OK with Sec IIA

g.tex

@@ -206,6 +206,7 @@ In Subsec.~\ref{sec:Green}, both the time- and energy-dependent Green's function
 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.
+
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 \section{Diffusion Monte Carlo}
 \label{sec:DMC}
@@ -262,25 +263,25 @@ Each path is associated with a weight $\prod_{k=0}^{N-1} T_{i_{k} i_{k+1}}$ and
 
 This expression allows a simple and vivid interpretation of the solution. 
 In the limit $N \to \infty$, the $i_N$th component of the ground-state wave function (obtained as $\lim_{N \to \infty} G^{(N)}_{i_0 i_N})$ is the weighted sum over all possible paths arriving at vector $\ket{i_N}$. 
-This result is independent of the initial vector $\ket{i_0}$, apart from some irrelevant global phase factor. We illustrate this fundamental property pictorially in 
+This result is independent of the initial vector $\ket{i_0}$, apart from some irrelevant global phase factor. 
-Fig.\ref{fig1}.
+We illustrate this fundamental property pictorially in Fig.~\ref{fig:paths}.
 When the size of the linear space is small, the explicit calculation of the full sums involving $M^N$ terms (where $M$ is the size of the Hilbert space) can be performed. 
-In such a case, we are in the realm of what one would call the ``deterministic'' power methods, such as the Lancz\`os or Davidson approaches. 
+In such a case, we are in the realm of what one would call ``deterministic'' power methods, such as the Lancz\`os or Davidson approaches. 
 If not, probabilistic techniques for generating only the paths contributing significantly to the sums are to be preferred.
 This is the central theme of the present work.
 
-%%% FIG 0 %%%
+%%% FIG 1 %%%
-\begin{figure}
+\begin{figure*}
-\includegraphics[width=\columnwidth]{fig1}
+\includegraphics[width=0.7\textwidth]{fig1}
+\label{fig:paths}
 \caption{
-\titou{$\Psi$ or $\Phi$?}
+\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 
+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}]. 
-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.
+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 $\braket{i_0}{\Psi_0}$ is non-zero.
+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. 
 }
-\label{fig1}
+\end{figure*}
-\end{figure}
 
 
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