\sl \label{th:Lq} Let $V_1,V_2\in L^1(\mathbb{R}^n)\cap L^q(\mathbb{R}^n)$ be time-independent potentials, where $q>1$ if $n=2$ and $q\ge n/2$ if $n\ge 3$. Let $\mathcal U_T^1$ and $\mathcal U_T^2$ denote the corresponding initial-to-final-state maps. Then
\[
\mathcal U_T^1=\mathcal U_T^2 \quad\Longrightarrow\quad V_1=V_2.
\]

\begin{theorem}\sl \label{th:Lq} Let $V_1,V_2\in L^1(\R^n)\cap L^q(\R^n)$ be time-independent potentials, where $q>1$ if $n=2$ and $q\ge n/2$ if $n\ge 3$. Let $\mathcal U_T^1$ and $\mathcal U_T^2$ denote the corresponding initial-to-final-state maps. Then
\[
\mathcal U_T^1=\mathcal U_T^2 \quad\Longrightarrow\quad V_1=V_2.
\]
\end{theorem}

In this paper, we study an inverse problem for the Schr\"odinger equation in which the available data consist of the map that sends any initial state $f$ at time $t=0$ to the solution at a fixed final time $t=T$.

To make this precise, let us consider the initial-value problem for the Schr\"odinger equation with a time-independent potential
\begin{equation}\label{eq:Schrodinger}
\begin{cases}
i \partial_t u = -\Delta u + V u & \text{for } (t,x)\in (0, T)\times \mathbb{R}^n \eqqcolon \Sigma,
\vspace{.6em}
\\
u(0,x) = f(x) & \text{for } x\in \mathbb{R}^n.
\end{cases}
\end{equation}
We assume throughout the paper that $V=V(x)\in L^q(\mathbb{R}^n)$, where $q\ge n/2$ if $n\ge 3$ or $q>1$ if $n=2$. Then, by \cites{zbMATH02204588,zbMATH00179225}, this direct problem is well posed, and for every $f\in L^2(\mathbb{R}^n)$ there exists a unique solution
\[
u\in C\left([0,T];L^2(\mathbb{R}^n)\right).
\]
The evolution associated with \eqref{eq:Schrodinger} therefore defines a bounded linear operator
\[
\mathcal U : f\in L^2(\mathbb{R}^n)\mapsto u\in C\left([0,T];L^2(\mathbb{R}^n)\right).
\]
Consequently, for any fixed time $t\in[0,T]$, the operator
\[
\mathcal U_t : f \in L^2(\mathbb{R}^n) \mapsto u(t,\centerdot)\in L^2(\mathbb{R}^n)
\]
is also bounded, uniformly in $t$. Solutions of the form $u=\mathcal U f$, with $f\in L^2(\mathbb{R}^n)$, will be referred to as \emph{physical solutions}, while we call $\mathcal U_T$ the \emph{initial-to-final-state map}.

The main question addressed in this paper is whether the initial-to-final-state map $\mathcal U_T$ uniquely determines the Hamiltonian $-\Delta+V$. This inverse problem was first studied for \emph{time-dependent} potentials in \cite{zbMATH07801151}. There, the authors show that if the potentials $V_1,V_2 \in L^1((0,T);L^\infty(\mathbb{R}^n))$ satisfy a \emph{super-exponential decay} condition at infinity, and if $\mathcal U_T^j$ denotes the initial-to-final-state map associated with $-\Delta+V_j$, then there holds:
\[
\mathcal U_T^1=\mathcal U_T^2 \quad\Longrightarrow\quad V_1=V_2.
\]
More recently, this uniqueness result was extended in \cite{caro2025initialtofinalstateinverseproblemunbounded} to unbounded time-dependent potentials that are allowed to exhibit local $L^q$-type singularities, but still requiring the super-exponential decay assumption at infinity.

The case of time-independent potentials was previously considered in \cite{zbMATH08122191}, where uniqueness was established under comparatively weaker decay assumptions than in the time-dependent setting, namely assuming only super-linear decay at infinity. The purpose of the present paper is to relax these assumptions even further. Specifically, we prove uniqueness for time-independent potentials that may exhibit singularities of $L^q$-type in sets of finite measure and satisfy only $L^1$-integrability in sets of infinite measure; in the time-independent setting, this represents a substantial improvement over the decay and integrability assumptions in \cite{zbMATH08122191}, \cite{zbMATH07801151}, and \cite{caro2025initialtofinalstateinverseproblemunbounded}.

\begin{equation}\label{eq:Schrodinger}
\begin{cases}
i \partial_t u = -\Delta u + V u & \text{for } (t,x)\in (0, T)\times \R^n \eqqcolon \Sigma,
\vspace{.6em}
\\
u(0,x) = f(x) & \text{for } x\in \R^n.
\end{cases}
\end{equation}

The case of time-independent potentials was previously considered in \cite{zbMATH08122191}, where uniqueness was established under comparatively weaker decay assumptions than in the time-dependent setting, namely assuming only super-linear decay at infinity. The purpose of the present paper is to relax these assumptions even further. Specifically, we prove uniqueness for time-independent potentials that may exhibit singularities of $L^q$-type in sets of finite measure and satisfy only $L^1$-integrability in sets of infinite measure; in the time-independent setting, this represents a substantial improvement over the decay and integrability assumptions in \cite{zbMATH08122191}, \cite{zbMATH07801151}, and \cite{caro2025initialtofinalstateinverseproblemunbounded}.

\medskip
\noindent\textbf{Outline of the proof.}
The proof of Theorem~\ref{th:Lq} proceeds by extracting information on the difference $V_1-V_2$ from $\mathcal U_T^1 = \mathcal U_T^2$, by testing it against suitable families of solutions.

Returning to the solutions in \eqref{eq:steadysol}, we can now use the invertibility of $\mathrm{Id}-P_\lambda\circ V$ to complete the construction of the correction term $w^{\mathrm{cor}}$. In the statement of the corollary below, the additional assumption $V\in L^1(\R^n)$ is used only to ensure that $Vw^{(0)}\in L^{p'}(\R^n)$, so that $P_\lambda(Vw^{(0)})$ is well defined and belongs to $L^p(\R^n)$; the invertibility of $\mathrm{Id}-P_\lambda\circ V$ on $L^p(\R^n)$ relies solely on the $L^q$ assumption.
\begin{corollary}\label{cor:solutions}\sl Let $n\geq 2$ and consider a time-independent potential $V\in L^1(\R^n)\cap L^q(\R^n)$ for some $n/2<q\leq (n+1)/2$, and let $p$ be defined via $\frac{1}{q} = 1 - \frac{2}{p} $. For $\lambda > 2\lambda_V$ with $\lambda_V$ as in the statement of Lemma~\ref{lem:I-P inverse in Lp}, the functions
\[
w(x)\coloneqq e^{-i\lambda\omega\cdot x}+w^{\mathrm{cor}}(x)= w^{(0)}(x)+w^{\mathrm{cor}}(x),\qquad w^{\mathrm{cor}}(x)\coloneqq (\mathrm{Id}-P_\lambda\circ V)^{-1}[P_\lambda(Vw^{(0)})],
\]
with $ \omega\in\mathbb S^{n-1}$ and $x\in\R^n$,
are weak solutions of the time-independent Schr\"odinger equation
\[
(\Delta+\lambda^2-V)w=0\quad\mathrm{in}\quad\R^n.
\]
Additionally, for $\lambda>2\lambda_V$ we have the estimate
\[
\left \|w^{\mathrm{cor}}\right\|_{L^p(\R^n)}\lesssim \frac{1}{\lambda^{2n\left(\frac1p-\frac{1}{p_n}\right)}} \|V\|_{L^{p'}(\R^n)}\leq \frac{1}{\lambda^{2n\left(\frac1p-\frac{1}{p_n}\right)}} \|V\|_{L^1(\R^n)\cap L^q(\R^n)}.
\]
In particular, the function
\[
\psi_V ^{\lambda,\omega}(t,x)\coloneqq e^{-i\lambda^2 t}w(x)= e^{-i\lambda^2 t}(w^{(0)}(x)+w^{\mathrm{cor}}(x)),\qquad (t,x)\in\R\times \R^n,
\]
is a weak solution of \eqref{sch_timeind}.
\end{corollary}

\begin{corollary}\label{cor:solutionsVcrit}\sl Let $n\geq 3$ and consider a time-independent potential $V\in L^1(\R^n)\cap L^{n/2}(\R^n)$. Let $X_\lambda,X_\lambda ^*$ and $\|V\|_\lambda$ as defined above. There exists a constant $\lambda_V$ depending only on $V$ and the dimension such that for $\lambda > \lambda_V$, the functions
\[
w(x)\coloneqq e^{-i\lambda\omega\cdot x} +w^{\mathrm{cor}} (x)\eqqcolon w^{(0)}(x)+w^{\mathrm{cor}}(x),\qquad w^{\mathrm{cor}}(x)\coloneqq (\mathrm{Id}-P_\lambda\circ V)^{-1}[P_\lambda(Vw^{(0)})],
\]
with $\omega\in\mathbb S^{n-1} $ and $ x\in\R^n$,
are weak solutions of the time-independent Schr\"odinger equation
\[
(\Delta+\lambda^2-V)w=0\quad\mathrm{in}\quad\R^n.
\]
Additionally, the following estimates are satisfied
\[
\left\|w^{\mathrm{cor}} \right\|_{X_\lambda ^*}\lesssim \|V\|_{X_\lambda}\leq \lambda^{-\frac{1}{n+1}} \|V\|_{L^{q_n '}(\R^n)}\leq \lambda^{-\frac{1}{n+1}} \|V\|_{L^1(\R^n)\cap L^{n/2}(\R^n)}
\]
In particular, the function
\[
\psi_V ^{\lambda,\omega}(t,x)\coloneqq e^{-i\lambda^2 t}w(x)= e^{-i\lambda^2 t}(w^{\mathrm{cor}}(x)+w^{(0)}(x)),\qquad (t,x)\in\R\times \R^n,
\]
is a weak solution of \eqref{sch_timeind}.
\end{corollary}

\begin{proposition}\label{pr:int_by_parts_time-harmonic_solutions}\sl Let $V_1,V_2\in L^1(\R^n)\cap L^q(\R^n)$, where $q\in(1,(n+1)/2]$ if $n=2$, or $q\in[n/2,(n+1)/2]$ if $n\geq 3$, and define $p$ by
\[
 \frac{1}{q}+\frac{2}{p}=1.
\]
Let $\mathcal U_T^1,\mathcal U_T^2$ denote the initial-to-final-state maps corresponding to $V_1,V_2$, respectively.

