text:
\section{Introduction}
\label{SecIntro}
Let $\mathcal{G}=(V,E)$ be a connected undirected graph, with $V$ at most countable and each vertex $x\in V$ of finite degree. We do not allow self-loops, however the edges might be multiple. Given $e\in E$ an edge, we will denote
$e_{+}$ and $e_{-}$ its end-vertices, even though $e$ is non-oriented and one can interchange $e_{+}$ and $e_{-}$.
Each edge $e\in E$ is endowed with a conductance $W_{e}>0$. There may be a killing measure $\kappa=(\kappa_{x})_{x\in V}$ on vertices.

We consider $(X_{t})_{t\ge0}$ the \textit{Markov jump processes} on $V$ which being in $x\in V$, jumps along an adjacent edge $e$ with rate $W_{e}$. Moreover if
$\kappa_{x}\neq 0$, the process is killed at $x$ with rate $\kappa_{x}$ (the process is not defined after that time). $\zeta$ will denote the time up to which $X_{t}$ is defined. If $\zeta<+\infty$, then either the process has been killed by the killing measure $\kappa$ (and $\kappa \not\equiv 0$) or it has gone off to infinity in finite time
(and $V$ infinite). We will assume that the process $X$ is transient, which means, if $V$ is finite, that
$\kappa\not\equiv 0$. $\mathbb{P}_{x}$ will denote the law of $X$ started from $x$.
Let $(G(x,y))_{x,y\in V}$ be the Green function of $X_{t}$:
\begin{displaymath}
G(x,y)=G(y,x)=\mathbb{E}_{x}\left[\int_{0}^{\zeta} 1_{\{X_{t}=y\}} dt\right].
\end{displaymath}
Let $\mathcal{E}$ be the Dirichlet form defined on functions $f$ on $V$ with finite support:
\begin{eqnarray}\label{Dirichlet-form}
\mathcal{E}(f,)=\sum_{x\in V}\kappa_{x} f(x)^{2}+
\sum_{e\in E}W_e(f(e_{+})-f(e_{-}))^{2}.
\end{eqnarray}
$P_{\varphi}$ will be the law of $(\varphi_{x})_{x\in V}$ the centred
\textit{Gaussian free field} (GFF) on $V$ with covariance
$E_{\varphi}[\varphi_{x}\varphi_{y}]=G(x,y)$. In case $V$ is finite, the density of $P_{\varphi}$ is
\begin{displaymath}
\dfrac{1}{(2\pi)^{\frac{\vert V\vert}{2}}\sqrt{\det G}}
\exp\left(-\dfrac{1}{2}\mathcal{E}(f,f)\right)\prod_{x\in V} df_{x}.
\end{displaymath}
Given $U$ a finite subset of $V$, and $f$ a function on $U$, $P^{U,f}_{\varphi}$ will denote the law of
the GFF $\varphi$ conditioned to equal $f$ on $U$.
$(\ell_{x}(t))_{x\in V, t\in [0,\zeta]}$ will denote the family of local times of $X$:
\begin{displaymath}
\ell_{x}(t)=\int_{0}^{t}1_{\{X_{s}=x\}} ds.
\end{displaymath}
For all $x\in V$, $u>0$, let
\begin{displaymath}
\tau_{u}^{x}=\inf\lbrace t\geq 0; \ell_{x}(t)>u\rbrace.
\end{displaymath}
Recall the generalized second Ray-Knight theorem on discrete graphs by Eisenbaum, Kaspi, Marcus, Rosen and Shi \cite{ekmrs} (see also
\cite{MarcusRosen2006MarkovGaussianLocTime,Sznitman2012LectureIso}):
\begin{2ndRK}
For any $u>0$ and $x_{0}\in V$,
\begin{center}
$\left(\ell_{x}(\tau_{u}^{x_{0}})+\dfrac{1}{2}\varphi_{x}^{2}\right)_{x\in V}$
under $\mathbb{P}_{x_{0}}(\cdot \vert \tau_{u}^{x_{0}}<\zeta)\otimes P^{\lbrace x_{0}\rbrace,0}_{\varphi}$
\end{center}
has the same law as
\begin{center}
$\left(\dfrac{1}{2}\varphi_{x}^{2}\right)_{x\in V}$
under $P^{\lbrace x_{0}\rbrace,\sqrt{2u}}_{\varphi}$.
\end{center}
\end{2ndRK}

Sabot and Tarrès showed in \cite{SabotTarres2015RK} that the so-called ``magnetized'' reverse Vertex-Reinforced Jump Process provides an inversion of the generalized second Ray-Knight theorem, in the sense that it enables to retrieve the law of $(\ell_x(\tau_u^{x_0}), \varphi^2_x)_{x\in V}$ conditioned on $\left(\ell_x(\tau_u^{x_0})+\frac{1}{2}\varphi^2_x\right)_{x\in V}$. The jump rates of that latter process can be interpreted as the two-point functions of the Ising model associated to the time-evolving weights.

However in \cite{SabotTarres2015RK} the link with the Ising model is only implicit, and a natural question is whether Ray-Knight inversion can be described in a simpler form if we enlarge the state space of the dynamics, and in particular include the ``hidden'' spin variables.

The answer is positive, and goes through an extension  of the Ray-Knight isomorphism introduced by Lupu \cite{Lupu2014LoopsGFF}, which couples the sign of the GFF to the path of the Markov chain. The Ray-Knight inversion will turn out to take a rather simple form  in Theorem \ref{thm-Poisson} of the present paper, where it will be defined not only through the spin variables but also random currents associated to the field though an extra Poisson Point Process.

The paper is organised as follows.

In Section \ref{sec:srk} we recall some background on loop soup isomorphisms and on related couplings and state and prove a signed version of generalized second Ray-Knight theorem. We begin in Section \ref{sec:lejan} by a statement of Le Jan's isomorphism which couples the square of the Gaussian Free Field to the loop soups, and recall how the generalized second Ray-Knight theorem can be seen as its Corollary: for more details see \cite{lejan4}. In Subsection \ref{sec:lupu} we state Lupu's isomorphism which extends Le Jan's isomorphism and couples the  sign of the GFF to the loop soups, using  a  cable graph extension of the GFF and Markov Chain. Lupu's isomorphism yields an interesting realisation of the well-known  FK-Ising coupling,  and provides as well a ``Current+Bernoulli=FK'' coupling lemma \cite{lupu-werner}, which occur in the relationship between the discrete and cable graph versions. We briefly recall those couplings in Sections \ref{fkising} and \ref{randomcurrent}, as they are implicit in this paper. In Section \ref{sec:glupu} we state and prove the generalized second Ray-Knight ``version'' of Lupu's isomorphism, which we aim to invert.

Section \ref{sec:inversion} is devoted to the statements of inversions of those isomorphisms. We state in Section \ref{sec_Poisson} a signed version of the inversion of the generalized second Ray-Knight theorem through an extra Poisson Point Process, namely Theorem \ref{thm-Poisson}. In Section \ref{sec_dicr_time} we provide a discrete-time description of the process, whereas in Section \ref{sec_jump} we yield  an alternative version of that process through jump rates, which can be seen as an annealed version of the first one. We deduce a signed inversion of Le Jan's isomorphism for loop soups in Section \ref{sec:lejaninv}, and an inversion of the coupling of random current with FK-Ising in Section \ref{sec:coupinv}

Finally Section \ref{sec:proof} is devoted to the proof of Theorem \ref{thm-Poisson}: Section \ref{sec:pfinite} deals with the case of a finite graph without killing measure, and Section \ref{sec:pgen} deduces the proof in the general case.

\section{Le Jan's and Lupu's isomorphisms}
\label{sec:srk}
\subsection{Loop soups and Le Jan's isomorphism}
\label{sec:lejan}
The \textit{loop measure} associated to the Markov jump process
$(X_{t})_{0\leq t<\zeta}$ is defined as follows. Let $\mathbb{P}^{t}_{x,y}$ be the bridge probability measure from
$x$ to $y$ in time $t$ (conditionned on $t<\zeta$). Let $p_{t}(x,y)$ be the transition probabilities of
$(X_{t})_{0\leq t<\zeta}$.

Let $\mu_{\rm loop}$ be the measure on time-parametrised nearest-neighbour based loops (i.e. loops with a starting site)
\begin{displaymath}
\mu_{\rm loop}=\sum_{x\in V}\int_{t>0}\mathbb{P}^{t}_{x,x} p_{t}(x,x) \dfrac{dt}{t}.
\end{displaymath}
The loops will be considered here up to a rotation of parametrisation (with the corresponding pushforward measure induced by $\mu_{\rm loop}$), that is to say a loop $(\gamma(t))_{0\leq t\leq t_{\gamma}}$ will be the same as
$(\gamma(T+t))_{0\leq t\leq t_{\gamma}-T}\circ (\gamma(T+t-t_{\gamma}))_{t_{\gamma}-T\leq t\leq t_{\gamma}}$, where $\circ$ denotes the concatenation of paths.
A \textit{loop soup} of intensity $\alpha>0$, denoted
$\mathcal{L}_{\alpha}$, is a Poisson random measure of intensity
$\alpha \mu_{\rm loop}$. We see it as a random collection of loops in $\mathcal{G}$. Observe that a.s. above each vertex
$x\in V$, $\mathcal{L}_{\alpha}$ contains infinitely many trivial "loops" reduced to the vertex $x$. There are also with positive probability non-trivial loop that visit several vertices.

Let  $L_{.}(\mathcal{L}_{\alpha})$ be the \textit{occupation field} of $\mathcal{L}_{\alpha}$ on $V$ i.e., for all $x\in V$,
\begin{displaymath}
L_x(\mathcal{L}_{\alpha})=
\sum_{(\gamma(t))_{0\leq t\leq t_{\gamma}}\in\mathcal{L}_{\alpha}}
\int_{0}^{t_{\gamma}}1_{\{\gamma(t)=x\}} dt.
\end{displaymath}
In \cite{LeJan2011Loops} Le Jan shows that for transient Markov jump processes, $L_x(\mathcal{L}_{\alpha})<+\infty$ for all $x\in V$ a.s. For $\alpha=\frac{1}{2}$ he identifies the law of
$L_.(\mathcal{L}_{\alpha})$:
\begin{IsoLeJan}
$L_.(\mathcal{L}_{1/2})=\left(L_x(\mathcal{L}_{1/2})\right)_{x\in V}$ has the same law as
$\dfrac{1}{2}\varphi^2=\left(\dfrac{1}{2}\varphi_{x}^{2}\right)_{x\in V}$
under $P_{\varphi}$.
\end{IsoLeJan}

Let us briefly recall how Le Jan's isomorphism enables one to retrieve the generalized second Ray-Knight theorem stated in Section \ref{SecIntro}: for more details, see for instance \cite{lejan4}. We assume that $\kappa$ is supported by $x_0$: the general case can be dealt with by an argument similar to the proof of Proposition \ref{PropKillingCase}.
Let $D=V\setminus\{x_0\}$, and note that the isomorphism in particular implies that $L_.(\mathcal{L}_{1/2})$ conditionally on $L_{x_0}(\mathcal{L}_{1/2})=u$ has the same law as $\varphi^2/2$ conditionally on $\varphi_{x_0}^2/2=u$.

On the one hand, given the classical energy decomposition, we have $\varphi=\varphi^D+\varphi_{x_0}$, with $\varphi^D$ the GFF associated to the restriction of $\mathcal{E}$ to $D$, where $\varphi^D$ and $\varphi_{x_0}$ are independent. Now $\varphi^2/2$ conditionally on $\varphi_{x_0}^2/2=u$ has the law of $(\varphi^D+\eta\sqrt{2u})^2/2$, where $\eta$ is the sign of $\varphi_{x_0}$, which is independent of $\varphi^D$. But $\varphi^D$ is symmetric, so that the latter also has the law of $(\varphi^D+\sqrt{2u})^2/2$.

On the other hand, the loop soup $\mathcal{L}_{1/2}$ can be decomposed into the two independent loop soups $\mathcal{L}_{1/2}^D$ contained in $D$ and $\mathcal{L}_{1/2}^{(x_0)}$ hitting $x_0$. Now $L_.(\mathcal{L}_{1/2}^D)$ has the law of $(\varphi^D)^2/2$ and  $L_.(\mathcal{L}_{1/2}^{(x_0)})$ conditionally on $L_{x_0}(\mathcal{L}_{1/2}^{(x_0)})=u$ has the law of the occupation field of the Markov chain $\ell(\tau_{u}^{x_{0}})$
under $\mathbb{P}_{x_{0}}(\cdot \vert \tau_{u}^{x_{0}}<\zeta)$, which enables us to conclude.

\subsection{Lupu's isomorphism}
\label{sec:lupu}
As in \cite{Lupu2014LoopsGFF}, we consider the \textit{metric graph} $\tilde{\mathcal{G}}$ associated to $\mathcal{G}$. Each edge $e$ is replaced by a continuous line of length
$\frac{1}{2}W_{e}^{-1}$.

The GFF $\varphi$ on $\mathcal{G}$ with law $P_\varphi$ can be extended to a GFF $\tilde{\varphi}$ on $\tilde{\mathcal{G}}$ as follows. Given $e\in E$, one considers inside $e$ a conditionally independent Brownian bridge, actually a bridge of a $\sqrt{2} \times$
\textit{standard Brownian motion}, of length $\frac{1}{2}W_{e}^{-1}$, with end-values
$\varphi_{e_{-}}$ and $\varphi_{e_{+}}$. This provides a continuous field on the metric graph which satisfies the spatial Markov property.

Similarly one can define a standard Brownian motion $(B^{\tilde{\mathcal{G}}})_{0\le t\le \tilde{\zeta}}$ on $\tilde{\mathcal{G}}$, whose trace on $\mathcal{G}$ indexed by the local times at $V$ has the same law as the Markov process $(X_t)_{t\ge0}$ on $V$ with jump rate $W_e$ to an adjacent edge $e$ up to time $\zeta$, as explained in Section 2 of \cite{Lupu2014LoopsGFF}. One can associate a measure on time-parametrized continuous loops $\tilde{\mu}$, and let $\tilde{\mathcal{L}}_{\frac{1}{2}}$ be the Poisson Point Process of loops of intensity $\tilde{\mu}/2$: the discrete-time loops $\mathcal{L}_{\frac{1}{2}}$ can be obtained from $\tilde{\mathcal{L}}_{\frac{1}{2}}$ by taking the print of the latter on $V$.

Lupu introduced in \cite{Lupu2014LoopsGFF} an isomorphism linking  the GFF $\tilde{\varphi}$ and the loop soup $\tilde{\mathcal{L}}_{\frac{1}{2}}$ on $\tilde{\mathcal{G}}$.

\begin{theorem}[Lupu's Isomorphism,\cite{Lupu2014LoopsGFF}]
\label{thm:Lupu}
There is a coupling between the Poisson ensemble of loops $\tilde{\mathcal{L}}_{\frac{1}{2}}$ and $(\tilde{\varphi}_y)_{y\in\tilde{\mathcal{G}}}$ defined above, such that the two following constraints hold:
\begin{itemize}
\item For all $y\in\tilde{\mathcal{G}}$, $L_y(\tilde{{\mathcal{L}}}_{\frac{1}{2}})=\frac{1}{2}\tilde{\varphi}_y^2$
\item The clusters of loops of $\tilde{\mathcal{L}}_{\frac{1}{2}}$ are exactly the sign clusters of $(\tilde{\varphi}_y)_{y\in\tilde{\mathcal{G}}}$.
\end{itemize}
Conditionally on $(|\tilde{\varphi}_y|)_{y\in\tilde{\mathcal{G}}}$, the sign of $\tilde{\varphi}$ on each of its connected components is distributed independently and uniformly in $\{-1,+1\}$.
\end{theorem}
Lupu's isomorphism and the idea of using metric graphs were applied in \cite{Lupu2015ConvCLE} to show that on the discrete half-plane $\mathbb{Z}\times\mathbb{N}$, the scaling limits of outermost boundaries of clusters of loops in loop soups are the Conformal Loop Ensembles $\hbox{CLE}$.

