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almost OK with Sec IIIB

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Pierre-Francois Loos 2022-10-03 13:50:23 +02:00
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@ -543,29 +543,19 @@ This probability is given by
= \sum_{\ket{i_1} \in \cD_{I_0}} \cdots \sum_{\ket{i_{n-1}} \in \cD_{I_0}} = \sum_{\ket{i_1} \in \cD_{I_0}} \cdots \sum_{\ket{i_{n-1}} \in \cD_{I_0}}
p_{I_0 \to i_1} \ldots p_{i_{n-2} \to i_{n-1}} p_{i_{n-1} \to I}. p_{I_0 \to i_1} \ldots p_{i_{n-2} \to i_{n-1}} p_{i_{n-1} \to I}.
\ee \ee
Since the sums are restricted to states belonging to the domain it is convenient to introduce a projector for over each domain Since the sums are restricted to states belonging to the domain, it is convenient to introduce a projector over each domain
\be \be
\label{eq:pi} \label{eq:pi}
P_I = \sum_{\ket{i} \in \cD_I} \dyad{i}{i}. P_I = \sum_{\ket{i} \in \cD_I} \dyad{i}{i},
\ee \ee
and, also, as well as the projection of $T^+$ over $\cD_I$, \ie, $T^+_I = P_I T^+ P_I$, which governs the dynamics of the walkers trapped in this domain.
the projection of the operator $T^+$ over the domain, \ie, %see Eq.~\eqref{eq:pij} where $T^+$ is now restricted to the domain.
\be
T^+_I= P_I T^+ P_I.
\ee
Th operator $T^+_I$ governs the dynamics of the walkers trapped in the domain $\cD_{I}$,
see Eq.(\ref{eq:pij}) where $T^+$ is now restricted to the domain.
Using Eqs.~\eqref{eq:pij} and \eqref{eq:eq1C}, the probability can be rewritten as Using Eqs.~\eqref{eq:pij} and \eqref{eq:eq1C}, the probability can be rewritten as
\be \be
\label{eq:eq3C} \label{eq:eq3C}
\cP_{I_0 \to I}(n) = \frac{1}{\PsiG_{I_0}} \mel{I_0}{\qty(T^+_{I_0})^{n-1} F^+_{I_0}}{I} \PsiG_{I}, \cP_{I_0 \to I}(n) = \frac{1}{\PsiG_{I_0}} \mel{I_0}{\qty(T^+_{I_0})^{n-1} F^+_{I_0}}{I} \PsiG_{I},
\ee \ee
where the operator where the operator $F^+_I = P_I T^+ (1-P_I)$, corresponding to the last move connecting the inside and outside regions of the domain, has the following matrix elements:
\be
\label{eq:Fi}
F^+_I = P_I T^+ (1-P_I),
\ee
corresponding to the last move connecting the inside and outside regions of the domain, has the following matrix elements:
\be \be
(F^+_I)_{ij} = (F^+_I)_{ij} =
\begin{cases} \begin{cases}
@ -582,9 +572,7 @@ the probability of being trapped $n$ times in $\cD_{I}$, just by summing over al
\label{eq:PiN} \label{eq:PiN}
P_{I}(n) = \frac{1}{\PsiG_{I}} \mel{ I }{ \qty(T^+_{I})^{n-1} F^+_{I} }{ \PsiG }. P_{I}(n) = \frac{1}{\PsiG_{I}} \mel{ I }{ \qty(T^+_{I})^{n-1} F^+_{I} }{ \PsiG }.
\ee \ee
The normalization to one of this probability can be verified The normalization of this probability can be verified using the fact that
by using
the fact that
\be \be
\label{eq:relation} \label{eq:relation}
\qty(T^+_{I})^{n-1} F^+_I = \qty(T^+_{I})^{n-1} T^+ - \qty(T^+_I)^n, \qty(T^+_{I})^{n-1} F^+_I = \qty(T^+_{I})^{n-1} T^+ - \qty(T^+_I)^n,