\subsection{Proof of Theorem~\ref{th:Lq}: the endpoint case}\label{subsec: proof_uniqueness_endpoint} Let $V_1,V_2\in L^1(\R^n)\cap L^{n/2}(\R^n)$ and define $F\coloneqq V_1 - V_2$. Given $\xi \in \R^n$ we consider $\nu \in \Sph^{n-1}$ such that $\xi \cdot \nu = 0$ and $\lambda \geq \max\{\lambda_{V_1},\lambda_{V_2},|\xi|/2\}$, where $\lambda_{V_1},\lambda_{V_2}$ are the constants corresponding to $V_1,V_2$ from Corollary~\ref{cor:solutionsVcrit}. As in \S\ref{subsec:proof_uniqueness_unbounded}, we consider the vectors
\[
\omega_1 \coloneqq \frac{1}{\lambda} \frac{\xi}{2} + \left( 1 - \frac{|\xi|^2}{4 \lambda^2} \right)^{1/2} \nu,\qquad \omega_2 \coloneqq -\frac{1}{\lambda} \frac{\xi}{2} + \left( 1 - \frac{|\xi|^2}{4 \lambda^2} \right)^{1/2} \nu,
\]
and note that $\omega_1,\omega_2\in\mathbb S^{n-1}$ and $\omega_1 - \omega_2 = \xi/\lambda$.
We now construct the stationary state solutions as in Corollary~\ref{cor:solutionsVcrit}
\[
\psi_j\coloneqq \psi_{V_j} ^{\lambda,\omega_j}=e^{-i\lambda^2t}w_j(x)=e^{-i\lambda^2t}\left(w^{(0)}_j(x)+w^{\mathrm{cor}}_j(x)\right),\qquad j\in\{1,2\},\qquad (t,x)\in\R\times \R^n,
\]
and we recall the estimates
\begin{equation}\label{eq:psiestimates}
\left\|w^{\mathrm{cor}}_j\right\|_{X_\lambda ^*}\lesssim \|V_j\|_{X_\lambda}\leq \lambda^{-\frac{1}{n+1}} \|V_j\|_{L^1(\R^n)\cap L^{n/2}(\R^n)},\qquad j\in\{1,2\},
\end{equation}
also proved in Corollary~\ref{cor:solutionsVcrit}. Since $ \mathcal{U}^1_T=\mathcal{U}^2_T$, plugging the solutions $\psi_1,\psi_2$ into the orthogonality relation of
Proposition~\ref{pr:int_by_parts_time-harmonic_solutions} and repeating the calculations from \S\ref{subsec:proof_uniqueness_unbounded}, we have
\[
\begin{split}
|\wh{F}(\xi)| &\leq \frac{1}{(2\pi)^{n/2}} \int_{\R^n} \left|F w^{(0)}_1 \overline{w^{\mathrm{cor}}_2}\right| + \int_{\R^n} \left|F \overline{w^{(0)}_2} w^{\mathrm{cor}}_1\right| + \int_{\R^n} \left|F w^{\mathrm{cor}}_1 \overline{w^{\mathrm{cor}}_2}\right|
\\
&\lesssim \left\|Fw^{(0)}_1\right\|_{X_\lambda} \left\|w^{\mathrm{cor}}_2\right\|_{\X_\lambda ^*}+\left\|Fw^{(0)}_2\right\|_{X_\lambda} \left\|w^{\mathrm{cor}}_1\right\|_{\X_\lambda ^*}+\|F\|_\lambda\|w^{\mathrm{cor}}_1\|_{\X_\lambda ^*} \left \|w^{\mathrm{cor}}_2\right \|_{\X_\lambda ^*},
\end{split}
\]
where we have used the duality of $X_\lambda$ and $X_\lambda ^*$ and the H\"older-type inequality of Lemma~\ref{lem:multiplication_V_endpoint_X_norm}. Using the estimates \eqref{eq:psiestimates} and the fact that $|w^{(0)}_1|=|w^{(0)}_2|=1$ we get
\begin{align*}
|\wh{F}(\xi)| &\lesssim
 \|F\|_{\X_\lambda} \lambda^{-\frac{1}{n+1}}(\|V_1\|_{L^1(\R^n)\cap L^{n/2}(\R^n)} + \|V_2\|_{L^1(\R^n)\cap L^{n/2}(\R^n)}) \\
&+ \|F\|_{\lambda} \lambda^{-\frac{2}{n+1}} \|V_1\|_{L^1(\R^n)\cap L^{n/2}(\R^n)} \|V_2\|_{L^1(\R^n)\cap L^{n/2}(\R^n)} \\
&\lesssim\lambda^{-\frac{2}{n+1}} \left(\|V_1\|_{L^1(\R^n)\cap L^{n/2}(\R^n)} +\|V_2\|_{L^1(\R^n)\cap L^{n/2}(\R^n)}\right) ^2
(1 + \|F\|_{\lambda}) \to 0
\end{align*}
as $\lambda \to \infty$. Hence $\wh{F}(\xi) = 0$, and since $\xi\in\R^n$ was arbitrary we conclude that $F=0$ and hence that $V_1=V_2$ almost everywhere in $\R^n$, thus completing the proof of the endpoint case of Theorem~\ref{th:Lq}.

\medskip
\noindent\textbf{Outline of the proof.}
The proof of Theorem~\ref{th:Lq} proceeds by extracting information on the difference $V_1-V_2$ from $\mathcal U_T^1 = \mathcal U_T^2$, by testing it against suitable families of solutions.

The first step is to show that the equality $\mathcal U_T^1=\mathcal U_T^2$ yields an \emph{Alessandrini-type orthogonality relation}, in the sense of \cite{Aless}, of the form
\begin{equation}\label{eq:orthogonality_intro}
\int_{\Sigma} (V_1-V_2)\,u_1\,\overline{v_2}=0,
\end{equation}
valid for pairs of solutions associated with the potentials $V_1$ and $\overline{V_2}$. This identity is initially available only for \emph{physical solutions}, namely solutions belonging to $C([0,T];L^2(\mathbb{R}^n))$.

At this stage, our first obstruction is that in order to recover pointwise information on $V_1-V_2$, we need to test \eqref{eq:orthogonality_intro} against special \emph{time-harmonic} solutions. Such solutions can be constructed as perturbations of time-harmonic solutions of the  free Schr\"odinger equation; we refer to such solutions as \emph{stationary states} associated with the given potential. However, these stationary states are not in $C([0, T]; L^2(\mathbb{R}^n))$, and using them as test functions requires a substantial extension of the Alessandrini-type orthogonality relation beyond physical solutions.

More precisely, stationary states are obtained by inverting a resolvent-type operator and constructing correction terms via Neumann series. Concretely, they are of the form
\[
u(t,x)=e^{-i|\kappa|^2 t}\left(e^{-i\kappa\cdot x}+w^{\mathrm{cor}}(x)\right),
\]
where the leading term \(e^{-i|\kappa|^2 t}e^{-i\kappa\cdot x}\) is a time-harmonic solution of the free Schr\"odinger equation, and the correction \(w^{\mathrm{cor}}\) accounts for the presence of the potential \(V\).

The construction of \(w^{\mathrm{cor}}\) reduces to inverting an operator of the form \(\mathrm{Id}-P_\lambda\circ V\), where \(P_\lambda\) denotes a solution operator for the Helmholtz equation \((\Delta+\lambda^2)u=f\). This inversion is carried out on suitable function spaces, chosen so that the operator \(P_\lambda\circ V\) is small, in a suitable sense.

In following the proof strategy outlined above, we encounter a second important obstruction. In the non-endpoint regime $V\in L^q$ with $q>n/2$, decay in the energy parameter $\lambda$ follows from the classical Kenig--Ruiz--Sogge resolvent estimate. The latter ensures that $P_\lambda\circ V$ is small on the relevant function spaces for large $\lambda$. At the critical endpoint $q=n/2$, however, this estimate no longer yields decay in $\lambda$, and the smallness of $P_\lambda\circ V$ cannot be obtained from the standard resolvent bound alone.

To address this issue, we introduce a new scale of Banach spaces that allows us to recover a decaying factor in the energy parameter, also at the endpoint. Within this framework, we establish an improved form of the Kenig--Ruiz--Sogge resolvent estimate, sufficient to construct stationary states for critical potentials despite the absence of quantitative decay at the endpoint. A similar idea has appeared in \cite{CaroGarcia} for a related scattering problems with critically singular potentials; see also \cite{zbMATH07867333}. When inserted into the extended Alessandrini-type orthogonality relation, these stationary states allow us to isolate the Fourier phase and hence recover the Fourier transform of $V_1-V_2$, proving uniqueness.

Throughout the argument, the assumption of a stationary potential plays a crucial role. Indeed, in the time-dependent setting, the uniqueness results in \cites{zbMATH07801151,caro2025initialtofinalstateinverseproblemunbounded} rely on constructing complex geometrical optics solutions whose leading terms are of the form
\[
(t,x) \longmapsto e^{it|\kappa|^2} e^{\kappa\cdot x}, \qquad \kappa \in \mathbb{R}^n,
\]
supplemented by correction terms depending on the potential. These complex exponentials grow in certain directions.  Using such solutions in an orthogonality relation of the form \eqref{eq:orthogonality_intro} and controlling the corresponding correction terms naturally leads to assuming super-exponential decay of the potential at infinity.

However, in the time-independent case considered here, we work with time-harmonic solutions whose main term is given in the form
\[
(t,x) \longmapsto e^{-i|\kappa|^2 t} e^{-i\kappa\cdot x}, \qquad \kappa \in \mathbb{R}^n,
\]
ignoring again the perturbative terms correcting for the potential. Now the exponents are imaginary, so the leading term is purely oscillatory and unimodular. This allows us to restrict attention to a more regular class of solutions, which however is still sufficient to establish uniqueness in the case of time-independent potentials.

Inverse problems associated with the dynamical Schrödinger equation have been extensively studied; see, for example, \cites{zbMATH01886353,zbMATH06733553,zbMATH05549395,zbMATH05655673,zbMATH05839237,zbMATH06864429}. Many of these works have also considered time-dependent Hamiltonians \cites{zbMATH06769718,zbMATH06516179,zbMATH05379127,zbMATH07033617,zbMATH07242805}. A common feature of these investigations is the use of a dynamical Dirichlet-to-Neumann map, recorded on the boundary of a domain that contains the non-constant portions of the Hamiltonian.

This formulation stands in contrast to the one presented in our work. Here, the variable part of the Hamiltonian is not localized to a bounded region but is possibly present throughout the whole space. Moreover, our inverse problem is formulated with a distinct data requirement: knowledge only of the system's initial state and its corresponding state at a final time.

The rest of the paper is organized as follows. Section~\ref{sec:aux} collects background material and auxiliary analytic tools, including well–posedness results and resolvent estimates. In Section~\ref{sec:timeindependent_solutions} we construct stationary states for the Schr\"odinger equation with time-independent potentials, treating both the non-endpoint case $q>n/2$ and the endpoint case $q=n/2$ in dimensions $n\ge3$. Section~\ref{sec:orthorelation} is devoted to extending the Alessandrini-type orthogonality relation beyond physical solutions, and in particular to stationary-state solutions. Finally, in Section~\ref{sec:unbounded_endpoint} we prove Theorem~\ref{th:Lq} by applying the extended orthogonality relation to the stationary states constructed earlier.

The proof of Theorem~\ref{th:Lq} \emph{extracts information on} $V_1-V_2$ from $\mathcal U_T^1=\mathcal U_T^2$ by testing against suitable families of solutions.

1. From $\mathcal U_T^1=\mathcal U_T^2$, first derive an \emph{Alessandrini-type orthogonality relation} (initially for \emph{physical solutions} in $C([0,T];L^2(\mathbb{R}^n))$):
\[
\int_{\Sigma} (V_1-V_2)\,u_1\,\overline{v_2}=0,
\]
for pairs of solutions associated with $V_1$ and $\overline{V_2}$.

2. To recover pointwise/Fourier information on $V_1-V_2$, test the orthogonality identity with special \emph{time-harmonic} solutions (\emph{stationary states}), constructed as perturbations of free time-harmonic Schr\"odinger solutions. These have the form
\[
 u(t,x)=e^{-i|\kappa|^2 t}\big(e^{-i\kappa\cdot x}+w^{\mathrm{cor}}(x)\big),
\]
where $w^{\mathrm{cor}}$ accounts for the potential. Since these stationary states are \emph{not} in $C([0,T];L^2)$, one must \emph{extend the Alessandrini-type orthogonality relation beyond physical solutions} to allow using them as test functions.

3. Construct $w^{\mathrm{cor}}$ by inverting a resolvent-type operator and building correction terms via Neumann series: reduce to inverting
\(\mathrm{Id}-P_\lambda\circ V\), where $P_\lambda$ solves the Helmholtz equation $(\Delta+\lambda^2)u=f$, on function spaces where $P_\lambda\circ V$ is small.

4. Handle the endpoint obstruction: for $q>n/2$, smallness for large $\lambda$ comes from the classical Kenig--Ruiz--Sogge resolvent estimate, but at the critical endpoint $q=n/2$ it yields no decay in $\lambda$. To fix this, \emph{introduce a new scale of Banach spaces} that recovers a decaying factor in $\lambda$ and establish an \emph{improved} Kenig--Ruiz--Sogge resolvent estimate, enabling stationary-state construction even at the endpoint.