Let $\mathcal{O}(\tilde{\varphi})$ (resp. $\mathcal{O}(\tilde{\mathcal{L}}_{\frac{1}{2}})$) be the set of edges $e\in E$ such that $\tilde{\varphi}$ (resp. $\tilde{\mathcal{L}}_{\frac{1}{2}}$) does not touch $0$ on $e$, in other words such that all the edge $e$ remains in the same sign cluster of $\tilde{\varphi}$ (resp. $\tilde{\mathcal{L}}_{\frac{1}{2}}$). Let $\mathcal{O}(\mathcal{L}_{\frac{1}{2}})$ be the set of edges $e\in E$ that are crossed (i.e. visited consecutively) by the trace of the loops $\mathcal{L}_{\frac{1}{2}}$ on $V$.

In order to translate Lupu's isomorphism back onto the initial graph $\mathcal{G}$, one needs to describe on one hand the distribution of $\mathcal{O}(\tilde{\varphi})$ conditionally on the values of $\varphi$, and on the other hand the distribution of $\mathcal{O}(\tilde{\mathcal{L}}_{\frac{1}{2}})$  conditionally on $\mathcal{L}_{\frac{1}{2}}$ and the cluster of loops $\mathcal{O}(\mathcal{L}_{\frac{1}{2}})$ on the discrete graph $G$. These two distributions are described respectively in Subsections \ref{fkising} and \ref{randomcurrent}, and provide realisations of the  FK-Ising coupling and the ``Current+Bernoulli=FK'' coupling lemma \cite{lupu-werner}.

\subsection{The FK-Ising distribution of $\mathcal{O}(\tilde{\varphi})$ conditionally on $|\varphi|$}
\label{fkising}
\begin{lemma}
\label{lem:fki}
Conditionally on $(\varphi_{x})_{x\in V}$, $(1_{e\in \mathcal{O}(\tilde{\varphi})})_{e\in E}$
is a family of independent random variables and
\begin{displaymath}
\mathbb{P}\left(e\not\in \mathcal{O}(\tilde{\varphi})\vert \varphi\right)=
\left\lbrace
\begin{array}{ll}
1 & \text{if}~ \varphi_{e_{-}}\varphi_{e_{+}}<0,\\
\exp\left(-2W_{e}\varphi_{e_{-}}\varphi_{e_{+}}\right)
& \text{if}~ \varphi_{e_{-}}\varphi_{e_{+}}>0.
\end{array}
\right.
\end{displaymath}
\end{lemma}
\begin{proof}
Conditionally on $(\varphi_{x})_{x\in V}$, are constructed as independent Brownian bridges on each edge, so that $(1_{e\in \mathcal{O}(\tilde{\varphi})})_{e\in E}$ are independent random variables, and it follows from the reflection principle that, if $\varphi_{e_{-}}\varphi_{e_{+}}>0$, then
$$\mathbb{P}\left(e\not\in \mathcal{O}(\tilde{\varphi})\vert \varphi\right)=\dfrac{\exp\left(-\frac{1}{2}W_{e}(\varphi_{e_{-}}+\varphi_{e_{+}})^{2}\right)}
{\exp\left(-\frac{1}{2}W_{e}(\varphi_{e_{-}}-\varphi_{e_{+}})^{2}\right)}=\exp\left(-2W_{e}\varphi_{e_{-}}\varphi_{e_{+}}\right).$$
\end{proof}

Let us now recall how the conditional probability in Lemma \ref{lem:fki} yields a realisation of the FK-Ising coupling.

 Assume $V$ is finite. Let $(J_{e})_{e\in E}$ be a family of positive weights. An \textit{Ising model} on $V$ with interaction constants $(J_{e})_{e\in E}$ is a
probability on configuration of spins $({\sigma}_{x})_{x\in V}\in \{+1,-1\}^V$ such that
\begin{displaymath}
\mathbb{P}^{\rm Isg}_{J}((\sigma_x)_{x\in V})=
\dfrac{1}{\mathcal{Z}^{\rm Isg}_{J}}\exp\left(\sum_{e\in E}
J_{e}\sigma_{e_{+}}\sigma_{e_{-}}\right).
\end{displaymath}
An \textit{FK-Ising random cluster model} with weights
$(1-e^{-2J_{e}})_{e\in E}$ is a random configuration of open (value $1$) and closed
edges (value $0$) such that
\begin{displaymath}
\mathbb{P}^{\rm FK-Isg}_{J}((\omega_{e})_{e\in E})=
\dfrac{1}{\mathcal{Z}^{\rm FK-Isg}_{J}}
2^{\sharp~\text{clusters}}
\prod_{e\in E}(1-e^{-2J_{e}})^{\omega_{e}}(e^{-2J_{e}})^{1-\omega_{e}},
\end{displaymath}
where "$\sharp~\text{clusters}$" denotes the number of clusters created by open edges.

The well-known FK-Ising and Ising coupling reads as follows.
\begin{proposition}[FK-Ising and Ising coupling]
\label{FK-Ising}
Given an FK-Ising model, sample on each cluster an independent uniformly distributed spin. The spins are then distributed according to the Ising model. Conversely, given a spins configuration
$\hat{\sigma}$ following the Ising distribution, consider each edge $e$, such that
$\hat{\sigma}_{e_{-}}\hat{\sigma}_{e_{+}}<0$, closed, and each edge $e$, such that
$\hat{\sigma}_{e_{-}}\hat{\sigma}_{e_{+}}>0$ open with probability
$1-e^{-2J_{e}}$. Then the open edges are distributed according to the FK-Ising model.
The two couplings between FK-Ising and Ising are the same.
\end{proposition}

Consider the GFF $\varphi$ on $\mathcal{G}$ distributed according to $P_{\varphi}$. Let
$J_{e}(\vert\varphi\vert)$ be the random interaction constants
\begin{displaymath}
J_{e}(\vert\varphi\vert)=W_{e}\vert\varphi_{e_{-}}\varphi_{e_{+}}\vert.
\end{displaymath}

Conditionally on $\vert\varphi\vert$,
$(\operatorname{sign}(\varphi_{x}))_{x\in V}$ follows an Ising distribution with interaction constants $(J_{e}(\vert\varphi\vert))_{e\in E}$:
indeed, the Dirichlet form (\ref{Dirichlet-form}) can be written as
\begin{displaymath}
\mathcal{E}(\varphi,\varphi)=\sum_{x\in V}\kappa_{x} \varphi(x)^{2}+
\sum_{x\in V}(\varphi(x))^2(\sum_{y\sim x} W_{x,y})-
2\sum_{e\in E}J_e(\vert\varphi\vert) \operatorname{sign}(\varphi(e_{+}))\operatorname{sign}(\varphi(e_{-})).
\end{displaymath}
Similarly, when $\varphi\sim P_{\varphi}^{\{x_0\},\sqrt{2u}}$ has boundary condition $\sqrt{2u}\ge 0$ on $x_0$, then $(\operatorname{sign}(\varphi_{x}))_{x\in V}$ has an Ising distribution
with interaction $(J_{e}(\vert\varphi\vert))_{e\in E}$ and conditioned on $\sigma_{x_0}=+1$.

Now, conditionally on $\varphi$, $\mathcal{O}(\tilde{\varphi})$ has FK-Ising distribution  with weights
$(1-e^{-2J_{e}(\vert\varphi\vert)})_{e\in E}$. Indeed, the probability for $e\in\mathcal{O}(\tilde{\varphi})$ conditionally on $\varphi$ is $1-e^{-2J_{e}(\vert\varphi\vert)}$,  by Lemma \ref{lem:fki}, as in Proposition \ref{FK-Ising}.

Note that, given that $\mathcal{O}(\tilde{\varphi})$ has FK-Ising distribution, the fact that the sign of on its connected components is distributed independently and uniformly in $\{-1,1\}$ can be seen either as a consequence of Proposition \ref{FK-Ising}, or from Theorem \ref{thm:Lupu}.

Given $\varphi=(\varphi_x)_{x\in V}$ on the discrete graph $\mathcal{G}$, we introduce in Definition \ref{def_FK-Ising} as the random set of edges which has the distribution of $\mathcal{O}(\tilde{\varphi})$ conditionally on $\varphi=(\varphi_x)_{x\in V}$.
\begin{definition}\label{def_FK-Ising}
We let $\mathcal{O}(\varphi)$ be a random set of edges which has the distribution of $\mathcal{O}(\tilde{\varphi})$ conditionally on $\varphi=(\varphi_x)_{x\in V}$ given by Lemma \ref{lem:fki}.
\end{definition}
\subsection{Distribution of $\mathcal{O}(\tilde{\mathcal{L}}_{\frac{1}{2}})$  conditionally on $\mathcal{L}_{\frac{1}{2}}$ }
\label{randomcurrent}
The distribution of $\mathcal{O}(\tilde{\mathcal{L}}_{\frac{1}{2}})$  conditionally on $\mathcal{L}_{\frac{1}{2}}$ can be retrieved by  Corollary 3.6 in \cite{Lupu2014LoopsGFF}, which reads as follows.
\begin{lemma}[Corollary 3.6 in \cite{Lupu2014LoopsGFF}]
\label{36} Conditionally on $\mathcal{L}_{\frac{1}{2}}$, the events $e\not\in\mathcal{O}(\tilde{\mathcal{L}}_{\frac{1}{2}})$, $e\in E\setminus\mathcal{O}(\mathcal{L}_{\frac{1}{2}})$, are independent and have probability
\begin{equation}
\label{cp}
\exp\left(-2 W_{e} \sqrt{L_{e_{+}}(\mathcal{L}_{\frac{1}{2}})L_{e_{-}}(\mathcal{L}_{\frac{1}{2}})}\right).
\end{equation}
\end{lemma}
This result gives rise, together with Theorem \ref{thm:Lupu}, to the following discrete version of Lupu's isomorphism, which is stated without any recourse to the cable graph induced by $\mathcal{G}$.
\begin{definition}
\label{def:out}
Let $(\omega_{e})_{e\in E}\in\lbrace 0,1\rbrace^{E}$ be a percolation defined as follows: conditionally on $\mathcal{L}_{\frac{1}{2}}$, the random variables
$(\omega_{e})_{e\in E}$ are independent, and $\omega_{e}$ equals $0$ with conditional probability given by \eqref{cp}.

Let $\mathcal{O}_{+}(\mathcal{L}_{\frac{1}{2}})$ the set of edges:
\begin{displaymath}
\mathcal{O}_{+}(\mathcal{L}_{\frac{1}{2}})=\mathcal{O}(\mathcal{L}_{\frac{1}{2}})
\cup \lbrace e\in E\vert \omega_{e}=1\rbrace.
\end{displaymath}
\end{definition}
\begin{proposition}[Discrete version of Lupu's isomorphism, Theorem 1 bis in \cite{Lupu2014LoopsGFF}]
\label{PropIsoLupuLoops}
Given a loop soup $\mathcal{L}_{\frac{1}{2}}$, let $\mathcal{O}_{+}(\mathcal{L}_{\frac{1}{2}})$ be as in Definition \ref{def:out}.
Let $(\sigma_{x})_{x\in V}\in\lbrace -1,+1\rbrace^{V}$ be random spins taking constant values on
clusters induced by $\mathcal{O}_{+}(\mathcal{L}_{\frac{1}{2}})$
($\sigma_{e_{-}}=\sigma_{e_{+}}$ if $e\in \mathcal{O}_{+}(\mathcal{L}_{\frac{1}{2}})$) and such that the values on each cluster, conditional on $\mathcal{L}_{\frac{1}{2}}$ and $\mathcal{O}_{+}(\mathcal{L}_{\frac{1}{2}})$, are independent and uniformly distributed. Then
\begin{displaymath}
\left(\sigma_{x}\sqrt{2 L_{x}(\mathcal{L}_{\frac{1}{2}})}\right)_{x\in V}
\end{displaymath}
is a Gaussian free field distributed according to $P_{\varphi}$.
\end{proposition}

Proposition \ref{PropIsoLupuLoops}  induces the following coupling between FK-Ising and random currents.

If $V$ is finite, a \textit{random current model} on $\mathcal{G}$ with weights
$(J_{e})_{e\in E}$ is a random assignment to each edge $e$ of a non-negative integer
$\hat{n}_{e}$ such that for all $x\in V$,
\begin{displaymath}
\sum_{e~\text{adjacent to}~x}\hat{n}_{e}
\end{displaymath}
is even, which is called the \textit{parity condition}. The probability of a configuration
$(n_{e})_{e\in E}$ satisfying the parity condition is
\begin{displaymath}
\mathbb{P}^{\rm RC}_{J}(\forall e\in E, \hat{n}_{e}=n_{e})=
\dfrac{1}{\mathcal{Z}^{\rm RC}_{J}}\prod_{e\in E}\dfrac{(J_{e})^{n_{e}}}{n_{e}!},
\end{displaymath}
where actually $\mathcal{Z}^{\rm RC}_{J}=\mathcal{Z}^{\rm Isg}_{J}$. Let
\begin{displaymath}
\mathcal{O}(\hat{n})=\lbrace e\in E\vert \hat{n}_{e}>0\rbrace.
\end{displaymath}
The open edges in $\mathcal{O}(\hat{n})$ induce clusters on the graph $\mathcal{G}$.

Given a loop soup $\mathcal{L}_{\alpha}$, we denote by $N_{e}(\mathcal{L}_{\alpha})$ the number of times the loops in $\mathcal{L}_{\alpha}$ cross the nonoriented edge $e\in E$. The transience of the Markov jump process $X$ implies that
$N_{e}(\mathcal{L}_{\alpha})$ is a.s. finite for all $e\in E$. If $\alpha=\frac{1}{2}$, we have the following identity (see for instance \cite{Werner2015}):

\begin{LoopsRC}
Assume $V$ is finite and consider the loop soup $\mathcal{L}_{\frac{1}{2}}$. Conditionally on the occupation field $(L_{x}(\mathcal{L}_{\frac{1}{2}}))_{x\in V}$,
$(N_{e}(\mathcal{L}_{\frac{1}{2}}))_{e\in E}$ is distributed as a random current model with weights
$\left(2W_{e}\sqrt{L_{e_{-}}(\mathcal{L}_{\frac{1}{2}})L_{e_{+}}
(\mathcal{L}_{\frac{1}{2}})}\right)_{e\in E}$. If $\varphi$ is the GFF on $\mathcal{G}$ given by Le Jan's or Lupu's isomorphism, then these weights are
$(J_{e}(\vert\varphi\vert))$.
\end{LoopsRC}

Conditionally on the occupation field
$(L_{x}(\mathcal{L}_{\frac{1}{2}}))_{x\in V}$,
$\mathcal{O}(\mathcal{L}_{\frac{1}{2}})$ are the edges occupied by a random current and
$\mathcal{O}_{+}(\mathcal{L}_{\frac{1}{2}})$ the edges occupied by FK-Ising.
Lemma \ref{lem:fki} and Proposition \ref{PropIsoLupuLoops} imply the following coupling, as noted by Lupu and Werner in
\cite{lupu-werner}.

\begin{proposition}[Random current and FK-Ising coupling,
\cite{lupu-werner}]
\label{RCFKIsing}
Assume $V$ is finite. Let $\hat{n}$ be a random current on $\mathcal{G}$ with weights
$(J_{e})_{e\in E}$. Let $(\omega_{e})_{e\in E}\in\lbrace 0,1\rbrace^{E}$ be an independent percolation, each edge being opened (value $1$) independently with probability
$1-e^{-J_{e}}$. Then
\begin{displaymath}
\mathcal{O}(\hat{n})\cup\lbrace e\in E\vert \omega_{e}=1\rbrace
\end{displaymath}
is distributed like the open edges in an FK-Ising with weights
$(1-e^{-2 J_{e}})_{e\in E}$.
\end{proposition}

\subsection{Generalized second Ray-Knight ``version'' of Lupu's isomorphism}\label{sec:glupu}
We are now in a position to state the coupled version of the second Ray-Knight theorem.
\begin{theorem}
\label{Lupu}
Let $x_{0}\in V$. Let $(\varphi_{x}^{(0)})_{x\in V}$ with distribution $P_{\varphi}^{\lbrace x_{0}\rbrace,0}$, and define $\mathcal{O}(\varphi^{(0)})$ as in Definition \ref{def_FK-Ising}.
Let $X$ be an independent Markov jump process started from $x_{0}$.

