\begin{thm}\label{thm-main11}
    For any complex curve in a K\"ahler surface $M^4$, the first eigenvalue $\Lambda_1$ of its area Jacobi operator $\mathcal{L}$ satisfies
    $$\Lambda_1\geq 2\,\mathfrak{Ric}.$$
\end{thm}

The first eigenvalue of the Laplace-Beltrami operator on a compact Riemannian manifold encodes crucial geometric information, including curvature bounds. A fundamental result in this direction is the Lichnerowicz theorem~\cite{Lichnerowicz}, which states that if the Ricci curvature of an \(n\)-dimensional compact manifold satisfies \(\mathrm{Ric} \ge (n-1)K\) for some \(K>0\), then the first positive eigenvalue \(\lambda_1\) obeys \(\lambda_1 \ge nK\). This estimate is sharp, and the Obata theorem~\cite{Obata1962} characterizes the equality case: \(\lambda_1 = nK\) holds iff the manifold is isometric to a sphere of constant sectional curvature \(K\). In the K\"ahler setting, Futaki~\cite{Futaki} proved that the first eigenvalue of the complex Laplacian is bounded below by \(k\) for a compact K\"ahler manifold satisfying \(\mathrm{Ric} \ge k\) with \(k>0\). The equality case was later characterized by Tam and Yu~\cite{Tam-Yu}.

The area Jacobi operator $\mathcal{L}$, which governs the second variation of the area functional for minimal submanifolds, is a second-order, self-adjoint elliptic operator. As such, it can be regarded as the extrinsic counterpart to the intrinsic Laplace–Beltrami operator. Unlike the latter, it is not positive (or even nonnegative) in general. We denote by $\Lambda_1$ the smallest nonzero eigenvalue of $\mathcal{L}$, referred to as the first eigenvalue. Since $\mathcal{L}$ serves as the stability operator, $\Lambda_1$ governs the infinitesimal behavior of the minimal submanifold under deformations. A positive $\Lambda_1$ means the submanifold is locally minimizing, whereas a negative one indicates instability.
In recent decades, considerable attention has been devoted to estimating the first eigenvalue of the area Jacobi operator for minimal surfaces in various ambient spaces. In \cite{Perdomo}, Perdomo proved that if $\Sigma^2$ is a compact oriented minimal surface which is not totally geodesic in $\mathbb{S}^3$, then the first eigenvalue of its area Jacobi operator satisfies $\Lambda_1 \leq -4$, with equality if and only if $\Sigma^2$ is isometric to the Clifford torus. This result sparked a series of further estimates for $\Lambda_1$ on closed hypersurfaces of constant mean curvature or conatnt scalar curvature (see \cite{Alias,Ambrozio,BatistaCavalcanteMelo,Batista,Bray,Chen-Cheng,Cheng,Dung,Li-Wang,Merono,Zhu} and references therein).
Micallef and Wolfson~\cite{MW} obtained fundamental results on the stability of minimal surfaces in 4-dimensional manifolds by considering a complex version of the second variation of area. Subsequently, for superminimal surfaces with negative spin in self-dual Einstein $4$-dimensional manifolds, Montiel and Urbano~\cite{MU} proved that the first eigenvalue of the area Jacobi operator satisfies $\Lambda_1\geq-2\mathfrak{c}$, where $\mathfrak{c}$ denotes the Einstein constant of the ambient space, and they computed exactly the index explicitly in terms of the genus and area of the surface.

This paper focuses on another important class of codimension-2 minimal surfaces, i.e., complex curves in Kähler surfaces, which are superminimal surfaces of positive spin. It is a classical fact that such surfaces are absolutely area‑minimizing and thus stable. Consequently, their associated area Jacobi operator $\mathcal{L}$ is a non‑negative self‑adjoint elliptic operator, exactly analogous to the Laplace–Beltrami operator on a Riemannian manifold. While the non-negativity of $\mathcal{L}$ is well understood, obtaining sharp quantitative estimates for its spectrum in terms of the ambient curvature is a more subtle problem, and the focus of our investigation. We established a universal lower bound for the first eigenvalue $\Lambda_1$ of $\mathcal{L}$. To state our result, we denote by
\[
\mathfrak{Ric} \triangleq \inf_{\substack{p \in \Sigma, \\ u \in UT_p M^4}} \operatorname{Ric}(u, u),
\]
the infimum of the Ricci curvature over all unit tangent vectors at all points of $M^4$.

If we restrict the ambient space to a Kähler–Einstein surface with positive scalar curvature, then the estimate established in Theorem \ref{thm-main11} is sharp. In this setting, the curvature invariant $\mathfrak{Ric}$ simplifies to the Einstein constant $\mathfrak{c}>0$. We explore the equality case and are able to compute exactly the dimension of the first eigenspace. This is accomplished by relating it to the space of Jacobi fields of $\mathcal{W}^+$, which corresponds precisely to the preimage of the space of holomorphic sections of a certain line bundle, and then applying the Riemann-Roch theorem in conjunction with the Dolbeault theorem.
\begin{thm}\label{thm-main2}
 Let \(M^{4}\) be a K\"ahler-Einstein surface with Einstein constant \(\mathfrak{c}>0\), and let \(x\colon\Sigma \rightarrow M^{4}\) be a complex curve. Then
 \begin{itemize}
 \item[{\rm(1)}] the first eigenvalue $\Lambda_1$ of its area Jacobi operator is at least $2\mathfrak{c}$,
 \item[{\rm(2)}] if $\Sigma$ has genus $g\leq 1$, then $\Lambda_1=2\mathfrak{c}$ and the corresponding eigenspace has dimension
 \[
 \mathfrak{c}\frac{\operatorname{Area}}{2\pi}+1-g+\dim H^{\,0}(N^*_\Sigma\otimes K_\Sigma^2),
 \]
 where $N^*_\Sigma$ is the dual of the normal bundle, $K_\Sigma$ is the canonical line bundle of $\Sigma$, and $H^{\,0}(N^*_\Sigma\otimes K_\Sigma^2)$ denotes the space of holomorphic sections of $N^*_\Sigma\otimes K_\Sigma^2$.
\end{itemize}
\end{thm}

Now we assume that $x$ is a complex curve, then \eqref{eq-sfm1} yields
$$ a \equiv 0,~~~ b \equiv 0,$$
which implies $ x $ is a superminimal surface with positive spin. Consequently, $x$ minimizes the functional $\mathcal{W}^+$ within its homotopy class. Without loss of generality, we assume $\lambda\equiv 1$ and $\mu\equiv 0$. Denote by $\theta_{12}$ (resp. $\theta_{34}$) the connection form of the tangent (resp. normal) bundle of $x$, It follows from \eqref{eq-pull}$\sim$\eqref{eq-conn} that
\begin{equation}\label{eq-o1100}
 x^*\omega_{1\bar{1}} = -i\theta_{12}, ~~~x^*\omega_{0\bar{0}} = -i\theta_{34}.
\end{equation}
Differentiating \eqref{eq-o1100} yields the Gauss-Ricci equation
\begin{equation}\label{eq-Gauss1}
 (R_{1212}+R_{1234})\,\theta\wedge\bar\theta = -2x^*(\Omega_{0\bar0}+\Omega_{1\bar1}),
\end{equation}
where \(R_{1212}\) (resp. \(R_{1234}\)) denotes the curvature of the tangent (resp. normal) bundle, defined by
\[
d\theta_{12} = \frac{R_{1212}}{2i}\,\theta\wedge\bar\theta, \qquad
d\theta_{34} = \frac{R_{1234}}{2i}\,\theta\wedge\bar\theta.
\]
Note that
$-i(\Omega_{0\bar0}+ \Omega_{1\bar1})$ is exactly the Ricci form of $M^4$, which implies
\begin{equation}\label{eq-Ric}
x^*(\Omega_{0\bar0}+ \Omega_{1\bar1})=-\mathrm{Ric}(e_1, \bar e_1)\theta\wedge\bar\theta.
\end{equation}
Therefore, we can also reformulate the Gauss-Ricci equation \eqref{eq-Gauss1} as
\begin{equation}\label{eq-Gauss2}
 R_{1212}+R_{1234}=2\mathrm{Ric}(e_1,\bar e_1).
\end{equation}

Using \eqref{eq-W+3} and \eqref{eq-lambda1}, we obtain that along the variation determined by $V$,
\begin{equation}\label{eq-varW}
 \frac{\partial^{2}}{\partial t^{2}} \bigg|_{t=0} \mathcal{W}^{+}
\leq 8 \int_{\Sigma} \Bigl( |\nu_{\bar11}|^2-\frac{1}{2}\mathfrak{Ric}\,|\nu_{\bar1}|^2\Bigr) dA \le \frac{1}{2} \int_{\Sigma} \bigl( \Lambda_{1}^{2} - 2 \Lambda_{1}\mathfrak{Ric} \bigr) |\nu|^{2} dA,
\end{equation}
where the first inequality follows from
 $$\operatorname{Ric}(e_{1}, \bar e_{1})=\operatorname{Ric}(\mathrm{Re}(e_{1}), \mathrm{Re}(e_{1}))+\operatorname{Ric}(\mathrm{Im}(e_{1}), \mathrm{Im}(e_{1}))\geq \frac{1}{2}\mathfrak{Ric},$$
with $\mathrm{Re}(e_{1})$ and $\mathrm{Im}(e_{1})$ denoting the real and imaginary parts of $e_1$. The fact that any complex curve minimizes $\mathcal{W}^+$ implies the left-hand side of \eqref{eq-varW} is nonnegative. Together with $\Lambda_1>0$, this yields the lower bound
$$\Lambda_1\geq 2\,\mathfrak{Ric}.$$
\end{proof}
\section{The proof of Theorem 2}\label{sec5}
We denote by ${E}_0^{\mathcal{W}^+}$ (resp. $E_0^{\mathcal{A}}$) the space of Jacobi fields associated with $\mathcal{W}^+$ (resp. $\mathcal{A}$). It follows from \eqref{eq-Area} and \eqref{eq-Will} that
\begin{equation}\label{eq-Jaco}
 E_0^{\mathcal{W}^+}=\{V=\nu e_0+\bar\nu\bar e_0 \mid \nu_{\bar1\bar1}=0\},~~~~~~E_0^{\mathcal{A}}=\{V=\nu e_0+\bar\nu\bar e_0\mid \nu_{\bar1}=0\}.
\end{equation}
Obviously, there holds
$$E_0^{\mathcal{A}}\subset E_0^{\mathcal{W}^+}.$$

{\bf Claim 1.} {\em If $E_0^{\mathcal{W}^+}\setminus E_0^{\mathcal{A}}\neq\emptyset$, then $\Lambda_1=2\mathfrak{c}$ and $E_0^{\mathcal{W}^+}\setminus E_0^{\mathcal{A}}$ is exactly the first eigenspace of $\mathcal{L}$}.
\vskip 0.2cm
\noindent To prove this claim, first note that $\mathcal{L}$ is a nonnegative self-adjoint operator. Hence, for each eigenvalue $\Lambda_j$ of $\mathcal{L}$, we can choose a basis $\{\mathrm{V}_{j_1}, \mathrm{V}_{j_2}, \cdots \mathrm{V}_{j_{k_j}}\}$ of the corresponding eigenspace $E_j^{\mathcal{A}}$ such that the collection
$$\{\mathrm{V}_{j_l}\mid 1\leq l\leq {k_j}, j\in \mathbb{Z}^{\geq 0}\}$$
forms a complete orthonormal system of \( L^2(\Gamma(N_\Sigma)) \).
Let $V\in E_0^{\mathcal{W}^+}\setminus E_0^{\mathcal{A}}$ be a Jacobi field of $\mathcal{W}^+$. Write
\[V = \sum_{j=0}^{+\infty}\sum_{l=1}^{k_j} \varepsilon_{j_l} \mathrm{V}_{j_l}.\]
Then we have
\[\mathcal{L}(V) = -\sum_{j=1}^{+\infty}\sum_{l=1}^{k_j} \Lambda_j \varepsilon_{j_l} \mathrm{V}_{j_l}.\]
and then
\[\int_{\Sigma}|\mathcal{L}(V)|^2 dA = \sum_{j=1}^{+\infty}\sum_{l=1}^{k_j} \varepsilon_{j_l}^2\Lambda_j^2, \quad\quad \int_{\Sigma}\langle \mathcal{L}(V), V \rangle dA= -\sum_{j=1}^{+\infty}\sum_{l=1}^{k_j} \varepsilon_{j_l}^2 \Lambda_j.\]
Substituting these equalities into \eqref{eq-W+5}, we derive
\[\sum_{j=1}^{+\infty}\sum_{l=1}^{k_j} \varepsilon_{j_l}^2 \Lambda_j(\Lambda_j - 2\mathfrak{c}) = 0.\]
It follows from
$$\sum_{j=1}^{+\infty}\sum_{l=1}^{k_j}\varepsilon_{j_l}^2\neq0,~~\text{and}~~\Lambda_{j}>\Lambda_1\geq 2\mathfrak{c},~j\geq 2$$
that
$$\Lambda_1=2c,~~\text{and}~~\varepsilon_{j_l}=0,~j\geq 2,$$
which implies $V$ is an eigensection associated with the first eigenvalue $\Lambda_1=2\mathfrak{c}$. Therefore, we obtain
$$E_0^{\mathcal{W}^+}\setminus E_0^{\mathcal{A}}\subset E_1^{\mathcal{A}}.$$
When $\Lambda_1=2\mathfrak{c}$, the converse inclusion follows directly from \eqref{eq-W+5}. Thus Claim $1$ is proved.