5. Insert these stationary states into the \emph{extended} orthogonality relation; this \emph{isolates the Fourier phase} and thus \emph{recovers the Fourier transform of} $V_1-V_2$, yielding $V_1=V_2$ (uniqueness).

6. The argument uses crucially that the potential is time-independent: one can use time-harmonic solutions with leading term $e^{-i|\kappa|^2 t}e^{-i\kappa\cdot x}$, which is \emph{purely oscillatory and unimodular}, avoiding the growth issues of complex geometrical optics solutions used in the time-dependent setting.

The proof of the main theorem extracts information on $V_1-V_2$ from $\mathcal U_T^1=\mathcal U_T^2$ by testing against suitable families of solutions.

1. From $\mathcal U_T^1=\mathcal U_T^2$, first derive an \emph{Alessandrini-type orthogonality relation} (initially for \emph{physical solutions} in $C([0,T];L^2(\mathbb{R}^n))$):
\[
\int_{\Sigma} (V_1-V_2)\,u_1\,\overline{v_2}=0,
\]
for pairs of solutions associated with $V_1$ and $\overline{V_2}$.

2. To recover pointwise/Fourier information on $V_1-V_2$, test the orthogonality identity with special \emph{time-harmonic} solutions (\emph{stationary states}), constructed as perturbations of free time-harmonic Schr\"odinger solutions. These have the form
\[
 u(t,x)=e^{-i|\kappa|^2 t}\big(e^{-i\kappa\cdot x}+w^{\mathrm{cor}}(x)\big),
\]
where $w^{\mathrm{cor}}$ accounts for the potential. Since these stationary states are \emph{not} in $C([0,T];L^2)$, one must \emph{extend the Alessandrini-type orthogonality relation beyond physical solutions} to allow using them as test functions.

3. Construct $w^{\mathrm{cor}}$ by inverting a resolvent-type operator and building correction terms via Neumann series: reduce to inverting
\(\mathrm{Id}-P_\lambda\circ V\), where $P_\lambda$ solves the Helmholtz equation $(\Delta+\lambda^2)u=f$, on function spaces where $P_\lambda\circ V$ is small.

4. Handle the endpoint obstruction: for $q>n/2$, smallness for large $\lambda$ comes from the classical Kenig--Ruiz--Sogge resolvent estimate, but at the critical endpoint $q=n/2$ it yields no decay in $\lambda$. To fix this, \emph{introduce a new scale of Banach spaces} that recovers a decaying factor in $\lambda$ and establish an \emph{improved} Kenig--Ruiz--Sogge resolvent estimate, enabling stationary-state construction even at the endpoint.

5. Insert these stationary states into the \emph{extended} orthogonality relation; this \emph{isolates the Fourier phase} and thus \emph{recovers the Fourier transform of} $V_1-V_2$, yielding $V_1=V_2$ (uniqueness).

6. The argument uses crucially that the potential is time-independent: one can use time-harmonic solutions with leading term $e^{-i|\kappa|^2 t}e^{-i\kappa\cdot x}$, which is \emph{purely oscillatory and unimodular}, avoiding the growth issues of complex geometrical optics solutions used in the time-dependent setting.

Question:
Fix $n\ge 2$ and a final time $T>0$. For $j=1,2$, let $V_j=V_j(x)$ be a time-independent potential on $\mathbb{R}^n$ such that $V_j\in L^1(\mathbb{R}^n)\cap L^q(\mathbb{R}^n)$, where $q>1$ if $n=2$ and $q\ge n/2$ if $n\ge 3$. For each initial datum $f\in L^2(\mathbb{R}^n)$, let $u_j$ be the solution of
\[
\begin{cases}
i\partial_t u = -\Delta u + V_j u & \text{for } (t,x)\in(0,T)\times \mathbb{R}^n,\\
u(0,x)=f(x) & \text{for } x\in \mathbb{R}^n,
\end{cases}
\]
and define the initial-to-final-state map $\mathcal U_T^j:L^2(\mathbb{R}^n)\to L^2(\mathbb{R}^n)$ by $\mathcal U_T^j f=u_j(T,\cdot)$. Which statement holds for every such pair of potentials?

If $\mathcal U_T^1=\mathcal U_T^2$ as operators from $L^2(\mathbb{R}^n)$ to $L^2(\mathbb{R}^n)$, then $V_1=V_2$ almost everywhere on $\mathbb{R}^n$.

Choices:
B. If $\mathcal U_T^1=\mathcal U_T^2$ as operators from $L^2(\mathbb{R}^n)$ to $L^2(\mathbb{R}^n)$, then $V_1=V_2$ almost everywhere on $\mathbb{R}^n$, provided in addition that $q>n/2$ when $n\ge 3$.
C. If $V_1=V_2$ almost everywhere on $\mathbb{R}^n$, then $\mathcal U_T^1=\mathcal U_T^2$ as operators from $L^2(\mathbb{R}^n)$ to $L^2(\mathbb{R}^n)$.
D.
If for every pair of physical solutions $u_1,u_2\in C([0,T];L^2(\mathbb{R}^n))$ associated with $V_1$ and $V_2$ one has
\[
\int_0^T\!\int_{\mathbb{R}^n} (V_1-V_2)(x)\,u_1(t,x)\,\overline{u_2(t,x)}\,dx\,dt=0,
\]
then $\mathcal U_T^1=\mathcal U_T^2$, and hence $V_1=V_2$ almost everywhere on $\mathbb{R}^n$.
E. If $\mathcal U_T^1=\mathcal U_T^2$ as operators from $L^2(\mathbb{R}^n)$ to $L^2(\mathbb{R}^n)$, then the Fourier transform of $V_1-V_2$ vanishes almost everywhere on $\mathbb{R}^n$, and therefore $V_1=V_2$ almost everywhere on $\mathbb{R}^n$.

reverse implication only; drops uniqueness direction from map equality to potential equality

need to extend orthogonality beyond physical solutions and use conjugate-potential pairing

Fourier recovery is pointwise in frequency via stationary-state tests, not merely almost everywhere

The stem does not explicitly reveal the correct option. It states the setup and asks which claim is universally valid; the correct answer is not leaked directly, though option A closely matches the likely theorem-level conclusion.

This is close to a theorem-recall item: the correct choice is essentially the main uniqueness statement under the stated hypotheses. However, it is not a pure tautology because the alternatives modify endpoints, directions of implication, and related consequences.

Some reasoning is needed to distinguish the exact valid statement from nearby variants, especially the endpoint case in B and the overclaims in D/E. Still, the item primarily tests recognition of the theorem rather than substantial generative mathematical reasoning.

The distractors are mathematically sophisticated and plausible: B tampers with the sharp range, C gives only the converse, D confuses an orthogonality condition with the full uniqueness mechanism, and E overstates Fourier-recovery consequences. They are distinct and target realistic failure modes.

A solid MCQ with strong, nuanced distractors and little answer leakage, but it leans heavily on theorem recognition and is only moderately effective at testing genuine generative reasoning.


\begin{theorem}		\label{thm:highT}
		Fix $\beta<1$. Let $M=M(N)$ satisfy $M/N\to0$, and let $k=k(N)\to\infty$
		with
		\[
		\frac{k(N)M(N)}{N}\longrightarrow 0
		\]
		(which in particular allows $k=o(N)$ when $M$ is fixed).

		Then
		\[
		\TV\big(\mu_N^{(k)},\pi^{\otimes k}\big)
		\longrightarrow 0
		\qquad\text{in }\Pp_\xi\text{-probability}.
		\]
	\end{theorem}

Propagation of chaos is a central concept in the study of interacting
	particle systems and mean-field models.
	Roughly speaking, it describes the phenomenon that,
	as the system size $N$ tends to infinity,
	finite collections of particles behave asymptotically independently,
	with a common limiting law.
	Propagation of chaos originated in Kac's Markovian models for gas dynamics
	\cite{Kac_foundations,Kac_probability}, an attempt to justify
	Boltzmann's ''Sto\ss zahlenansatz''.
	Propagation of chaos has since become an important
	object of study
	in probability, statistical mechanics, and mathematical physics.
	The original approach by Kac was that,
	if at time 0 the finite marginal distributions of a system are product measures in the
	thermodynamic limit, then this should carry over to the time-evolved system. In equilibrium settings for mean-field
	Gibbs measures (where the energy function is a
	function of the empirical measure) with a unique minimizer of the Helmholtz free energy, however, this was shown to
	follow from the fact the extremal Gibbs measures locally look like product measures, i.e.\
	that any finite subset of spins forms a family of independent random variables in the thermodynamic limit \cite[Theorem 3]{BAZ_chaos}.
	Such results provide a rigorous justification of mean-field
	approximations and explain why macroscopic behavior
	can often be described by effective one-particle models. In the present note, we remain entirely within a static framework, and study propagation of chaos in the sense of asymptotic factorization of finite-dimensional marginals for families of (random)
	mean-field Gibbs measures.

	Moreover, for mean-field spin systems, propagation of chaos
	is closely tied to the high-temperature regime.
	In classical models such as the Curie--Weiss Ising model,
	chaos holds when the inverse temperature is below the critical value,
	while it breaks down in the low-temperature phase. There one has to replace
	the product measure by a mixture of the (several) extremal limiting Gibbs measures.
	An additional difficulty may arise when the Gibbs measures in question are random. To the best of our knowledge comparatively little is known in such situations (with the exception of of \cite{BG98book} and \cite{KL24}). The present paper contributes to the study of propagation of chaos for random mean-field Gibbs measures in the high-temperature and near-critical regimes.

	Importantly, in \cite{BAZ_chaos}, the authors also introduced the concept of \textit{increasing} propagation of chaos and showed that in the true high temperature regime of many mean-field models one can let the size of the marginals $k$ grow with the system size $N$ as long as $k=o(N)$.
	Understanding not only whether propagation of chaos holds,
	but also the precise scales at which it breaks down,
	has become an important theme in recent work on mean-field Gibbs measures \cite{BAZ_chaos,Lacker22, JKLM23,KL24,RS25,JKL25}.

	While increasing propagation of chaos has been established in several genuinely high-temperature regimes of ordered mean-field models, much less is known in the presence of random mean-field Gibbs measures, where already the analysis of critical fluctuations can become delicate (\cite{gentzloewe, gentzloewe2, talagrand_critical_hopf}).
	In the present setting of random mean-field Gibbs measures, we show that this high-temperature scaling persists for $\beta <1$ (the critical value),
	whereas at criticality
	$\beta=1$, the admissible growth drops to $k=o(\sqrt N)$. Moreover,
	these bounds on
	$k$ are optimal for the class of random mean-field Gibbs measures considered here.

	In this article we study propagation of chaos
	in the Hopfield model, a paradigmatic example
	of a disordered mean-field spin system.
	The Hopfield model has (at least) two distinct origins. It was first introduced by
	Pastur and Figotin \cite{PS84} as a solvable model of a disordered system. At about the same time it
	was independently invented by Hopfield as
	a model of associative memory \cite{Hopfield1982}.
	Both aspects have been intensively studied. For the disordered systems facet see e.g. \cite{BGP94, BG97} or \cite{talagrand}, while
	the neural network aspect has found a renewed interest through two recent papers \cite{KrotovHopfield2016,DHLUV17}.

	For the purposes of the present paper, the
	probabilistic viewpoint of seeing the Hopfield model
	as a mean-field Ising model with random, structured interactions is more appropriate.
	The disorder, i.e.\ the random interactions, is generated by a collection of random patterns,
	which induces a random quadratic Hamiltonian
	and leads to a rich interplay between thermal fluctuations
	and quenched randomness.
	While the thermodynamic properties of the Hopfield model
	are well understood, much less is known about
	the fine structure of its finite-dimensional marginals
	and their asymptotic independence properties. There is only one result concerning the propagation of chaos in the Hopfield model, see \cite[Theorem 8.15]{BG98}. However, this result addresses fixed-dimensional marginals in the low-temperature regime, and relies on mechanisms different from those considered here.

