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OK with Sec IIIA

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Pierre-Francois Loos 2022-10-03 11:37:25 +02:00
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@ -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. \sum_j p_{i \to j}=1.
\ee \ee
To build the transition probability density, the following operator is introduced 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 \be
\label{eq:T+} \label{eq:T+}
T^+= \Id - \tau \qty( H^+ - \EL^+ \Id ), 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. 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 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 $\text{Prob}_{i_0}$. 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. 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. 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}. We shall not insist here on these practical details that are discussed, for example, in Refs.~\onlinecite{Foulkes_2001,Kolorenc_2011}.