From \eqref{eq-E0W} and Claim $2$, we derive
$$\dim E_0^{\mathcal{W}^+}=\dim \ker{\bar\partial_1}+\dim \ker{\bar\partial}= \dim H^0(\Sigma, N_\Sigma \otimes \overline{K_\Sigma})+\dim E_0^{\mathcal{A}}.$$
By Riemann-Roch Theorem, we have
$$\begin{aligned}\dim H^0(\Sigma, N_\Sigma \otimes \overline{K_\Sigma})=&\deg(N_\Sigma \otimes \overline{K_\Sigma})+1-g+\dim H^0(\Sigma, N_\Sigma^* \otimes K_\Sigma^2)\\
=&\,\mathfrak{c}\frac{\mathrm{Area}(\Sigma)}{2\pi}+1-g+\dim H^0(\Sigma, N_\Sigma^* \otimes K_\Sigma^2).
\end{aligned}$$
So when $g\leq 1$, there holds
$$\dim E_0^{\mathcal{W}^+}-\dim E_0^{\mathcal{A}}=\dim H^0(\Sigma, N_\Sigma \otimes \overline{K_\Sigma})\geq 1.$$
Together with Claim 1, this yields the second conclusion of this theorem.
\end{proof}
\begin{remark}
 It follows from the proof of Claim 1 that for complex curves in K\"ahler-Einstein surfaces of nonpositive scalar curvature, $${E}^{\mathcal{W}^+}_0={E}^{\mathcal{A}}_0,$$
 i.e., the Jacobi fields of $\mathcal{W}^+$ are exactly the same as those of $\mathcal{A}$.
\end{remark}

Somewhat surprisingly, the proof of this theorem relies on computing the second variation of the following  conformally invariant functional
$$\mathcal{W}^+(x) \triangleq \int_{\Sigma} (|\vec H|^2 + K(\mathfrak{e}_1, \mathfrak{e}_2, \mathfrak{e_1}, \mathfrak{e_2}) + K(\mathfrak{e}_1, \mathfrak{e}_2, \mathfrak{e_3}, \mathfrak{e_4}))dA,$$
introduced by Montiel and Urbano in \cite{Montiel-Urbano} for closed oriented surfaces in an oriented $4$-dimensional Riemannian manifold. Here $\vec H$ represents the mean curvature vector, $K$ denotes the ambient  curvature tensor, and $\{\mathfrak{e}_1, \mathfrak{e}_2, \mathfrak{e}_3, \mathfrak{e}_4\}$ is an orthonormal basis with $\{\mathfrak{e}_1, \mathfrak{e}_2\}$ tangent to the surface $x$ and $\{\mathfrak{e}_3, \mathfrak{e}_4\}$ spanning its normal space. Associated with $\mathcal{W}^+$, Montiel and Urbano \cite{Montiel-Urbano} also introduced
$$\mathcal{W}^-(x) \triangleq \int_{\Sigma} (|\vec H|^2 + K(\mathfrak{e}_1, \mathfrak{e}_2, \mathfrak{e_1}, \mathfrak{e_2}) - K(\mathfrak{e}_1, \mathfrak{e}_2, \mathfrak{e_3}, \mathfrak{e_4}))dA.$$
The average of these two functionals is precisely the Willmore functional,
$$\mathcal{W}(x) = \int_{\Sigma} (|\vec H|^2 + K_{1212}) dA,$$
an important global conformal invariant for surfaces in general curved Riemannian manifolds that has attracted considerable attention recently (see \cite{Marques1,Mondino,Mondino1,Wang-Xie} and the references therein). In \cite{Montiel-Urbano}, Montiel and Urbano studied $\mathcal{W}^+$ and $\mathcal{W}^-$ from the perspective of twistor theory and showed that a superminimal surface with positive (resp. negative) spin minimize $\mathcal{W}^+$ (resp. $\mathcal{W}^-$) among its homotopy class. In \cite{Wang-Xie}, we investigated these functionals from the viewpoint of conformal geometry and derived the following expressions,
\begin{equation}\label{eq-W+}
\mathcal{W}^+(x) = 4 \int_{\Sigma} |\psi|^2 dA + 2\pi (\chi + \chi^{\perp}),~~~\mathcal{W}^-(x) = 4 \int_{\Sigma} |\phi|^2 dA + 2\pi (\chi - \chi^{\perp}),
\end{equation}
where
$\psi$ and $\phi$ are two local conformal invariants arising from the normal-bundle-valued Hopf differential, and $\chi$ (resp. $\chi^{\perp}$) denotes the Euler characteristics of the tangent bundle (resp. normal bundle). When the ambient space is a Kähler surface, complex curves are superminimal surfaces with positive spin and hence are minimizers of $\mathcal{W}^+$.

If we restrict the ambient space to a Kähler–Einstein surface with positive scalar curvature, then the estimate established in Theorem \ref{thm-main11} is sharp. In this setting, the curvature invariant $\mathfrak{Ric}$ simplifies to the Einstein constant $\mathfrak{c}>0$. We explore the equality case and are able to compute exactly the dimension of the first eigenspace. This is accomplished by relating it to the space of Jacobi fields of $\mathcal{W}^+$, which corresponds precisely to the preimage of the space of holomorphic sections of a certain line bundle, and then applying the Riemann-Roch theorem in conjunction with the Dolbeault theorem.
\begin{thm}\label{thm-main2}
 Let \(M^{4}\) be a K\"ahler-Einstein surface with Einstein constant \(\mathfrak{c}>0\), and let \(x\colon\Sigma \rightarrow M^{4}\) be a complex curve. Then
 \begin{itemize}
    \item[{\rm(1)}] the first eigenvalue $\Lambda_1$ of its area Jacobi operator is at least $2\mathfrak{c}$,
    \item[{\rm(2)}] if $\Sigma$ has genus $g\leq 1$, then $\Lambda_1=2\mathfrak{c}$ and the corresponding eigenspace has dimension
    \[
    \mathfrak{c}\frac{\operatorname{Area}}{2\pi}+1-g+\dim H^{\,0}(N^*_\Sigma\otimes K_\Sigma^2),
    \]
    where $N^*_\Sigma$ is the dual of the normal bundle, $K_\Sigma$ is the canonical line bundle of $\Sigma$, and $H^{\,0}(N^*_\Sigma\otimes K_\Sigma^2)$ denotes  the space of holomorphic sections of $N^*_\Sigma\otimes K_\Sigma^2$.
\end{itemize}
\end{thm}

This paper is organized as follows. In Section~\ref{sec2}, we review the necessary background on surface theory in  Kähler surfaces. Section~\ref{sec3} is devoted to compute the variation of geometric quantities for a complex curve in a K\"ahler surface. Theorem~\ref{thm-main11} is established in Section~\ref{sec4}. Finally, in Section~\ref{sec5}, we focus on the Kähler-Einstein setting and prove Theorem \ref{thm-main2}.

The post-theorem introduction says that, “Somewhat surprisingly, the proof of this theorem relies on computing the second variation of the following conformally invariant functional”
\[
\mathcal{W}^+(x) \triangleq \int_{\Sigma} (|\vec H|^2 + K(\mathfrak{e}_1, \mathfrak{e}_2, \mathfrak{e_1}, \mathfrak{e_2}) + K(\mathfrak{e}_1, \mathfrak{e}_2, \mathfrak{e_3}, \mathfrak{e_4}))\,dA.
\]
It recalls that in a K"ahler surface “complex curves are superminimal surfaces with positive spin and hence are minimizers of $\mathcal{W}^+$.” It then points to where the argument is carried out: “Theorem~\ref{thm-main11} is established in Section~\ref{sec4},” after “Section~\ref{sec3} is devoted to compute the variation of geometric quantities for a complex curve in a K"ahler surface.”

The post-theorem introduction says that, “Somewhat surprisingly, the proof of this theorem relies on computing the second variation of the following conformally invariant functional”
\[
\mathcal{W}^+(x) \triangleq \int_{\Sigma} (|\vec H|^2 + K(\mathfrak{e}_1, \mathfrak{e}_2, \mathfrak{e_1}, \mathfrak{e_2}) + K(\mathfrak{e}_1, \mathfrak{e}_2, \mathfrak{e_3}, \mathfrak{e_4}))\,dA.
\]
It recalls that in a K"ahler surface “complex curves are superminimal surfaces with positive spin and hence are minimizers of $\mathcal{W}^+$.” It then points to where the argument is carried out: to prove the main theorem, the paper first devotes the next part to computing the variation of geometric quantities for a complex curve in a K"ahler surface, and later carries out the proof.

Question:
Let \(M^4\) be a K\"ahler surface, and let \(x\colon \Sigma \to M^4\) be a complex curve in \(M^4\). Let \(\mathcal L\) denote the area Jacobi operator of \(\Sigma\) (the self-adjoint elliptic operator governing the second variation of area), and let \(\Lambda_1\) be its first eigenvalue. Define
\[
\mathfrak{Ric}:=\inf\{\operatorname{Ric}(u,u): p\in M^4,\ u\in T_pM^4,\ |u|=1\},
\]
the infimum of the ambient Ricci curvature over all unit tangent vectors of \(M^4\). Which statement holds for every such complex curve?

The first eigenvalue of the area Jacobi operator satisfies \(\Lambda_1 \ge 2\,\mathfrak{Ric}\).

Choices:
B. The first eigenvalue of the area Jacobi operator satisfies \(\Lambda_1 \ge \mathfrak{Ric}\).
C. The first eigenvalue of the area Jacobi operator satisfies \(\Lambda_1 \ge 0\).
D. The first eigenvalue of the area Jacobi operator satisfies \(\Lambda_1 \ge 2\,\operatorname{Ric}(u,u)\) for every unit tangent vector \(u\in T_pM^4\) at every point \(p\in M^4\).
E. If \(\mathfrak{Ric}>0\), then the first eigenvalue of the area Jacobi operator satisfies \(\Lambda_1 > 2\,\mathfrak{Ric}\).

sharp factor 2 from second-variation identity

ambient-curvature-dependent lower bound dropped

global infimum over Ricci replaced by pointwise uniform lower bound

non-strict lower bound strengthened to strict inequality


\begin{thm}\label{thm:ubiquity:antiraysRays}
    Let $D$ be a digraph.
    If there is a sequence $(\cX_n)_{n\in\N}$ such that, for every $n\in\N$, the set $\cX_n$ consists of $n$ rays and $n$ anti-rays of~$D$ that are all pairwise disjoint and such that the sequence $(\cX_n)_{n\in\N}$ does not concentrate in any end of~$D$, then $D$ contains a set of infinitely many rays and infinitely many anti-rays every two of which are disjoint.
\end{thm}

\label{sec:intro}

The property of a graph $H$ to be \emph{ubiquitous} describes the concept that the existence of $n$ pairwise disjoint copies for all $n\in\ensuremath{\mathbb N}$ of a graph $H$ in a graph $G$ implies that $G$ already contains $\aleph_0$ many pairwise disjoint copies of~$H$.
Originally, this was considered with respect to the subgraph relation and Halin showed that the ray is ubiquitous~\cite{Halin65}*{Satz 1} and that the double ray is ubiquitous~\cite{Halin70}*{Satz 3}.
Later, counterexamples \cites{A77, A02} regarding being ubiquitous were found, which even include trees.
Hence, ubiquity was also considered with respect to other relations such as the minor and topological minor relation, and we refer to \cites{A79,A80,BEEGHPTI,BEEGHPTII,BEEGHPTIII} for ubiquity results regarding these notions.
Many of these results leverage that finite graphs (resp.~trees) are well-quasi ordered under the minor (resp.~topological minor) relation.

It should be noted that Andreae~\cite{A02} posed the major and still open conjecture whether all locally finite connected graphs are ubiquitous under the minor relation. In the same paper he also provided an example of an uncountable non-ubiquitous graph under the minor relation, whose construction exploits that uncountable graphs are not well-quasi ordered under minors.

Further modified questions regarding ubiquity have been raised, e.g.~where disjointness is replaced by edge-disjointness.
In this setting it turned out to be complicated to even verify that double rays are ubiquitous regarding the subgraph relation~\cite{BCP15}.

For digraphs, much less is known, even for the subdigraph relation, and for the rest of this article, we shall only consider the property of being ubiquitous under that relation.
While Zuther~\cite{Z1997}*{Theorem 2.17} proved that the consistently oriented ray is ubiquitous, Gut et al.~\cite{GKR2024}*{Theorem 1.1} proved the same for all possible orientations of the undirected ray.
Later, Gut et al.~\cite{GKR2025}*{Theorem 1.3} classified almost all orientations of the undirected double ray that are ubiquitous.
The only orientation they left out was the consistently oriented one.
Due to that, they posed the following problem.

\begin{problem}\cite{GKR2024}*{Problem 1.3}\label{prob:ubiquity:doubleRays}
    Is the consistently oriented double ray ubiquitous?
\end{problem}

Our main result (Theorem~\ref{thm:main:intro:ReductionToOneEnd:doubleRays}) states that it suffices to verify Problem~\ref{prob:ubiquity:doubleRays} for one-ended digraphs.
(We refer to Section~\ref{sec:prelims} for the definition of ends of digraphs.)

