OK for Titou

g.tex

@@ -119,7 +119,7 @@
 \newcommand{\LCT}{Laboratoire de Chimie Th\'eorique, Sorbonne-Universit\'e, Paris, France}
 
 \begin{document}	
-\title{Diffusion Monte Carlo using Domains in Configuration Space}
+\title{Diffusion Monte Carlo using domains in configuration space}
 \author{Roland Assaraf}
         \email{assaraf@lct.jussieu.fr}
         \affiliation{\LCT}
@@ -509,17 +509,17 @@ 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 label exit states with upper-case letters, while lower-case letters denote elementary states $\ket{i_k}$.
+In what follows, we shall systematically label exit states with upper-case letters $\ket{I_k}$, 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. 
+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 describe how the path drawn in Fig.~\ref{fig:domains} evolves in time.
+The walker starts at $\ket{i_0}$ in the domain $\cD_{i_0}$. 
+Then, it performs two steps from $\ket{i_0}$ to $\ket{i_2}$ and has left the domain at $\ket{i_3}$, which is thus the first exit state, \ie, $\ket{I_1} = \ket{i_3}$. 
+The trapping time in $\cD_{i_0}$ is $n_0=3$ since three states in $\cD_{i_0}$ 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.}
+%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 
@@ -601,7 +601,7 @@ Note that it is possible only if the dimension of the domains is not too large (
 In this section we generalize the path-integral expression of the Green's matrix, Eq.~\eqref{eq:G}, to the case where domains are used.
 To do so, we introduce the Green's matrix associated with each domain as follows:
 \be
-	G^{(N),\cD}_{ij}= \mel{ i }{ \titou{T_i^N} }{ j }.
+	G^{(N),\cD}_{ij}= \mel{ i }{ T_i^N }{ j }.
 \ee
 %Starting from Eq.~\eqref{eq:cn}
 %\be
@@ -662,7 +662,7 @@ and using the effective transition probability, Eq.~\eqref{eq:eq3C}, we get
 	\qty( \prod_{k=0}^{p-1} W_{I_k I_{k+1}} ) \qty( \prod_{k=0}^{p-1}\cP_{I_k \to I_{k+1}}(n_k)  )
 	 \frac{1}{\PsiG_{I_p}} {G}^{(n_p-1), \cD}_{I_p i_N} },
 \end{multline}
-\titou{where, for clarity, $\sum_{{(I,n)}_{p,N}}$ is used as a short-hand notation for the sum, $\sum_{\ket{I_1} \notin \cD_{I_0}, \ldots , \ket{I_p} \notin \cD_{I_{p-1}}} \sum_{n_0 \ge 1} \cdots \sum_{n_p \ge 1}$ with the constraint $\sum_{k=0}^p n_k=N+1$.}
+where, for clarity, $\sum_{{(I,n)}_{p,N}}$ is used as a short-hand notation for $\sum_{\ket{I_1} \notin \cD_{I_0}, \ldots , \ket{I_p} \notin \cD_{I_{p-1}}} \sum_{n_0 \ge 1} \cdots \sum_{n_p \ge 1}$ with the constraint $\sum_{k=0}^p n_k=N+1$.
 