Fix $u>0$. If $\tau_{u}^{x_{0}}<\zeta$, we
let $\mathcal{O}_{u}$ be the random subset of $E$ which contains $\mathcal{O}(\varphi_{x}^{(0)})$, the edges used by the path $(X_{t})_{0\leq t\leq \tau_{u}^{x_{0}}}$, and additional edges $e$ opened conditionally independently with probability
\begin{displaymath}
1-e^{W_{e}\vert\varphi_{e_{-}}^{(0)}\varphi_{e_{+}}^{(0)}\vert -
W_{e}\sqrt{(\varphi_{e_{-}}^{(0)2}+2\ell_{e_{-}}(\tau_{u}^{x_{0}}))
(\varphi_{e_{+}}^{(0)2}+2\ell_{e_{+}}(\tau_{u}^{x_{0}}))}}.
\end{displaymath}
We let $\sigma\in\lbrace -1,+1\rbrace^{V}$ be random spins sampled uniformly independently on each cluster induced by
$\mathcal{O}_{u}$, pinned at $x_0$, i.e. $\sigma_{x_0}=1$, and define
\begin{displaymath}
\varphi_{x}^{(u)}:=\sigma_{x}\sqrt{\varphi_{x}^{(0)2}+2\ell_{x}(\tau_{u}^{x_{0}})}.
\end{displaymath}

Then, conditionally on $\tau_{u}^{x_{0}}<\zeta$, $\varphi^{(u)}$ has  distribution $P_{\varphi}^{\lbrace x_{0}\rbrace,\sqrt{2u}}$, and
$\mathcal{O}_{u}$ has distribution $\mathcal{O}(\varphi^{(u)})$ conditionally on $\varphi^{(u)}$.
\end{theorem}
\begin{remark}
One consequence of that coupling is that the path $(X_{s})_{s\le \tau_{u}^{x_{0}}}$ stays in the
positive connected component of ${x_0}$ for $\varphi^{(u)}$. This yields a coupling between the range
of the Markov chain and the sign component of $x_{0}$ inside a GFF $P_{\varphi}^{\lbrace x_{0}\rbrace,\sqrt{2u}}$.
\end{remark}

\noindent{\it Proof of Theorem \ref{Lupu}:~}
The proof is based on
\cite{Lupu2014LoopsGFF}.  Let $D=V\setminus\{x_0\}$, and let $\tilde{\mathcal{L}}_{\frac{1}{2}}$ be the loop soup of intensity $1/2$ on the cable graph $\tilde{\mathcal{G}}$, which we decompose into $\tilde{\mathcal{L}}_{\frac{1}{2}}^{(x_0)}$ (resp. $\tilde{\mathcal{L}}_{\frac{1}{2}}^{D}$) the loop soup hitting (resp. not hitting) $x_0$, which are independent. We let $\mathcal{L}_{\frac{1}{2}}$ and $\mathcal{L}_{\frac{1}{2}}^{(x_0)}$ (resp.  $\mathcal{L}_{\frac{1}{2}}^{D}$) be the prints of these loop soups on $V$ (resp. on $D=V\setminus\{x_0\}$). We condition on $L_{x_0}(\mathcal{L}_{\frac{1}{2}})=u$.

Theorem \ref{thm:Lupu} implies (recall also Definition \ref{def_FK-Ising}) that we can couple $\tilde{\mathcal{L}}_{\frac{1}{2}}^{D}$ with $\varphi^{(0)}$ so that
$L_x(\mathcal{L}_{\frac{1}{2}}^{D})=\varphi_x^{(0)2}/2$ for all $x\in V$, and
$\mathcal{O}(\tilde{\mathcal{L}}_{\frac{1}{2}})=\mathcal{O}(\varphi^{(0)})$.

Define
$\varphi^{(u)}=(\varphi^{(u)}_x)_{x\in V}$ from
$\tilde{\mathcal{L}}_{\frac{1}{2}}$ by, for all $x\in V$,
\begin{equation*}
\label{abs}
|\varphi_x^{(u)}|=\sqrt{2L_x(\mathcal{L}_{\frac{1}{2}})}
\end{equation*}
and $\varphi_x^{(u)}=\sigma_x|\varphi_x^{(u)}|$, where $\sigma\in\{-1,+1\}^V$ are random spins sampled uniformly independently on each cluster induced by $\mathcal{O}(\tilde{\mathcal{L}}_{\frac{1}{2}})$, pinned at $x_0$,  i.e. $\sigma_{x_0}=1$. Then, by Theorem \ref{thm:Lupu}, $\varphi^{(u)}$ has distribution $P_{\varphi}^{\lbrace x_{0}\rbrace,\sqrt{2u}}$.

For all $x\in V$, we have
$$L_x(\tilde{\mathcal{L}}_{\frac{1}{2}})=\frac{\varphi_x^{(0)2}}{2}+L_x(\mathcal{L}_{\frac{1}{2}}^{(x_0)}).$$

On the other hand, conditionally on $L_.(\mathcal{L}_{\frac{1}{2}})$,
\begin{align*}
&\mathbb{P}(e\not\in\mathcal{O}(\tilde{\mathcal{L}}_{\frac{1}{2}})\,|\,
e\not\in\mathcal{O}(\tilde{\mathcal{L}}_{\frac{1}{2}}^D)\cup\mathcal{O}(\mathcal{L}_{\frac{1}{2}}))
=\frac{\mathbb{P}(e\not\in\mathcal{O}(\tilde{\mathcal{L}}_{\frac{1}{2}}))}{\mathbb{P}(e\not\in\mathcal{O}(\tilde{\mathcal{L}}_{\frac{1}{2}}^D)\cup\mathcal{O}(\mathcal{L}_{\frac{1}{2}}))}=
\frac{\mathbb{P}(e\not\in\mathcal{O}(\tilde{\mathcal{L}}_{\frac{1}{2}})\,|\,
e\not\in\mathcal{O}(\mathcal{L}_{\frac{1}{2}}))}{\mathbb{P}(e\not\in\mathcal{O}(\tilde{\mathcal{L}}_{\frac{1}{2}}^D)\,|\,
e\not\in\mathcal{O}(\mathcal{L}_{\frac{1}{2}}))}\\
&=\frac{\mathbb{P}(e\not\in\mathcal{O}(\tilde{\mathcal{L}}_{\frac{1}{2}})\,|\,
e\not\in\mathcal{O}(\mathcal{L}_{\frac{1}{2}}))}{\mathbb{P}(e\not\in\mathcal{O}(\tilde{\mathcal{L}}_{\frac{1}{2}}^D)\,|\,
e\not\in\mathcal{O}(\mathcal{L}_{\frac{1}{2}}^D))}
=\exp\left(-W_e\sqrt{L_{e_-}(\mathcal{L}_{\frac{1}{2}})L_{e_+}(\mathcal{L}_{\frac{1}{2}})}
+W_e\sqrt{L_{e_-}(\mathcal{L}_{\frac{1}{2}}^D)L_{e_+}(\mathcal{L}_{\frac{1}{2}}^D)}\right),
\end{align*}
where we use in the third equality that the event $e\not\in\mathcal{O}(\tilde{\mathcal{L}}_{\frac{1}{2}}^D)$ is measurable with respect to the $\sigma$-field generated by $\tilde{\mathcal{L}}_{\frac{1}{2}}^D$, which is independent of $\tilde{\mathcal{L}}_{\frac{1}{2}}^{(x_0)}$, and where we use Lemma \ref{36} in the fourth equality, for $\tilde{\mathcal{L}}_{\frac{1}{2}}$ and for $\tilde{\mathcal{L}}_{\frac{1}{2}}^D$.

We conclude the proof by observing that  $\mathcal{L}_{\frac{1}{2}}^{(x_0)}$ conditionally on $L_{x_0}(\mathcal{L}_{\frac{1}{2}}^{(x_0)})=u$ has the law of the occupation field of the Markov chain $\ell(\tau_{u}^{x_{0}})$
under $\mathbb{P}_{x_{0}}(\cdot \vert \tau_{u}^{x_{0}}<\zeta)$.
{\hfill $\Box$}
\section{Inversion of the signed isomorphism}
\label{sec:inversion}
In \cite{SabotTarres2015RK}, Sabot and Tarrès give a new proof of the generalized second Ray-Knight theorem together with a construction that inverts the coupling between the square of a GFF conditioned by its value at a vertex $x_{0}$ and the excursions of the jump process $X$ from and to $x_{0}$.
In this paper we are interested in inverting the coupling of Theorem \ref{Lupu} with the signed GFF : more precisely, we want to describe the law of
$(X_t)_{0\le t\le \tau_u^{x_0}}$ conditionally on $\varphi^{(u)}$.

We present in section \ref{sec_Poisson}
an inversion involving an extra Poisson process. We provide in Section \ref{sec_dicr_time} a discrete-time description of the process and in Section \ref{sec_jump} an alternative description via jump rates.  Sections \ref{sec:lejaninv} and \ref{sec:coupinv} are respectively dedicated to a signed inversion of Le Jan's isomorphism for loop soups, and to an inversion of the coupling of random current with FK-Ising.

\subsection{A description via an extra Poisson point process}\label{sec_Poisson}

Let $(\check \varphi_x)_{x\in V}$ be a real function on $V$ such that
$\check\varphi_{x_0}=+\sqrt{2u}$ for some $u>0$. Set
$$
\check \Phi_x=\vert\check\varphi_x\vert, \;\;\sigma_x=\operatorname{sign}(\check\varphi_x).
$$
We define a self-interacting process $(\check X_t, (\check n_e(t))_{e\in E})$ living on $V\times {\mathbb{N}}^E$ as follows.
The process $\check X$ starts at $\check X(0)=x_0$.
For $t\ge 0$, we set
$$
\check\Phi_x(t)=\sqrt{(\check\Phi_x)^2-2\check\ell_x(t)},\;\;\forall x\in V,\;\;\;\hbox{ and }\;
J_e(\check\Phi(t))=W_e \check\Phi_{e-}(t)\check\Phi_{e+}(t), \;\; \forall e\in E.
$$
where $\check\ell_x(t)=\int_0^t{{\mathbbm 1}}_{\{\check X_s=x\}}ds$ is the local time of the process $\check X$ up to time $t$.
Let $(N_e(u))_{u\ge 0}$ be an independent Poisson Point Processes on ${\mathbb R}_+$ with intensity 1, for each edge $e\in E$.
We set
$$
\check n_e(t)=
\begin{cases}
N_e(2J_e(t)), &\hbox{ if } \sigma_{e-}\sigma_{e+}=+1,
\\
0, &\hbox{ if } \sigma_{e-}\sigma_{e+}=-1.
\end{cases}
$$
We also denote by $\check {\mathcal C}(t)\subset E$ the configuration of edges such that $\check n_e(t)>0$.
As time increases, the interaction parameters $J_{e}(\check\Phi(t))$ decreases for the edges neighboring $\check X_t$, and at some random times $\check n_e(t)$ may drop
by 1.
The process $(\check X_t)_{t\ge 0}$ is defined as the process that jumps only at the times when one of the $\check n_e(t)$ drops by 1, as follows:
\begin{itemize}
\item
if $\check n_e(t)$ decreases by 1 at time $t$, but does not create a new cluster in $\check {\mathcal C}_t$, then $\check X_t$ crosses the edge
$e$ with probability ${1/2}$ or does not move with probability ${1/2}$,
\item
if $\check n_e(t)$ decreases by 1 at time $t$, and does create a new cluster in $\check {\mathcal C}_t$,
 then $\check X_t$ moves/or stays with probability 1 on the unique extremity
of $e$ which is in the cluster of the origin $x_0$ in the new configuration.
\end{itemize}
We set
$$
\check T:=\inf\{t\ge 0,\;\;  \exists x\in V, \hbox{ s. t. } \check\Phi_x(t)=0\},
$$
clearly, the process is well-defined up to time $\check T$.
\begin{proposition}
For all $0\le t\le \check T$, $\check X_t$ is in the connected component of $x_0$ of the configuration $\check {\mathcal C}(t)$. If $V$ is finite,
the process ends at $x_0$, i.e. $\check X_{\check T}=x_0$.
\end{proposition}

\begin{theorem}
\label{thm-Poisson}
Assume that $V$ is finite.
With the notation of Theorem \ref{Lupu}, conditioned on $\varphi^{(u)}=\check\varphi$, $(X_{t})_{t\le \tau_{u}^{x_{0}}}$ has the law
of $(\check X_{\check T-t})_{0\le t\le \check T}$.

Moreover, conditioned on  $\varphi^{(u)}=\check\varphi$, $(\varphi^{(0)},\mathcal{O}(\varphi^{(0)}))$ has the law of
$(\sigma'_x\check\Phi_x(\check T), \check{\mathcal C}(\check T))$ where $(\sigma'_x)_{x\in V}\in \lbrace -1,+1\rbrace^{V}$ are random spins sampled uniformly independently on
each cluster induced by $\check{\mathcal C}(\check T)$,
with the condition that $\sigma'_{x_0}=+1$.

