OK with Sec IIIA

g.tex

@@ -418,43 +418,42 @@ We shall not here insist on these practical details that are discussed, for exam
 
 During the simulation, walkers move from state to state with the possibility of being trapped a certain number of times on the same state before 
 exiting to a different state. This fact can be exploited in order to integrate out some part of the dynamics, thus leading to a reduction of the statistical 
-fluctuations. This idea was proposed some time ago \cite{Assaraf_1999A,Assaraf_1999B,Caffarel_2000} and applied to the SU(N) one-dimensional Hubbard model.
+fluctuations. This idea was proposed some time ago \cite{Assaraf_1999A,Assaraf_1999B,Caffarel_2000} and applied to the SU($N$) one-dimensional Hubbard model.
 
-Let us consider a given state $|i\rangle$. The probability that the walker remains exactly $n$ times on $|i\rangle$ ($n$ from                                      
-1 to $\infty$) and then exits to a different state $j$ is
+Considering a given state $\ket{i}$, the probability that a walker remains exactly $n$ times in $\ket{i}$ (with $1 \le n < \infty$) and then exits to a different state $j$ (with $j \neq i$) is
 \be
-	\cP_{i \to j}(n) = \qty(p_{i \to i})^{n-1}  p_{i \to j} \qq{$j \ne i$.}
+	\cP_{i \to j}(n) = \qty(p_{i \to i})^{n-1}  p_{i \to j}.
 \ee
 Using the relation 
 \be
 	\sum_{n=1}^{\infty} (p_{i \to i})^{n-1}=\frac{1}{1-p_{i \to i}}
 \ee
-and the normalization of the $p_{i \to j}$, Eq.~\eqref{eq:sumup}, we verify that the probability is normalized to one 
+and the normalization of the $p_{i \to j}$'s [see Eq.~\eqref{eq:sumup}], one can check that the probability is properly normalized, \ie,
 \be
 	\sum_{j \ne i} \sum_{n=1}^{\infty} \cP_{i \to j}(n) = 1.
 \ee
 
-The probability of being trapped during $n$ steps is obtained by summing over all possible exit states
+Naturally, the probability of being trapped during $n$ steps is obtained by summing over all possible exit states
 \be
-P_i(n)=\sum_{j \ne i} \cP_{i \to j}(n) = \qty(p_{i \to i})^{n-1} \qty( 1 - p_{i \to i} ).
+P_i(n)=\sum_{j \ne i} \cP_{i \to j}(n) = \qty(p_{i \to i})^{n-1} \qty( 1 - p_{i \to i} ),
 \ee
-This probability defines a Poisson law with an average number $\bar{n}_i$ of trapping events given by
+and this defines a Poisson law with an average number of trapping events
 \be
 	\bar{n}_i= \sum_{n=1}^{\infty} n P_i(n) = \frac{1}{1 -p_{i \to i}}.
 \ee
-Introducing the continuous time $t_i=n_i\tau$ the average trapping time is given by
+Introducing the continuous time $t_i = n \tau$,  the average trapping time is thus given by
 \be
 \bar{t_i}= \frac{1}{H^+_{ii}-(\EL^+)_{i}}.
 \ee
-Taking the limit $\tau \to 0$, the Poisson probability takes the usual form
+In the limit $\tau \to 0$, the Poisson probability takes the usual form
 \be
-	P_{i}(t) = \frac{1}{\bar{t}_i} \exp(-\frac{t}{\bar{t}_i})
+	P_{i}(t) = \frac{1}{\bar{t}_i} \exp(-\frac{t}{\bar{t}_i}).
 \ee
-The time-averaged contribution of the \titou{on-state} weight can be easily calculated to be
+The time-averaged contribution of the on-state weight can be easily calculated to be
 \be
 	\bar{w}_i= \sum_{n=1}^{\infty} w^n_{ii} P_i(n)= \frac{T_{ii}}{T^+_{ii}} \frac{1-T^+_{ii}}{1-T_{ii}}
 \ee
-Details of the implementation of the effective dynamics can be in found in Refs.~\onlinecite{Assaraf_1999B} and \onlinecite{Caffarel_2000}.
+Details of the implementation of this effective dynamics can be in found in Refs.~\onlinecite{Assaraf_1999B} and \onlinecite{Caffarel_2000}.
 
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 \subsection{General domains}