minor corrections

g.tex

@@ -619,7 +619,7 @@ The time-dependent Green's matrix, Eq.~\eqref{eq:cn}, is then rewritten as
 	\prod_{k=0}^{N-1} \mel{ i_k }{ T }{ i_{k+1} } .
 \ee
 $\cC_p$ can be partitioned further by considering the subset of paths, denoted by $\cC_p^{I,n}$, 
-having $(\ket{I_k}, n_k)$ for $k=0$ to $p$ as exit states and trapping times.
+having $(\ket{I_k}, n_k)$, for $0 \le k \le p$, as exit states and trapping times.
 We recall that, by definition of the exit states, $\ket{I_k} \notin \cD_{I_{k-1}}$. The total time must be conserved, so the relation $\sum_{k=0}^p n_k= N+1$ 
 must be fulfilled.
 Now, the contribution of $\cC_p^{I,n}$ to the path integral is obtained by summing over all elementary paths $(i_1,...,i_{N-1})$ 
@@ -1009,7 +1009,7 @@ The two variational parameters of the trial vector have been optimized and fixed
 variational energy $E_\text{v}=-0.495361\ldots$. 
 In what follows, $\ket{I_0}$ is systematically chosen as one of the two N\'eel states, \eg, $\ket{I_0} = \ket{\uparrow,\downarrow, \uparrow,\ldots}$. 
 
-Figure \ref{fig3} shows the convergence of $H^{DMC}_p$, Eq.~\eqref{eq:defHp}, as a function of $p$ for different values of the reference energy $E$.
+Figure \ref{fig3} shows the convergence of $H^\text{DMC}_p$, Eq.~\eqref{eq:defHp}, as a function of $p$ for different values of the reference energy $E$.
 We consider the simplest case where a single-state domain is associated to each state.
 Five different values of $E$ have been chosen, namely $E=-1.6$, $-1.2$, $-1.0$, $-0.9$, and $-0.8$. 
 Only $H_0$ is computed analytically ($p_\text{ex}=0$). 
@@ -1129,7 +1129,7 @@ The extrapolated values obtained from the five values of the energy with three d
 Fitting the data using a simple linear function leads to an energy of $-0.7680282(5)$ (to be compared with the exact value of $-0.768068\ldots$). 
 A small bias of about $4 \times 10^{-5}$ is observed. 
 This bias vanishes within the statistical error when resorting to more flexible fitting functions, such as a quadratic function of $E$ or the
-two-component representation given by Eq.(\ref{eq:2comp}).
+two-component representation given by Eq.~\eqref{eq:2comp}.
 Our final value is in full agreement with the exact value with about six decimal places.
 
 Table \ref{tab4} shows the evolution of the average trapping times and extrapolated energies as a function of $U$ when using $\cD(0,1) \cup \cD(1,0)$ as the main domain. 
@@ -1154,7 +1154,7 @@ All these aspects will be considered in a forthcoming work.
 %%% TABLE II %%%
 \begin{table}
 \caption{One-dimensional Hubbard model with $N=4$, $U=12$, and $E=-1$. 
-Dependence of the statistical error on the energy with the number of $p$-components $p_{ex}$ calculated analytically in the expression 
+Dependence of the statistical error on the energy with the number of $p$-components $p_\text{ex}$ calculated analytically in the expression 
 of the energy, $\cE^\text{DMC}(E,p_\text{ex},p_\text{max})$, Eq.~\eqref{eq:E_pex}.
 The simulation is performed the same way as in Table \ref{tab1}. 
 Results are presented when a single-state domain is used for all states and when $\cD(0,1) \cup \cD(1,0)$ is chosen as the main domain.}