If $V$ is infinite, then $P_{\varphi}^{\lbrace x_{0}\rbrace, \sqrt{2u}}$-a.s.,
$\check X_t$ (with the initial condition $\check\varphi=\varphi^{(u)}$)
ends at $x_0$, i.e. $\check T<+\infty$ and $\check X_{\check T}=x_0$.
All previous conclusions for the finite case still hold.
\end{theorem}

\subsection{Discrete time description of the process}
\label{sec_dicr_time}

We give a discrete time description of the process
$(\check X_t, (\check n_e(t))_{e\in E})$
that appears in the previous section.
Let $t_{0}=0$ and $0<t_{1}<\dots<t_{j}$ be the stopping times when one of the
stacks $n_e(t)$ decreases by $1$, where $t_{j}$ is the time when one of the stacks is completely depleted. It is elementary to check the following:

\begin{proposition}
\label{PropDiscrTime}
The discrete time process
$(\check X_{t_{i}}, (\check n_e(t_{i}))_{e\in E})_{0\leq i\leq j}$ is a stopped Markov process. The transition from time $i-1$ to $i$ is the following:
\begin{itemize}
\item first chose $e$ an edge adjacent to the vertex $\check X_{t_{i-1}}$
according to a probability proportional to $\check n_e(t_{i-1})$,
\item decrease the stack $\check n_e(t_{i-1})$ by 1,
\item
if decreasing $\check n_e(t_{i-1})$ by 1 does not create a new cluster in
$\check {\mathcal C}_{t_{i-1}}$, then $\check X_{t_{i-1}}$ crosses the edge
$e$ with probability ${1/2}$ or does not move with probability ${1/2}$,
\item
if decreasing $\check n_e(t_{i-1})$ by 1 does create a new cluster in
$\check {\mathcal C}_{t_{i-1}}$,
 then $\check X_{t_{i-1}}$ moves/or stays with probability 1 on the unique extremity of $e$ which is in the cluster of the origin $x_0$ in the new configuration.
\end{itemize}
\end{proposition}

\subsection{An alternative description via jump rates}\label{sec_jump}
We provide an alternative description of the process $(\check X_t, \check {\mathcal C}(t))$ that appears in Section \ref{sec_Poisson}.

\begin{proposition}\label{prop-jump}
The process $(\check X_t, \check{\mathcal C}(t))$ defined in section \ref{sec_Poisson} can be alternatively described by its jump rates :
conditionally on its past at time $t$, if $\check X_t=x$, $y\sim x$ and $\lbrace x,y\rbrace\in \check{\mathcal{C}}(t)$, then
\begin{itemize}
\item[(1)]  $\check X$ jumps to $y$ without modification of $\check{\mathcal C}(t)$ at rate
\begin{displaymath}
W_{x,y}\dfrac{\check\Phi_{y}(t)}{\check\Phi_{x}(t)}
\end{displaymath}
\item[(2)]  the edge $\lbrace x,y\rbrace$ is closed in $\check{\mathcal C}(t)$ at rate
\begin{displaymath}
2W_{x,y}\dfrac{\check\Phi_{y}(t)}{\check\Phi_{x}(t)}
\left(e^{2W_{x,y}\check\Phi_{x}(t)\check\Phi_{y}(t)}-1\right)^{-1}
\end{displaymath}
and, conditionally on that last event:

- if $y$ is connected to
$x$ in the configuration $\check {\mathcal C}(t)\setminus\{x,y\}$, then $\check X$ simultaneously jumps to $y$ with probability $1/2$ and stays at $x$ with probability $1/2$

- otherwise $\check X_t$ moves/or stays with probability 1 on the unique extremity
of $\{x,y\}$ which is in the cluster of the origin $x_0$ in the new configuration.
\end{itemize}
\end{proposition}
\begin{remark}
It is clear from this description that the joint process $(\check X_t, \check {\mathcal C}(t), \check \Phi(t))$ is Markov process, and well defined up to the time
$$
\check T:=\inf\{t\ge 0:\;\;  \exists x\in V, \hbox{ s.t. } \check\Phi_x(t)=0\}.
$$
\end{remark}

\begin{remark}
One can also retrieve the process in Section \ref{sec_Poisson} from the representation in Proposition \ref{prop-jump} as follows.
Consider the representation of Proposition \ref{prop-jump} on the graph where each edge $e$ is replaced by a large number $N$ of
parallel edges with conductance $W_e/N$. Consider now $\check n^{(N)}_{x,y}(t)$ the number of parallel edges that are open in the configuration
$\check {\mathcal C}(t)$ between $x$ and $y$. Then, when $N\to\infty$, $(\check n^{(N)}(t))_{t\ge0}$,  converges in law to
$(\check n(t))_{t\ge0}$,  defined in section \ref{sec_Poisson}.
\end{remark}

\noindent {\it Proof of Proposition \ref{prop-jump}:~}
Assume $\check X_t=x$, fix $y\sim x$ and let $e=\{x,y\}$. Recall that $\{x,y\}\in\check{\mathcal C}(t)$ iff $\check n_e(t)\ge1$.

Let us first prove (1):
\begin{align*}
&\mathbb{P}\left(\check X\text{ jumps to $y$ on time interval $[t,t+\Delta t]$ without modification of }\check{\mathcal C}(t)\,|\,\{x,y\}\in\check{\mathcal C}(t)\right)\\
&=\frac{1}{2}\mathbb{P}\left(\check n_e(t)-\check n_e(t+\Delta t)=1,\,\check n_e(t+\Delta t)\ge1\,|\,\check n_e(t)\ge1\right)\\
&=\frac{1}{2}(2J_e(t)-2J_e(t+\Delta t))+o(\Delta t)=W_{xy}\dfrac{\check\Phi_{y}(t)}{\check\Phi_{x}(t)}\Delta t+o(\Delta t).
\end{align*}
Similarly, (2) follows from the following computation:
\begin{align*}
&\mathbb{P}\left(\{x,y\}\text{ closed in }\check{\mathcal C}(t+\Delta t)\,|\,\{x,y\}\in\check{\mathcal C}(t)\right)
=\mathbb{P}\left(n_e(t+\Delta t)=0\,|\,\check n_e(t)\ge1\right)\\
&=\frac{\mathbb{P}\left(\check n_e(t)=1,\,\check n_e(t+\Delta t)=0\right)}{\mathbb{P}\left(\check n_e(t)\ge1\right)}
=\frac{e^{-2J_e(t)}2J_e(t)}{1-e^{-2J_e(t)}}(J_e(t)-J_e(t+\Delta t))+o(\Delta t)
\end{align*}
{\hfill $\Box$}

We easily deduce from the Proposition \ref{prop-jump} and Theorem \ref{thm-Poisson2} the following alternative inversion of the coupling in Theorem \ref{Lupu}.
\begin{theorem}\label{thm-jump-rates}
With the notation of Theorem \ref{Lupu}, conditionally on $(\varphi^{(u)},\mathcal{O}_{u})$, $(X_{t})_{t\le \tau_{u}^{x_{0}}}$ has the law
of self-interacting process $(\check X_{\check T-t})_{0\le t\le \check T}$ defined by jump rates of Proposition \ref{prop-jump}
starting with
$$
\check \Phi_x=\sqrt{(\varphi_{x}^{(0)})^2+2\ell_{x}(\tau_{u}^{x_{0}})} \hbox{ and } \check{\mathcal C}(0)=\mathcal{O}_{u}.
$$
Moreover $(\varphi^{(0)},\mathcal{O}(\varphi^{(0)}))$ has the same law as
$(\sigma'\check\Phi(T), \check{\mathcal C}(\check T))$ where $(\sigma'_x)_{x\in V}$ is a configuration of signs obtained by picking a sign at random independently on
each connected component of $\check{\mathcal C}(T)$, with the condition that the component of $x_0$ has a + sign.

\end{theorem}

\subsection{A signed version of Le Jan's isomorphism for loop soup}
\label{sec:lejaninv}
Let us first recall how the loops in $\mathcal{L}_{\alpha}$ are connected to the excursions of the jump process $X$.
\begin{proposition}[From excursions to loops]
\label{PropPD}
Let $\alpha>0$ and $x_{0}\in V$.
$L_{x_{0}}(\mathcal{L}_{\alpha})$ is distributed according to a Gamma
$\Gamma(\alpha, G(x_{0},x_{0}))$ law, where $G$ is the Green's function. Let $u>0$, and consider the path $(X_{t})_{0\leq t\leq \tau_{u}^{x_{0}}}$ conditioned on $\tau_{u}^{x_{0}}<\zeta$. Let $(Y_{j})_{j\geq 1}$ be an independent Poisson-Dirichlet partition $PD(0,\alpha)$ of $[0,1]$. Let $S_{0}=0$ and
\begin{displaymath}
S_{j}=\sum_{i=1}^{j}Y_{i}.
\end{displaymath}
Let
\begin{displaymath}
\tau_{j}= \tau_{u S_{j}}^{x_{0}}.
\end{displaymath}
Consider the family of paths
\begin{displaymath}
\left((X_{\tau_{j-1}+t})_{0\leq t\leq \tau_{j}-\tau_{j-1}}\right)_{j\geq 1}.
\end{displaymath}
It is a countable family of loops rooted in $x_{0}$. It has the same law as the family of all the loops in $\mathcal{L}_{\alpha}$ that visit $x_{0}$, conditioned on $L_{x_0}(\mathcal{L}_{\alpha})=u$.
\end{proposition}

Next we describe how to invert the discrete version fo Lupu's isomorphism Proposition \ref{PropIsoLupuLoops} for the loop-soup in the same way as in Theorem \ref{thm-Poisson}.

Let $(\check \varphi_x)_{x\in V}$ be a real function on $V$ such that
$\check\varphi_{x_0}=+\sqrt{2u}$ for some $u>0$. Set
$$
\check \Phi_x=\vert\check\varphi_x\vert, \;\;\sigma_x=\operatorname{sign}(\check\varphi_x).
$$

Let $(x_{i})_{1\leq i\leq\vert V\vert}$ be an enumeration of $V$ (which may be infinite).
We define by induction the self interacting processes
$((\check X_{i,t})_{1\leq i\leq\vert V\vert},
(\check n_e(t))_{e\in E})$.
$\check{T}_{i}$ will denote the end-time for $\check X_{i,t}$, and
$\check{T}^{+}_{i}=\sum_{1\leq j\leq i}\check{T}_{j}$.
By definition, $\check{T}^{+}_{0}=0$.
$L(t)$ will denote
\begin{displaymath}
L_{x}(t):=\sum_{1\leq i\leq\vert V\vert}
\check{\ell}_{x}(i,0\vee(t-\check{T}^{+}_{i})),
\end{displaymath}
where $\check{\ell}_{x}(i,t)$ are the occupation times for
$\check X_{i,t}$.
For $t\ge 0$, we set
$$
\check\Phi_x(t)=\sqrt{(\check\Phi_x)^2-2L_x(t)},\;\;\forall x\in V,\;\;\;\hbox{ and }\;
J_e(\check\Phi(t))=W_e \check\Phi_{e-}(t)\check\Phi_{e+}(t), \;\; \forall e\in E.
$$
The end-times $\check{T}_{i}$ are defined by inductions as
\begin{displaymath}
\check{T}_{i}=\inf\lbrace t\geq 0\vert
\check{\Phi}_{\check{X}_{i,t}}(t+\check{T}^{+}_{i-1})=0\rbrace.
\end{displaymath}
Let $(N_e(u))_{u\ge 0}$ be independent Poisson Point Processes on ${\mathbb R}_+$ with intensity 1, for each edge $e\in E$.
We set
$$
\check n_e(t)=
\begin{cases}
N_e(2J_e(t)), &\hbox{ if } \sigma_{e-}\sigma_{e+}=+1,
\\
0, &\hbox{ if } \sigma_{e-}\sigma_{e+}=-1.
\end{cases}
$$
We also denote by $\check {\mathcal C}(t)\subset E$ the configuration of edges such that $\check n_e(t)>0$.
$\check X_{i,t}$ starts at $x_{i}$.
For $t\in[\check{T}^{+}_{i-1},\check{T}^{+}_{i}]$,
\begin{itemize}
\item
if $\check n_e(t)$ decreases by 1 at time $t$, but does not create a new cluster in $\check {\mathcal C}_t$, then $\check X_{i,t-\check{T}^{+}_{i-1}}$ crosses the edge
$e$ with probability ${1/2}$ or does not move with probability ${1/2}$,
\item
if $\check n_e(t)$ decreases by 1 at time $t$, and does create a new cluster in $\check {\mathcal C}_t$,
 then $\check X_{i,t-\check{T}^{+}_{i-1}}$ moves/or stays with probability 1 on the unique extremity
of $e$ which is in the cluster of the origin $x_i$ in the new configuration.
\end{itemize}

By induction, using Theorem \ref{thm-Poisson}, we deduce the following:

\begin{theorem}
\label{ThmPoissonLoopSoup}
Let $\varphi$ be a GFF on $\mathcal{G}$ with the law $P_{\varphi}$.
If one sets $\check{\varphi}=\varphi$ in the preceding construction, then
for all $i\in \lbrace 1,\dots,\vert V\vert\rbrace$,
$\check{T}_{i}<+\infty$,
$\check{X}_{i,\check{T}_{i}} = x_{i}$ and the
path $(\check{X}_{i,t})_{t\leq\check{T}_{i}}$ has the same law as a concatenation in $x_{i}$ of all the loops in a loop-soup
$\mathcal{L}_{1/2}$ that visit $x_{i}$, but none of the
$x_{1},\dots,x_{i-1}$. To retrieve the loops out of each path
$(\check{X}_{i,t})_{t\leq\check{T}_{i}}$, on has to partition it according to
a Poisson-Dirichlet partition as in Proposition \ref{PropPD}.
The coupling between the GFF $\varphi$ and the loop-soup obtained from
$((\check X_{i,t})_{1\leq i\leq\vert V\vert},
(\check n_e(t))_{e\in E})$ is the same as in Proposition
\ref{PropIsoLupuLoops}.
\end{theorem}

\subsection{Inverting the coupling of random current with FK-Ising}
\label{sec:coupinv}
By combining Theorem \ref{ThmPoissonLoopSoup} and the discrete time
description of Section \ref{sec_dicr_time}, and by conditionning on the occupation field of the loop-soup, one deduces an inversion of the coupling
of Proposition \ref{RCFKIsing} between the random current and FK-Ising.

We consider that the graph $\mathcal{G}=(V,E)$ and that the edges are endowed with weights $(J_{e})_{e\in E}$. Let
$(x_{i})_{1\le i\le \vert V\vert}$ be an enumeration of $V$.
Let $\check{\mathcal{C}}(0)$ be a subset of open edges of $E$.
Let $(\check{n}_{e}(0))_{e\in E}$ be a family of
random integers such that
$\check{n}_{e}(0)=0$ if $e\not\in\check{\mathcal{C}}(0)$, and
$(\check{n}_{e}(0)-1)_{e\in\check{\mathcal{C}}(0)}$
are independent Poisson random variables, where
$\mathbb{E}[\check{n}_{e}(0)-1]=2J_{e}$.

We will consider a family of discrete time self-interacting processes
$((\check X_{i,j})_{1\leq i\leq \vert V\vert},
(\check{n}_{e}(j))_{e\in E})$. $\check X_{i,j}$ starts at $j=0$ at
$x_{i}$ and is defined up to a integer time $\check{T}_{i}$.
Let $\check{T}_{i}^{+}=\sum_{1\leq k\leq i}\check{T}_{k}$, with
$\check{T}_{0}^{+}=0$. The end-times $\check{T}_{i}$ are defined by induction as
\begin{displaymath}
\check{T}_{i}=
\inf\Big\lbrace j\geq 0\Big\vert
\sum_{e~\text{edge adjacent to}~\check X_{i,j}}
\check{n}_{e}(j+\check{T}_{i-1}^{+})=0\Big\rbrace.
\end{displaymath}
For $j\geq 1$, $\check{\mathcal{C}}(j)$ will denote
\begin{displaymath}
\check{\mathcal{C}}(j)=\lbrace e\in E\vert \check{n}_{e}(j)\geq 1\rbrace,
\end{displaymath}
which is consistent with the notation $\check{\mathcal{C}}(0)$.

The evolution is the following. For
$j\in \lbrace \check{T}_{i-1}^{+}+1,\dots, \check{T}_{i}^{+}\rbrace$, the transition from time $j-1$ to time $j$ is the following:
\begin{itemize}
\item first chose an edge $e$ adjacent to the vertex
$\check{X}_{i,j-1-\check{T}_{i-1}^{+}}$ with probability proportional to
$\check{n}_{e}(j-1)$,
\item decrease the stack $\check{n}_{e}(j-1)$ by 1,
\item if decreasing $\check{n}_{e}(j-1)$ by 1 does not create a new cluster in  $\check{\mathcal{C}}(j-1)$, then
$\check{X}_{i,\cdot}$ crosses $e$ with probability $1/2$ and
does not move with probability $1/2$.
\item if decreasing $\check{n}_{e}(j-1)$ by 1 does create a new cluster in  $\check{\mathcal{C}}(j-1)$, then $\check{X}_{i,\cdot}$
moves/or stays with probability 1 on the unique extremity of $e$ which is in the cluster of the origin $x_{i}$ in the new configuration.
\end{itemize}

Denote $\hat{n}_{e}$ the number of times the edge $e$ has been crossed, in both directions, by all the walks
$((\check{X}_{i,j})_{0\le j\le \check{T}_{i}})_{1\le i\le\vert V\vert}$.

\begin{proposition}
A.s., for all $i\in\lbrace 1,\dots,\vert V\vert\rbrace$,
$\check{T}_{i}<+\infty$ and $\check{X}_{i,\check{T}_{i}}=x_{i}$. If the initial configuration of open edges
$\check{\mathcal{C}}(0)$ is random and follows an FK-Ising distribution
with weights $(1-e^{-2 J_{e}})_{e\in E}$, then the family of integers
$(\hat{n}_{e})_{e\in E}$ is distributed like a random current with weights
$(J_{e})_{e\in E}$. Moreover, the coupling between the random current and the FK-Ising obtained this way is the same as the one given by
Proposition \ref{RCFKIsing}.
\end{proposition}

\section{Proof of theorem \ref{thm-Poisson} }
\label{sec:proof}
\subsection{Case of finite graph without killing measure}
\label{sec:pfinite}
Here we will assume that $V$ is finite and that the killing measure
$\kappa\equiv 0$.