	Our main goal is to analyze propagation of chaos
	for the Hopfield Gibbs measure in the high-temperature
	and critical regimes, with particular emphasis
	on the size of the marginals.
	We identify precise conditions under which propagation of chaos holds,
	as well as sharp thresholds for its breakdown.
	In the high-temperature regime, we prove propagation of chaos
	for growing marginals, provided their size grows sublinearly in the system size $N$.
	At criticality, we show that chaos breaks down in a critical window,
	whose scale matches that of the dominant collective fluctuations.
	reveal a clear transition between asymptotic independence and regimes of partial or complete breakdown of chaos,
	and highlight the role played by disorder-induced fluctuations
	in determining these regimes.

	\subsection{The Model}
	Let us next describe the central model for the purpose of this note.
	Let $\xi_i=(\xi_i^1,\dots,\xi_i^M)\in\{-1,+1\}^M$, $i=1,\dots,N$, be i.i.d.\ random vectors with independent coordinates,
	\[
	\mathbb E[\xi_i^\nu]=0, \qquad \mathbb E[(\xi_i^\nu)^2]=1 .
	\]
	(In the interpretation of an associative memory, the vectors $(\xi^\mu)_{\mu=1}^M=((\xi_i^\mu)_{i=1}^N)_{\mu=1}^M$ are called images or patterns).
	In what follows we will always assume  that $M=M(N)$ may depend on $N$, but in such a way, that $M=o(N)$, which is the natural regime in which the Hopfield model exhibits mean-field behavior.
	For fixed, i.e.\ quenched, patterns $(\xi^\mu)_\mu$	define and for $\sigma\in\{-1,+1\}^N$, define the overlap vector
	\[
	m_N(\sigma) = \big(m_N^1(\sigma),\dots,m_N^M(\sigma)\big),
	\qquad
	m_N^\nu(\sigma)=\frac1N\sum_{i=1}^N \sigma_i\xi_i^\nu .
	\]
	The Hopfield Gibbs measure at inverse temperature $\beta>0$ is
	\[
	\mu_N(\sigma)=\frac{1}{Z_N}
	\exp\Big(\frac{\beta N}{2}\|m_N(\sigma)\|^2\Big).
	\]
	The Gibbs measure is fully determined by $m_N(\sigma)$ (plus the inverse temperature), it thus is natural to consider
	the behaviour of the overlap under the Gibbs measure.
	This has been done in numerous papers:
	In \cite{BGP94} it was shown that -- similar to the Curie-Weiss model -- in the Hopfield model the critical (inverse) temperature is $\beta=1$. While for $\beta \le 1$ the overlap vector  gets concentrated in the $M$-dimensional 0-vector, for larger $\beta$ the limit points of $m_N$ are associated with the 2$M$ vectors $\pm z(\beta) e_\mu$ where
	$e_\mu$ is the $\mu$'th unit vector and $z(\beta)$ is the largest solution of the equation
	$$
	z=\tanh(\beta z).
	$$
	Central Limit theorems for $\sqrt N m_N(\cdot)$ were proven by Gentz \cite{gentz_annals} or Bovier and Gayrard \cite{BG_CLT}. All these results hold true for almost all realizations of the patterns and only need $M=o(N)$
	(for \cite{gentz_annals} this is only true in the high temperature regime, which, however, is the most relevant for us in the present note.) Importantly, as in the Curie-Weiss model, at $\beta=1$ the fluctuations are non-Gaussian. As was shown in \cite{gentzloewe, gentzloewe2, talagrand_critical_hopf} at $\beta=1$
	the rescaled overlap vector $N^{1/4} m_N$ converges in distribution to a random limit (while in the other limit theorems the limit was deterministic). Also large and moderate deviations for the overlap vector are available (see \cite{BG_LDP, EL_hopf}).

	\subsection{Statement of the results}
	In this subsection we state our main results on increasing propagation of chaos
	for the Hopfield model in the high-temperature and critical regimes. In particular, we identify regimes in which propagation of chaos breaks down
	at explicit scales of the marginals. For $\beta<1$, we allow the number of patterns $M=M(N)$ to diverge with $N$,
	and establish both increasing propagation of chaos and sharp breakdown results.
	At criticality $\beta=1$, we restrict attention to fixed $M$, and identify the
	critical window in which propagation of chaos fails.

	In order to formulate our results, let us agree on the following notation:
	For $k\le N$, let $\mu_N^{(k)}$ denote the marginal of $\mu_N$ on
	$(\sigma_1,\dots,\sigma_k)$.
	Let $\pi$ be the Rademacher law on $\{-1,+1\}$ with $\pi(\pm1)=\frac12$.

	Then we will prove:

Propagation of chaos is a central concept in the study of interacting
 particle systems and mean-field models.
 Roughly speaking, it describes the phenomenon that,
 as the system size $N$ tends to infinity,
 finite collections of particles behave asymptotically independently,
 with a common limiting law.
 Propagation of chaos originated in Kac's Markovian models for gas dynamics
 \cite{Kac_foundations,Kac_probability}, an attempt to justify
 Boltzmann's ''Sto\ss zahlenansatz''.
 Propagation of chaos has since become an important
 object of study
 in probability, statistical mechanics, and mathematical physics.
 The original approach by Kac was that,
 if at time 0 the finite marginal distributions of a system are product measures in the
 thermodynamic limit, then this should carry over to the time-evolved system. In equilibrium settings for mean-field
 Gibbs measures (where the energy function is a
 function of the empirical measure) with a unique minimizer of the Helmholtz free energy, however, this was shown to
 follow from the fact the extremal Gibbs measures locally look like product measures, i.e.\
 that any finite subset of spins forms a family of independent random variables in the thermodynamic limit \cite[Theorem 3]{BAZ_chaos}.
 Such results provide a rigorous justification of mean-field
 approximations and explain why macroscopic behavior
 can often be described by effective one-particle models. In the present note, we remain entirely within a static framework, and study propagation of chaos in the sense of asymptotic factorization of finite-dimensional marginals for families of (random)
 mean-field Gibbs measures.

Importantly, in \cite{BAZ_chaos}, the authors also introduced the concept of \textit{increasing} propagation of chaos and showed that in the true high temperature regime of many mean-field models one can let the size of the marginals $k$ grow with the system size $N$ as long as $k=o(N)$.
 Understanding not only whether propagation of chaos holds,
 but also the precise scales at which it breaks down,
 has become an important theme in recent work on mean-field Gibbs measures \cite{BAZ_chaos,Lacker22, JKLM23,KL24,RS25,JKL25}.

\subsection{The Model}
 Let us next describe the central model for the purpose of this note.
 Let $\xi_i=(\xi_i^1,\dots,\xi_i^M)\in\{-1,+1\}^M$, $i=1,\dots,N$, be i.i.d.\ random vectors with independent coordinates,
 \[
 \mathbb E[\xi_i^\nu]=0, \qquad \mathbb E[(\xi_i^\nu)^2]=1 .
 \]
 (In the interpretation of an associative memory, the vectors $(\xi^\mu)_{\mu=1}^M=((\xi_i^\mu)_{i=1}^N)_{\mu=1}^M$ are called images or patterns).
 In what follows we will always assume that $M=M(N)$ may depend on $N$, but in such a way, that $M=o(N)$, which is the natural regime in which the Hopfield model exhibits mean-field behavior.
 For fixed, i.e.\ quenched, patterns $(\xi^\mu)_\mu$ define and for $\sigma\in\{-1,+1\}^N$, define the overlap vector
 \[
 m_N(\sigma) = \big(m_N^1(\sigma),\dots,m_N^M(\sigma)\big),
 \qquad
 m_N^\nu(\sigma)=\frac1N\sum_{i=1}^N \sigma_i\xi_i^\nu .
 \]
 The Hopfield Gibbs measure at inverse temperature $\beta>0$ is
 \[
 \mu_N(\sigma)=\frac{1}{Z_N}
 \exp\Big(\frac{\beta N}{2}\|m_N(\sigma)\|^2\Big).
 \]
 The Gibbs measure is fully determined by $m_N(\sigma)$ (plus the inverse temperature), it thus is natural to consider
 the behaviour of the overlap under the Gibbs measure.
 This has been done in numerous papers:
 In \cite{BGP94} it was shown that -- similar to the Curie-Weiss model -- in the Hopfield model the critical (inverse) temperature is $\beta=1$. While for $\beta \le 1$ the overlap vector gets concentrated in the $M$-dimensional 0-vector, for larger $\beta$ the limit points of $m_N$ are associated with the 2$M$ vectors $\pm z(\beta) e_\mu$ where
 $e_\mu$ is the $\mu$'th unit vector and $z(\beta)$ is the largest solution of the equation
 $$
 z=\tanh(\beta z).
 $$
 Central Limit theorems for $\sqrt N m_N(\cdot)$ were proven by Gentz \cite{gentz_annals} or Bovier and Gayrard \cite{BG_CLT}. All these results hold true for almost all realizations of the patterns and only need $M=o(N)$
 (for \cite{gentz_annals} this is only true in the high temperature regime, which, however, is the most relevant for us in the present note.) Importantly, as in the Curie-Weiss model, at $\beta=1$ the fluctuations are non-Gaussian. As was shown in \cite{gentzloewe, gentzloewe2, talagrand_critical_hopf} at $\beta=1$
 the rescaled overlap vector $N^{1/4} m_N$ converges in distribution to a random limit (while in the other limit theorems the limit was deterministic). Also large and moderate deviations for the overlap vector are available (see \cite{BG_LDP, EL_hopf}).

\subsection{Statement of the results}
 In this subsection we state our main results on increasing propagation of chaos
 for the Hopfield model in the high-temperature and critical regimes. In particular, we identify regimes in which propagation of chaos breaks down
 at explicit scales of the marginals. For $\beta<1$, we allow the number of patterns $M=M(N)$ to diverge with $N$,
 and establish both increasing propagation of chaos and sharp breakdown results.
 At criticality $\beta=1$, we restrict attention to fixed $M$, and identify the
 critical window in which propagation of chaos fails.

In order to formulate our results, let us agree on the following notation:
 For $k\le N$, let $\mu_N^{(k)}$ denote the marginal of $\mu_N$ on
 $(\sigma_1,\dots,\sigma_k)$.
 Let $\pi$ be the Rademacher law on $\{-1,+1\}$ with $\pi(\pm1)=\frac12$.

Then we will prove:

In order to formulate our results, let us agree on the following notation:
 For $k\le N$, let $\mu_N^{(k)}$ denote the marginal of $\mu_N$ on
 $(\sigma_1,\dots,\sigma_k)$.
 Let $\pi$ be the Rademacher law on $\{-1,+1\}$ with $\pi(\pm1)=\frac12$.

\begin{theorem} \label{thm:stop_macro_Msmall}
 Fix $\beta\in(0,1)$ and assume $M=M(N)\to\infty$ with $M=o(\sqrt N)$.
 Let $k=k(N)$ satisfy $k/N\to\rho\in(0,1)$.
 Then, in $\P_\xi$-probability,
 \[
 \TV\big(\mu_N^{(k)},\pi^{\otimes k}\big)\longrightarrow 1.
 \]
 \end{theorem}

\begin{theorem} \label{thm:crit_poc}
 Fix $\beta=1$ and let $M\in\mathbb N$ be fixed. Assume $k=k(N)\to\infty$ satisfies
 \[
 \frac{k(N)}{N^{1/2}}\longrightarrow 0.
 \]
 Then
 \[
 d_{\mathrm{TV}}\big(\mu_N^{(k)},\pi^{\otimes k}\big)\longrightarrow 0
 \qquad\text{in }\mathbb P_\xi\text{-probability}.
 \]
 \end{theorem}

And finally, we show
 \begin{theorem}[Critical-window breakdown at $\beta=1$ (fixed $M$)]
 \label{thm:crit_break_sqrtN}
 Fix $\beta=1$ and let $M\in\mathbb N$ be fixed.
 Let $k=k(N)$ satisfy
 \[
 \frac{k}{\sqrt N}\longrightarrow c\in(0,\infty).
 \]
 Then there exists a deterministic constant $b=b(c,M)>0$ such that
 \[
 \liminf_{N\to\infty} \ d_{\mathrm{TV}}\big(\mu_N^{(k)},\pi^{\otimes k}\big)\ \ge\ b
 \qquad\text{in }\mathbb P_\xi\text{-probability}.
 \]
 In particular, propagation of chaos fails in the critical window $k\asymp \sqrt N$,
 showing that the scaling in Theorem~\ref{thm:crit_poc} is optimal.