\label{sec:intro}

The property of a graph $H$ to be \emph{ubiquitous} describes the concept that the existence of $n$ pairwise disjoint copies for all $n\in\ensuremath{\mathbb N}$ of a graph $H$ in a graph $G$ implies that $G$ already contains $\aleph_0$ many pairwise disjoint copies of~$H$.
Originally, this was considered with respect to the subgraph relation and Halin showed that the ray is ubiquitous~\cite{Halin65}*{Satz 1} and that the double ray is ubiquitous~\cite{Halin70}*{Satz 3}.
Later, counterexamples \cites{A77, A02} regarding being ubiquitous were found, which even include trees.
Hence, ubiquity was also considered with respect to other relations such as the minor and topological minor relation, and we refer to \cites{A79,A80,BEEGHPTI,BEEGHPTII,BEEGHPTIII} for ubiquity results regarding these notions.
Many of these results leverage that finite graphs (resp.~trees) are well-quasi ordered under the minor (resp.~topological minor) relation.

It should be noted that Andreae~\cite{A02} posed the major and still open conjecture whether all locally finite connected graphs are ubiquitous under the minor relation. In the same paper he also provided an example of an uncountable non-ubiquitous graph under the minor relation, whose construction exploits that uncountable graphs are not well-quasi ordered under minors.

For digraphs, much less is known, even for the subdigraph relation, and for the rest of this article, we shall only consider the property of being ubiquitous under that relation.
While Zuther~\cite{Z1997}*{Theorem 2.17} proved that the consistently oriented ray is ubiquitous, Gut et al.~\cite{GKR2024}*{Theorem 1.1} proved the same for all possible orientations of the undirected ray.
Later, Gut et al.~\cite{GKR2025}*{Theorem 1.3} classified almost all orientations of the undirected double ray that are ubiquitous.
The only orientation they left out was the consistently oriented one.
Due to that, they posed the following problem.

\begin{problem}\cite{GKR2024}*{Problem 1.3}\label{prob:ubiquity:doubleRays}
 Is the consistently oriented double ray ubiquitous?
\end{problem}

Our main result (Theorem~\ref{thm:main:intro:ReductionToOneEnd:doubleRays}) states that it suffices to verify Problem~\ref{prob:ubiquity:doubleRays} for one-ended digraphs.
(We refer to Section~\ref{sec:prelims} for the definition of ends of digraphs.)

\begin{problem}\cite{GKR2024}*{Problem 1.3}\label{prob:ubiquity:doubleRays}
    Is the consistently oriented double ray ubiquitous?
\end{problem}

\begin{theorem}\label{thm:main:intro:ReductionToOneEnd:doubleRays}
    The consistently oriented double ray is ubiquitous if and only if it is ubiquitous restricted to the class of one-ended digraphs.
\end{theorem}

\begin{abstract}
We prove that the consistently oriented double ray is ubiquitous if and only if it is ubiquitous restricted to the class of one-ended digraphs.
Additionally, we prove the same equivalence for the disjoint union of a consistently forward and a consistently backward oriented ray.
Furthermore, we discuss the connection between these two ubiquity problems.
\end{abstract}

For digraphs, much less is known, even for the subdigraph relation, and for the rest of this article, we shall only consider the property of being ubiquitous under that relation.
While Zuther~\cite{Z1997}*{Theorem 2.17} proved that the consistently oriented ray is ubiquitous, Gut et al.~\cite{GKR2024}*{Theorem 1.1} proved the same for all possible orientations of the undirected ray.
Later, Gut et al.~\cite{GKR2025}*{Theorem 1.3} classified almost all orientations of the undirected double ray that are ubiquitous.
The only orientation they left out was the consistently oriented one.
Due to that, they posed the following problem.

\begin{problem}\cite{GKR2024}*{Problem 1.3}\label{prob:ubiquity:doubleRays}
 Is the consistently oriented double ray ubiquitous?
\end{problem}

Our main result (Theorem~\ref{thm:main:intro:ReductionToOneEnd:doubleRays}) states that it suffices to verify Problem~\ref{prob:ubiquity:doubleRays} for one-ended digraphs.
(We refer to Section~\ref{sec:prelims} for the definition of ends of digraphs.)

As a major step in the proof of Theorem~\ref{thm:main:intro:ReductionToOneEnd:doubleRays}, we verify Problem~\ref{prob:ubiquity:doubleRays} for a large, but technical, class of digraphs, see Theorem~\ref{thm:ubiquity:doubleRays}.

\begin{problem}\label{prob:ubiquity:antiraysRays}
 Is the disjoint union of a consistently forward and a consistently backward oriented ray ubiquitous?
\end{problem}

\begin{theorem}\label{thm:main:intro:ReductionToOneEnd:RaysAntiRays}
 The disjoint union of a consistently forward and a consistently backward oriented ray is ubiquitous if and only if it is ubiquitous restricted to the class of one-ended digraphs.
\end{theorem}

\begin{proof}[Proof of Theorem~\ref{thm:main:intro:ReductionToOneEnd:doubleRays}.]
 Let us assume that Problem~\ref{prob:ubiquity:doubleRays} holds for the class of one-ended digraphs and let $D$ be a digraphs and $(\cR_n)_{n\in\N}$ be a valid sequence in~~$D$.
 By Theorem~\ref{thm:ubiquity:doubleRays}, we may assume that $(\cR_n)_{n\in\N}$ concentrates in finitely many ends.
 Then there is a valid sequence $(\cQ_n)_{n\in\N}$ induced by $(\cR_n)_{n\in\N}$ such that all tails from all elements of $\bigcup_{n\in\N}\cQ_n$ are equivalent.
 By Lemma~\ref{lem:reduction:doubleRays:oneEnd}, there exists a one-ended subdigraph $D'$ of~$D$ and a valid sequence $(\cP_n)_{n\in\N}$ in~$D'$ induced by $(\cQ_n)_{n\in\N}$.
 Thus, there are infinitely many pairwise disjoint double rays in~$D'$ and hence in~$D$.
\end{proof}

\begin{problem}\cite{GKR2024}*{Problem 1.3}\label{prob:ubiquity:doubleRays}
    Is the consistently oriented double ray ubiquitous?
\end{problem}

\begin{theorem}\label{thm:main:intro:ReductionToOneEnd:doubleRays}
    The consistently oriented double ray is ubiquitous if and only if it is ubiquitous restricted to the class of one-ended digraphs.
\end{theorem}

\begin{thm}\label{thm:ubiquity:doubleRays}
    Let $D$ be a digraph.
    If there is a valid sequence $(\cR_n)_{n\in\N}$ that does not concentrate in finitely many ends, then $D$ contains infinitely many pairwise disjoint double rays.
\end{thm}

As a major step in the proof of Theorem~\ref{thm:main:intro:ReductionToOneEnd:doubleRays}, we verify Problem~\ref{prob:ubiquity:doubleRays} for a large, but technical, class of digraphs, see Theorem~\ref{thm:ubiquity:doubleRays}.

Additionally, we consider the following more general version of \cite{HH2024+}*{Problem 4.2}.

\begin{problem}\label{prob:ubiquity:antiraysRays}
    Is the disjoint union of a consistently forward and a consistently backward oriented ray ubiquitous?
\end{problem}

We also verify Problem~\ref{prob:ubiquity:antiraysRays} for a large, but technical, class of digraphs (Theorem~\ref{thm:ubiquity:antiraysRays}) and show that Problem~\ref{prob:ubiquity:antiraysRays} holds if it holds for all one-ended digraphs.

\begin{theorem}\label{thm:main:intro:ReductionToOneEnd:RaysAntiRays}
    The disjoint union of a consistently forward and a consistently backward oriented ray is ubiquitous if and only if it is ubiquitous restricted to the class of one-ended digraphs.
\end{theorem}

\medskip

This paper is structured as follows.
After introducing some terminology in Section~\ref{sec:prelims}, we prove Theorem~\ref{thm:main:intro:ReductionToOneEnd:RaysAntiRays} in Section~\ref{sec:ubiquity:antiraysRays}.
In Section~\ref{sec:ubiquity:DoubleRays}, we prove Theorem~\ref{thm:main:intro:ReductionToOneEnd:doubleRays}.
In Section~\ref{sec:equivalence}, we discuss connections between Problems~\ref{prob:ubiquity:doubleRays} and \ref{prob:ubiquity:antiraysRays}.

As a major step in the proof of Theorem~\ref{thm:main:intro:ReductionToOneEnd:doubleRays}, the authors "verify Problem~\ref{prob:ubiquity:doubleRays} for a large, but technical, class of digraphs" (see Theorem~\ref{thm:ubiquity:doubleRays}).

Question: A digraph $H$ is called ubiquitous under the subdigraph relation if, whenever a digraph $G$ contains $n$ pairwise disjoint copies of $H$ for every $n\in\mathbb{N}$, the digraph $G$ already contains countably infinitely many pairwise disjoint copies of $H$. Let $C$ be the consistently oriented double ray, that is, the two-way infinite directed path $\cdots \to v_{-1} \to v_0 \to v_1 \to \cdots$. Which of the following statements is equivalent to saying that $C$ is ubiquitous under the subdigraph relation?

The consistently oriented double ray $C$ is ubiquitous when restricted to one-ended digraphs; that is, for every one-ended digraph $G$ (a digraph with exactly one end), if $G$ contains $n$ pairwise disjoint copies of $C$ for every $n\in\mathbb{N}$, then $G$ contains countably infinitely many pairwise disjoint copies of $C$.

Choices:
B. The consistently oriented double ray $C$ is ubiquitous when restricted to digraphs with at most one end; that is, for every digraph $G$ with at most one end, if $G$ contains $n$ pairwise disjoint copies of $C$ for every $n\in\mathbb{N}$, then $G$ contains countably infinitely many pairwise disjoint copies of $C$.
C. For every one-ended digraph $G$, if $G$ contains countably infinitely many pairwise disjoint copies of $C$, then in particular $G$ contains $n$ pairwise disjoint copies of $C$ for every $n\in\mathbb{N}$.
D. There exists a one-ended digraph $G$ such that, whenever $G$ contains $n$ pairwise disjoint copies of $C$ for every $n\in\mathbb{N}$, every digraph containing $G$ as a subdigraph contains countably infinitely many pairwise disjoint copies of $C$.
E. The consistently oriented double ray $C$ is ubiquitous when restricted to finitely-ended digraphs; that is, for every digraph $G$ with finitely many ends, if $G$ contains $n$ pairwise disjoint copies of $C$ for every $n\in\mathbb{N}$, then $G$ contains countably infinitely many pairwise disjoint copies of $C$.


\label{ergodic-th}
If Assumption B holds, then $\{u(t,x)\}_{x\in \mathbb{R}}$ is ergodic, for any $t>0$. Consequently, by the mean ergodic theorem,
\[
\frac{1}{R}F_R(t) \to 0 \quad \mbox{a.s and in $L^2(\Omega)$}, \quad \mbox{as $R \to \infty$}.
\]

\begin{theorem}
\label{ergodic-th}
If Assumption B holds, then $\{u(t,x)\}_{x\in \bR}$ is ergodic, for any $t>0$. Consequently, by the mean ergodic theorem,
\[
\frac{1}{R}F_R(t) \to 0 \quad \mbox{a.s and in $L^2(\Omega)$}, \quad \mbox{as $R \to \infty$}.
\]
\end{theorem}

\begin{equation}
\label{Gauss-color}
W_t(\varphi)=B_t(\varphi*\k) \quad \mbox{for all $t>0$ and $\varphi \in \cS(\bR^d)$},
\end{equation}

\begin{equation}
\label{def-cov-W}
\bE[W_t(\varphi) W_s(\psi)]=(t \wedge s) \int_{\bR^d} \int_{\bR^d}\varphi(x)\psi(y) f(x-y)dxdy,
\end{equation}

\begin{proof}
By direct calculation, we see that $\tilde{f} \star_{k}^{1} \tilde{g} =0$ implies that $\tilde{f} \star_{k}^{\ell} \tilde{g} =0$ for all $\ell=2,\ldots,k$. Hence, all the terms in the product formula are zero, except those corresponding to $\ell=0$ and $\ell=1$:
\begin{align*}
I_n(f) I_m(g) &=I_n(\tilde{f}) I_m(\tilde{g})=I_{n+m}(\tilde{f} \otimes \tilde{g})+
\sum_{k=1}^{n\wedge m}k! \binom{n}{k}\binom{m}{k}I_{n+m-k}(\tilde{f} \star_{k}^{0} \tilde{g} ).
\end{align*}
Note that $I_{n+m}(\tilde{f} \otimes \tilde{g})=I_{n+m}(f \otimes g)$ since $\tilde{f} \otimes \tilde{g}$ and $f \otimes g$ have the same symmetrization.
\end{proof}

\section{Inequalities for Riesz potentials}
\label{appB}

In this section, we include some inequalities for Riesz potentials which were used for the proof of the QCLT in the case when $\k$ is the Riesz kernel.

We recall that the Riesz kernel of order $\alpha \in (0,d)$ is defined by
\[
R_{d,\alpha}(x)=C_{d,\alpha}|x|^{-(d-\alpha)} \quad \mbox{with} \quad C_{d,\alpha}=\pi^{-\frac{d}{2}}2^{-\alpha} \frac{\Gamma(\frac{d-\alpha}{2})}{\Gamma(\frac{\alpha}{2})}.
\]
Its Fourier transform (in the sense of distributions) is $\cF R_{d,\alpha}(\xi)=|\xi|^{-\alpha}$, since
\[
\int_{\bR^d}\varphi(x)|x|^{-\alpha}dx=C_{d,\alpha} \int_{\bR^d}\cF \varphi(\xi)|\xi|^{-(d-\alpha)}d\xi \quad
\mbox{for any $\varphi \in \cS(\bR^d)$},
\]
where $\cS(\bR^d)$ is the set of rapidly decreasing functions on $\bR^d$.