In order to prove Theorem \ref{thm-Poisson}, we first enlarge the state space of the process $(X_t)_{t\ge 0}$. We define a process
$(X_t,(n_e(t)))_{t\ge 0}$ living on the space $V\times {\mathbb N}^E$ as follows. Let
$\varphi^{(0)}\sim P_{\varphi}^{\{x_0\},0}$ be a GFF pinned at $x_0$.
Let $\sigma_x=\hbox{sign}(\varphi^{(0)}_x)$ be the signs of the GFF with the convention that $\sigma_{x_0}=+1$.
The process $(X_t)_{t\ge 0}$ is as usual the Markov Jump process starting at $x_0$ with jump rates $(W_e)$. We set
\begin{equation}
\label{Phi-J}
\Phi_x=\vert\varphi^{(0)}_x\vert, \;\; \Phi(t)=\sqrt{\Phi_x^2+2\ell_x(t)}, \;\;\;\forall x\in V, \;\;\; J_e(\Phi(t))=W_e \Phi_{e-}(t)\Phi_{e+}(t), \;\;\; \forall e\in E.
\end{equation}
The initial values $(n_e(0))$ are choosen independently on each edge with distribution
$$
n_e(0)\sim
\begin{cases}
0,& \hbox{ if $\sigma_{e-}\sigma_{e+}=-1$}
\\
\mathcal{P}(2J_e(\Phi)),& \hbox{ if $\sigma_{e-}\sigma_{e+}=+1$}
\end{cases}
$$
where ${\mathcal{P}}(2J_e(\Phi))$ is a Poisson random variable with parameter $2J_e(\Phi)$. Let $((N_e(u))_{u\ge 0})_{e\in E}$ be independent Poisson point processes
on ${\mathbb R}_+$ with intensity 1. We define the process $(n_e(t))$ by
$$
n_e(t)=n_e(0)+N_e(J_e(\Phi(t)))-N_e(J_e(\Phi))+K_e(t),
$$
where $K_e(t)$ is the number of crossings of the edge $e$ by the Markov jump process $X$ before time $t$.
\begin{remark}
Note that compared to the process defined in Section \ref{sec_Poisson}, the speed of the Poisson process is related to $J_e(\Phi(t))$ and not $2J_e(\Phi(t))$.
\end{remark}
We will use the following notation
$$
{\mathcal C}(t)=\{e\in E, \;\; n_e(t)>0\}.
$$
Recall that $\tau_u^{x_0}=\inf\{t\ge 0, \; \ell_{x_0}(t)=u\}$ for $u>0$. To simplify notation, we will write $\tau_u$ for $\tau_u^{x_0}$ in the sequel.
We define $\varphi^{(u)}$ by
$$
\varphi^{(u)}_x=\sigma_x\Phi(\tau_u), \;\;\; \forall x\in V,
$$
where  $(\sigma_x)_{x\in V}\in \lbrace -1,+1\rbrace^{V}$ are random spins sampled uniformly independently on
each cluster induced by $\check{\mathcal C}(\check T)$ with the condition that $\sigma_{x_0}=+1$.

\begin{lemma}
\label{end-distrib}
The random vector $(\varphi^{(0)}, {\mathcal C}(0), \varphi^{(u)}, {\mathcal C}(\tau_u^{x_0}))$ thus defined has the same distribution
 as $(\varphi^{(0)}, {\mathcal{O}}(\varphi^{(0)}), \varphi^{(u)}, {\mathcal{O}}_u)$ defined in Theorem \ref{Lupu}.
\end{lemma}
\begin{proof}
It is clear from construction, that ${\mathcal C}(0)$ has the same law as ${\mathcal{O}}(\varphi^{(0)})$ (cf Definition \ref{def_FK-Ising}), the FK-Ising configuration coupled with the signs of
$\varphi^{(0)}$ as in Proposition \ref{FK-Ising}. Indeed, for each edge $e\in E$ such that $\varphi^{(0)}_{e-}\varphi^{(0)}_{e+}>0$, the probability that
$n_e(0)>0$ is $1-e^{-2J_e(\Phi)}$.
Moreover, conditionally on ${\mathcal C}(0)={\mathcal{O}}(\varphi^{(0)})$, ${\mathcal C}(\tau_u^{x_0})$ has the same law as ${\mathcal{O}}_u$ defined in Theorem \ref{Lupu}. Indeed, ${\mathcal C}(\tau_u^{x_0})$
is the union of the set ${\mathcal C}(0)$, the set of edges crossed by the process $(X_u)_{u\le \tau_u^{x_0}}$, and the additional edges such that $N_e(J_e(\tau_u^{x_0}))-N_e(J_e(\Phi))>0$.
Clearly $N_e(J_e(\tau_u^{x_0}))-N(J_e(\Phi))>0$ independently with probability $1-e^{-(J_e(\Phi(\tau_u^{x_0}))-J_e(\Phi))}$ which coincides with the probability given in
Theorem \ref{Lupu}.
\end{proof}

We will prove the following theorem that, together with Lemma \ref{end-distrib}, contains the statements of both Theorem \ref{Lupu} and \ref{thm-Poisson}.
\begin{theorem}\label{thm-Poisson2}
The random vector $\varphi^{(u)}_x$ is a GFF distributed according to $P_{\varphi}^{\{x_0\},\sqrt{2u}}$.
Moreover, conditionally on $\varphi^{(u)}_x=\check \varphi$, the process
$$(X_{t},(n_{e}(t))_{e\in E})_{t\le \tau_u^{x_0}}$$
has the law of the process $(\check X_{\check T-t },(\check n_e(\check T -t))_{e\in E})_{t\le \check T}$
described in section \ref{sec_Poisson}.
\end{theorem}
\begin{proof}
{\bf Step 1 :}
We start by a simple lemma.
\begin{lemma}\label{distrib-phi-n}
The distribution of $(\Phi:=\vert \varphi^{(0)}\vert, n_e(0))$ is given by the following formula for any bounded measurable test function $h$
\begin{multline*}
{\mathbb{E}}\left(h(\Phi, n(0))\right)= \\\sum_{(n_e)\in {\mathbb N}^E} \int_{{\mathbb R}_+^{V\setminus\{x_0\}}} d\Phi  h(\Phi, n)
e^{-{1\over 2} \sum_{x\in V} W_x(\Phi_x)^2-\sum_{e\in E} J_e(\Phi)}
\left(\prod_{e\in E}{\frac{(2J_e(\Phi))^{n_e}}{n_e!}}\right)
2^{\#{\mathcal C}(n_e)-1}.
\end{multline*}
where the integral is on the set $\{(\Phi_x)_{x\in V}, \;\; \Phi_x>0\; \forall x\neq x_0,\; \Phi_{x_0}=0\}$ and
$d\Phi={\frac{\prod_{x\in V\setminus\{x_0\}} d\Phi_x}{\sqrt{2\pi}^{\vert V\vert -1}}}$
and $\#{\mathcal C}(n)$ is the number of clusters
induced by the edges such that $n_e>0$.
\end{lemma}
\begin{proof}
Indeed, by construction, summing on possible signs of $\varphi^{(0)}$, we have
\begin{eqnarray}
\nonumber
&&{\mathbb{E}}\left(h(\Phi, n(0))\right)
\\
\label{int-eee}&=&\sum_{\sigma_x}
 \sum_{n\ll \sigma_x}
 \int_{{\mathbb R}_+^{V\setminus\{x_0\}}} d\Phi  h(\Phi, n) e^{-{1\over 2} {\mathcal E}(\sigma\Phi)}\left(\prod_{e\in E, \; \sigma_{e+}\sigma_{e-}=+1} {e^{-2J_e(\Phi)} (2J_e(\Phi))^{n_e}\over n_e!}\right).
 \end{eqnarray}
where the first sum is on the set $\{\sigma_x\in \{+1,-1\}^V, \; \sigma_{x_0}=+1\}$ and the second sum is on the set of
$\{(n_e)\in {\mathbb N}^E, \; n_e=0\hbox{ if $\sigma_{e-}\sigma_{e+}=-1$}\}$ (we write $n\ll \sigma$ to mean that $n_e$ vanishes on the edges
such that  $\sigma_{e-}\sigma_{e+}=-1$). Since
\begin{eqnarray*}
{1\over 2}{\mathcal E}(\sigma \Phi)&=& {1\over 2}\sum_{x\in V} W_x (\Phi_x)^2-\sum_{e\in E} J_e(\Phi)\sigma_{e-}\sigma_{e+}.
\\
&=&
 {1\over 2}\sum_{x\in V} W_x (\Phi_x)^2+\sum_{e\in E} J_e(\Phi)
 -\sum_{\substack{e\in E\\\sigma_{e-}\sigma_{e+}=+1}} 2J_e(\Phi),
\end{eqnarray*}
we deduce that the integrand in (\ref{int-eee}) is equal to
\begin{eqnarray*}
 && h(\Phi,n) e^{-{1\over 2} {\mathcal E}(\sigma\Phi)}\left(\prod_{e\in E, \; \sigma_{e+}\sigma_{e-}=+1} {e^{-2J_e(\Phi)} (2J_e(\Phi))^{n_e}\over n_e!}\right)
 \\
 &=&
h(\Phi,n)   e^{-{1\over 2} {\mathcal E}(\sigma\Phi)}e^{-\sum_{e\in E, \; \sigma_{e+}\sigma_{e-}=+1} 2J_e(\Phi)}\left(\prod_{e\in E} {(2J_e(\Phi))^{n_e}\over n_e!}\right)
 \\
  &=&
 h(\Phi,n)  e^{-{1\over 2}\sum_{x\in V} W_x (\Phi_x)^2-\sum_{e\in E} J_e(\Phi)}\left(\prod_{e\in E} {(2J_e(\Phi))^{n_e}\over n_e!}\right)
\end{eqnarray*}
where we used in the first equality that $n_e=0$ on the edges such that
$\sigma_{e+}\sigma_{e-}=-1$.
Thus,
\begin{eqnarray*}
&&{\mathbb{E}}\left(h(\Phi, n(0))\right)
\\
&=&
\sum_{\sigma_x}\sum_{n_e\ll \sigma_x}
 \int_{{\mathbb R}_+^{V\setminus\{x_0\}}} d\Phi h(\Phi, n) e^{-{1\over 2}\sum_{x\in V} W_x (\Phi_x)^2-\sum_{e\in E} J_e(\Phi)}\left(\prod_{e\in E} {(2J_e(\Phi))^{n_e}\over n_e!}\right).
\end{eqnarray*}
Inverting the sum on $\sigma$ and $n$ and summing on the number of possible signs which are constant on  clusters induced by the configuration of edges
$\{e\in E, \; n_e>0\}$,
we deduce Lemma \ref{distrib-phi-n}.
\end{proof}
\noindent{\bf Step 2 :} We denote by $Z_t=(X_t, \Phi(t), n_e(t))$ the process defined previously and by
$E_{x_0, \Phi, n_0}$ its law with initial condition $(x_0, \Phi, n_0)$.

We now introduce a process $\tilde Z_t$, which is a "time reversal" of the process $Z_t$. This process will be related to the
process defined in section \ref{sec_Poisson} in Step 4, Lemma \ref{RN}.

For $(\tilde n_e)\in {\mathbb N}^E$ and $(\tilde \Phi_x)_{x\in V}$ such that
$$
\tilde \Phi_{x_0}=u, \;\; \tilde \Phi_x>0, \;\; \forall x\neq x_0,
$$
we define the process $\tilde Z_t=(\tilde X_t, \tilde\Phi(t), \tilde n_e(t))$  with values in $V\times {\mathbb R}_+^V\times {\mathbb Z}^E$  as follows.
The process $(\tilde X_t)$ is a Markov jump process with jump rates $(W_e)$ (so that $\tilde X\stackrel{\text{law}}{=} X$), and
$\tilde\Phi(t)$, $\tilde n_e(t)$ are defined by
\begin{eqnarray}\label{tildePhi}
\tilde \Phi_x(t)=\sqrt{\tilde \Phi_x^2-2\tilde \ell_x(t)},\;\;\;\forall x\in V,
\end{eqnarray}
where $(\tilde\ell_x(t))$ is the local time of the process $\check X$ up to time $t$,
\begin{eqnarray}\label{tilden}
\tilde n_e(t)= \tilde n_e-\left(N_e(J_e(\tilde\Phi))-N_e(J_e(\tilde\Phi(t)))\right)-\tilde K_e(t)
\end{eqnarray}
where $((N_e(u))_{u\ge 0})_{e\in E}$ are independent Poisson point process on ${\mathbb R}_+$ with intensity 1 for each edge $e$, and
$\tilde K_e(t)$ is the number of crossings of the edge $e$ by the process $\tilde X$ before time $t$.
We set
\begin{eqnarray}\label{tildeZ}
\tilde Z_t=(\tilde X_t, (\tilde \Phi_x(t)), (\tilde n_e(t))),
\end{eqnarray}
This process is well-defined up to time
$$
\tilde T=\inf\left\{t\ge 0, \;\; \exists x\in V\; \tilde \Phi_x(t)=0\right\}.
$$
We denote by $\tilde E_{x_0, \tilde\Phi, \tilde n_0}$ its law. Clearly $\tilde Z_t=(\tilde X_t, \tilde\Phi(t), \tilde n_e(t))$ is a Markov process, we will later on make explicit its generator.

We have the following change of variable lemma.
\begin{lemma}\label{change-var}
For all bounded measurable test functions $F,G,H$
\begin{multline*}
\sum_{(n_e)\in {\mathbb N}^E} \int d\Phi F(\Phi, n)E_{x_0,\Phi,n}
\left( G((Z_{\tau_u^{x_0}-t})_{0\le t\le\tau_u^{x_0}})
H(\Phi(\tau_u^{x_0}), n(\tau_u^{x_0}))\right)=
\\
\sum_{(\tilde n_e)\in {\mathbb N}^E} \int d\tilde\Phi H(\tilde\Phi, \tilde n)
\tilde E_{x_0,\tilde \Phi,\tilde n}
\Big({{\mathbbm 1}}_{\{\tilde X_{\tilde T}=x_0,\tilde n_e(\tilde T)\ge 0\; \forall e\in E\}}
G((\tilde Z_{t})_{t\le\check T})
F(\tilde\Phi(\tilde T), \tilde n(\tilde T))\prod_{x\in V\setminus\{x_0\}} {\tilde \Phi_x\over \tilde\Phi_x(\tilde T) }\Big)
\end{multline*}
where the integral on the l.h.s. is on the set $\{(\Phi_x)\in {\mathbb R}_+^V, \;\; \Phi_{x_0}=0\}$ with $d\Phi= {\prod_{x\in V\setminus\{x_0\}} d\Phi_x\over \sqrt{2\pi}^{\vert V\vert -1}}$
and the integral on the r.h.s. is on the set $\{(\tilde\Phi_x)\in {\mathbb R}_+^V, \;\; \tilde\Phi_{x_0}=u\}$ with
$d\tilde\Phi= {\prod_{x\in V\setminus\{x_0\}} d\tilde\Phi_x\over \sqrt{2\pi}^{\vert V\vert -1}}$
\end{lemma}
\begin{proof}
We start from the left-hand side, i.e. the process, $(X_t, n_e(t))_{0\le\le \tau_u^{x_0}}$.
We define
$$
\tilde X_{t}=X_{\tau_u-t},\;\;\; \tilde n_e(t)=n_e(\tau_u-t),
$$
and
$$
\tilde \Phi_x=\Phi_x(\tau_u),\;\;\;, \tilde\Phi_x(t)=\Phi_x({\tau_u-t}),
$$
(The law of the processes such defined will later be identified with the law of the processes ($\tilde X_t, \tilde \Phi(t),\tilde n(t))$ defined at the beginning of step 2, cf (\ref{tildePhi}) and (\ref{tilden})).
We also set
$$
\tilde K_e(t)= K_e(\tau_u)-K_e(t),
$$
which is also the number of crossings of the edge $e$ by the process $\tilde X$, between time 0 and $t$. With these notations we clearly have
$$
\tilde \Phi_x(t)=\sqrt{\tilde \Phi_x^2-2\tilde \ell_x(t)},
$$
where $\tilde \ell_x(t)=\int_{0}^t{{\mathbbm 1}}_{\{\tilde X_u=x\}} du$ is the local time of $\tilde X$ at time $t$, and
$$
\tilde n_e(t)= \tilde n_e(0)+(N_e(J_e(\tilde \Phi(t)))-N_e(J_e(\tilde\Phi(0))))-\tilde K_e(t).
$$
By time reversal, the law of $(\tilde X_t)_{0\le s\le \tilde \tau_u}$ is the same as the law of the Markov Jump process $(X_t)_{0\le t\le \tau_u}$, where
$\tilde \tau_u=\inf\{t\ge 0, \; \tilde\ell_{x_0}(t)=u\}$. Hence, we see that up to the time $\tilde T=\inf\{t\ge 0, \; \exists x\; \tilde\Phi_x(t)=0\}$, the process
$(\tilde X_t, (\tilde \Phi_x(t))_{x\in V}, (\tilde n_e(t))_{t\le \tilde T}$ has the same law as the process defined at the beginning of step 2.

