OK with intro

g.tex

@@ -162,7 +162,7 @@ DMC can be used either for systems defined in a continuous configuration space (
 Here, we shall consider only the discrete case, that is, the general problem of calculating the lowest eigenvalue and/or eigenstate of a (very large) matrix.
 The generalization to continuous configuration spaces presents no fundamental difficulty.
 
-In essence, DMC is based on \textit{stochastic} power methods, a family of old and widely employed numerical approaches able to extract the largest or smallest eigenvalues of a matrix (see, \eg, Ref.~\onlinecite{Golub_2012}). 
+In essence, DMC is based on \textit{stochastic} power methods, a family of well established numerical approaches able to extract the largest or smallest eigenvalues of a matrix (see, \eg, Ref.~\onlinecite{Golub_2012}). 
 These approaches are particularly simple as they merely consist in applying a given matrix (or some simple function of it) as many times as required on some arbitrary vector belonging to the linear space. 
 Thus, the basic step of the corresponding algorithm essentially reduces to successive matrix-vector multiplications.
 In practice, power methods are employed under more sophisticated implementations, such as, \eg, the Lancz\`os algorithm (based on Krylov subspaces) \cite{Golub_2012} or Davidson's method where a diagonal preconditioning is performed. \cite{Davidson_1975} 
@@ -179,7 +179,7 @@ Another possibility, at the heart of the present work, is to integrate out exact
 
 In previous works,\cite{Assaraf_1999B,Caffarel_2000} it has been shown that the probability for a walker to stay a certain amount of time in the same state obeys a Poisson law and that the on-state dynamics can be integrated out to generate an effective dynamics connecting only different states with some renormalized estimators for the properties.
 Numerical applications have shown that the statistical errors can be very significantly decreased.
-Here, we extend this idea to the general case where a walker remains a certain amount of time within a finite domain no longer restricted to a single state. 
+Here, we extend this idea to the general case where a walker remains a certain amount of time in a finite domain no longer restricted to a single state. 
 It is shown how to define an effective stochastic dynamics describing walkers moving from one domain to another. 
 The equations of the effective dynamics are derived and a numerical application for a model (one-dimensional) problem is presented. 
 In particular, it shows that the statistical convergence of the energy can be greatly enhanced when domains associated with large average trapping times are considered.
@@ -187,28 +187,22 @@ In particular, it shows that the statistical convergence of the energy can be gr
 It should be noted that the use of domains in quantum Monte Carlo is not new. 
 In their pioneering work, \cite{Kalos_1974} Kalos and collaborators introduced the so-called domain Green's function Monte Carlo approach in continuous space that they applied to a system of bosons with hard-sphere interaction. 
 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. 
+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 one of us in Ref.~\onlinecite{Assaraf_1999B}.
+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}.
 
-Finally, from a general perspective, it is interesting to mention that the method proposed here is an example 
-of how combining stochastic and deterministic techniques can be valuable.
-%Here, a new stochastic process is built by taking advantage of a partial exact (deterministic) summation of local quantities.
+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.
-Let us mention, for example, 
-the semi-stochastic approach of Petruzielo {\it et al.},\cite{Petruzielo_2012} 
-two different methods for evaluating stochastically the second-order perturbation energy of selected CI methods\cite{Garniron_2017b,Sharma_2017}, 
-the stochastic approach of Willow {\it et al.} for the second-order many-body correction to the Hartree-Fock energy,\cite{Willow_2012} 
-or the zero variance Monte Carlo scheme for evaluating the two-electron integrals of quantum chemistry.\cite{Caffarel_2019}
+Let us mention, for example, the semi-stochastic approach of Petruzielo {\it et al.}, \cite{Petruzielo_2012} two different hybrid algorithms for evaluating the second-order perturbation energy in selected configuration interaction methods, \cite{Garniron_2017b,Sharma_2017} the approach of Willow \textit{et al.} for computing stochastically second-order many-body perturbation energies, \cite{Willow_2012} or the zero variance Monte Carlo scheme for evaluating two-electron integrals in quantum chemistry. \cite{Caffarel_2019}
 
 The paper is organized as follows. 
 Section \ref{sec:DMC} presents the basic equations and notations of DMC. 
 First, the path integral representation of the Green's function is given in Subsec.~\ref{sec:path}. 
-The probabilistic framework allowing the Monte Carlo calculation of the Green's function is presented in Subsec.~\ref{sec:proba}. 
+Second, the probabilistic framework allowing the Monte Carlo calculation of the Green's function is presented in Subsec.~\ref{sec:proba}. 
 Section \ref{sec:DMC_domains} is devoted to the use of domains in DMC. 
-First, we recall in Subsec.~\ref{sec:single_domains} the case of a domain consisting of a single state. \cite{Assaraf_1999B} 
+We recall in Subsec.~\ref{sec:single_domains} the case of a domain consisting of a single state. 
 The general case is then treated in Subsec.~\ref{sec:general_domains}. 
-In Subsec.~\ref{sec:Green}, both the time- and energy-dependent Green's function using domains is derived. 
+In Subsec.~\ref{sec:Green}, both the time- and energy-dependent Green's function using domains are derived. 
 Section \ref{sec:numerical} presents the application of the approach to the one-dimensional Hubbard model. 
 Finally, in Sec.\ref{sec:conclu}, some conclusions and perspectives are given.
 Atomic units are used throughout.
@@ -278,8 +272,10 @@ This is the central theme of the present work.
 %%% FIG 0 %%%
 \begin{figure}
 \includegraphics[width=\columnwidth]{fig1}
-\caption{Path integral representation of the exact coefficient $c_i=\braket{i}{\Psi_0}$ of the ground-state wavefunction obtained as an infinite sum of paths starting 
-from $\ket{i_0}$ and ending at $\ket{i}$, Eq.(\ref{eq:G}). Each path carries a weight $\prod_k T_{i_{k} i_{k+1}}$ computed along the path.
+\caption{
+\titou{$\Psi$ or $\Phi$?}
+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}]. Each path carries a weight $\prod_k T_{i_{k} i_{k+1}}$ computed along the path.
 The result is independent of the choice of the initial state  $\ket{i_0}$, provided that $\braket{i_0}{\Psi_0}$ is non-zero.
 Here, only four paths of infinite length have been represented. 
 }
@@ -546,10 +542,9 @@ the fact that each possible path can be decomposed in this way.
 \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 
-states $\ket{i_k}$ along the path are represented with small black solid circles and the exit states, $\ket{I_k}$, with larger black solid squares. 
-By convention the initial state, $\ket{i_0}$ is denoted using a capital letter, $\ket{I_0}$ since it will be the first state of the effective dynamics 
-involving only exit states. 
-See text, for some comments on the time evolution of the path.}
+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}
 \end{figure}