\medskip

For the next result, we refer to Theorem 1, p.119 of \cite{stein70}.

\begin{theorem}[Hardy-Littlewood-Sobolov inequality]
\label{HLS}
For any $\alpha \in (0,d)$ and $p>\frac{d}{d-\alpha}$,
\[
\| \varphi *R_{d,\alpha}\|_{L^p(\bR^d)} \leq A_{d.\alpha,p}\|\varphi\|_{L^{q}(\bR^d)} \quad \mbox{for any $\varphi \in L^q(\bR^d)$},
\]
where $A_{d,\alpha,p}>0$ is a constant depending on $(d,\alpha,p)$, and $q$ is defined by:
\[
\frac{1}{q}=\frac{1}{p}+\frac{\alpha}{d}.
\]
(Condition $p>\frac{d}{d-\alpha}$ is equivalent to $q>1$.)
\end{theorem}

As a consequence of Theorem \ref{HLS}, we obtain the following lemma.

\begin{lemma}
\label{Hol-HLS}
For any $\alpha \in (0,d)$, $q \in (1,\frac{d}{\alpha})$, $\varphi \in L^q(\bR^d)$ and $\psi \in L^{q'}(\bR^d)$,
\[
\left|\int_{\bR^d}\int_{\bR^d}\varphi(x)\psi(y) R_{d,\alpha}(x-y) dxdy \right| \leq A_{d,\alpha,p} \|\varphi\|_{L^q(\bR^d)} \|\psi\|_{L^{q'}(\bR^d)},
\]
where $A_{d,\alpha,p}$ is the constant  from Theorem \ref{HLS}, and $(p,q')$ are defined by
\[
\frac{1}{p}+\frac{1}{q}=1 \quad \mbox{and} \quad \frac{1}{q}+\frac{1}{q'}=1+\frac{\alpha}{d}.
\]
\end{lemma}

\begin{proof}
We apply H\"older's inequality, letting $\frac{1}{p}+\frac{1}{q}=1$, and obtain that
\begin{align*}
\left|\int_{\bR^d}\int_{\bR^d}\varphi(x)\psi(y) R_{d,\alpha}(x-y) dxdy \right|
& =\left| \int_{\bR^d}\varphi(x) \big(\psi * R_{d,\alpha} \big) dx\right| \leq \|\varphi\|_{L^q(\bR^d)} \|\psi * R_{d,\alpha}\|_{L^p(\bR^d)}.
\end{align*}
Note that $q<\frac{d}{\alpha}$ is equivalent to $p>\frac{d}{d-\alpha}$.
Therefore, by Theorem \ref{HLS}
\[
 \|\psi * R_{d,\alpha}\|_{L^p(\bR^d)} \leq A_{d,\alpha,p}  \|\psi \|_{L^{q'}(\bR^d)},
\]
where $\frac{1}{q'}=\frac{1}{p}+\frac{\alpha}{d}$. Finally, we note that $
1-\frac{1}{q}=\frac{1}{p}=\frac{1}{q'}-\frac{\alpha}{d}$, and hence, $\frac{1}{q}+\frac{1}{q'}=1+\frac{\alpha}{d}$.

\end{proof}

\label{appA}

In this appendix section, we include some auxiliary results from Malliavin calculus which were used in the sequel.
\medskip

Let $(Z,\cZ,\fm)$ be a $\sigma$-finite measure space, and $N

\begin{align}
\label{HAM}
	\begin{cases}
		\dfrac{\partial^2 u}{\partial^2 t} (t,x)
		=  \dfrac{\partial^2 u}{\partial x^2} (t,x)+u(t,x) \dot{X}(t,x), \
		t>0, \ x \in \bR, \\
		u(0,x) = 1, \dfrac{\partial u}{\partial t} (0,x)=0, \quad x \in \bR,
	\end{cases}
\end{align}

The theory of stochastic partial differential equations (SPDEs) has grown consistently in the last three decades, and now constitutes a central area of stochastic analysis. SPDEs model the evolution of random systems across multiple scales, ranging from microscopic particle dynamics to macroscopic continuum limits, and arise in diverse contexts, such as interface growth, disordered media, turbulence, and transport in random environments. They provide a rigorous mathematical framework for universality phenomena observed in physics. Motivated by these physical models and mathematical challenges, systematic theories for SPDEs began to emerge in the late 1980s and 1990s. Prominent developments include Walsh's \cite{walsh86} probabilistic approach based on martingale measures and random fields, and the semigroup approach introduced by Da Prato and Zabczyk \cite{DZ92}. Basic equations which have been studied are the stochastic heat equation and the stochastic wave equation, and their particular cases, the parabolic Anderson model (pAm), respectively the hyperbolic Anderson model (hAm). However, comprehensive solution theories for numerous other SPDEs
remained beyond the grasp of mathematical analysis, until the early 21st century. These challenges
stemmed from the singular nature of such equations, a singularity intricately linked to the irregularities inherent in the random data involved. A significant breakthrough arrived in 2014 when Hairer \cite{hairer14}
constructed a well-posedness framework for the Kardar-Parisi-Zhang (KPZ) equation, using concepts from Lyons' rough paths theory \cite{lyons98}. In dimension 1, the solution of the KPZ equation is directly related to the solution of (pAm
) via the Cole-Hopf transformation.

\medskip

Traditionally, SPDEs are perturbed by a space-time Gaussian white noise in dimension $d=1$, which is a zero-mean Gaussian process $\{B_t(\varphi);t> 0,\varphi \in L^2(\mathbb{R})\}$ with covariance:
\[
\mathbb{E}[B_t(\varphi) B_s(\psi)]=(t \wedge s) \langle \varphi,\psi \rangle_{L^2(\mathbb{R})}.
\]
In higher dimensions, Dalang introduced in \cite{dalang99}, a spatially-homogeneous (or colored) noise, as a zero-mean Gaussian process $\{W_t(\varphi);t\geq 0,\varphi \in \mathcal{S}(\mathbb{R})\}$ with covariance:
\begin{equation}
\label{def-cov-W}
\mathbb{E}[W_t(\varphi) W_s(\psi)]=(t \wedge s) \int_{\mathbb{R}^d} \int_{\mathbb{R}^d}\varphi(x)\psi(y) f(x-y)dxdy,
\end{equation}
where $\mathcal{S}(\mathbb{R}^d)$ is the set of all rapidly decreasing functions on $\mathbb{R}^d$, and $f:\mathbb{R}^d \to [0,\infty]$ is the Fourier transform of a tempered measure $\mu$. This noise can also be defined as:
\begin{equation}
\label{Gauss-color}
W_t(\varphi)=B_t(\varphi*\kappa) \quad \mbox{for all $t>0$ and $\varphi \in \mathcal{S}(\mathbb{R}^d)$},
\end{equation}
where the kernel $\kappa$ is chosen such that $k * \widetilde{k}=f$, and $\widetilde{k}(x):=k(-x)$ for all $x \in \mathbb{R}^d$.

\medskip

One of the problems which has received a lot of interest in the literature in the recent years is the study of the asymptotic behaviour as $R \to \infty$ of the spatial integral
\[
F_R(t)=\int_{|x|<R} \big(u(t,x)-\mathbb{E}[u(t,x)]\big)dx
\]
of the random-field solution $u$ of an SPDE. The major result is the {\em Quantitative Central Limit Theorem} (QCLT), which shows that $F_R(t)/\sqrt{{\rm Var}\big(F_R(t)\big)}$ converges in distribution as $R \to \infty$ to $Z \sim N(0,1)$, and gives an estimate for the rate of this convergence in the total variation distance. This was achieved for the first time in the seminal article \cite{HNV20} for the stochastic heat equation driven by space-time Gaussian white noise, by combining tools from Malliavin calculus with Stein's method for normal approximations. Since then, this method was extended to various other models, each with its own challenges. We refer the reader to \cite{BNZ,CKNP-delta,CKNP22,DNZ20,HNVZ,NSZ20,NXZ22,NZ20-1,NZ20-2,NZ22} for a sample of relevant references.

\medskip

In the recent article \cite{BZ24}, the QCLT problem was considered for the first time for an SPDE, namely (hAm),
driven by a L\'evy white noise, by using Poisson-Malliavin calculus tools (in this case, with respect to the underlying Poisson random measure of the noise), which have been developed for instance in \cite{Last16,LPS16}.  On the other hand, the recent article \cite{BJ25} has introduced a coloration procedure similar to \eqref{Gauss-color} for the L\'evy noise, and analyzed various SPDEs with this noise. Therefore, a natural question is whether the QCLT continues to hold for an SPDE driven by the L\'evy colored noise of \cite{BJ25}. We propose to address this question in the present article, by focusing on (hAm) in dimension $d=1$, as in \cite{BZ24}.

\medskip

More precisely, we consider the (hAm):
\begin{align}
\label{HAM}
	\begin{cases}
		\dfrac{\partial^2 u}{\partial^2 t} (t,x)
		=  \dfrac{\partial^2 u}{\partial x^2} (t,x)+u(t,x) \dot{X}(t,x), \
		t>0, \ x \in \mathbb{R}, \\
		u(0,x) = 1, \dfrac{\partial u}{\partial t} (0,x)=0, \quad x \in \mathbb{R},
	\end{cases}
\end{align}
where $X=\{X_t(\varphi);t>0,\varphi \in \mathcal{S}(\mathbb{R})\}$ is the L\'evy colored noise, given by:
\[
X_t(\varphi)=L_t(\varphi *\kappa).
\]
with $L_t(\varphi)=L(1_{[0,t]} \varphi)$ for all $\varphi \in L^2(\mathbb{R})$, and $L=\{L(\varphi);\varphi \in L^2(\bR_{+} \times \mathbb{R})\}$ being the {\em L\'evy white noise}:
\[
L(\varphi)=\int_{\bR_{+} \times \mathbb{R} \times \bR_0} \varphi(t,x)z \widehat{N}(dt,dx,dz) \quad \mbox{for all} \quad \varphi \in L^2(\bR_{+} \times \mathbb{R}).
\]

Here $N$ is a Poisson random measure on $Z=\bR_{+}\times \mathbb{R} \times \bR_0$ with intensity
\[
\mathfrak{m}(dt,dx,dz)=dt dx \nu(dz),
\]
defined on a complete probability space $(\Omega,\mathcal{F},\mathbb{P})$, and  $\widehat{N}(A)=N(A)-\mathfrak{m}(A)$ is the compensated version of $N$.
The space $\bR_0=\mathbb{R} \verb2\2 \{0\}$ is equipped with the distance $d(x,y)=|x^{-1}-y^{-1}|$, so that the bounded subsets of $\bR_0$ are those that are bounded away from 0.
We assume that the measure $\nu$ satisfies the following condition:
\[
m_2:=\int_{\bR_0}|z|^2 \nu(dz)<\infty.
\]
Under this condition, the following isometry property holds: for any $\varphi,\psi \in L^2(\bR_{+}\times \mathbb{R})$,
\[
\mathbb{E}[L(\varphi)L(\psi)]=m_2 \int_{\bR_+ \times \mathbb{R}} \varphi(t,x)\psi(t,x)dtdx.
\]

We denote by $\mathcal{S}'(\mathbb{R})$ the set of tempered distributions on $\mathbb{R}$.
As in \cite{BJ25}, we assume that the kernel $\kappa$ satisfies the following assumption:

\medskip

\noindent {\bf Assumption A.}
$\kappa:\mathbb{R} \to [0,\infty]$ is a continuous, symmetric, and tempered function such that:\\
(a) $\mathcal{F}\kappa=h$ in $\mathcal{S}'(\mathbb{R})$ is a tempered non-negative function, and $h^2$ is tempered;\\
(b) $f=\kappa* \widetilde{\kappa}$ is a continuous, symmetric, and tempered function ($\mathcal{F} f=h^2$ in $\mathcal{S}'(\mathbb{R}))$;\\
(c)  $\kappa(x)<\infty$ for all $x\not=0$, and $f(x)<\infty$ for all $x\not=0$.

\medskip

We will impose Assumption A throughout this article.
Under this assumption, the noise $X$ is well-defined, since $\varphi*\kappa \in \L^2(\mathbb{R})$ for any $\varphi \in \mathcal{S}(\mathbb{R})$. Moreover, for any $\varphi,\psi \in \mathcal{S}(\mathbb{R})$,
\[
\int_{\mathbb{R}} \int_{\mathbb{R}}\varphi(x)\psi(y) f(x-y)dxdy=\int_{\mathbb{R}} \mathcal{F} \varphi(\xi) \overline{\mathcal{F} \psi(\xi)} \mu(d\xi),
\]
where $\mathcal{F} \varphi(\xi)=\int_{\mathbb{R}}e^{-i \xi x} \varphi(x) dx$ is the Fourier transform of $\varphi$, and
\begin{equation}
\label{def-mu}
\mu(d\xi)=\frac{1}{2\pi} |\mathcal{F} \kappa(\xi)|^2 d\xi.
\end{equation}

Since $d=1$, $\mu$ satisfies {\em Dalang's condition}: (see Remark 10.(b) of \cite{dalang99})
\[
\int_{\mathbb{R}}\frac{1}{1+|\xi|^2}\mu(d\xi)<\infty.
\]

\medskip

The stochastic integral with respect to $X$ is defined using Walsh's theory \cite{walsh86},
since $X$ induces a martingale measure. This integral shares many properties with the integral defined in \cite{dalang99} for the Gaussian colored noise, except that its moments are estimated in a different way.