\begin{lemma}
 \label{lem:covconcentration}
 Let $\xi_1,\dots,\xi_N\in\mathbb R^M$ be i.i.d.\ centered subgaussian random vectors
 with covariance $\Sigma:=\E[\xi_1\xi_1^\top]$.
 Let
 \[
 \widehat\Sigma_N := \frac1N\sum_{i=1}^N \xi_i\xi_i^\top .
 \]
 Then there exists an absolute constant $C>0$ (depending only on the subgaussian
 norm of $\xi_1$) such that for every $u\ge 0$,
 \begin{equation}\label{eq:conc}
 \Big\|\widehat\Sigma_N-\Sigma\Big\|_{\mathrm{op}}
 \le
 C\Bigg(\sqrt{\frac{M+u}{N}}+\frac{M+u}{N}\Bigg)\,\|\Sigma\|_{\mathrm{op}}
 \qquad\text{with probability at least }1-2e^{-u}.
 \end{equation}
 In particular, if $\Sigma=I_M$ (the $M$-dimensional identity matrix) and $M/N\to 0$, then for every fixed $\varepsilon>0$,
 \[
 \Pp\Big(\big\|\widehat\Sigma_N-I_M\big\|_{\mathrm{op}}\le \varepsilon\Big)\longrightarrow 1,
 \qquad\text{and hence}\qquad
 \Pp\Big(\|\widehat\Sigma_N\|_{\mathrm{op}}\le 1+\varepsilon\Big)\longrightarrow 1.
 \]
 \end{lemma}

\begin{lemma} \label{lem:UI_LN_correct}
 Assume $k/N\to\rho\in(0,1)$ and $M=o(k)$, and set
 $\lambda_N:=\tau^2 k/N$ with $\tau^2=\beta/(1-\beta)$, so $\lambda_N\to\lambda>0$.
 Let $L_N$ be given on $\mathcal G_N$ by
 \[
 L_N
 =
 (1+\lambda_N)^{-M/2}
 \exp\!\Big(\alpha_N\|S_{N,k}\|^2\Big),
 \qquad \text{with}\quad
 \alpha_N:=\frac{\lambda_N}{2(1+\lambda_N)}.
 \]
 Then there exists $\eta>0$ such that
 \[
 \sup_N \E_{\pi^{\otimes k}}\big[L_N^{1+\eta}\mathbf 1_{\mathcal G_N}\big] <\infty.
 \]
 In particular, $(L_N)_N$ is uniformly integrable under $\pi^{\otimes k}$ on $\mathcal G_N$.
 \end{lemma}

Let $r_\ast^2:=\frac{M(1+\lambda)\log(1+\lambda)}{\lambda}$be the unique radius where the two densities cross.
 Define the (optimal) set is $$A_\ast:=\{\|x\|\ge r_\ast\}.$$ Then,
 \[
 \TV(\mathcal N(0,I_M),\mathcal N(0,(1+\lambda)I_M))
 =
 \P(Y\in A_\ast)-\P(X\in A_\ast).
 \]
 Since $\chi^2_M/M\to1$ in probability and concentrates at scale $\sqrt M$,
 one has $\|X\|^2/M\to1$ while $\|Y\|^2/M\to 1+\lambda$,
 and $r_\ast^2/M$ lies strictly between $1$ and $1+\lambda$.
 Therefore $\P(X\in A_\ast)\to0$ and $\P(Y\in A_\ast)\to1$, hence
 \[
 \TV(\mathcal N(0,I_M),\mathcal N(0,(1+\lambda)I_M))\longrightarrow 1
 \qquad (M\to\infty).
 \]
 Combining the previous steps yields $\TV(\mu_N^{(k)},\pi^{\otimes k})\to1$ in $\P_\xi$-probability.
 \end{proof}

\begin{proof}[Proof of Theorem~\ref{thm:crit_poc}]
 Let
 \[
 L_N(\sigma_{1:k})
 :=\frac{d\mu_N^{(k)}}{d\pi^{\otimes k}}(\sigma_{1:k}).
 \]
 Then, by the Cauchy--Schwarz inequality
 \[
 d_{\mathrm{TV}}(\mu_N^{(k)},\pi^{\otimes k})
 =\frac12\,\E_{\pi^{\otimes k}}\big[|L_N-1|\big]
 \le \frac12\,\Big(\E_{\pi^{\otimes k}}\big[(L_N-1)^2\big]\Big)^{1/2}
 =\frac12\,\Big(\E_{\pi^{\otimes k}}[L_N^2]-1\Big)^{1/2}.
 \]
 Thus it suffices to show that
 \begin{equation}\label{eq:chi2_to_0_beta1}
 \E_{\pi^{\otimes k}}[L_N^2]\longrightarrow 1
 \qquad\text{in }\mathbb P_\xi\text{-probability}.
 \end{equation}

\begin{theorem}		\label{thm:crit_poc}
		Fix $\beta=1$ and let $M\in\mathbb N$ be fixed. Assume $k=k(N)\to\infty$ satisfies
		\[
		\frac{k(N)}{N^{1/2}}\longrightarrow 0.
		\]
		Then
		\[
		d_{\mathrm{TV}}\big(\mu_N^{(k)},\pi^{\otimes k}\big)\longrightarrow 0
		\qquad\text{in }\mathbb P_\xi\text{-probability}.
		\]
	\end{theorem}

\begin{theorem}		\label{thm:stop_macro_Msmall}
		Fix $\beta\in(0,1)$ and assume $M=M(N)\to\infty$ with $M=o(\sqrt N)$.
		Let $k=k(N)$ satisfy $k/N\to\rho\in(0,1)$.
		Then, in $\P_\xi$-probability,
		\[
		\TV\big(\mu_N^{(k)},\pi^{\otimes k}\big)\longrightarrow 1.
		\]
	\end{theorem}

	\begin{remark}
		The restriction $M=o(\sqrt N)$ in Theorem~\ref{thm:stop_macro_Msmall}
		is stronger than in Theorem~\ref{thm:highT}.
		Whether this condition can be relaxed remains an open problem.
	\end{remark}

	\begin{theorem}		\label{thm:crit_poc}
		Fix $\beta=1$ and let $M\in\mathbb N$ be fixed. Assume $k=k(N)\to\infty$ satisfies
		\[
		\frac{k(N)}{N^{1/2}}\longrightarrow 0.
		\]
		Then
		\[
		d_{\mathrm{TV}}\big(\mu_N^{(k)},\pi^{\otimes k}\big)\longrightarrow 0
		\qquad\text{in }\mathbb P_\xi\text{-probability}.
		\]
	\end{theorem}

	And finally, we show
	\begin{theorem}[Critical-window breakdown at $\beta=1$ (fixed $M$)]
		\label{thm:crit_break_sqrtN}
		Fix $\beta=1$ and let $M\in\mathbb N$ be fixed.
		Let $k=k(N)$ satisfy
		\[
		\frac{k}{\sqrt N}\longrightarrow c\in(0,\infty).
		\]
		Then there exists a deterministic constant $b=b(c,M)>0$ such that
		\[
		\liminf_{N\to\infty} \ d_{\mathrm{TV}}\big(\mu_N^{(k)},\pi^{\otimes k}\big)\ \ge\ b
		\qquad\text{in }\mathbb P_\xi\text{-probability}.
		\]
		In particular, propagation of chaos fails in the critical window $k\asymp \sqrt N$,
		showing that the scaling in Theorem~\ref{thm:crit_poc} is optimal.

	\end{theorem}
	\subsection{Outline of the proofs}

	A central idea in our proofs is that the Hopfield Gibbs measure admits,
	via a Hubbard--Stratonovich transformation, an explicit representation
	in terms of a mixture of product measures.

	\medskip
	\noindent\textit{Step 1: Mixture-of-products representation.}
	For fixed (quenched) patterns $(\xi^\mu)_\mu$, the quadratic Hamiltonian
	$\frac{\beta N}{2}\|m_N(\sigma)\|^2$ can be linearized by introducing an
	auxiliary Gaussian field. This standard technqie for quadratic Hamiltonians yields a representation of the form
	\[
	\mu_N(d\sigma)\;=\;\int \nu_{N,\beta}(d y)\,
	\bigotimes_{i=1}^N \mu_{y,i}(d\sigma_i),
	\]
	where $y\in\mathbb R^M$ is the Hubbard--Stratonovich field, $\nu_{N,\beta}$ is an
	explicit probability measure on $\mathbb R^M$, and $\mu_{y,i}$ is a Bernoulli law
	with bias depending on $y\cdot \xi_i$.
	Consequently, the $k$-spin marginal $\mu_N^{(k)}$ is a mixture of product
	measures on $\{-1,+1\}^k$.

	\medskip
	\noindent\textit{Step 2: Propagation of chaos reduces to stability of the mixture.}
	Conditionally on $y$, the spins are independent.
	Thus propagation of chaos for $\mu_N$ is controlled by how strongly the
	random field $y$ fluctuates under $\nu_{N,\beta}$ and by how sensitively
	the single-spin biases $\mu_{y,i}$ depend on $y$.
	Quantitatively, we compare the mixture to the unbiased product law
	$\pi^{\otimes k}$ in total variation distance.

	\medskip
	\noindent\textit{Step 3: High temperature $\beta<1$.}
	In the regime $\beta<1$, the Hubbard--Stratonovich field typically remains
	of order one and concentrates near the origin.
	A Taylor expansion of the single-spin biases and a control of
	$\nu_{N,\beta}$ show that correlations between $k$ spins are of order
	$kM/N$, yielding Theorem~\ref{thm:highT} under the condition $kM/N\to0$.
	For macroscopic $k$ (with $k/N\to\rho$), the same representation allows us
	to exhibit order-one correlations induced by the random field, which leads
	to strong breakdown in total variation (Theorem~\ref{thm:stop_macro_Msmall})
	under the stated assumptions on $M$.

	\medskip
	\noindent\textit{Step 4: Critical temperature $\beta=1$ and the critical window.}
	At criticality, the mixing measure $\nu_{N,1}$ develops non-Gaussian
	fluctuations on the scale $N^{-1/4}$, reflecting the well-known
	critical behavior of the overlap.
	This amplification of the mixing fluctuations reduces the admissible growth
	of the marginals to $k=o(N^{1/4})$ (Theorem~\ref{thm:crit_poc}).
	Moreover, when $k\asymp \sqrt N$, the mixture retains a nontrivial
	amount of randomness that produces correlations bounded away from zero,
	yielding a breakdown of propagation of chaos in the critical window
	(Theorem~\ref{thm:crit_break_sqrtN}) and hence the optimality of the scaling.

A central idea in our proofs is that the Hopfield Gibbs measure admits, via a Hubbard--Stratonovich transformation, an explicit representation in terms of a mixture of product measures.

\medskip
\noindent\textit{Step 1: Mixture-of-products representation.}
For fixed (quenched) patterns $(\xi^\mu)_\mu$, the quadratic Hamiltonian $\frac{\beta N}{2}\|m_N(\sigma)\|^2$ is linearized by introducing an auxiliary Gaussian field (Hubbard--Stratonovich), yielding
\[
\mu_N(d\sigma)=\int \nu_{N,\beta}(d y)\, \bigotimes_{i=1}^N \mu_{y,i}(d\sigma_i),
\]
where $y\in\R^M$, $\nu_{N,\beta}$ is an explicit probability measure on $\R^M$, and $\mu_{y,i}$ is a Bernoulli law with bias depending on $y\cdot \xi_i$. Consequently, the $k$-spin marginal $\mu_N^{(k)}$ is a mixture of product measures on $\{-1,+1\}^k$.