 Under this form, ${G}^{(N)}_{i_0 i_N}$ is now amenable to Monte Carlo calculations 
 by generating paths using the transition probability matrix $\cP_{I \to J}(n)$. 
@@ -678,7 +678,7 @@ which can be rewritten probabilistically as
         \frac{ {G}^{(N),\cD}_{i_0 i_N} + {\PsiG_{i_0}}  \sum_{p=1}^{N}  \expval{ \qty( \prod_{k=0}^{p-1} W_{I_k I_{k+1}} ) {\cal H}_{n_p,I_p} }_p}
              { {G}^{(N),\cD}_{i_0 i_N} + {\PsiG_{i_0}} \sum_{p=1}^{N}  \expval{ \qty( \prod_{k=0}^{p-1} W_{I_k I_{k+1}} ) {\cal S}_{n_p,I_p} }_p},
 \ee
-\titou{where $\expval{\cdots}_p$ is the probabilistic average defined over the set of paths $p$ exit events of probability $\prod_{k=0}^{p-1} \cP_{I_k \to I_{k+1}}(n_k)$} and 
+where $\expval{\cdots}_p$ is the probabilistic average defined over the set of paths with $p$ exit events of probability $\prod_{k=0}^{p-1} \cP_{I_k \to I_{k+1}}(n_k)$, and 
 \begin{align}
 	\cH_{n_p,I_p} & = \frac{1}{\PsiG_{I_p}} \sum_{i_N} {G}^{(n_p-1),\cD}_{I_p i_N} (H \Psi_T)_{i_N},
 	\\
@@ -740,7 +740,7 @@ Using the fact that $G^E_{ij}= \tau \sum_{N=0}^{\infty} G^{(N)}_{ij}$, where $G^
 	\qty[ \prod_{k=0}^{p-1} \mel{ I_k }{ {\qty[ P_k \qty( H-E \Id ) P_k ] }^{-1} (-H)(\Id-P_k) }{ I_{k+1} } ]
        {G}^{E,\cD}_{I_p i_N},
 \end{multline}
-where, ${G}^{E,\cD}$ is the energy-dependent domain's Green matrix defined as ${G}^{E,\cD}_{ij} = \tau \sum_{N=0}^{\infty} \mel{ i }{ \titou{T^N_i} }{ j}$.
+where, ${G}^{E,\cD}$ is the energy-dependent domain's Green matrix defined as ${G}^{E,\cD}_{ij} = \tau \sum_{N=0}^{\infty} \mel{ i }{ T^N_i }{ j}$.
 
 As a didactical example, Appendix \ref{app:A} reports the exact derivation of this formula in the case of a two-state system.
 
@@ -822,7 +822,7 @@ of the non-linear equation $\cE(E)= E$ in the vicinity of $E_0$.
 In practical Monte Carlo calculations, the DMC energy is obtained by computing a finite number of components $H_p$ and $S_p$ defined as follows
 \be
 \cE^\text{DMC}(E,p_{max})= \frac{ H_0+ \sum_{p=1}^{p_\text{max}} H^\text{DMC}_p }
-                         {S_{\titou{0}} +\sum_{p=1}^{p_\text{max}} S^\text{DMC}_p }.
+                         {S_0 +\sum_{p=1}^{p_\text{max}} S^\text{DMC}_p }.
 \ee
 For $ p\ge 1$, Eq.~\eqref{eq:final_E} gives
 \begin{align}
@@ -860,7 +860,8 @@ Finally, $\cE^\text{DMC}(E,p_\text{ex},p_\text{max})$ is evaluated in practice a
                          {                 \sum_{p=0}^{p_\text{ex}-1} S_p +\sum_{p=p_\text{ex}}^{p_\text{max}} S^\text{DMC}_p },
 \ee
 where $p_\text{ex}$ is the number of components computed exactly. 
-\titou{Let us emphasize that, since the magnitude of $H_p$ and $S_p$ decreases as a function of $p$, to remove the statistical error for the first most important contributions can lead to important gains, as illustrated in the numerical application to follow.}
+Let us emphasize that, since the magnitude of $H_p$ and $S_p$ decreases as a function of $p$, most of the statistical error is removed by computing the dominant terms analytically.
+This will be illustrated in the numerical application presented below.
 
 It is easy to check that, in the vicinity of the exact energy $E_0$, $\cE(E)$ is a linear function of $E - E_0$.
 Therefore, in practice, we compute the value of $\cE(E)$ for several values of $E$, and fit these data using a linear, quadratic or a more complicated function of $E$ in order to obtain, via extrapolation, an estimate of $E_0$.