Let $G$ be the fundamental solution of the wave equation in dimension $d=1$, given by:
\begin{equation}
\label{def-G}
G_t(x)=\frac{1}{2}1_{\{|x|<t\}} \quad \mbox{for all $t>0$ and $x \in \mathbb{R}$.}
\end{equation}

Inspired by D'Alembert formula, we introduce the following definition.

\begin{definition}
{\rm A predictable process $\{u(t,x);t\geq 0,x\in \mathbb{R}\}$ is called a {\em (mild) solution} of \eqref{HAM}
if it satisfies the integral equation:
\begin{equation}
\label{def-sol}
u(t,x)=1+\int_0^t \int_{\mathbb{R}}G_{t-s}(x-y)u(s,y)X(ds,dy).
\end{equation}
}
\end{definition}

We recall that a random field $\{\Phi(t,x);t\geq 0,x\in \mathbb{R}\}$ is {\em predictable} if it is measurable with respect to the predictable $\sigma$-field on $\Omega \times \bR_{+} \times \mathbb{R}$, which is the minimal $\sigma$-field with respect to which all elementary processes are measurable. An {\em elementary process} is a linear combination of processes of the form
\[
\Phi(t,x)=Y 1_{(a,b]}(t) 1_{A}(x)
\]
where  $0\leq a<b$, $A \in \cB_b(\mathbb{R})$ and $Y$ is $\cF_a$-measurable. Predictable processes
$\{\Phi(t,x,z);t\geq 0,x\in \mathbb{R},z\in \bR_0\}$ are defined similarly. Here $(\cF_t)_{t\geq 0}$ is the filtration induced by $N$, i.e.
\[
\cF_t=\sigma(\{N([0,s] \times A \times B); s \in [0,t],A\in \cB_b(\mathbb{R}), B \in \cB_b(\bR_0)\}) \vee \mathcal{N},
\]
where $\mathcal{N}$ is the class of $\mathbb{P}$-null sets, and $\mathcal{B}(\mathbb{R})$, $\cB_b(\bR_0)$ are the classes of bounded Borel subsets of $\mathbb{R}$, respectively $\bR_0$.

\medskip

By Theorem 5.2 of \cite{BJ25}, equation \eqref{HAM} has a unique solution $u$, which satisfies:
\[
\sup_{(t,x)\in [0,T] \times \mathbb{R}}\mathbb{E}|u(t,x)|^2<\infty \quad \mbox{for all} \quad T>0.
\]

By Lemma 5.5 of \cite{BJ25}, the process $\{u(t,x)\}_{x\in \mathbb{R}}$ is strictly stationary, for any $t>0$.
Moreover, by Theorem 5.6 of \cite{BJ25}, if for some $p\geq 2$, we have
\begin{equation}
\label{def-mp}
m_p:=\int_{\bR_0}|z|^p \nu(dz)<\infty,
\end{equation}
and
\begin{equation}
\label{def-Mp}
\cM_p(t):=\int_0^t \int_{\mathbb{R}} \big|(G_{t-s}*\kappa)(x)\big|^p dx ds<\infty \quad \mbox{for all} \quad t>0,
\end{equation}
then
\begin{equation}
\label{mom-u}
K_p(T):=\sup_{(t,x)\in [0,T] \times \mathbb{R}}\|u(t,x)\|_p<\infty \quad \mbox{for all} \quad T>0,
\end{equation}
where $\|\cdot\|_p$ denotes the norm in $L^p(\Omega)$. The goal of the present paper is to complement this analysis by showing that the {\em spatial average} of the (centered) solution:
\[
F_R(t)=\int_{-R}^{R}\big(u(t,x)-1 \big)dx.
\]
has asymptotic Gaussian fluctuations when $R \to \infty$, with a precise rate of convergence which depends on the noise.  We denote $\sigma_R^2(t)={\rm Var}(F_R(t))$.

\medskip

\medskip

Recall that the {\em Riesz kernel of order $\alpha \in (0,1)$} is given by:
\[
R_{1,\alpha}(x)=C_{1,\alpha}|x|^{-(1-\alpha)} \quad \mbox{where} \quad C_{1,\alpha}=\pi^{-1/2}2^{-\alpha}\frac{\Gamma(\frac{1-\alpha}{2})}{\Gamma(\frac{\alpha}{2})}.
\]
Moreover, $\mathcal{F} R_{1,\alpha}(\xi)=|\xi|^{-\alpha}d\xi$, and $R_{1,\alpha}*R_{1,\beta}=R_{1,\alpha+\beta}$ for any $\alpha,\beta>0$ with $\alpha+\beta<1$.

\medskip

We introduce the following assumption.

\medskip

\noindent {\bf Assumption B.}
(i) $\kappa\in L^1(\mathbb{R})$; or (ii) $\kappa=R_{1,\alpha/2}$ for some $\alpha \in (0,1)$.

\begin{remark}
\label{rem-M}
{\rm Suppose that Assumption B holds. Applying Young's inequality if $\kappa \in L^1(\mathbb{R})$, and Theorem \ref{HLS} if $\kappa=R_{1,\alpha/2}$, we see that:
\[
\|G_t* \kappa\|_{L^p(\mathbb{R})} \leq
\left\{
\begin{array}{ll}
\|\kappa\|_{L^1(\mathbb{R})} \|G_t\|_{L^p(\mathbb{R})} & \mbox{if $\kappa \in L^1(\mathbb{R})$ and $p\geq 1$} \\
A_{1,\alpha,p}\|G_t\|_{L^q(\mathbb{R})} & \mbox{if $\kappa=R_{1,\alpha/2}$, $\alpha \in (0,1)$, $p>\frac{2}{2-\alpha}$ and $\frac{1}{q}=\frac{1}{p}+\frac{\alpha}{2}$}
\end{array} \right.
\]
Since $\|G_t\|_{L^p(\mathbb{R})}^p=2^{1-p}t$ for any $p>0$, we infer that condition \eqref{def-Mp} holds for any $p\geq 2$. Therefore, \eqref{mom-u} holds for any $p \geq 2$ such that $m_p<\infty$.
}
\end{remark}

We recall the following definitions. Let $X$ and $Y$ be random variables defined on the same probability space.
The {\em $1$-Wasserstein distance} between $X$ and $Y$ is:
\[
d_{W}(X,Y)=\sup_{{\rm Lip}(h) \leq 1} \big|\mathbb{E}[h(X)]-\mathbb{E}[h(Y)]\big|, \quad
\mbox{where} \quad
{\rm Lip}(h):=\sup_{x \not=y}\frac{|h(x)-h(y)|}{|x-y|}.
\]
The {\em Fortet-Mourier distance} between $X$ and $Y$ is:
\[
d_{FM}(X,Y)=\sup_{\|h\|_{\infty}+{\rm Lip}(h) \leq 1} \big|\mathbb{E}[h(X)]-\mathbb{E}[h(Y)]\big|,
\]
The {\em Kolmogorov distance} between $X$ and $Y$ is:
\[
d_{K}(X,Y)=\sup_{x \in \mathbb{E}} \big|P(X \leq x)-P(Y \leq x)\big|.
\]

We are now ready to state the main results of this article.

Traditionally, SPDEs are perturbed by a space-time Gaussian white noise in dimension $d=1$, which is a zero-mean Gaussian process $\{B_t(\varphi);t> 0,\varphi \in L^2(\mathbb{R})\}$ with covariance:
\[
\mathbb{E}[B_t(\varphi) B_s(\psi)]=(t \wedge s) \langle \varphi,\psi \rangle_{L^2(\mathbb{R})}.
\]
In higher dimensions, Dalang introduced in \cite{dalang99}, a spatially-homogeneous (or colored) noise, as a zero-mean Gaussian process $\{W_t(\varphi);t\geq 0,\varphi \in \mathcal{S}(\mathbb{R})\}$ with covariance:
\begin{equation}
\label{def-cov-W}
\mathbb{E}[W_t(\varphi) W_s(\psi)]=(t \wedge s) \int_{\mathbb{R}^d} \int_{\mathbb{R}^d}\varphi(x)\psi(y) f(x-y)dxdy,
\end{equation}
where $\mathcal{S}(\mathbb{R}^d)$ is the set of all rapidly decreasing functions on $\mathbb{R}^d$, and $f:\mathbb{R}^d \to [0,\infty]$ is the Fourier transform of a tempered measure $\mu$. This noise can also be defined as:
\begin{equation}
\label{Gauss-color}
W_t(\varphi)=B_t(\varphi*\kappa) \quad \mbox{for all $t>0$ and $\varphi \in \mathcal{S}(\mathbb{R}^d)$},
\end{equation}
where the kernel $\kappa$ is chosen such that $k * \widetilde{k}=f$, and $\widetilde{k}(x):=k(-x)$ for all $x \in \mathbb{R}^d$.

Let $G$ be the fundamental solution of the wave equation in dimension $d=1$, given by:
\begin{equation}
\label{def-G}
G_t(x)=\frac{1}{2}1_{\{|x|<t\}} \quad \mbox{for all $t>0$ and $x \in \mathbb{R}$.}
\end{equation}

By Lemma 5.5 of \cite{BJ25}, the process $\{u(t,x)\}_{x\in \mathbb{R}}$ is strictly stationary, for any $t>0$.
Moreover, by Theorem 5.6 of \cite{BJ25}, if for some $p\geq 2$, we have
\begin{equation}
\label{def-mp}
m_p:=\int_{\bR_0}|z|^p \nu(dz)<\infty,
\end{equation}
and
\begin{equation}
\label{def-Mp}
\cM_p(t):=\int_0^t \int_{\mathbb{R}} \big|(G_{t-s}*\kappa)(x)\big|^p dx ds<\infty \quad \mbox{for all} \quad t>0,
\end{equation}
then
\begin{equation}
\label{mom-u}
K_p(T):=\sup_{(t,x)\in [0,T] \times \mathbb{R}}\|u(t,x)\|_p<\infty \quad \mbox{for all} \quad T>0,
\end{equation}
where $\|\cdot\|_p$ denotes the norm in $L^p(\Omega)$. The goal of the present paper is to complement this analysis by showing that the {\em spatial average} of the (centered) solution:
\[
F_R(t)=\int_{-R}^{R}\big(u(t,x)-1 \big)dx.
\]
has asymptotic Gaussian fluctuations when $R \to \infty$, with a precise rate of convergence which depends on the noise. We denote $\sigma_R^2(t)={\rm Var}(F_R(t))$.

\begin{remark}
\label{rem-M}
{\rm Suppose that Assumption B holds. Applying Young's inequality if $\kappa \in L^1(\mathbb{R})$, and Theorem \ref{HLS} if $\kappa=R_{1,\alpha/2}$, we see that:
\[
\|G_t* \kappa\|_{L^p(\mathbb{R})} \leq
\left\{
\begin{array}{ll}
\|\kappa\|_{L^1(\mathbb{R})} \|G_t\|_{L^p(\mathbb{R})} & \mbox{if $\kappa \in L^1(\mathbb{R})$ and $p\geq 1$} \\
A_{1,\alpha,p}\|G_t\|_{L^q(\mathbb{R})} & \mbox{if $\kappa=R_{1,\alpha/2}$, $\alpha \in (0,1)$, $p>\frac{2}{2-\alpha}$ and $\frac{1}{q}=\frac{1}{p}+\frac{\alpha}{2}$}
\end{array} \right.
\]
Since $\|G_t\|_{L^p(\mathbb{R})}^p=2^{1-p}t$ for any $p>0$, we infer that condition \eqref{def-Mp} holds for any $p\geq 2$. Therefore, \eqref{mom-u} holds for any $p \geq 2$ such that $m_p<\infty$.
}
\end{remark}

We recall the following definitions. Let $X$ and $Y$ be random variables defined on the same probability space.
The {\em $1$-Wasserstein distance} between $X$ and $Y$ is:
\[
d_{W}(X,Y)=\sup_{{\rm Lip}(h) \leq 1} \big|\mathbb{E}[h(X)]-\mathbb{E}[h(Y)]\big|, \quad
\mbox{where} \quad
{\rm Lip}(h):=\sup_{x \not=y}\frac{|h(x)-h(y)|}{|x-y|}.
\]
The {\em Fortet-Mourier distance} between $X$ and $Y$ is:
\[
d_{FM}(X,Y)=\sup_{\|h\|_{\infty}+{\rm Lip}(h) \leq 1} \big|\mathbb{E}[h(X)]-\mathbb{E}[h(Y)]\big|,
\]
The {\em Kolmogorov distance} between $X$ and $Y$ is:
\[
d_{K}(X,Y)=\sup_{x \in \mathbb{E}} \big|P(X \leq x)-P(Y \leq x)\big|.
\]

We are now ready to state the main results of this article.