Then, following \cite{SabotTarres2015RK},  we make the following change of variables conditionally on the processes $(X_t, (N_e(t)))$
\begin{eqnarray*}
({\mathbb R}_+^*)^V\times {\mathbb N}^E&\mapsto& ({\mathbb R}_+^*)^V\times {\mathbb N}^E\\
((\Phi_x), (n_e)_{e\in E})&\mapsto&
((\tilde \Phi_x), (\tilde n_e)_{e\in E})
\end{eqnarray*}
which is bijective onto the set
\begin{multline*}
\{\tilde\Phi_x, \;\; \tilde\Phi_{x_0}=\sqrt{2u}, \; \check\Phi_x>\sqrt{2\ell_x(\tau_u^{x_0})}\;\;\forall x\neq x_0\}
\\\times \{(\tilde n_e),\;\; \tilde n_e\ge K_e(\tau_u)+(N_e(J_e(\tilde \Phi(\tau_u)))-N_e(J_e(\Phi)))\}
\end{multline*}
(Note that we always have $\tilde \Phi_{x_0}=\sqrt{2u}$.) The last conditions on $\tilde \Phi$ and $\tilde n_e$ are equivalent to
the conditions $\tilde X_{\tilde T}=x_0$ and $\tilde n_e(\tilde T)\ge 0$.
The Jacobian of the change of variable is given by
$$
\prod_{x\in V\setminus\{x_0\}} d\Phi_x=\left({\prod_{x\in V\setminus\{x_0\}} {\check\Phi_x\over \Phi_x} }\right)\prod_{x\in V\setminus\{x_0\}} d\check\Phi_x.
$$

\end{proof}

\noindent
{\bf Step 3:}
With the notations of Theorem \ref{thm-Poisson2}, we consider the following expectation for $g$ and $h$ bounded measurable test functions
\begin{eqnarray}\label{test-functions}
{\mathbb{E}}\left( g\left(\left(X_{\tau_u-t}, n_e(\tau_u-t)\right)_{0\le t\le \tau_u}\right)h(\varphi^{(u)})\right)
\end{eqnarray}
By definition, we have
$$
\varphi^{(u)}=\sigma \Phi(\tau_u),
$$
where $(\sigma_x)_{x\in V}\in \{\pm 1\}^V$ are random signs sampled uniformly independently on clusters induced by
$\{e\in E, \; n_e(\tau_u)>0\}$ and conditioned on the fact that $\sigma_{x_0}=+1$.
Hence, we define for $(\Phi_x)\in {\mathbb R}_+^V$ and $(n_e)\in {\mathbb N}^E$
\begin{eqnarray}\label{h}
H(\Phi, n)=2^{-\#{\mathcal C}(n)+1} \sum_{\sigma\ll n} h(\sigma \Phi),
\end{eqnarray}
where $\sigma\ll n$ means that the signs $(\sigma_x)$ are constant on clusters of $\{ e\in E, \; n_e>0\}$ and such that $\sigma_{x_0}=+1$.
Hence, setting
$$
F(\Phi, n)=e^{-{1\over 2} \sum_{x\in V} W_x (\Phi_x)^2-\sum_{e\in E} J_e(\Phi) }\left(\prod_{e\in E} {(2J_e(\Phi))^{n_e}\over n_e!}\right)
2^{\#{\mathcal C}(n_e)-1},
$$
$$
G\left((Z_{\tau_u-t})_{t\le\tau_u}\right)= g\left(\left(X_{\tau_u-t}, n_e(\tau_u-t)\right)_{t\le \tau_u}\right),
$$
using lemma \ref{distrib-phi-n} in the first equality and lemma \ref{change-var} in the second equality, we deduce that
(\ref{test-functions}) is equal to
\begin{multline}
\label{eq-3.3}
{\mathbb{E}}\left( G\left((Z_{\tau_u-t})_{0\le t\le \tau_u}\right)H(\Phi(\tau_u), n(\tau_u)))\right)=
\\
\sum_{(n_e)\in {\mathbb N}^E} \int
d\Phi
F(\Phi, n) E_{x_0, \Phi,n}\left(G\left((Z_{\tau_u-t})_{t\le\tau_u}\right)H\left(\Phi(\tau_u, n(\tau_u))\right)\right)
d\Phi =
\\
\sum_{(\tilde n_e)\in {\mathbb N}^E} \int d\tilde\Phi
H\left(\tilde \Phi,\tilde n\right)
\tilde E_{x_0, \tilde \Phi, \tilde n}\Big({{\mathbbm 1}}_{\{\tilde X_{\tilde T}=x_0,\tilde n_e(\tilde T)\ge 0\; \forall e\in E\}}
F\left(\tilde \Phi(\tilde T) , \tilde n(\tilde T)\right) G\left((\tilde Z_{t})_{t\le\tilde T}\right) \prod_{x\in V\setminus\{x_0\}} {\tilde \Phi_x\over \tilde\Phi_x(\tilde T) }
\Big)
\end{multline}
with notations of Lemma \ref{change-var}.

Let $\tilde{\mathcal F}_t=\sigma\{\tilde X_s, \; s\le t\}$ be the filtration generated by $\tilde X$. We define the $\tilde {\mathcal F}$-adapted process
$\tilde M_t$, defined up to time $\tilde T$ by
\begin{multline}
\label{Mart}
\tilde M_t
= {F(\tilde \Phi(t), \tilde n(t))\over \prod_{V\setminus\{\tilde X_t\}} \tilde\Phi_x(t) }{{\mathbbm 1}}_{\{\tilde X_t\in {\mathcal C}(x_0,\tilde n)\}}{{\mathbbm 1}}_{\{\tilde n_e(t)\ge 0\; \forall e\in E\}}=
\\
e^{-{1\over 2} \sum_{x\in V} W_x(\tilde \Phi_x(t))^2-\sum_{e\in E} J_e(\tilde\Phi(t)) }
\Big(\prod_{e\in E} {(2J_e(\tilde \Phi(t)))^{\tilde n_e(t)}\over \tilde n_e(t) !}\Big)
{2^{\#{\mathcal C}(\tilde n_e(t))-1}
\over \prod_{x\in V\setminus\{\tilde X_t\}} \tilde\Phi_x(t) }{{\mathbbm 1}}_{\{\tilde X_t\in {\mathcal C}(x_0,\tilde n(t)),\tilde n_e(t)\ge 0\; \forall e\in E\}}
\end{multline}
where ${\mathcal C}(x_0,\tilde n(t))$ denotes the cluster of the origin $x_0$ induced by the configuration ${\mathcal C}(\tilde n(t))$.
Note that at time $t=\tilde T$, we also have
\begin{eqnarray}\label{M-T}
\tilde M_{\tilde T}= {F(\tilde \Phi(\tilde T), \tilde n(\tilde T))\over \prod_{V\setminus\{\tilde x_0\}} \tilde\Phi_x(\tilde T) }{{\mathbbm 1}}_{\{\tilde X_{\tilde T}=x_0}{{\mathbbm 1}}_{\tilde n_e(t)\ge 0\; \forall e\in E\}}
\end{eqnarray}
since $\tilde M_{\tilde T}$ vanishes on the event where $\{\tilde X_{\tilde T}=x\}$, with $x\neq x_0$. Indeed, if $\tilde X_{\tilde T}=x\neq x_0$, then
$\tilde\Phi_x(\tilde T)=0$ and $J_e(\tilde\Phi(\tilde T))=0$ for $e\in E$ such that $x\in e$. It means that $\tilde M_{\tilde T}$ is equal to
0 if $\tilde n_{e}(\tilde T)>0$ for some edge $e$ neighboring $x$. Thus, $\tilde M_{\tilde T}$ is null unless $\{x\}$ is a cluster in ${\mathcal C}(\tilde n(\tilde T))$.
Hence, $\tilde M_{\tilde T}=0$ if $x\neq x_0$ since $\tilde M_{\tilde T}$ contains the indicator of the event that $\tilde X_{\tilde T}$ and $x_0$ are in the same cluster.

Hence, using identities (\ref{eq-3.3}) and (\ref{M-T})
we deduce that (\ref{test-functions}) is equal to
\begin{eqnarray}
\label{equ-M}
(\ref{test-functions})&=&
\sum_{(\tilde n_e)\in {\mathbb N}^E} \int d\tilde\Phi
H\left(\tilde \Phi,\tilde n\right) F\left(\tilde \Phi,\tilde n\right)
\tilde E_{x_0, \tilde \Phi, \tilde n}\left(
{\tilde M_{\tilde T}\over \tilde M_0}
 G\left((\tilde Z_{t})_{t\le\tilde T}\right)
\right)
\end{eqnarray}

\noindent {\bf Step 4 :}
We denote by $\check Z_t=(\check X_t, \check \Phi_t, \check n(t))$ the process defined in section \ref{sec_Poisson}, which is well defined up to
stopping time $\check T$, and $\check Z^T_t=\check Z_{t\wedge \check T}$. We denote by $\check E_{x_0, \check \Phi, \check n}$
the law of the process $\check Z$ conditionnally on the initial value $\check n(0)$, i.e. conditionally on $(N_e(2J(\check\Phi)))=(\check n_e)$.
The last step of the proof goes through the following lemma.
\begin{lemma}\label{RN}
i) Under $\check E_{x_0,\check\Phi,\check n}$, $\check X$ ends at $\check X_{\check T}=x_0$ a.s. and
$\check n_e(\check T)\ge 0$ for all $e\in E$.

ii) Let $\tilde P^{\le t}_{x_0,\tilde\Phi,\tilde n}$ and $\check P^{\le t}_{x_0,\check\Phi,\check n}$ be the law
of the process $(\tilde Z^T_s)_{s\le t}$ and $(\check Z^T_s)_{s\le t}$, then
$$
{d\check P^{\le t}_{x_0,\tilde \Phi,\tilde n}\over d\tilde P^{\le t}_{x_0,\tilde \Phi,\check n}}={\tilde M_{t\wedge \tilde T}\over \tilde M_0}.
$$
\end{lemma}
Using this lemma we obtain that in the right-hand side of (\ref{equ-M})
$$
\tilde E_{x_0, \tilde \Phi , \tilde n}\left(
{\tilde M_{\tilde T}\over \tilde M_0}
 G\left((\tilde Z_{t})_{t\le\tilde T}\right)\right)=
 \check E_{x_0, \tilde \Phi , \tilde n}
 \left(
 G\left((\check Z_{t})_{t\le\check T}\right)\right)
$$
Hence, we deduce, using formula (\ref{h}) and proceeding as in lemma \ref{distrib-phi-n}, that (\ref{test-functions}) is equal to
\begin{multline*}
\label{final}
\int_{{\mathbb R}^{V\setminus\{x_0\} }} d\tilde\varphi
e^{-{1\over 2} {\mathcal E}(\tilde\varphi)}  h(\tilde \varphi)
\sum_{(\tilde n_e)\ll (\tilde \varphi_x)} \left(\prod_{e\in E, \; \tilde\varphi_{e-}\tilde\varphi_{e+}\ge 0}
{e^{-2J_e(\vert \tilde \varphi\vert)}(2J_e(\vert \tilde \varphi\vert ))^{\tilde n_e}\over \tilde n_e !}\right)
\\\tilde E_{x_0, \vert \tilde \varphi\vert , \tilde n}\left({\tilde M_{\tilde T}\over \tilde M_0}
 G\left((\tilde Z_{t})_{t\le\tilde T}\right)\right),
\end{multline*}
where the last integral is on the set $\{(\tilde\varphi_x)\in {\mathbb R}^V, \;\; \varphi_{x_0}=u\}$,
$d\tilde\varphi={\prod_{x\in V\setminus\{x_0\}} d\tilde\varphi_x\over \sqrt{2\pi}^{\vert V\vert -1}}$, and where $(n_e)\ll (\varphi_x)$ means that
$(\tilde n_e)\in {\mathbb N}^E$ and $\tilde n_e=0$ if $\tilde\varphi_{e-}\tilde\varphi_{e+}\le 0$.
Finally, we conclude that
\begin{eqnarray*}
{\mathbb{E}}\left[ g\left(\left(X_{\tau_u^{x_0}-t}, n_e(\tau_u^{x_0}-t)\right)_{0\le t\le \tau_u^{x_0}}\right)h(\varphi^{(u)})\right]=
{\mathbb{E}}\left[ g\left(\left(\check X_{t}, \check n_e(t)\right)_{0\le t\le \check T}\right)h(\check \varphi)\right]
\end{eqnarray*}
where in the right-hand side
$\check \varphi\sim P_{\varphi}^{\{x_0\}, \sqrt{2u}} $
is a GFF and $(\check X_t, \check n(t))$
is the process defined in section \ref{sec_Poisson} from the
GFF $\check \varphi$.
This exactly means that
$\varphi^{(u)} \sim P_{\varphi}^{\{x_0\}, \sqrt{2u}}$
and that
$$
{\mathcal L}\left(\left(X_{\tau_u^{x_0}-t}, n_e(\tau_u^{x_0}-t)\right)_{0\le t\le \tau_u^{x_0}}\;
\Big| \; \varphi^{(u)}=\check\varphi\right)= {\mathcal L}\left(\left(\check X_t, \check n(t)\right)_{t\le \check T}\right).
$$
This concludes the proof of Theorem \ref{thm-Poisson2}.
\end{proof}
\begin{proof}[Proof of lemma \ref{RN}]
The generator of the process $\tilde Z_t$ defined in (\ref{tildeZ}) is given, for any bounded and $\mathcal{C}^{1}$ for the second component test function $f$, by
\begin{equation}
\label{tildeL2}
\begin{split}
&(\tilde L f)(x,\tilde\Phi,\tilde n)=
-{1\over \tilde \Phi_x} ({\partial\over \partial \tilde\Phi_x}f)
(x,\tilde\Phi, \tilde n) +\\
&
\sum_{y, \; y\sim x} \left(W_{x,y} \left(f(y,\tilde\Phi,\tilde n-\delta_{\{x,y\}})-f(x,\tilde\Phi,n)\right)+
W_{x,y} {\tilde \Phi_{y}\over \tilde \Phi_x} \left(f(x,\tilde\Phi, n-\delta_{\{x,y\}})-f(x,\tilde\Phi,n)\right)\right).
\end{split}
\end{equation}
where $ n-\delta_{\{x,y\}}$ is the value obtained by removing 1 from $n$ at edge $\{x,y\}$.
Indeed, since $\tilde \Phi_x(t)=
\sqrt{\tilde\Phi_{x}(0)^{2} -2\tilde \ell_x(t)}$, we have
\begin{eqnarray}
\label{deriv-Phi}
{\partial\over\partial t} \tilde \Phi_x(t)=
-{{\mathbbm 1}}_{\{\tilde X_t=x\}}{1\over \tilde \Phi_x(t)},
\end{eqnarray}
which is explains the first term in the expression. The second term is obvious from the definition of $\tilde Z_t$, and corresponding to the term induced by jumps
of the Markov process $\tilde X_t$. The last term corresponds to the decrease of $\tilde n$ due to the increase in the process
$\tilde N_e(\tilde \Phi)-\tilde N_e(\tilde \Phi(t))$. Indeed, on the interval $[t,t+dt]$, the probability that
$\tilde N_{e}(\tilde \Phi(t))-\tilde N_{e}(\tilde \Phi(t+dt))$
is equal to 1 is of order
$$-{\partial\over \partial t} \tilde N_e(\tilde \Phi(t))dt={{\mathbbm 1}}_{\{\tilde X_t\in e\}}
 {W_e \tilde \Phi_{\underline e}(t)\tilde\Phi_{\overline e}(t)\over \Phi_{\tilde X_t}(t)^2}dt
 $$
using identity (\ref{deriv-Phi}).