\medskip
\noindent\textit{Step 2: Propagation of chaos reduces to stability of the mixture.}
Conditionally on $y$, the spins are independent, so propagation of chaos is controlled by how strongly $y$ fluctuates under $\nu_{N,\beta}$ and by how sensitively the single-spin biases $\mu_{y,i}$ depend on $y$. Quantitatively, one compares the mixture to the unbiased product law $\pi^{\otimes k}$ in total variation.

\medskip
\noindent\textit{Step 3: High temperature $\beta<1$ (Theorem~\ref{thm:highT}).}
In the regime $\beta<1$, the Hubbard--Stratonovich field “typically remains of order one and concentrates near the origin.” Using “a Taylor expansion of the single-spin biases and a control of $\nu_{N,\beta}$” shows that “correlations between $k$ spins are of order $kM/N$,” which yields Theorem~\ref{thm:highT} under the condition $kM/N\to0$.

A central idea in our proofs is that the Hopfield Gibbs measure admits, via a Hubbard--Stratonovich transformation, an explicit representation in terms of a mixture of product measures.

\medskip
\noindent\textit{Step 1: Mixture-of-products representation.}
For fixed (quenched) patterns $(\xi^\mu)_\mu$, the quadratic Hamiltonian $\frac{\beta N}{2}\|m_N(\sigma)\|^2$ is linearized by introducing an auxiliary Gaussian field (Hubbard--Stratonovich), yielding
\[
\mu_N(d\sigma)=\int \nu_{N,\beta}(d y)\, \bigotimes_{i=1}^N \mu_{y,i}(d\sigma_i),
\]
where $y\in\R^M$, $\nu_{N,\beta}$ is an explicit probability measure on $\R^M$, and $\mu_{y,i}$ is a Bernoulli law with bias depending on $y\cdot \xi_i$. Consequently, the $k$-spin marginal $\mu_N^{(k)}$ is a mixture of product measures on $\{-1,+1\}^k$.

\medskip
\noindent\textit{Step 2: Propagation of chaos reduces to stability of the mixture.}
Conditionally on $y$, the spins are independent, so propagation of chaos is controlled by how strongly $y$ fluctuates under $\nu_{N,\beta}$ and by how sensitively the single-spin biases $\mu_{y,i}$ depend on $y$. Quantitatively, one compares the mixture to the unbiased product law $\pi^{\otimes k}$ in total variation.

\medskip
\noindent\textit{Step 3: High temperature $\beta<1$ (main theorem).}
In the regime $\beta<1$, the Hubbard--Stratonovich field “typically remains of order one and concentrates near the origin.” Using “a Taylor expansion of the single-spin biases and a control of $\nu_{N,\beta}$” shows that “correlations between $k$ spins are of order $kM/N$.” Under the condition $kM/N\to0$, this yields the conclusion needed to establish the main theorem.

Question:
Let \(\xi_i=(\xi_i^1,\dots,\xi_i^M)\in\{-1,+1\}^M\), \(i=1,\dots,N\), be i.i.d. random vectors whose coordinates are independent and satisfy \(\mathbb E[\xi_i^\nu]=0\) and \(\mathbb E[(\xi_i^\nu)^2]=1\). For \(\sigma\in\{-1,+1\}^N\), define the overlap vector
\[
m_N(\sigma)=\big(m_N^1(\sigma),\dots,m_N^M(\sigma)\big),
\qquad m_N^\nu(\sigma)=\frac1N\sum_{i=1}^N \sigma_i\xi_i^\nu.
\]
For fixed inverse temperature \(\beta>0\), the quenched Hopfield Gibbs measure on \(\{-1,+1\}^N\) is
\[
\mu_N(\sigma)=\frac1{Z_N}\exp\!\Big(\frac{\beta N}{2}\|m_N(\sigma)\|^2\Big).
\]
Let \(\mu_N^{(k)}\) denote the marginal law of the first \(k\) spins under \(\mu_N\), and let \(\pi\) be the symmetric Bernoulli measure on \(\{-1,+1\}\), so that \(\pi^{\otimes k}\) is the product law of \(k\) i.i.d. fair \(\pm1\) spins. Assume that \(\beta<1\), that \(M=M(N)\) satisfies \(M/N\to0\), and that \(k=k(N)\to\infty\) with
\[
\frac{k(N)M(N)}{N}\longrightarrow 0.
\]
Here total variation distance is \(\mathrm{TV}(\nu,\eta)=\sup_A |\nu(A)-\eta(A)|\), and \(\mathbb P_\xi\) denotes probability with respect to the random patterns \((\xi_i)\). Under these assumptions, which statement about the \(k\)-spin marginal is valid?

\[\mathrm{TV}\big(\mu_N^{(k)},\pi^{\otimes k}\big)\longrightarrow 0\qquad\text{in }\mathbb P_\xi\text{-probability}.\]

Choices:
B. \[\mathrm{TV}\big(\mu_N^{(k)},\pi^{\otimes k}\big)\longrightarrow 0\qquad\text{almost surely in }\mathbb P_\xi.\]
C. \[\mu_N^{(k)}\xrightarrow[]{\ d\ }\pi^{\otimes k}\qquad\text{in }\mathbb P_\xi\text{-probability}.\]
D.
If \(M/N\to0\) and \(k=o(N)\), then for every fixed \(\beta<1\),
\[\mathrm{TV}\big(\mu_N^{(k)},\pi^{\otimes k}\big)\longrightarrow 0\qquad\text{in }\mathbb P_\xi\text{-probability}.\]
E.
Assume \(\beta<1\), \(M/N\to0\), and \(k(N)M(N)/N\to0\). Then
\[N\,\mathrm{TV}\big(\mu_N^{(k)},\pi^{\otimes k}\big)\longrightarrow 0\qquad\text{in }\mathbb P_\xi\text{-probability}.\]

The stem does not explicitly state the conclusion, and none of the notation or setup directly gives away choice A. The learner still has to know or infer the correct scaling and mode of convergence.

This is essentially a direct theorem-recall item: the stem lists the exact hypotheses and asks which limiting statement holds, with the correct option being the theorem’s conclusion almost verbatim.

There is some reasoning pressure because the student must distinguish between probability vs almost sure convergence, fixed-r vs growing-k marginals, and the sharp condition kM/N -> 0 vs the weaker k/N -> 0. But the item mainly tests recognition of the stated result rather than generating a new conclusion.

The distractors are mathematically meaningful and target realistic errors: overstrengthening the convergence mode (B), choosing a weaker true statement instead of the strongest one (C), missing the critical scale kM/N (D), or asserting the opposite asymptotic behavior (E).

A mathematically well-constructed theorem-recognition MCQ with strong distractors, but it is largely tautological and only moderately tests generative reasoning.


Question:
Let \(\xi_i=(\xi_i^1,\dots,\xi_i^M)\in\{-1,+1\}^M\), \(i=1,\dots,N\), be i.i.d. random vectors whose coordinates are independent and satisfy \(\mathbb E[\xi_i^\nu]=0\) and \(\mathbb E[(\xi_i^\nu)^2]=1\). For \(\sigma\in\{-1,+1\}^N\), define the overlap vector
\[
m_N(\sigma)=\big(m_N^1(\sigma),\dots,m_N^M(\sigma)\big),
\qquad m_N^\nu(\sigma)=\frac1N\sum_{i=1}^N \sigma_i\xi_i^\nu.
\]
For fixed inverse temperature \(\beta>0\), the quenched Hopfield Gibbs measure on \(\{-1,+1\}^N\) is
\[
\mu_N(\sigma)=\frac1{Z_N}\exp\!\Big(\frac{\beta N}{2}\|m_N(\sigma)\|^2\Big).
\]
Let \(\mu_N^{(k)}\) denote the marginal law of the first \(k\) spins under \(\mu_N\), and let \(\pi\) be the symmetric Bernoulli measure on \(\{-1,+1\}\), so that \(\pi^{\otimes k}\) is the product law of \(k\) i.i.d. fair \(\pm1\) spins. Assume that \(\beta<1\), that \(M=M(N)\) satisfies \(M/N\to0\), and that \(k=k(N)\to\infty\) with
\[
\frac{k(N)M(N)}{N}\longrightarrow 0.
\]
Here total variation distance is \(\mathrm{TV}(\nu,\eta)=\sup_A |\nu(A)-\eta(A)|\), and \(\mathbb P_\xi\) denotes probability with respect to the random patterns \((\xi_i)\). Under these assumptions, which statement about the \(k\)-spin marginal is valid?

Choices:
B. \[\mathrm{TV}\big(\mu_N^{(k)},\pi^{\otimes k}\big)\longrightarrow 0\qquad\text{almost surely in }\mathbb P_\xi.\]
C. \[\mu_N^{(k)}\xrightarrow[]{\ d\ }\pi^{\otimes k}\qquad\text{in }\mathbb P_\xi\text{-probability}.\]
D.
If \(M/N\to0\) and \(k=o(N)\), then for every fixed \(\beta<1\),
\[\mathrm{TV}\big(\mu_N^{(k)},\pi^{\otimes k}\big)\longrightarrow 0\qquad\text{in }\mathbb P_\xi\text{-probability}.\]
E.
Assume \(\beta<1\), \(M/N\to0\), and \(k(N)M(N)/N\to0\). Then
\[N\,\mathrm{TV}\big(\mu_N^{(k)},\pi^{\otimes k}\big)\longrightarrow 0\qquad\text{in }\mathbb P_\xi\text{-probability}.\]

The stem gives the model, assumptions, and notation but does not state the conclusion or obviously reveal the correct option. The correct answer is not leaked explicitly or trivially.

The item is close to a theorem-identification question: the correct choice is essentially the exact theorem conclusion under the stated assumptions. However, it is not a pure restatement because the alternatives vary in strength, mode of convergence, and scaling assumptions.

Some reasoning is required to distinguish total variation from weak convergence, probability from almost sure convergence, and the sharp condition kM/N -> 0 from weaker-looking but false variants. Still, the main task is recognizing the precise theorem statement rather than generating a new conclusion from scratch.

The distractors are mathematically plausible and well-targeted: one is stronger than the true result (almost sure), one is weaker but true, one relaxes the key scaling incorrectly, and one asserts an unjustified rate. These reflect realistic failure modes.

A solid theorem-discrimination MCQ with strong distractors and no answer leakage, but it leans more toward precise recall/recognition of the result than deep generative reasoning.


\label{thm:main}
Let $\mu$ be a translation-invariant, ergodic, finite-energy edge percolation measure on $\mathbb{Z}^2$.  Then one of the following holds:
\begin{itemize}
    \item Almost surely $\mu_{\hp}$ has a unique infinite cluster and $\mu^*_{\hp}$ has no infinite cluster.
    \item Almost surely $\mu^*_{\hp}$ has a unique infinite cluster and $\mu_{\hp}$ has no infinite cluster.
    \item Almost surely neither $\mu_{\hp}$ nor $\mu^*_{\hp}$ has an infinite cluster.
\end{itemize}

\begin{corollary}\label{cor:self-dual}
Let $\mu$ be a self-dual, translation-invariant, ergodic edge percolation measure on $\Z^2$ that almost surely has only finitely many infinite clusters.  Then $\mu_{\hp}$ almost surely has no infinite cluster.

In particular, if $\mu$ is a self-dual, translation-invariant, ergodic, finite-energy edge percolation measure on $\Z^2$, then $\mu_{\hp}$ almost surely has no infinite cluster.
\end{corollary}

\subsection{Background}
This note concerns non-coexistence phenomena for edge (bond) percolation on  the square lattice $\mathbb{Z}^2$ and the upper half-plane $\mathbb{Z} \times \Z_{\geq 0}$. On the full plane $\mathbb{Z}^2$, a celebrated argument of Zhang shows that (almost surely) infinite primal and dual clusters cannot coexist in any translation-invariant, finite-energy, and positively-associated (FKG) edge percolation measure.  See \cite[page 289]{grimmett2012percolation} for an exposition and \cite{bollobas2008percolation, glazman2025planar}, \cite[Theorem 9.3.1]{sheffield2005random} for generalizations.
Here, as in many arguments in percolation theory, the positive-association hypothesis is essential, and without this condition there can be pathological examples (as in, e.g., \cite{haggstrom2009some}).It is nonetheless desirable to see what can be said in the absence of positive-association, since many planar percolation models of interest, such as the loop $O(n)$ model, the random-cluster model with $q<1$, the arboreal gas, and other constrained percolation models, do not satisfy the FKG inequality.