\begin{theorem}[Hardy-Littlewood-Sobolov inequality]
\label{HLS}
For any $\alpha \in (0,d)$ and $p>\frac{d}{d-\alpha}$,
\[
\| \varphi *R_{d,\alpha}\|_{L^p(\bR^d)} \leq A_{d.\alpha,p}\|\varphi\|_{L^{q}(\bR^d)} \quad \mbox{for any $\varphi \in L^q(\bR^d)$},
\]
where $A_{d,\alpha,p}>0$ is a constant depending on $(d,\alpha,p)$, and $q$ is defined by:
\[
\frac{1}{q}=\frac{1}{p}+\frac{\alpha}{d}.
\]
(Condition $p>\frac{d}{d-\alpha}$ is equivalent to $q>1$.)
\end{theorem}

\begin{equation}
\label{def-Mp}
\cM_p(t):=\int_0^t \int_{\bR} \big|(G_{t-s}*\k)(x)\big|^p dx ds<\infty \quad \mbox{for all} \quad t>0,
\end{equation}

\begin{equation}
\label{mom-u}
K_p(T):=\sup_{(t,x)\in [0,T] \times \bR}\|u(t,x)\|_p<\infty \quad \mbox{for all} \quad T>0,
\end{equation}

By Lemma 5.5 of \cite{BJ25}, the process $\{u(t,x)\}_{x\in \bR}$ is strictly stationary, for any $t>0$.
Moreover, by Theorem 5.6 of \cite{BJ25}, if for some $p\geq 2$, we have
\begin{equation}
\label{def-mp}
m_p:=\int_{\bR_0}|z|^p \nu(dz)<\infty,
\end{equation}
and
\begin{equation}
\label{def-Mp}
\cM_p(t):=\int_0^t \int_{\bR} \big|(G_{t-s}*\k)(x)\big|^p dx ds<\infty \quad \mbox{for all} \quad t>0,
\end{equation}
then
\begin{equation}
\label{mom-u}
K_p(T):=\sup_{(t,x)\in [0,T] \times \bR}\|u(t,x)\|_p<\infty \quad \mbox{for all} \quad T>0,
\end{equation}
where $\|\cdot\|_p$ denotes the norm in $L^p(\Omega)$. The goal of the present paper is to complement this analysis by showing that the {\em spatial average} of the (centered) solution:
\[
F_R(t)=\int_{-R}^{R}\big(u(t,x)-1 \big)dx.
\]
has asymptotic Gaussian fluctuations when $R \to \infty$, with a precise rate of convergence which depends on the noise. We denote $\sigma_R^2(t)={\rm Var}(F_R(t))$.

\begin{remark}
\label{rem-M}
{\rm Suppose that Assumption B holds. Applying Young's inequality if $\k \in L^1(\bR)$, and Theorem \ref{HLS} if $\k=R_{1,\alpha/2}$, we see that:
\[
\|G_t* \k\|_{L^p(\bR)} \leq
\left\{
\begin{array}{ll}
\|\k\|_{L^1(\bR)} \|G_t\|_{L^p(\bR)} & \mbox{if $\k \in L^1(\bR)$ and $p\geq 1$} \\
A_{1,\alpha,p}\|G_t\|_{L^q(\bR)} & \mbox{if $\k=R_{1,\alpha/2}$, $\alpha \in (0,1)$, $p>\frac{2}{2-\alpha}$ and $\frac{1}{q}=\frac{1}{p}+\frac{\alpha}{2}$}
\end{array} \right.
\]
Since $\|G_t\|_{L^p(\bR)}^p=2^{1-p}t$ for any $p>0$, we infer that condition \eqref{def-Mp} holds for any $p\geq 2$. Therefore, \eqref{mom-u} holds for any $p \geq 2$ such that $m_p<\infty$.
}
\end{remark}

We are now ready to state the main results of this article.

\begin{theorem}[Limiting covariance]
\label{cov-th}
If Assumption B holds, then
for any $t,s>0$,
\[
\lim_{R \to \infty}\frac{1}{R^{\beta}}\bE[F_R(t) F_R(s)]=K(t,s) \quad \mbox{is finite},
\]
where
\begin{equation}
\label{def-beta}
\beta:=\left\{
\begin{array}{ll} 1 & \mbox{if $\k \in L^1(\bR)$,} \\
\alpha+1 & \mbox{if $\k=R_{1,\alpha/2}$ for some $\alpha \in (0,1)$. }
\end{array} \right.
\end{equation}
In particular, $R^{-\beta} \sigma_R^2(t) \to K(t,t)$ as $R \to \infty$, for any $t>0$.
\end{theorem}

\begin{theorem}[Quantitative Central Limit Theorem]
\label{QCLT}
Suppose that Assumption B holds, and there exists $p \in (1,2]$ such that
\begin{equation}
\label{mp-m2p}
m_{p}<\infty \quad \mbox{and} \quad m_{2p}<\infty.
\end{equation}

\begin{theorem}[Functional Central Limit Theorem]
\label{FCLT}
Under the hypotheses of Theorem \ref{QCLT}, for any $R>0$,
the process $\{F_R(t)\}_{t\geq 0}$ has a $\gamma$-H\"older continuous modification (denoted also $F_R$), for any $\gamma \in (0,\frac{\beta}{2})$, where $\beta$ is given by \eqref{def-beta}.
Moreover,
\[
\frac{1}{R^{\beta/2}}F_R(\cdot) \stackrel{d}{\to} \cG(\cdot) \quad \mbox{in $C[0,\infty)$ as $R \to \infty$},
\]
where $\{\cG(t)\}_{t\geq 0}$ is a zero-mean Gaussian process with covariance
$\bE[\cG(t) \cG(s)]=K(t,s)$, and $K(t,s)$ given by Theorem \ref{cov-th}. Here $\stackrel{d}{\to}$ denotes the convergence in distribution, and $C[0,\infty)$ is equipped with the uniform convergence on compact sets.
\end{theorem}

$\bullet$
Any random variable $F \in L^2(\Omega)$ which is $\cF^N$-measurable has the {\em Poisson-chaos expansion}:
\begin{equation}
\label{Poisson-chaos}
F=\bE(F)+\sum_{n\geq 1}I_n(f_n), \quad \mbox{for some $f_n \in \cH^{\odot n}$},
\end{equation}
and the series is orthogonal in $L^2(\Omega)$. Here $I_n$ is the multiple integral with respect to $\widehat{N}$ and $\cH^{\odot n}$ is the set of symmetric functions in $\cH^{\otimes n}$.
For any $f \in \cH^{\otimes n}$,
\[
\bE[I_n(f)]=0 \quad \mbox{and} \quad \bE|I_n(f)|^2=n! \|\widetilde{f}\|_{\cH^{\otimes n}}^2,
\]
where $\widetilde{f}$ is the symmetrization of $f$:
\[
\widetilde{f}(\xi_1,\ldots,\xi_n)=\frac{1}{n!}\sum_{\rho \in S_n}f(\xi_{\rho(1)},\ldots,\xi_{\rho(n)}),
\]
$S_n$ being the set of all permutations of $1,\ldots,n$.
Moreover, $I_n(f)=I_n(\widetilde{f})$ for any $f \in \cH^{\otimes n}$.

{\em Step 1.} (finite-dimensional convergence) We will prove that for any integer $m\geq 1$ and form any $t_1,\ldots,t_m \in [0,T]$
\[
\Big(\frac{1}{R^{\beta/2}} F_R(t_1),\ldots, \frac{1}{R^{\beta/2}}F_R(t_m) \Big)
\stackrel{d}{\to} \Big(\cG(t_1),\ldots, \cG(t_m) \Big) \quad \mbox{as $R \to \infty$}.
\]
By Cram\'er-Wold theorem, this is equivalent to showing that for any $b_1,\ldots,b_m \in \bR$,
\[
X_R:=\frac{1}{R^{\beta/2}}\sum_{j=1}^{m}b_jF_R(t_j) \stackrel{d}{\to}\sum_{j=1}^{m}b_j \cG(t_j)
\quad \mbox{as $R \to \infty$}.
\]
Using the same argument as on page 4215 of \cite{BZ24}, it is enough to prove that $X_R/\tau_R \stackrel{d}{\to} Z$ as $R \to \infty$, where $\tau_R^2={\rm Var}(X_R)$ and $Z \sim N(0,1)$. By Proposition \ref{tara},
\[
d_{W}\left(\frac{X_R}{\tau_R},Z \right) \leq \gamma_1+\gamma_2+\gamma_3,
\]
where $\gamma_1,\gamma_2$ and $\gamma_3$ are defined as in \eqref{gamma17} with $F=X_R$. Using the same argument as in the proof of Theorem \ref{QCLT}, we infer that
\[
d_{W}\left(\frac{X_R}{\tau_R},Z \right) \les R^{1-p} \quad \mbox{if $\k \in L^1(\bR)$,}
\]
and
\[
d_{W}\left(\frac{X_R}{\tau_R},Z \right) \les R^{-\varepsilon} \quad \mbox{for any $\varepsilon \in \big(0,1-\frac{1}{p}-\frac{\alpha}{2}\big)$, \quad if $\k =R_{1,\alpha/2}$.}
\]

\begin{theorem}[Hardy-Littlewood-Sobolov inequality]
\label{HLS}
For any $\alpha \in (0,d)$ and $p>\frac{d}{d-\alpha}$,
\[
\| \varphi *R_{d,\alpha}\|_{L^p(\bR^d)} \leq A_{d.\alpha,p}\|\varphi\|_{L^{q}(\bR^d)} \quad \mbox{for any $\varphi \in L^q(\bR^d)$},
\]
where $A_{d,\alpha,p}>0$ is a constant depending on $(d,\alpha,p)$, and $q$ is defined by:
\[
\frac{1}{q}=\frac{1}{p}+\frac{\alpha}{d}.
\]
(Condition $p>\frac{d}{d-\alpha}$ is equivalent to $q>1$.)
\end{theorem}

\begin{theorem}[Quantitative Central Limit Theorem]
\label{QCLT}
Suppose that Assumption B holds, and there exists $p \in (1,2]$ such that
\begin{equation}
\label{mp-m2p}
m_{p}<\infty \quad \mbox{and} \quad m_{2p}<\infty.
\end{equation}

If $\k \in L^1(\bR)$, then
for any $t>0$,
\[
{\rm dist}\left( \frac{F_R(t)}{\sigma_R(t)},Z\right) \leq  C_t R^{-(1-\frac{1}{p})}
\]

If $\k=R_{1,\alpha/2}$ for some $\alpha \in (0,1)$, and $p>\frac{2}{2-\alpha}$, then
\[
{\rm dist}\left( \frac{F_R(t)}{\sigma_R(t)},Z\right) \leq
C_t R^{-\e}  \quad \mbox{for any $\varepsilon \in \big(0, 1-\frac{1}{p}-\frac{\alpha}{2}\big)$}.
\]

Here $C_t>0$ is a constant depending on $t$, and ${\rm dist}$ is the Fortet-Mourier distance, the 1-Wasserstein distance, or the Kolmogorov distance.
\end{theorem}

\begin{theorem}[Limiting covariance]
\label{cov-th}
If Assumption B holds, then
for any $t,s>0$,
\[
\lim_{R \to \infty}\frac{1}{R^{\beta}}\mathbb{E}[F_R(t) F_R(s)]=K(t,s) \quad \mbox{is finite},
\]
where
\begin{equation}
\label{def-beta}
\beta:=\left\{
\begin{array}{ll} 1 & \mbox{if $\kappa \in L^1(\mathbb{R})$,} \\
\alpha+1 & \mbox{if $\kappa=R_{1,\alpha/2}$ for some $\alpha \in (0,1)$. }
\end{array} \right.
\end{equation}
In particular, $R^{-\beta} \sigma_R^2(t) \to K(t,t)$ as $R \to \infty$, for any $t>0$.
\end{theorem}

\begin{theorem}[Quantitative Central Limit Theorem]
\label{QCLT}
Suppose that Assumption B holds, and there exists $p \in (1,2]$ such that
\begin{equation}
\label{mp-m2p}
m_{p}<\infty \quad \mbox{and} \quad m_{2p}<\infty.
\end{equation}

If $\kappa \in L^1(\mathbb{R})$, then
for any $t>0$,
\[
{\rm dist}\left( \frac{F_R(t)}{\sigma_R(t)},Z\right) \leq  C_t R^{-(1-\frac{1}{p})}
\]

If $\kappa=R_{1,\alpha/2}$ for some $\alpha \in (0,1)$, and $p>\frac{2}{2-\alpha}$, then
\[
{\rm dist}\left( \frac{F_R(t)}{\sigma_R(t)},Z\right) \leq
C_t R^{-\varepsilon}  \quad \mbox{for any $\varepsilon \in \big(0, 1-\frac{1}{p}-\frac{\alpha}{2}\big)$}.
\]

Here $C_t>0$ is a constant depending on $t$, and ${\rm dist}$ is the Fortet-Mourier distance, the 1-Wasserstein distance, or the Kolmogorov distance.
\end{theorem}

\begin{theorem}[Functional Central Limit Theorem]
\label{FCLT}
Under the hypotheses of Theorem \ref{QCLT}, for any $R>0$,
the process $\{F_R(t)\}_{t\geq 0}$ has a $\gamma$-H\"older continuous modification (denoted also $F_R$), for any $\gamma \in (0,\frac{\beta}{2})$, where $\beta$ is given by \eqref{def-beta}.
Moreover,
\[
\frac{1}{R^{\beta/2}}F_R(\cdot) \stackrel{d}{\to} \mathcal{G}(\cdot) \quad \mbox{in $C[0,\infty)$ as $R \to \infty$},
\]
where $\{\mathcal{G}(t)\}_{t\geq 0}$ is a zero-mean Gaussian process with covariance
$\mathbb{E}[\mathcal{G}(t) \mathcal{G}(s)]=K(t,s)$, and $K(t,s)$ given by Theorem \ref{cov-th}. Here $\stackrel{d}{\to}$ denotes the convergence in distribution, and $C[0,\infty)$ is equipped with the uniform convergence on compact sets.
\end{theorem}

The proofs of these theorems are based on a key estimate  which shows that the moments of the Malliavin derivative of the solution of \eqref{HAM} can be bounded, up to a constant, by a deterministic function (see relation \eqref{key-D} below). Similar estimates appear in all references dedicated to QCLT, where they play a crucial role. To prove this key estimate, we proceed as in \cite{BZ24}, developing a connection to (hAm) with delta initial velocity by using the form \eqref{def-G} of the fundamental solution $G$.