Let $\check L$ be the generator of the Markov jump process $\check Z_t=(\check X_t, (\check \Phi_x(t)), (\check n_e(t)))$.
We have that the generator is equal, for any smooth test function $f$, to

\begin{eqnarray*}
&&(\check L f)(x,\Phi, n)=
-{1\over  \Phi_x} ({\partial\over \partial \Phi_x}f)(x,\Phi,  n) +\\
&&{1\over 2} \sum_{y, \; y\sim x}{ n_{x,y} \over  \Phi_x^2}
{{{\mathbbm 1}}_{\mathcal{A}_1(x,y)}} \left(f(y,\tilde\Phi,n-\delta_{\{x,y\}})+f(x,\tilde\Phi,n-\delta_{\{x,y\}})- 2f(x,\tilde\Phi,n)\right)
\\
&&+ \sum_{y, \; y\sim x}{ n_{x,y} \over \Phi_x^2}  {{\mathbbm 1}}_{\mathcal{A}_2(x,y)} \left( f(y,\tilde\Phi,n-\delta_{\{x,y\}})- f(x,\tilde\Phi,n)) \right)
\\
&&+\sum_{y, \; y\sim x}{n_{x,y} \over \Phi_x^2}
 {{\mathbbm 1}}_{\mathcal{A}_3(x,y)} \left(f(x,\tilde\Phi,n-\delta_{\{x,y\}}) - f(x,\tilde\Phi,n) \right)
\end{eqnarray*}
where
$\mathcal{A}_{i}(x,y)$ correspond to the following disjoint events
\begin{itemize}
\item
$\mathcal{A}_1(x,y)$ if the numbers of connected clusters induced by $n-\delta_{\{x,y\}}$ is the same as that of $\check n$.
\item
$\mathcal{A}_2(x,y)$ if a new cluster is created in $ n-\delta_{\{x,y\}}$ compared with $\check n$ and if $y$ is in the connected component
of $x_0$ in the cluster induced by $ n-\delta_{\{x,y\}}$.
\item
$\mathcal{A}_3(x,y)$ if a new cluster is created in $ n-\delta_{\{x,y\}}$ compared with $n$ and if $x$ is in the connected component
of $x_0$ in the cluster induced by $ n-\delta_{\{x,y\}}$.
\end{itemize}
Indeed, conditionally on the value of $\check n_e(t)=N_e(2J_e(\check\Phi(t)))$ at time $t$, the point process $N_e$ on the interval $[0,  2J_e(\check\Phi(t)))]$ has the law of
$n_e(t)$ independent points with uniform distribution on $[0,  2J_e(\check\Phi(t)))]$. Hence, the probability that a point lies in the interval
$[2J_e(\check\Phi(t+dt))),  2J_e(\check\Phi(t)))]$ is of order
$$
-\check n_e(t) {1\over J_e(\check\Phi(t)))}{\partial\over \partial t} J_e(\check\Phi(t)))  dt= {{\mathbbm 1}}_{\{X_t\in e\}}\;\check n_e(t){1\over \check\Phi_{X_t}(t)^2}dt.
$$
We define the function
\begin{multline}
\nonumber\Theta(x,(\Phi_x),(n_e))=\\
e^{-{1\over 2} \sum_{x\in V}W_x (\Phi_x)^2-\sum_{e\in E} J_e(\Phi) }
\left(\prod_{e\in E} {(2J_e(\Phi))^{n_e}\over n_e !}\right)
{2^{\#{\mathcal C}(n_e)-1}
\over \prod_{V\setminus\{x\}} \Phi }{{\mathbbm 1}}_{\{x\in {\mathcal C}(x_0,n),
n_e\ge 0\; \forall e\in E\}},
\end{multline}
so that
$$
\tilde M_{t\wedge \tilde T}= \Theta(\tilde Z_{t\wedge\tilde T}).
$$
To prove the lemma it is sufficient to prove (\cite{ChungWalsh05MP}, Chapter 11) that for any bounded smooth test function $f$
\begin{eqnarray}\label{LcheckL}
{1\over \Theta}\tilde L\left(\Theta f\right)= \check L\left(f\right)
 \end{eqnarray}
Let us first consider the first term in (\ref{tildeL2}).
Direct computation gives
$$
\left({1\over \Theta}{1\over \Phi_x}\left({\partial\over\partial \Phi_x} \Theta\right)\right) (x,\Phi,n)= -W_x+\sum_{y\sim x} \left(- W_{x,y}{\Phi_y\over\Phi_x}+n_{x,y}{1\over \Phi_x^2}\right).
$$
For the second part, remark that the indicators ${{\mathbbm 1}}_{\{x\in {\mathcal C}(x_0,n)\}}$ and ${{\mathbbm 1}}_{\{n_e\ge 0\; \forall e\in E\}}$ imply that
$
\Theta(y,\Phi, n-\delta_{x,y})
$
vanishes if $n_{x,y}=0$ or if $y\not\in {\mathcal C}(x_0,n-\delta_{x,y})$.
By inspection of the expression of $\Theta$, we obtain for $x\sim y$,
\begin{eqnarray*}
\Theta (y,\Phi, n-\delta_{x,y})&=& \left({{\mathbbm 1}}_{\{n_{x,y}>0\}} ({{\mathbbm 1}}_{\mathcal{A}_1}+2{{\mathbbm 1}}_{\mathcal{A}_2}) {n_{x,y}\over 2J_{x,y}(\Phi)}{\Phi_y\over \Phi_x}\right)\Theta(x,\Phi, n)
\\
&=&\left(({{\mathbbm 1}}_{\mathcal{A}_1}+2{{\mathbbm 1}}_{\mathcal{A}_2}) {n_{x,y}\over 2W_{x,y}}{1\over \Phi_x^2}\right)\Theta(x,\Phi, n).
\end{eqnarray*}
Similarly,  for $x\sim y$,
\begin{eqnarray*}
\Theta(x,\Phi, n-\delta_{x,y})&=& \left({{\mathbbm 1}}_{\{n_{x,y}>0\}}({{\mathbbm 1}}_{\mathcal{A}_1}+2{{\mathbbm 1}}_{\mathcal{A}_3}){n_{x,y}\over 2J_{x,y}}\right)\Theta(x,\Phi, n)\\
&=&
\left(({{\mathbbm 1}}_{\mathcal{A}_1}+2{{\mathbbm 1}}_{\mathcal{A}_3}) {n_{x,y}\over 2W_{x,y}\Phi_x\Phi_y}\right)\Theta(x,\Phi, n).
\end{eqnarray*}
Combining these three identities with the expression (\ref{tildeL2}) we deduce
\begin{eqnarray*}
&&{1\over \Theta}\tilde L\left(\Theta f\right)(x,\Phi,n)=\\
&&
-{1\over \Phi_x} {\partial\over\partial \Phi_x}f(x,\Phi,n)-\sum_{y\sim x} \left(n_{x,y}{1\over \Phi_x^2}\right)f(x,\Phi,n)
\\
&&  +\sum_{y\sim x} ({{\mathbbm 1}}_{\mathcal{A}_1}+2{{\mathbbm 1}}_{\mathcal{A}_2}) n_{x,y}{1\over 2\Phi_x^2} f(y,  n-\delta_{\{x,y\}},\Phi)+
\sum_{y\sim x}({{\mathbbm 1}}_{\mathcal{A}_1}+2{{\mathbbm 1}}_{\mathcal{A}_3}){1\over 2 \Phi_x^2} f(x, n-\delta_{\{x,y\}},\Phi).
\end{eqnarray*}
It exactly coincides with the expression for $\check L$ since $1={{\mathbbm 1}}_{\mathcal{A}_1}+{{\mathbbm 1}}_{\mathcal{A}_2}+{{\mathbbm 1}}_{\mathcal{A}_3}$.
\end{proof}

\subsection{General case}
\label{sec:pgen}

\begin{proposition}
\label{PropKillingCase}
The conclusion of
Theorem \ref{thm-Poisson} still holds
if the graph $\mathcal{G}=(V,E)$ is finite and the killing measure is non-zero ($\kappa\not\equiv 0$).
\end{proposition}

\begin{proof}
Let $h$ be the function on $V$ defined as
\begin{displaymath}
h(x)=\mathbb{P}_{x}(X~\text{hits}~x_{0}~\text{before}~\zeta).
\end{displaymath}
By definition $h(x_{0})=1$. Moreover, for all
$x\in V\setminus\lbrace x_{0}\rbrace$,
\begin{displaymath}
-\kappa_{x} h(x)+\sum_{y\sim x}W_{x,y}(h(y)-h(x))=0.
\end{displaymath}
Define the conductances
$W^{h}_{x,y}:=W_{x,y}h(x)h(y)$, and the corresponding jump process $X^{h}$, and the GFF $\varphi_{h}^{(0)}$ and $\varphi_{h}^{(u)}$ with conditions
$0$ respectively $\sqrt{2u}$ at $x_{0}$. The Theorem \ref{thm-Poisson}
holds for the graph $\mathcal{G}$ with conductances
$(W^{h}_{e})_{e\in E}$ and with zero killing measure. But the process
$(X^{h}_{t})_{t\leq \tau_{u}^{x_{0}}}$ has the same law as the process
$(X_{s})_{s\leq \tau_{u}^{x_{0}}}$, conditioned on
$\tau_{u}^{x_{0}}<\zeta$, after the change of time
\begin{displaymath}
dt = h(X_{s})^{-2}ds.
\end{displaymath}
This means in particular that for the occupation times,
\begin{equation}
\label{EqTimeChange}
\ell_{x}(t)=h(X_{s})^{-2}\ell_{x}(s).
\end{equation}
Moreover, we have the equalities in law
\begin{displaymath}
\varphi_{h}^{(0)}\stackrel{\text{law}}{=}h^{-1}\varphi^{(0)},\qquad
\varphi_{h}^{(u)}\stackrel{\text{law}}{=}h^{-1}\varphi^{(u)}.
\end{displaymath}
Indeed, at the level of energy functions, we have:
\begin{equation*}
\begin{split}
&\mathcal{E}(hf,hf)=
\sum_{x\in V}\kappa_{x} h(x)^{2}f(x)^{2}+
\sum_{e}W_{e}(h(e_{+})f(e_{+})-h(e_{-})f(e_{-}))^{2}\\&=
\sum_{x\in V}[\kappa_{x}h(x)^{2}f(x)^{2}+
\sum_{y\sim x}W_{x,y}h(y)f(y)(h(y)f(y)-h(x)f(x))]\\
&=
\sum_{x\in V}[\kappa_{x}h(x)^{2}f(x)^{2}-
\sum_{y\sim x}W_{x,y}(h(y)-h(x))h(x)f(x)^{2}]
-\sum_{\substack{x\in V\\y\sim x}}W_{x,y}h(x)h(y)(f(y)-f(x))f(x)
\\&=[\kappa_{x_{0}}-
\sum_{y\sim x_{0}}W_{x_{0},y}(h(y)-1)]f(x_{0})^{2}
+\sum_{e}W_{e}^{h}(h(e_{+})f(e_{+})-h(e_{-})f(e_{-}))^{2}
\\&= \text{Cste}(f(x_{0}))+\mathcal{E}^{h}(f,f),
\end{split}
\end{equation*}
where $\text{Cste}(f(x_{0}))$ means that this term does not depend of $f$
once the value of the function at $x_{0}$ fixed.

Let $\check{X}^{h}_{t}$ be the inverse process for the conductances
$(W_{e}^{h)_{e\in E}}$ and the initial condition for the field
$\varphi_{h}^{(u)}$, given by Theorem \ref{thm-Poisson}.
By applying the time change
\ref{EqTimeChange} to the process $\check{X}^{h}_{t}$, we obtain an inverse process for the conductances $W_{e}$ and the field $\varphi^{(u)}$.
\end{proof}

\begin{proposition}
\label{PropInfiniteCase}
Assume that the graph $\mathcal{G}=(V,E)$ is infinite. The killing measure $\kappa$ may be non-zero. Then the conclusion of
Theorem \ref{thm-Poisson} holds.
\end{proposition}

\begin{proof}
Consider an increasing sequence of connected sub-graphs
$\mathcal{G}_{i}=(V_{i},E_{i})$ of $\mathcal{G}$ which converges to the whole graph. We assume that $V_{0}$ contains $x_{0}$.
Let $\mathcal{G}_{i}^{\ast}=(V_{i}^{\ast},E_{i}^{\ast})$ be the graph obtained by adding to $\mathcal{G}_{i}$ an abstract vertex
$x_{\ast}$, and for every edge $\lbrace x,y\rbrace$, where $x\in V_{i}$ and
$y\in V\setminus V_{i}$, adding an edge $\lbrace x,x_{\ast}\rbrace$,
with the equality of conductances
$W_{x,x_{\ast}}=W_{x,y}$.
$(X_{i,t})_{t\geq 0}$ will denote the Markov jump process on
$\mathcal{G}_{i}^{\ast}$, started from $x_{0}$.
Let $\zeta_{i}$ be the first hitting time of $x_{\ast}$ or the first
killing time by the measure $\kappa{{\mathbbm 1}}_{V_{i}}$. Let
$\varphi^{(0)}_{i}$,
$\varphi^{(u)}_{i}$ will denote the GFFs on $\mathcal{G}_{i}^{\ast}$ with condition $0$ respectively $\sqrt{2u}$ at $x_{0}$, with condition $0$ at
$x_{\ast}$, and taking in account the possible killing measure
$\kappa{{\mathbbm 1}}_{V_{i}}$.
The limits in law of $\varphi^{(0)}_{i}$
respectively $\varphi^{(u)}_{i}$ are
$\varphi^{(0)}$
respectively $\varphi^{(u)}$.

