OK with Sec IIIA

g.tex

@@ -309,8 +309,6 @@ Here, we recall that a stochastic matrix is defined as a matrix with positive en
 	\sum_j p_{i \to j}=1.
 \ee
 To build the transition probability density, the following operator is introduced
-%As known, there is a natural way of associating a stochastic matrix to a matrix having a positive ground-state vector (here, a positive vector is defined here as 
-%a vector with all components positive). 
 \be
 \label{eq:T+}
 	T^+= \Id - \tau \qty( H^+ - \EL^+ \Id ), 
@@ -423,13 +421,6 @@ as it should.
 To calculate the probabilistic averages, an artificial (mathematical) ``particle'' called walker (or psi-particle) is introduced.
 During the Monte Carlo simulation, the walker moves in configuration space by drawing new states with 
 probability $p_{i_k \to i_{k+1}}$, thus realizing the path of probability  $\text{Prob}_{i_0}$. 
-%In this framework, the energy defined in Eq.~\eqref{eq:E0} is given by
-%\be
-%	E_0 = \lim_{N \to \infty } 
-%	\frac{ \expval{ \prod_{k=0}^{N-1} w_{i_{k}i_{k+1}} {(H\PsiT)}_{i_N}} }
-%	{ \expval{ \prod_{k=0}^{N-1} w_{i_{k}i_{k+1}} {\PsiT}_{i_N} }}.
-%\ee
-%A schematic algorithm is presented in Fig.\ref{scheme1B}.
 Note that, instead of using a single walker, it is common to introduce a population of independent walkers and to calculate the averages over this population.
 In addition, thanks to the ergodic property of the stochastic matrix (see, for example, Ref.~\onlinecite{Caffarel_1988}), a unique path of infinite length from which sub-paths of length $N$ can be extracted may also be used. 
 We shall not insist here on these practical details that are discussed, for example, in Refs.~\onlinecite{Foulkes_2001,Kolorenc_2011}.