Translation-invariance alone is certainly not sufficient to guarantee non-coexistence. For example, the dual of the uniform spanning tree is again a uniform spanning tree, so for this measure both the primal and the dual have infinite clusters \cite{PemantleUST}.  These infinite clusters are ``non-robustly connected'' in the sense that each vertex has a unique path to infinity.  A more troubling example is Häggström and Mester's construction \cite{haggstrom2009some} of a translation-invariant, finite-energy site percolation measure on $\mathbb{Z}^2$ with simultaneous infinite clusters of open and closed vertices which are robust under iid thinning.

In the present paper we show that the situation is better in the half-plane than in the full plane: Here, translation-invariance, finite-energy, and absence of infinitely many infinite clusters together suffice for non-coexistence.

\subsection{Main results}

We work with the square lattice $\mathbb{Z}^2$, namely, the Cayley graph of $\mathbb{Z}^2$ with respect to the generators $(1,0),(0,1)$.
Subgraphs of $\mathbb{Z}^2$ correspond to tuples $\omega \in  \{0,1\}^{E(\mathbb{Z}^2)}$,  where a coordinate $1$ represents an ``open'' edge that is present and a coordinate $0$ represents a ``closed'' edge that is absent. By an \emph{edge percolation measure} we mean a probability measure on $\{0,1\}^{E(\mathbb{Z}^2)}$ (with the product topology).

We need a few more notions before we can state our main results.  The natural translation action of $\mathbb{Z}^2$ on the square lattice extends to an action of $\mathbb{Z}^2$ on $\{0,1\}^{E(\mathbb{Z}^2)}$; we say that a percolation measure is \emph{translation-invariant} if it is invariant under this action.  Say that an edge percolation measure $\mu$ has \emph{finite-energy} if there is some $\delta>0$ such that for every edge $e \in E(\mathbb{Z}^2)$, the conditional probability $$\mu(\omega_e=1|\omega_{E(\mathbb{Z}^2) \setminus\{e\}})$$  has essential infimum at least $\delta$ and essential supremum at most $1-\delta$; this condition, which is sometimes called uniform finite-energy in the literature, is stronger than other common notions of finite-energy which have less uniformity (in either the edge $e$ or the bounds on the conditional probability).

Every edge percolation measure $\mu$ on $\mathbb{Z}^2$ induces an edge percolation measure $\mu^*$ on the planar dual of $\mathbb{Z}^2$, which is itself isomorphic to $\mathbb{Z}^2$; see \Cref{sec:duality} below for further discussion of duality.  It is easy to see that $\mu$ is ergodic (respectively, has finite-energy) if and only if $\mu^*$ is ergodic (respectively, has finite-energy).  Finally, we write $\mu_{\hp}$ for the marginal of $\mu$ on the upper half-plane $\mathbb{Z} \times \Z_{ \geq 0}$, and we define $\mu^*_{\hp}$ analogously.  Our first non-coexistence result goes as follows.

\subsection{Background}
This note concerns non-coexistence phenomena for edge (bond) percolation on the square lattice $\mathbb{Z}^2$ and the upper half-plane $\mathbb{Z} \times \Z_{\geq 0}$. On the full plane $\mathbb{Z}^2$, a celebrated argument of Zhang shows that (almost surely) infinite primal and dual clusters cannot coexist in any translation-invariant, finite-energy, and positively-associated (FKG) edge percolation measure. See \cite[page 289]{grimmett2012percolation} for an exposition and \cite{bollobas2008percolation, glazman2025planar}, \cite[Theorem 9.3.1]{sheffield2005random} for generalizations.
Here, as in many arguments in percolation theory, the positive-association hypothesis is essential, and without this condition there can be pathological examples (as in, e.g., \cite{haggstrom2009some}).It is nonetheless desirable to see what can be said in the absence of positive-association, since many planar percolation models of interest, such as the loop $O(n)$ model, the random-cluster model with $q<1$, the arboreal gas, and other constrained percolation models, do not satisfy the FKG inequality.

Translation-invariance alone is certainly not sufficient to guarantee non-coexistence. For example, the dual of the uniform spanning tree is again a uniform spanning tree, so for this measure both the primal and the dual have infinite clusters \cite{PemantleUST}. These infinite clusters are ``non-robustly connected'' in the sense that each vertex has a unique path to infinity. A more troubling example is Häggström and Mester's construction \cite{haggstrom2009some} of a translation-invariant, finite-energy site percolation measure on $\mathbb{Z}^2$ with simultaneous infinite clusters of open and closed vertices which are robust under iid thinning.

In the present paper we show that the situation is better in the half-plane than in the full plane: Here, translation-invariance, finite-energy, and absence of infinitely many infinite clusters together suffice for non-coexistence.

We work with the square lattice $\mathbb{Z}^2$, namely, the Cayley graph of $\mathbb{Z}^2$ with respect to the generators $(1,0),(0,1)$.
Subgraphs of $\mathbb{Z}^2$ correspond to tuples $\omega \in \{0,1\}^{E(\mathbb{Z}^2)}$, where a coordinate $1$ represents an ``open'' edge that is present and a coordinate $0$ represents a ``closed'' edge that is absent. By an \emph{edge percolation measure} we mean a probability measure on $\{0,1\}^{E(\mathbb{Z}^2)}$ (with the product topology).

We need a few more notions before we can state our main results. The natural translation action of $\mathbb{Z}^2$ on the square lattice extends to an action of $\mathbb{Z}^2$ on $\{0,1\}^{E(\mathbb{Z}^2)}$; we say that a percolation measure is \emph{translation-invariant} if it is invariant under this action. Say that an edge percolation measure $\mu$ has \emph{finite-energy} if there is some $\delta>0$ such that for every edge $e \in E(\mathbb{Z}^2)$, the conditional probability $$\mu(\omega_e=1|\omega_{E(\mathbb{Z}^2) \setminus\{e\}})$$ has essential infimum at least $\delta$ and essential supremum at most $1-\delta$; this condition, which is sometimes called uniform finite-energy in the literature, is stronger than other common notions of finite-energy which have less uniformity (in either the edge $e$ or the bounds on the conditional probability).

Every edge percolation measure $\mu$ on $\mathbb{Z}^2$ induces an edge percolation measure $\mu^*$ on the planar dual of $\mathbb{Z}^2$, which is itself isomorphic to $\mathbb{Z}^2$; see \Cref{sec:duality} below for further discussion of duality. It is easy to see that $\mu$ is ergodic (respectively, has finite-energy) if and only if $\mu^*$ is ergodic (respectively, has finite-energy). Finally, we write $\mu_{\hp}$ for the marginal of $\mu$ on the upper half-plane $\mathbb{Z} \times \Z_{ \geq 0}$, and we define $\mu^*_{\hp}$ analogously. Our first non-coexistence result goes as follows.

\begin{abstract}
For $\mu$ an edge percolation measure on the infinite square lattice, let $\mu_{\texorpdfstring{\hp}{hp}}$ (respectively, $\mu^*_{\texorpdfstring{\hp}{hp}}$) denote its marginal (respectively, the marginal of its planar dual process) on the upper half-plane. We show that if $\mu$ is translation-invariant and ergodic and almost surely has only finitely many infinite clusters, then either almost surely $\mu_{\texorpdfstring{\hp}{hp}}$ has no infinite cluster, or almost surely $\mu^*_{\texorpdfstring{\hp}{hp}}$ has no infinite cluster.
By the classical Burton--Keane argument, these hypotheses are satisfied if $\mu$ is translation-invariant and ergodic and has finite-energy.
In contrast to previous ``non-coexistence'' theorems, our result does not impose a positive-correlation (FKG) hypothesis on $\mu$.
Our arguments also apply to the random-cluster model (including the regime $q<1$, which lacks FKG), the uniform spanning tree, and the uniform odd subgraph.
\end{abstract}

Every edge percolation measure $\mu$ on $\Z^2$ induces an edge percolation measure $\mu^*$ on the planar dual of $\Z^2$, which is itself isomorphic to $\Z^2$; see \Cref{sec:duality} below for further discussion of duality. It is easy to see that $\mu$ is ergodic (respectively, has finite-energy) if and only if $\mu^*$ is ergodic (respectively, has finite-energy). Finally, we write $\mu_{\hp}$ for the marginal of $\mu$ on the upper half-plane $\Z \times \Z_{ \geq 0}$, and we define $\mu^*_{\hp}$ analogously. Our first non-coexistence result goes as follows.

If we drop the ergodicity hypothesis from this theorem, then we still obtain the conclusion that for $H$ sampled from $\mu_{\hp}$, almost surely the number of infinite clusters in $H$ plus the number of infinite clusters in $H^*$ is at most $1$; in particular, almost surely $H$ and $H^*$ do not simultaneously have infinite clusters. This statement follows from applications of the theorem to the individual components in the ergodic decomposition of $\mu$, which almost surely inherit the finite-energy property (see, e.g., \cite[page 307]{burton1991topological}).

\begin{theorem}\label{thm:main2}
Let $\mu$ be a translation-invariant, ergodic edge percolation measure on $\Z^2$ which has finitely many infinite clusters almost surely. Then the trichotomy of \Cref{thm:main} holds.
\end{theorem}

\begin{corollary}\label{cor:self-dual}
Let $\mu$ be a self-dual, translation-invariant, ergodic edge percolation measure on $\Z^2$ that almost surely has only finitely many infinite clusters. Then $\mu_{\hp}$ almost surely has no infinite cluster.

\begin{proposition}\label{lemma:uniqueness_implies_non_tenuousness}
 Let $\mu$ be a translation-invariant edge percolation measure on $\Z^2$ such that $\mu_{\hp}$ almost surely has at most $1$ infinite cluster. Then $\mu_{\hp}$ almost surely has no tenuous infinite cluster.
\end{proposition}

\begin{lemma}\label{lem:a.s.constant}
Let $\mu$ be a translation-invariant, ergodic edge percolation measure on $\Z^2$ that almost surely has only finitely many infinite clusters. Then the number of infinite clusters in $\mu_{\hp}$ is either almost surely $0$ or almost surely $1$.
\end{lemma}

\begin{proposition}\label{proposion31}
 Let $\mu$ be a translation-invariant edge percolation measure on $\Z^2$ such that $\mu_{\hp}$ almost surely has a unique infinite cluster. Then almost surely $\mu^*_{\hp}$ has no infinite cluster.
\end{proposition}

\begin{theorem}\label{thm:main}
Let $\mu$ be a translation-invariant, ergodic, finite-energy edge percolation measure on $\Z^2$.  Then one of the following holds:
\begin{itemize}
    \item Almost surely $\mu_{\hp}$ has a unique infinite cluster and $\mu^*_{\hp}$ has no infinite cluster.
    \item Almost surely $\mu^*_{\hp}$ has a unique infinite cluster and $\mu_{\hp}$ has no infinite cluster.
    \item Almost surely neither $\mu_{\hp}$ nor $\mu^*_{\hp}$ has an infinite cluster.
\end{itemize}
\end{theorem}

If we drop the ergodicity hypothesis from this theorem, then we still obtain the conclusion that for $H$ sampled from $\mu_{\hp}$, almost surely the number of infinite clusters in $H$ plus the number of infinite clusters in $H^*$ is at most $1$; in particular, almost surely $H$ and $H^*$ do not simultaneously have infinite clusters.  This statement follows from applications of the theorem to the individual components in the ergodic decomposition of $\mu$, which almost surely inherit the finite-energy property (see, e.g., \cite[page 307]{burton1991topological}).