We consider the process
$(\hat{X}_{i,t},(\check{n}_{i,e}(t))_{e\in E_{i}^{\ast}})
_{0\leq t\leq\check{T}_{i}}$ be the inverse process on
$\mathcal{G}_{i}^{\ast}$, with initial field $\varphi^{(u)}_{i}$.
$(X_{i,t})_{t\leq \tau_{i,u}^{x_{0}}}$, conditional on
$\tau_{i,u}^{x_{0}}$, has the same law as
$(\check{X}_{i,\check{T}_{i}-t})_{t\leq \check{T}_{i}}$.
Taking the limit in law as $i$ tends to infinity, we conclude that
$(X_{t})_{t\leq \tau_{u}^{x_{0}}}$, conditional on
$\tau_{u}^{x_{0}}<+\infty$, has the same law as
$(\check{X}_{\check{T}-t})_{t\leq \check{T}}$ on the infinite graph
$\mathcal{G}$. The same for the clusters.
In particular,
\begin{multline*}
\mathbb{P}(\check{T}\leq t, \check{X}_{[0,\check{T}]}~\text{stays in}~V_{j})=
\lim_{i\to +\infty}
\mathbb{P}(\check{T}_{i}\leq t, \check{X}_{i,[0,\check{T}_{i}]}~\text{stays in}~V_{j})
\\=
\lim_{i\to +\infty}
\mathbb{P}(\tau_{i,u}^{x_{0}}\leq t, X_{i,[0,\tau_{i,u}^{x_{0}}]}~\text{stays in}~V_{j}\vert \tau_{i,u}^{x_{0}}<\zeta_{i})=
\mathbb{P}(\tau_{u}^{x_{0}}\leq t, X_{[0,\tau_{u}^{x_{0}}]}
~\text{stays in}~V_{j}\vert \tau_{u}^{x_{0}} < \zeta),
\end{multline*}
where in the first two probabilities we also average by the values of the
free fields.
Hence
\begin{displaymath}
\mathbb{P}(\check{T}=+\infty~\text{or}~\check{X}_{\check{T}}\neq x_{0})=
1-\lim_{\substack{t\to +\infty\\ j\to +\infty}}
\mathbb{P}(\tau_{u}^{x_{0}}\leq t, X_{[0,\tau_{u}^{x_{0}}]}
~\text{stays in}~V_{j}\vert \tau_{u}^{x_{0}} < \zeta) = 0.
\end{displaymath}
\end{proof}

\section*{Acknowledgements}

TL acknowledges the support of Dr. Max Rössler, the Walter Haefner
Foundation and the ETH Zurich Foundation.

\bibliographystyle{plain}

redpajama_set_name: RedPajamaArXiv


Lua error in Module:Location_map at line 488: Unable to find the specified location map definition: "Module:Location map/data/India Punjab" does not exist.
Sri Harmandir Sahib is the official name of the Golden Temple. It is in the city of Amritsar, Punjab, India. The temple is surrounded by a sacred pool and group of buildings important to the Sikh religion. The complex is the most important pilgrimage site in Sikhism and it is also called Darbar Sahib. One of these buildings in the complex is the Akal Takht, the highest religious authority for Khalsa Sikhs. Another building is the langar, where a free simple vegetarian meal is served to all without any discrimination. The temple is a place of worship for men and women from all walks of life and all religions to come and worship God. The four entrances (representing the four directions) to get into the Harmandir Sahib also symbolise the openness of the Sikhs towards all people and religions.
The present-day gurdwara was rebuilt in 1764 by Jassa Singh Ahluwalia with the help of other Sikh Misls. In the early nineteenth century, Maharaja Ranjit Singh founded the Sikh Empire and covered the upper floors of the gurdwara with gold, which gives it its distinctive appearance and its English name. It contains the Guru Granth Sahib, the Sikh scripture. In the 1980s the Temple was the scene of fighting connected with the Khalistan movement.
↑ 1.0 1.1 Arvind-Pal Singh Mandair 2013, pp. 41-42.
↑ Eleanor Nesbitt (2016). Sikhism: A Very Short Introduction. Oxford University Press. pp. 64–65. ISBN 978-0-19-874557-0.
↑ Kerr, Ian J. (2015). "Harimandar". Encyclopaedia of Sikhism. Punjabi University Patiala.
This page was last changed on 16 January 2019, at 17:48.


text:
class Api::V7::AgentsController < Api::BaseController
  before_filter :authenticate_user_from_token!

  swagger_controller :agents, "Agents"

  swagger_api :index do
    summary 'Returns all agents, sorted by group'
    response :ok
    response :unprocessable_entity
    response :not_found
  end

  swagger_api :show do
    summary 'Returns agent by name'
    param :path, :name, :string, :required, "Agent name"
    response :ok
    response :unprocessable_entity
    response :not_found
  end

  def index
    collection = Agent.visible.includes(:group)
    @agents = collection.decorate
  end

  def show
    agent = Agent.where(name: params[:id]).first
    @agent = agent.decorate
  end
end

redpajama_set_name: RedPajamaGithub


text:
God's creations, Objects, Angels, Materials
Changes: Grace
Revision as of 09:57, December 3, 2018 (edit)
V3z4k2 (Talk | contribs)
(→Other uses)
Revision as of 18:57, December 3, 2018 (edit)(undo)
Mgdodl (Talk | contribs)
(→Removal)
==Removal==
{{Quote|Kinda hurts. Try cutting your kidney out with a butter knife. That kinda hurt.|[[Anna Milton]]|[[Heaven and Hell]]}}[[File:TheoGrace.gif|thumb|250px|Castiel absorbing Theo's Grace.]]
{{Quote|Kinda hurts. Try cutting your kidney out with a butter knife. That kinda hurt.|[[Anna Milton]]|[[Heaven and Hell]]}}[[File:TheoGrace.gif|thumb|left|250px|Castiel absorbing Theo's Grace.]][[File:CastielGrace.gif|thumb|right|Castiel "regains his wings".]]
Though it is agonizing for the angel, it is possible for an angel's grace to be removed, either voluntarily or by force and in either [[vessels]] or in their true forms. It is viewed as a serious crime for angels to remove their own grace, one that requires the angel in question to be punished either via execution or imprisonment and torture. Anna even compared it to first-degree murder for humans.<ref name="four10" /> The forcible removal of grace is viewed in a similar light, as Castiel was disgusted with himself when he did it to Theo, calling it "barbaric",<ref name="nine09">''[[Holy Terror]]''</ref> and later in the series likened it to cannibalism.<ref name="thirteen12">''[[Various & Sundry Villains]]''</ref>
Revision as of 18:57, December 3, 2018
The third component to the spell for Expelling Angels
Can be used to track angels
A component to a spell to open a rift between worlds (only archangel grace)
Can be used to grant powers
Supports Angelic Powers and Abilities.
Regenerates an Archangel's Power.
Location/Status
In angels
In Nephilim
Metatron's with Castiel
" It's... energy. "
— Anna Milton, on grace
in Heaven and Hell
Grace is celestial energy integral to angels. It is essentially what differentiates an angel and a human. If an angel removes their grace, they are either rendered mortal within their vessel, or, if they are not using a vessel, are reborn to human parents in their own body.
Grace gives angels access to Enochian Magic, which is how they use their abilities. Without it, they are rendered almost completely human.
" Dean: So, grace ground zero, it's not destruction, it's...
Anna: Pure creation. "
— Dean Winchester and Anna Milton
Grace is a type of energy angels and Nephilim are infused with. It resembles a shining bluish-white mist.
All angels are endowed with their own grace, even archangels, seraphim, ordinary angels, and even the weakest angels such as cupids have grace. Along with angels, their offspring the Nephilim have their own grace as well. When not infused in an angel, grace must be stored in a vial or it will fall to Earth. Humans who see falling grace may mistake it for a meteorite.[1] Where grace touches down on Earth, it vastly enriches its surroundings rather than destroys them; the presence of Anna's grace caused a barren field to flourish with vegetation and the spot where it landed created a massive oak tree that looked as though it were centuries old in the space of six months. It is suggested that Anna's grace resided in this tree until it was taken by Uriel.[1]
Grace is the source of most of angels' power, endowing them with Enochian magic, including immortality and teleportation. However, angels who had their grace removed may still be able to tap into varying powers, including the ability to tune into "angel radio", Supernatural Perception, and telekinesis.[1]
So long as a vessel is possessed by an angel, that vessel also contain that angel's grace. Even after an angel departs from their vessel, a portion of their grace lingers within that vessel. Containing traces of an angel's grace does not harm the human in question and may even help them, as Castiel notes that Gadreel's grace was keeping Sam Winchester mostly healed from the Trials of God. Once Castiel removed this grace, Sam fell ill once more.[2]
While possessing Adina's grace, when Castiel retrieved what was left of his own grace (a portion of it was used in Metatron's spell), he became a Seraph once again but his powers weren't completely restored and his wings were damaged. It is unknown if this was a result of Metatron's spell or the amount of the grace that was left.
After a great portion of Lucifer's grace was extracted from him by Michael and his powers greatly diminished, Lucifer claims to Castiel that his diminished grace will eventually replenish. Lucifer believed that taking in some of Castiel's grace would replenish his powers at least somewhat, but Castiel refused to do so as he didn't trust Lucifer. Feeding on other angels grace did help Lucifer recharge faster.
The archangel Gabriel confirms that an archangel's grace will recharge. The Prince of Hell Asmodeus had been extracting Gabriel's grace for years, injecting himself with vials of it to vastly improve his own powers.
" Kinda hurts. Try cutting your kidney out with a butter knife. That kinda hurt. "
— Anna Milton
Castiel absorbing Theo's Grace.
Castiel "regains his wings".
Though it is agonizing for the angel, it is possible for an angel's grace to be removed, either voluntarily or by force and in either vessels or in their true forms. It is viewed as a serious crime for angels to remove their own grace, one that requires the angel in question to be punished either via execution or imprisonment and torture. Anna even compared it to first-degree murder for humans.[1] The forcible removal of grace is viewed in a similar light, as Castiel was disgusted with himself when he did it to Theo, calling it "barbaric",[3] and later in the series likened it to cannibalism.[4]
Angels who lose their grace fall to Earth and essentially become humans. If the angel in question is possessing a vessel at the time they lose their grace, they are then locked into their vessel until death.[5] If, however, the angel does not have a vessel, they are reborn to human parents through a human body of their own. In the case of Anna Milton, who was reborn with her own body, she gradually lost her memories of being an angel as she grew up human, but it is unknown if she deliberately repressed her memories or if this happens over time to all angels who lose their grace.[1] Regardless, angels who lose their grace become subject to human emotions, urges, and weaknesses. Other angels may still recognize an angel who has lost their grace, but will be able to tell that they are no longer truly an angel.[6]
Anna regains her grace.
It is also possible for angels who have lost their grace to regain it in order to restore their angelic power and status. They can use either their own grace (if they can find it) or remove the grace of another angel and use it for themselves.[1][3] The effects of an angel absorbing another angel's grace are unknown—specifically, whether the angel regains all of his original powers or that of the other angel's. However its revealed that using another angels grace for too long will cause the grace to burn out and kill the angel in question, although the exact time line is unknown. Whether or not any other creature can also absorb grace is also unknown.
Uriel wore Anna's hijacked grace in a necklace.
The absorption of grace is a potentially dangerous process for humans and demons: when Anna absorbed her grace, she frantically warned Sam, Dean, and Ruby to close their eyes, and a powerful demon who did not heed her warning had his host body disintegrated in the ensuing blast. Castiel and Uriel, however, were able to watch without harm. Absorbing her grace caused Anna's body to glow white, her eyes to glow yellow, and her body to soon disintegrate in a burst of white light, requiring her to call in favors to have it restored for her to use as a vessel.[1] Castiel's absorption of Theo's grace did not cause a similar reaction.[3] It is unknown what happens to the angel's soul when he reabsorbs grace, although that Anna specifically feared the prospect of losing of her human emotion by regaining her grace may indicate that the angel loses his soul by regaining his grace. However, Anna seemed to continue feeling emotions.[1]
Angels who were rendered mortal have shown some supernatural abilities, most commonly being the ability to hear other angels' telepathic communications, colloquially known as "angel radio". The angels, Castiel and Anna Milton both showed the ability to hear "Angel radio", even after their grace was removed. However, while Anna still showed other powers such as Telekinesis and Astral Perception, hearing "Angel radio" was the only ability Castiel still showed after his grace was extracted.
In Stranger in a Strange Land, Castiel stated that Jack's grace would regenerate in time, implying that if only part of angel's or nephilim's grace removed, it will be regenerate to its original power over time. In Gods and Monsters, Castiel expressed a belief that Jack's human heritage is slowing the process of his grace recharging.
An angel's grace is the third part of the spell to banish all the angels. Metatron removes Castiel's for the spell by slitting his throat and collecting his grace in a vial.[5] He later claims to Hannah that there is some left, indicating that he didn't use it all up for the spell. This is shown to be the truth in Book of the Damned when Castiel locates his remaining grace inside a book in a library. Reabsorbing it, combined with the grace he took from Adina restores Castiel's full power and Seraphim status.
The Men of Letters theorized that an angel's grace could be used to locate that angel through a tracking spell, but did not live to test their theory. The spell requires a sufficient amount of grace, extracted and collected with a special syringe. Sam and Castiel tried to track Gadreel this way, but did not use enough grace in the spell for it to work.[2]
Kevin Tran has stated that archangel grace is a key ingredient to open a rift between worlds, so Michael extracts some of Lucifer's grace. After the spell was casted by Kevin, Lucifer seized the opportunity and hijacked the rift to escape back to his original world.[7]
Asmodeus has been injecting Gabriel's grace to power himself up, granting him abilities more advanced than any other demons. This shows grace, or possibly only Archangel grace, can power up other beings such as demons.
Lucifer began absorbing the power of various Angelic Soldiers from Apocalypse World to accelerate the regeneration process of his own Archangel grace, rejuvenating his strength, durability and powers.
Lucifer absorbed Jack's Grace to get stronger than ever before.
Alternate Michael used grace to try and enhance Monsters, although this failed as the creatures he tested on were ultimately killed by it. However, he later "cracked the code" and was able to enhance the Monsters' (Vampires, Werewolves etc.) powers and resilience, for example his Werewolves became immune to the Vampires races weakness: silver, while his Vampires became immune to the Vampires usual Weakness: Dead man's blood. He was also able to enhance the powers of the Djinn; Neil to the point where he was able to conjure the fears of his victims in to the world and be able to make the manifestations do whatever he wanted. He also sent out dozens of these creatures to destroy hunters to eliminate their interference in his machinations.
It is also revealed that grace keeps a Nephilim's human and angelic halves intact but if it is removed, their cells will attack each other causing the body to slowly shut down and kill the angelic hybrid.[8]
↑ 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 Heaven and Hell
↑ 2.0 2.1 First Born
↑ 3.0 3.1 3.2 Holy Terror
↑ Various & Sundry Villains
↑ 5.0 5.1 Sacrifice
↑ I'm No Angel
↑ War of the Worlds
↑ Unhuman Nature
Retrieved from "https://supernatural.fandom.com/wiki/Grace?oldid=332179"
God's creations

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Doctors & Staff \ Doctors \ Amy Lindsay Howe, MD
Amy Lindsay Howe, MD
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Dr. Amy Lindsay Howe is a Board Certified Neurologist who has been in private practice since 2007. Dr. Howe was born in Michigan but is practically a native Houstonian since she has lived here since she was nine years old. After graduating from Spring Woods High School in 1993 she attended Texas A&M and graduated with a Bachelor of Arts degree in Biology and a Bachelor of Science degree in Psychology in 1997.
She fulfilled her basic sciences training at Texas Tech University and completed her medical training at The University of Texas Medical School at Houston, where she also completed an internship in internal medicine and a residency in neurology. She went on to complete a fellowship in Neuromuscular Disease at the Baylor College of Medicine before starting her practice.
Dr. Howe is Board Certified by the American Board of Psychiatry and Neurology. Dr. Howe treats multiple neurological conditions, including headaches and migraines, pinched nerves or radiculopathy, neck and back pain, seizures, dizziness, concussions, balance problems, muscle diseases, and memory problems.
Dr. Howe is a mother of three boys and four stepchildren, and has two granddaughters. She enjoys working out, tennis, and is a 1st Dan black belt in Tae Kwon Do. She is also an avid Rockets fan! Dr. Howe is a follower of Christ and prays for her staff and patients every day. She has a passion for international missions work.
William M. Hayes, MD, FAAOS
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