Theorem \ref{ergodic-th} follows from the key estimate \eqref{key-D} and a criterion which may be of independent interest (Lemma \ref{lem-key-Z} below).
To prove Theorem \ref{cov-th} we show that the covariance of the solution of \eqref{HAM} coincides with the covariance of the solution of (hAm) driven by $\sqrt{m_2}W$, where $W$ is the Gaussian colored noise with covariance \eqref{def-cov-W}.

To prove Theorem \ref{QCLT}, we apply the recent result of Trauthwein \cite{trauthwein25}, which gives the optimal rates for the Wasserstein and Kolmogorov distances between $F/\sqrt{{\rm Var}(F)}$ and $Z \sim N(0,1)$, for a centered random variable $F$ with finite variance, which is Malliavin differentiable with respect to a compensated Poisson random measure. These estimates involve the first and second order Malliavin derivatives. Therefore, to  apply the result in \cite{trauthwein25}, we develop a similar key estimate for the second Malliavin derivative of the solution (see relation \eqref{key-D2} below). Different techniques are used for estimating the 7 quantities $\gamma_1,\ldots,\gamma_7$ which appear in the result of \cite{trauthwein25}, which rely on Young's inequality in the case when $k$ is integrable, respectively on Hardy-Littlewood-Sobolev inequality when $\kappa$ is the Riesz kernel. Finally, Theorem \ref{FCLT} follows by the classical method of finite dimensional convergence and tightness.

\begin{remark}
{\rm a) If $\kappa \in L^1(\mathbb{R})$, then $f=\kappa * \tilde{\kappa} \in L^1(\mathbb{R})$ by Young's inequality. In this case, the variance $\sigma_R^2(t)$ decays with rate $R$, as in the L\'evy white noise case and the Gaussian colored noise case with integrable kernel $f$ (see Theorem 1.2 of \cite{NZ22}). The decay rate $R^{-(1-\frac{1}{p})}$  in the QCLT (in the $d_{W}$, $d_{FM}$ or $d_K$ distances) is the same as in the L\'evy white noise case (see Theorem 1.1.(iii) of \cite{BZ24}), and depends on the parameter $p$ from \eqref{mp-m2p}. This can be explained since formally, the L\'evy white noise can be viewed as a L\'evy colored noise with integrable kernel $\kappa=\delta_0$. When $p=2$ (i.e. $m_4<\infty)$, this rate coincides with the rate $R^{-1/2}$ obtained in the case of the Gaussian colored noise case with integrable kernel $f$, for the QCLT in the total variation distance (see Theorem 1.2 of \cite{NZ22}).

\medskip

b) If $\kappa=R_{1,\alpha/2}$ for some $\alpha \in (0,1)$, then $f =R_{1,\alpha}$. In this case, using the parametrization $\alpha=2H-1$ with $H \in (\frac{1}{2},1)$, we see that the decay rate $R^{2H}$ of $\sigma_R^2(t)$ is the same as in the Gaussian colored noise case (see  Proposition 3.3 of \cite{DNZ20}). In this Gaussian case, Theorem 1.1 {\em ibid.} gives the rate $R^{-(1-H)}=R^{-\frac{1-\alpha}{2}}$ for the QCLT in the total variation distance, which is almost the same as the rate $R^{-\varepsilon}$ with $\varepsilon \in (0,\frac{1-\alpha}{2})$ obtained in Theorem \ref{QCLT} above when $p=2$ (i.e. $m_4<\infty$).
}
\end{remark}

This article is organized as follows. In Section \ref{section-prelim}, we include some preliminaries about Poisson-Malliavin calculus and moment inequalities with respect to noises $L$ and $X$. In Sections \ref{section-Mall1} and \ref{section-Mall2}, we prove some key estimates for the first, respectively the second,  Malliavin derivative of the solution. In Section \ref{section-proofs}, we present the proofs of Theorems \ref{ergodic-th}, \ref{cov-th}, \ref{QCLT} and \ref{FCLT}. Appendix \ref{appA} contains some auxiliary results about Poisson-Malliavin calculus. Appendix \ref{appB} presents some inequalities for Riesz potentials.

\medskip

We conclude the introduction with few words about the notation. We write $a\lesssim b$ to indicate that $a \leq C b$ for some positive constant $C>0$ that does not depend on $(a,b)$.
For any $p\geq 1$, we denote by $\|\cdot \|_p$ the norm in $L^p(\Omega)$.

The post-theorem introduction outlines the following proof strategy (centered around Theorem~\ref{ergodic-th}):

- The proofs are "based on a key estimate" showing that "the moments of the Malliavin derivative of the solution of \eqref{HAM} can be bounded, up to a constant, by a deterministic function" (relation \eqref{key-D}).

- To prove this key estimate, the authors "proceed as in \cite{BZ24}, developing a connection to (hAm) with delta initial velocity by using the form \eqref{def-G} of the fundamental solution $G$."

- Then, explicitly for the main theorem: "Theorem \ref{ergodic-th} follows from the key estimate \eqref{key-D} and a criterion which may be of independent interest (Lemma \ref{lem-key-Z} below)."

The post-theorem introduction outlines the following proof strategy (centered around the main theorem):

- The proofs are based on a key estimate showing that the moments of the Malliavin derivative of the solution of
\begin{align}
\label{HAM}
	\begin{cases}
		\dfrac{\partial^2 u}{\partial^2 t} (t,x)
		=  \dfrac{\partial^2 u}{\partial x^2} (t,x)+u(t,x) \dot{X}(t,x), \\
		t>0, \ x \in \bR, \\
		u(0,x) = 1, \dfrac{\partial u}{\partial t} (0,x)=0, \quad x \in \bR,
	\end{cases}
\end{align}
can be bounded, up to a constant, by a deterministic function, namely
\begin{equation}
\label{key-D}
\|D_{r,y,z}u(t,x)\|_p \leq C'_{T,p,\nu,\k}|z| \int_{\bR}G_{t-r}(x-y')\k(y-y')dy',
\end{equation}

- To prove this key estimate, the authors “proceed as in \cite{BZ24}”, developing a connection to (hAm) with delta initial velocity by using the form
\begin{equation}
\label{def-G}
G_t(x)=\frac{1}{2}1_{\{|x|<t\}} \quad \mbox{for all $t>0$ and $x \in \bR$.}
\end{equation}
of the fundamental solution $G$.

- Then, in establishing the main theorem: it follows from the key estimate (the equation above) and a criterion which may be of independent interest. We first record that criterion.
\begin{lemma}
\label{lem-key-Z}
Suppose that Assumption B holds.
Let $\{Z(t,x);t\geq 0,x\in \bR\}$ be an adapted random field such that
$\{Z(t,x)\}_{x\in \bR^d}$ is strictly stationary and $Z(t,x) \in {\rm dom}(D)$ for any $t\geq 0$ and $x \in \bR$. Assume that for any $0<r<t$, $x,y \in \bR$ and $z \in \bR_0$,
\begin{equation}
\label{key-Z}
\|D_{r,y,z}Z(t,x)\|_{2} \leq C_t |z| \int_{\bR} G_{t-r}(x-y')\k(y-y')dy',
\end{equation}
where $C_t>0$ is a constant depending on $t$. Then $\{Z(t,x)\}_{x\in \bR}$ is ergodic.
\end{lemma}

Question:
Let \(u(t,x)\) be the one-dimensional random field under Assumption B, and for each fixed \(t>0\) define the centered spatial average over \([-R,R]\) by
\[
F_R(t)=\int_{-R}^{R}(u(t,x)-1)\,dx.
\]
Under Assumption B, which statement about the spatial process \(\{u(t,x)\}_{x\in\mathbb{R}}\) and the asymptotic behavior of \(F_R(t)\) is valid for every \(t>0\)?

For every \(t>0\), the spatial process \(\{u(t,x)\}_{x\in\mathbb{R}}\) is ergodic. Consequently,
\[
\frac{1}{R}F_R(t)=\frac{1}{R}\int_{-R}^{R}(u(t,x)-1)\,dx \to 0
\quad \text{as } R\to\infty,
\]
both almost surely and in \(L^2(\Omega)\).

Choices:
B.
For every \(t>0\), the spatial process \(\{u(t,x)\}_{x\in\mathbb{R}}\) is mixing. Consequently,
\[
\frac{1}{R}F_R(t)=\frac{1}{R}\int_{-R}^{R}(u(t,x)-1)\,dx \to 0
\quad \text{as } R\to\infty,
\]
both almost surely and in \(L^2(\Omega)\).
C.
For every \(t>0\),
\[
\frac{1}{R}F_R(t)=\frac{1}{R}\int_{-R}^{R}(u(t,x)-1)\,dx \to 0
\quad \text{as } R\to\infty,
\]
in \(L^2(\Omega)\).
D.
For every \(t>0\), the spatial process \(\{u(t,x)\}_{x\in\mathbb{R}}\) is ergodic. Consequently,
\[
\frac{1}{\sqrt{R}}F_R(t)=\frac{1}{\sqrt{R}}\int_{-R}^{R}(u(t,x)-1)\,dx \to 0
\quad \text{as } R\to\infty,
\]
both almost surely and in \(L^2(\Omega)\).
E.
For every \(t>0\), the spatial process \(\{u(t,x)\}_{x\in\mathbb{R}}\) is ergodic. Consequently,
\[
\frac{1}{R}F_R(t)=\frac{1}{R}\int_{-R}^{R}(u(t,x)-1)\,dx \to 0
\quad \text{as } R\to\infty,
\]
almost surely, and the convergence is uniform in \(t\) on compact time intervals in \(L^2(\Omega)\).


\label{introthm:tameness}
    Suppose $\varphi\colon \A\to\A^*$ is a primitive substitution of length $k\geqslant 2$. Then, the following are equivalent:
    \begin{enumerate}
        \item $\ac(X_{\varphi})\in \{0,1\}$ (with $\ac(X_{\varphi})=0$ corresponding to $X_{\varphi}$ being  finite),
        \item $X_{\varphi}$ is null,
        \item $X_{\varphi}$ is tame.
    \end{enumerate}

\begin{remark}\label{rem:inf_tiling_and_beyond_mean_eq}
    In the context of more general group actions, the authors of \cite{BaakeGaehlerGohlke2025} study amorphic complexity—for which they suggest to use the name \emph{orbit separation dimension}—for actions induced by primitive self-similar inflation tilings.
    They determine it for several notable examples, including the recently discovered Hat tiling \cite{SmithMyersKaplanGoodman-Strauss2024}.

    To extend the analysis of asymptotic separation numbers beyond mean equicontinuous systems, Kasjan and Keller propose in \cite{KasjanKeler2025} to study (Besicovitch) covering numbers of symbolic orbits of generic points.
    Their approach applies to the wider class of symbolic systems with discrete spectrum and reduces to the study of asymptotic separation numbers in the mean equicontinuous case.
    It allows them to distinguish between several Toeplitz subshifts associated with $\mathcal{B}$-free systems, which were previously indistinguishable using classical topological invariants.
\end{remark}

\begin{remark}\label{rem:Baakeresult}
    In \cite{BaakeGaehlerGohlke2025}, the authors introduce the notion of a \emph{discrepancy inflation} for a self-similar, nonperiodic topological tiling dynamical system $(\mathbb{X},\mathbb{R}^d)$ with pure-point spectrum generated by a primitive inflation, using it to establish lower and upper bounds for the amorphic complexity of $(\mathbb{X},\mathbb{R}^d)$. Though their work focuses on tilings (rather than the symbolic setting), it is easy to see that, for a substitution $\varphi$ of constant length, their discrepancy inflation corresponds to our discrepancy substitution.

