advantage of the fact that each possible path can be decomposed in this way.}
 
 %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 
@@ -531,15 +527,16 @@ the fact that each possible path can be decomposed in this way.
 %%% FIG 0B %%%
 \begin{figure}
 \includegraphics[width=\columnwidth,angle=0]{fig2}
-\caption{Representation of a path in terms of exit states, $\ket{I_k}$ and trapping times, $\ket{n_k}$. The 
+\caption{Representation of a path in terms of exit states $\ket{I_k}$ and trapping times $\ket{n_k}$. The 
 states $\ket{i_k}$ along the path are represented by small black circles and the exit states, $\ket{I_k}$, by larger black squares. 
 By convention, the initial state is denoted using a capital letter, \ie,  $\ket{i_0} = \ket{I_0}$, since it is the first state of the effective dynamics involving only exit states. 
 See text for additional comments on the time evolution of the path.}
-\label{fig2}
+\label{fig:domains}
 \end{figure}
 
 
-Now, generalizing what has been done previously for a single-state domain, let us define the probability of remaining $n$ times in the domain of $\ket{I_0}$ and to exit at $\ket{I} \notin \cD_{I_0}$. This probability is given by
+Generalizing the single-state case treated previously, let us define the probability of remaining $n$ times in the domain of $\ket{I_0}$ and to exit at $\ket{I} \notin \cD_{I_0}$. 
+This probability is given by
 \be
 \label{eq:eq1C}
 	\cP_{I_0 \to I}(n) 
@@ -556,7 +553,7 @@ the projection of the operator $T^+$ over the domain, \ie,
 \be
 	T^+_I= P_I T^+ P_I.
 \ee
-Th operator $T^+_I$  governs the dynamics of the walkers trapped within the domain $\cD_{I}$,
+Th operator $T^+_I$  governs the dynamics of the walkers trapped in the domain $\cD_{I}$,
 see Eq.(\ref{eq:pij}) where $T^+$ is now restricted to the domain.
 Using Eqs.~\eqref{eq:pij} and \eqref{eq:eq1C}, the probability can be rewritten as
 \be