Classical work of Burton and Keane~\cite{burton1989density} shows that under the hypotheses of \Cref{thm:main}, the number of infinite clusters in $\mu$ is almost surely at most $1$.\footnote{Burton and Keane's uniqueness argument has also been adapted to several settings that fall slightly short of having  finite-energy; examples include the loop $O(n)$ model \cite[Proposition 4.11]{crawford2020macroscopic},  the random current representation of the Ising model \cite{aizenman2015random}, various constrained percolation models \cite{lima2020constrained,holroyd2021constrained}, and the 1-2 model \cite{li2014uniqueness}.}  Thus \Cref{thm:main} is an immediate consequence of the following theorem, our main technical result.

\begin{theorem}\label{thm:main2}
Let $\mu$ be a translation-invariant, ergodic edge percolation measure on $\mathbb{Z}^2$ which has finitely many infinite clusters almost surely.  Then the trichotomy of \Cref{thm:main} holds.
\end{theorem}

\Cref{thm:main2} applies to some models, such as the uniform spanning tree, which \Cref{thm:main} does not cover.
We remark that there exist translation-invariant edge percolation measures on $\mathbb{Z}^2$ that almost surely have multiple, but only finitely many, infinite clusters.\footnote{For an example that almost surely has $2$ infinite clusters, one can modify the uniform spanning tree as follows.  Sample a uniform spanning tree $H$ on the dilated infinite square grid $(2\mathbb{Z})^2$.  Let $H^*$ be its planar dual (also a spanning tree), interpreted as a subgraph of the square grid $(1+2\mathbb{Z})^2$.  By subdividing each edge in half, we can interpret $H \cup H^*$ as a subgraph of $\mathbb{Z}^2$; let $\mu'$ denote the resulting edge percolation measure on $\mathbb{Z}^2$.  Notice that $\mu'$ is invariant under translations by $(2\mathbb{Z})^2$ and always has exactly $2$ infinite clusters.  Finally, let $\mu$ be the average of $\mu'$ and its translates by $(1,0),(0,1),(1,1)$; this guarantees that $\mu$ is $\mathbb{Z}^2$-translation-invariant, and of course it still always has $2$ infinite clusters.  For $2<k<\infty$, we do not know of examples of translation-invariant edge-percolation measures that almost surely have $k$ infinite clusters.}

We record the following corollary for self-dual measures.

\begin{corollary}\label{cor:self-dual}
Let $\mu$ be a self-dual, translation-invariant, ergodic edge percolation measure on $\mathbb{Z}^2$ that almost surely has only finitely many infinite clusters.  Then $\mu_{\hp}$ almost surely has no infinite cluster.

In particular, if $\mu$ is a self-dual, translation-invariant, ergodic, finite-energy edge percolation measure on $\mathbb{Z}^2$, then $\mu_{\hp}$ almost surely has no infinite cluster.
\end{corollary}

The proof strategy for Theorem~\ref{thm:main2} has two steps.  The first step is showing that the number of infinite clusters in $\mu_{\hp}$ is either almost surely $0$ or almost surely $1$.  The main challenge is eliminating the possibility that an infinite cluster in the full plane ``splits'' into several infinite clusters in the half-plane.  The second step is showing that if $\mu_{\hp}$ almost surely has a unique infinite cluster, then almost surely $\mu^*_{\hp}$ has no infinite cluster; it suffices to show that almost surely the plaquette centered at $(1/2,1/2)$ does not connect to infinity in the dual.  Here, almost surely both the positive $x$-axis and the negative $x$-axis connect to the unique infinite primal cluster, and these paths to infinity together ``block'' the origin plaquette from connecting to infinity in the dual.  A recurring technical point is ruling out the existence of what we term ``tenuous'' infinite clusters.

Our methods are robust enough to establish some slight extensions of these theorems.  Instead of taking the upper half-plane, one could take the half-plane lying on one side of any given line with rational slope.  Also, one could replace the square lattice with a different planar graph admitting a $\mathbb{Z}^2$-translation action with only finitely many orbits.

\subsection{Applications}

Let us highlight four special cases of our results.  First, when $\mu$ is Bernoulli percolation with density $1/2$, Corollary~\ref{cor:self-dual} (via finite-energy) provides a new proof of the fact that almost surely there is no infinite cluster in the half-plane.  Of course it is a classical result of Harris~\cite{harris1960lower} that there is no infinite cluster even in the full plane; the novelty of our argument is that it avoids using FKG (or the Harris inequality).

Second, Theorem~\ref{thm:main} demonstrates non-coexistence in the half-plane for (a suitable limit of) the random-cluster model with all values of the parameter $q$; see \Cref{sec:rc} below for precise definitions.  The random-cluster model has FKG only when $q \geq 1$, and for $q<1$ little is known.

Third, the uniform spanning tree on $\mathbb{Z}^2$ is self-dual and ergodic (due to tail-triviality; see \cite[Theorem 10.18]{lyons2017probability}), and it has a unique infinite cluster.
Thus, Corollary~\ref{cor:self-dual} shows that the half-plane marginal of the uniform spanning tree almost surely has only finite clusters.  (Note that this model lacks finite-energy.).  The same holds for the Peano uniform spanning tree (UST) curve \cite{lyons2017probability} and for constructions of Häggström--Mester type \cite{haggstrom2009some} (if they can be adapted to edge percolation).

Fourth, we show that the uniform odd subgraph almost surely has no infinite cluster in the half-plane.  This does not follow from the statement of Corollary~\ref{cor:self-dual}, but our arguments from Theorem~\ref{thm:main2} can be adapted.  The uniform odd subgraph measure is self-complementary, and here the ``non-crossing'' property of the primal and the complement plays the role that planar duality plays in Theorem~\ref{thm:main2}.

\subsection{Organization of the paper}
We start by proving \Cref{thm:main2} in Section~\ref{sec:thm1.1}.  We then turn to applications: Section~\ref{sec:rc} contains our results on the $q<1$ random-cluster model, and Section~\ref{sec:odd} treats the uniform odd subgraph.  In Section~\ref{sec:conclusion} we make some concluding remarks and mention some  open problems.  \Cref{sec:appendix} contains an alternative proof of a variant of one of our technical lemmas (the absence of tenuous infinite clusters).

For \Cref{thm:main}, the text explains that (using Burton--Keane uniqueness) it is an immediate consequence of \Cref{thm:main2}, since under the hypotheses of \Cref{thm:main} “the number of infinite clusters in \mu is almost surely at most $1$.”

A proof sketch is then given for \Cref{thm:main2} (and hence for \Cref{thm:main}): “The proof strategy for Theorem~\ref{thm:main2} has two steps.” (1) “showing that the number of infinite clusters in \mu_{\hp} is either almost surely $0$ or almost surely $1$,” with the “main challenge” being “eliminating the possibility that an infinite cluster in the full plane ‘splits’ into several infinite clusters in the half-plane.” (2) “showing that if \mu_{\hp} almost surely has a unique infinite cluster, then almost surely \mu^*_{\hp} has no infinite cluster; it suffices to show that almost surely the plaquette centered at $(1/2,1/2)$ does not connect to infinity in the dual.” The mechanism is that “almost surely both the positive $x$-axis and the negative $x$-axis connect to the unique infinite primal cluster, and these paths to infinity together ‘block’ the origin plaquette from connecting to infinity in the dual.” A “recurring technical point” throughout is “ruling out the existence of what we term ‘tenuous’ infinite clusters.”

Separately, if ergodicity is dropped, the text notes one can still deduce (by “applications of the theorem to the individual components in the ergodic decomposition of \mu”) that for $H\sim\mu_{\hp}$ the number of infinite clusters in $H$ plus the number in $H^*$ is “at most $1$,” so “$H$ and $H^*$ do not simultaneously have infinite clusters.”

To prove the main theorem, the text explains that (using Burton--Keane uniqueness) it is an immediate consequence of the following theorem, since under the hypotheses of the main theorem “the number of infinite clusters in \mu is almost surely at most $1$.”

\begin{theorem}\label{thm:main2}
Let $\mu$ be a translation-invariant, ergodic edge percolation measure on $\Z^2$ which has finitely many infinite clusters almost surely.  Then the trichotomy of \\Cref{thm:main} holds.
\end{theorem}

A proof sketch is then given for the theorem above (and hence for the main theorem): “The proof strategy for Theorem~\ref{thm:main2} has two steps.” (1) “showing that the number of infinite clusters in \mu_{\hp} is either almost surely $0$ or almost surely $1$,” with the “main challenge” being “eliminating the possibility that an infinite cluster in the full plane ‘splits’ into several infinite clusters in the half-plane.” (2) “showing that if \mu_{\hp} almost surely has a unique infinite cluster, then almost surely \mu^*_{\hp} has no infinite cluster; it suffices to show that almost surely the plaquette centered at $(1/2,1/2)$ does not connect to infinity in the dual.” The mechanism is that “almost surely both the positive $x$-axis and the negative $x$-axis connect to the unique infinite primal cluster, and these paths to infinity together ‘block’ the origin plaquette from connecting to infinity in the dual.” A “recurring technical point” throughout is “ruling out the existence of what we term ‘tenuous’ infinite clusters.”

Separately, if ergodicity is dropped, the text notes one can still deduce (by “applications of the theorem to the individual components in the ergodic decomposition of \mu”) that for $H\sim\mu_{\hp}$ the number of infinite clusters in $H$ plus the number in $H^*$ is “at most $1$,” so “$H$ and $H^*$ do not simultaneously have infinite clusters.”

Question: Let μ be an edge percolation measure on the square lattice Z2, viewed as a probability measure on {0,1}^{E(Z2)}. Assume that μ is translation-invariant, ergodic, and finite-energy, meaning that there exists δ > 0 such that for every edge e, the conditional probability μ(ω_e = 1 | ω_{E(Z2)\setminus\{e\}}) has essential infimum at least δ and essential supremum at most 1 - δ. Let μ* denote the induced edge percolation measure on the planar dual lattice, let μ_hp and μ*_hp denote the marginals of μ and μ* on the upper half-plane Z × Z_{≥0}, and interpret an infinite cluster as an infinite connected component of open edges in the corresponding half-plane configuration. Which statement holds for every such μ?

One of the following three alternatives holds almost surely: (i) μ_hp has a unique infinite cluster and μ*_hp has no infinite cluster; (ii) μ*_hp has a unique infinite cluster and μ_hp has no infinite cluster; or (iii) neither μ_hp nor μ*_hp has an infinite cluster.

Choices:
B. Almost surely exactly one of bc_{\hp} and \mu^*_{\hp} has an infinite cluster, and whenever one does, it has a unique infinite cluster.
C. Almost surely \mu_{\hp} and \mu^*_{\hp} do not simultaneously have infinite clusters.
D. If \mu_{\hp} almost surely has an infinite cluster, then almost surely \mu^*_{\hp} has no infinite cluster; and if \mu^*_{\hp} almost surely has an infinite cluster, then almost surely \mu_{\hp} has no infinite cluster.
E. Almost surely one of the following holds: (i) \mu_{\hp} has at least one infinite cluster and \mu^*_{\hp} has no infinite cluster; (ii) \mu^*_{\hp} has at least one infinite cluster and \mu_{\hp} has no infinite cluster; or (iii) neither \mu_{\hp} nor \mu^*_{\hp} has an infinite cluster.

omits the third alternative where neither half-plane process percolates

drops the uniqueness/trichotomy and keeps only non-coexistence

replaces the almost-sure trichotomy by one-way implication statements about percolation events

weakens 'unique infinite cluster' to 'at least one infinite cluster' despite the sketch's step eliminating splitting in the half-plane

The stem does not reveal the correct option directly. It states the hypotheses and asks for the full almost-sure classification, but it does not explicitly encode the trichotomy in choice A.

This is essentially a theorem-recall item: the stem gives the exact setup and asks for the theorem's conclusion. The correct choice is basically the classification theorem restated.

Some reasoning is needed to distinguish the complete statement from weaker true or stronger false variants, especially uniqueness versus mere existence. However, the item mainly tests recognition of the known result rather than substantial generative mathematical reasoning.

The distractors are strong and well targeted: one is weaker-but-true/incomplete, one allows forbidden coexistence, and one adds an unjustified extra claim. They are distinct and reflect realistic mathematical failure modes.

A solid theorem-recall MCQ with strong distractors and little answer leakage, but it is largely tautological and only moderately tests genuine reasoning.