    By translating their main result (\cite[Thm.\ 4.13 \& Prop.\ 4.14]{BaakeGaehlerGohlke2025}) to our framework, we obtain the following bounds:
    \[
        \frac{\log k}{\log k - \log \lambda_{min}} \leq \oac(X_{\varphi}) \leq \frac{\log k}{\log k - \log \lambda_s},
    \]
    where $\lambda_s$ (resp.\ $\lambda_{min}$) denotes the maximal (resp.\ minimal) growth rate of letters in the discrepancy substitution $\varphi_s$, and $k$ is the length of the substitution. If $\lambda_s = \lambda_{min}$, this yields a closed-form expression for the amorphic complexity of $X_{\varphi}$. However, in general, one typically expects that $\lambda_{min} < \lambda_s$; see e.g.\ Example~\ref{ex:sep_substitution_not_irreducible}.
\end{remark}

\begin{proposition}\label{prop:denseness}
 Let $\A$ be an alphabet of size at least 2. In the class of minimal automatic systems over $\A$, amorphic complexity takes a dense set of values in $[1,\infty)$.
\end{proposition}

\begin{introtheorem}\label{introthm:ac_of_automatic_systems} Let $\varphi$ be a primitive substitution of length $k\geq 2$.  The amorphic complexity of $X_{\varphi}$ is given by
\[\ac(X_{\varphi})=\frac{\log k}{\log k - \log \lambda_s},\]
where $\lambda_s$ is the discrepancy rate of $\varphi$.\footnote{Assuming the usual conventions when dividing by 0 or $\infty$ and $-\log 0 = \infty$.}
\end{introtheorem}

\begin{introtheorem}\label{introthm:tameness}
    Suppose $\varphi\colon \A\to\A^*$ is a primitive substitution of length $k\geq 2$. Then, the following are equivalent:
    \begin{enumerate}
        \item $\ac(X_{\varphi})\in \{0,1\}$ (with $\ac(X_{\varphi})=0$ corresponding to $X_{\varphi}$ being  finite),
        \item $X_{\varphi}$ is null,
        \item $X_{\varphi}$ is tame.
    \end{enumerate}
\end{introtheorem}

\label{sec:introduction}

Inspired by the seminal work \cite{BourgainFremlinTalagrand1978} of Bourgain, Fremlin, and Talagrand, K\"ohler introduced in \cite{Koehler1995} the notion of \emph{tame} dynamical systems (originally she used the term \emph{regular}).
Later, by adopting one of the main results of \cite{BourgainFremlinTalagrand1978} to the dynamical context, Glasner and Megrelishvili \cite{GlasnerMegerlishvili2006} provided the following dichotomy: the enveloping semigroup of \emph{any} system is either ``large'' (containing the Stone--\v{C}ech compactification of the natural numbers) or ``small'' (with its topology determined by sequential convergence), the latter case corresponding to the system being tame.

A particularly useful alternative characterization of tameness is provided by the theory of topological independence \cite{KerrLi2007}. Here, the existence or absence of large independence sets forces high or low dynamical complexity. For example, positive topological entropy of a system $T:X\to X$ is tied to the independence of iterates  $T^n$, $n\in\Z$ along subsets of positive density. At a weaker level, the absence of infinite independence sets is linked to tameness, while \emph{nullness} in the sense of Goodman \cite{Goodman} (i.e.\ zero topological sequence entropy) requires the lack of arbitrarily large finite independence sets, see Section \ref{sec:tamenull} or \cite{KerrLi2007} for precise definitions and further information.

Since their introduction—and in particular in recent years—tame and null systems have been widely studied. This research has focused on characterizing their structural properties in various ways and developing more directly applicable criteria for identifying tame or null systems.
For more information on tame systems, see for instance \cite{Glasner2006,GlasnerMegerlishvili2006,Huang2006,Glasner2007,Glasner2017,GlasnerMegrelishvili2018,FuhrmannKwietniak2020,FuhrmannGlasnerJagerOertel2021,GlasnerMegrelishvili2022,FuhrmannKellendonkYassawi2024,Kellendonk2025}, and for null systems, see for example \cite{HuangLiShaoYe2003,MaassShao2007,HuangYe2009,GarciaRamos2017,FuhrmannKwietniak2020,QiuZhao2020,Leonard2025,LePavlovSchlortt2025}.

As part of these recent developments, an important focus has been on establishing concrete and computationally tractable conditions to identify tame and null systems within canonical classes of dynamical systems.
In this article, we provide a complete characterisation of tameness and nullness for the well-studied class of minimal automatic dynamical systems \cite{Fogg2002,bookQueffelec} in terms of a single numerical invariant.
Such systems can be equivalently described as those that are conjugate to a subshift generated by a primitive, constant length substitution (see Sections \ref{sec:substitution} and \ref{sec:subsstsems}).

The numerical quantity we will utilize is the notion of amorphic complexity, a relatively new topological invariant introduced for the study of zero entropy systems \cite{FuhrmannGroegerJaeger2016}. It is tailor-made for analysing minimal systems with discrete spectrum and continuous eigenfunctions \cite{FuhrmannGroeger2020}. Beyond $\Z$-actions, this class encompasses dynamical systems canonically induced by mathematical quasicrystals, such as Penrose and chair tilings. In recent work \cite{FuhrmannGroegerJaegerKwietniak2023}, amorphic complexity was extended to these more general group actions; the authors of \cite{BaakeGaehlerGohlke2025} used this to calculate amorphic complexity for some prominent examples, including the newly discovered Hat tiling \cite{SmithMyersKaplanGoodman-Strauss2024}, see also Remarks \ref{rem:inf_tiling_and_beyond_mean_eq} and \ref{rem:Baakeresult}. Very recently, \cite{Gaehler2026} provided numerical evidence that amorphic complexity, together with another finitary invariant—asymptotic composants—is strong enough to completely distinguish minimal one-dimensional inflation tilings with pure-point spectrum for a class of small ternary Pisot unit inflation factors.
Furthermore, building on the ideas of \cite{FuhrmannGroegerJaeger2016}, the authors of \cite{KasjanKeler2025} were able to distinguish between several Toeplitz subshifts associated with $\mathcal{B}$-free systems—subshifts that were previously indistinguishable using classical topological invariants.

Before stating our main results, we need to introduce some notation: we first fix an \emph{alphabet}, i.e.\ a finite set $\mathcal{A}$. A \emph{substitution} $\varphi$ is a map that assigns to each letter $a \in \mathcal{A}$ a finite word in $\mathcal{A}^*$, the set of all finite words over $\mathcal{A}$. It is said to be of \emph{(constant) length} $k \geqslant 2$ if the word $\varphi(a)$ has length $k$ for all $a \in \mathcal{A}$. By concatenation, $\varphi$ naturally extends to a map on $\mathcal{A}^*$. We call $\varphi$ \emph{primitive} if there exists an $n \geqslant 1$ such that, for each $a \in \mathcal{A}$, the word $\varphi^n(a)$ contains every letter of $\mathcal{A}$.

To each substitution $\varphi$, we can associate a canonical subset $X_{\varphi}\subset \A^{\Z}$ of the space $\A^{\Z}$  of all bi-infinite sequences with symbols in $\mathcal{A}$. Specifically, $X_{\varphi}$ is the collection of all $x \in \mathcal{A}^\Z$ such that every finite word appearing in $x$ also appears in $\varphi^n(a)$ for some $a \in \mathcal{A}$ and $n \geqslant 0$. Note that there is a natural action of the \emph{(left) shift} $T$ on $X_{\varphi}$, defined by
$T((x_n)_{n \in \Z}) = (x_{n+1})_{n \in \Z}$ for all $(x_n)_{n \in \Z} \in X_{\varphi}$.
In the following, when we say that $X_\varphi$ is tame or null, we mean that the \emph{dynamical system} $(X_\varphi, T)$ is tame or null, respectively.

For the sake of simplicity, we introduce amorphic complexity here only in the symbolic context, where it admits a direct fractal geometric interpretation. The general definition is provided in Section~\ref{sec:asymp sep numbers and ac}, along with a brief overview of some of its basic properties as a topological invariant. For a symbolic system, its \emph{upper} and \emph{lower amorphic complexity} can be defined simply as the lower and upper box dimension, respectively, of its maximal equicontinuous factor equipped with the Besicovitch (pseudo)metric (see Remark \ref{rem:different_version_Besicovitch}). While these two values need not coincide in general, we will see next that for a minimal automatic system $X_\varphi$ they do, and we denote their common value by $\ac(X_\varphi)$.

Inspired by the seminal work \cite{BourgainFremlinTalagrand1978} of Bourgain, Fremlin, and Talagrand, K\"ohler introduced in \cite{Koehler1995} the notion of \emph{tame} dynamical systems (originally she used the term \emph{regular}).
Later, by adopting one of the main results of \cite{BourgainFremlinTalagrand1978} to the dynamical context, Glasner and Megrelishvili \cite{GlasnerMegerlishvili2006} provided the following dichotomy: the enveloping semigroup of \emph{any} system is either ``large'' (containing the Stone--\v{C}ech compactification of the natural numbers) or ``small'' (with its topology determined by sequential convergence), the latter case corresponding to the system being tame.

A particularly useful alternative characterization of tameness is provided by the theory of topological independence \cite{KerrLi2007}. Here, the existence or absence of large independence sets forces high or low dynamical complexity. For example, positive topological entropy of a system $T:X\to X$ is tied to the independence of iterates $T^n$, $n\in\Z$ along subsets of positive density. At a weaker level, the absence of infinite independence sets is linked to tameness, while \emph{nullness} in the sense of Goodman \cite{Goodman} (i.e.\ zero topological sequence entropy) requires the lack of arbitrarily large finite independence sets, see Section \ref{sec:tamenull} or \cite{KerrLi2007} for precise definitions and further information.

As part of these recent developments, an important focus has been on establishing concrete and computationally tractable conditions to identify tame and null systems within canonical classes of dynamical systems.
In this article, we provide a complete characterisation of tameness and nullness for the well-studied class of minimal automatic dynamical systems \cite{Fogg2002,bookQueffelec} in terms of a single numerical invariant.
Such systems can be equivalently described as those that are conjugate to a subshift generated by a primitive, constant length substitution (see Sections \ref{sec:substitution} and \ref{sec:subsstsems}).

Before stating our main results, we need to introduce some notation: we first fix an \emph{alphabet}, i.e.\ a finite set $\mathcal{A}$. A \emph{substitution} $\varphi$ is a map that assigns to each letter $a \in \mathcal{A}$ a finite word in $\mathcal{A}^*$, the set of all finite words over $\mathcal{A}$. It is said to be of \emph{(constant) length} $k \geqslant 2$ if the word $\varphi(a)$ has length $k$ for all $a \in \mathcal{A}$. By concatenation, $\varphi$ naturally extends to a map on $\mathcal{A}^*$. We call $\varphi$ \emph{primitive} if there exists an $n \geqslant 1$ such that, for each $a \in \mathcal{A}$, the word $\varphi^n(a)$ contains every letter of $\mathcal{A}$.

To each substitution $\varphi$, we can associate a canonical subset $X_{\varphi}\subset \A^{\Z}$ of the space $\A^{\Z}$ of all bi-infinite sequences with symbols in $\mathcal{A}$. Specifically, $X_{\varphi}$ is the collection of all $x \in \mathcal{A}^\Z$ such that every finite word appearing in $x$ also appears in $\varphi^n(a)$ for some $a \in \mathcal{A}$ and $n \geqslant 0$. Note that there is a natural action of the \emph{(left) shift} $T$ on $X_{\varphi}$, defined by
$T((x_n)_{n \in \Z}) = (x_{n+1})_{n \in \Z}$ for all $(x_n)_{n \in \Z} \in X_{\varphi}$.
In the following, when we say that $X_\varphi$ is tame or null, we mean that the \emph{dynamical system} $(X_\varphi, T)$ is tame or null, respectively.

For the sake of simplicity, we introduce amorphic complexity here only in the symbolic context, where it admits a direct fractal geometric interpretation. The general definition is provided in Section~\ref{sec:asymp sep numbers and ac}, along with a brief overview of some of its basic properties as a topological invariant. For a symbolic system, its \emph{upper} and \emph{lower amorphic complexity} can be defined simply as the lower and upper box dimension, respectively, of its maximal equicontinuous factor equipped with the Besicovitch (pseudo)metric (see Remark \ref{rem:different_version_Besicovitch}). While these two values need not coincide in general, we will see next that for a minimal automatic system $X_\varphi$ they do, and we denote their common value by $\ac(X_\varphi)$.

\begin{remark}\label{rem:different_version_Besicovitch}
    For a fixed partition $\mathcal{P}$ we denote by $\fp$ the full set
    $\fp = \mathcal{P} \ast \mathcal{P} \setminus \mathcal{P}$
    of distinct unordered pairs of elements of $\mathcal{P}$. With this convention,
    the pseudometric $D_{\fp}$ defined in \eqref{eq:def_of_pseudo} corresponds to
    the standard \emph{Besicovitch pseudometric}\footnote{Often, this term also
    refers to the pseudometric defined in \eqref{def:Besicovitch_pseudometric}.
    In the symbolic context, $D_{\fp}$ and $D_B$ are uniformly equivalent
    (see, for instance, the proof of Proposition 6.8 in \cite{FuhrmannGroeger2020})
    but, in general, they do not need to be Lipschitz equivalent.} on a subshift.
    Clearly, $D_{\fp}$ coincides with $D_\delta$ from
    \eqref{eq:besicovitch metric and density of discrepancy} for $\delta=1$ and
    for any $\brt \subseteq (\mathcal{P} \ast \mathcal{P}) \backslash \mathcal{P}$,
    we have
    \[
        D_{\brt}(x,y) \leq D_{\fp}(x,y),
    \]
    for all $x,y \in X$.
\end{remark}

For the sake of simplicity, we introduce amorphic complexity here only in the symbolic context, where it admits a direct fractal geometric interpretation. The general definition is provided in Section~\ref{sec:asymp sep numbers and ac}, along with a brief overview of some of its basic properties as a topological invariant. For a symbolic system, its \emph{upper} and \emph{lower amorphic complexity} can be defined simply as the lower and upper box dimension, respectively, of its maximal equicontinuous factor equipped with the Besicovitch (pseudo)metric (see Remark \ref{rem:different_version_Besicovitch}). While these two values need not coincide in general, we will see next that for a minimal automatic system $X_\varphi$ they do, and we denote their common value by $\ac(X_\varphi)$.

In particular, Theorem~\ref{introthm:tameness} yields that a system $X_{\varphi}$ is null iff it is tame (recall that nullness always implies tameness); to the best of our knowledge, this is a new result even in case of a binary alphabet. Furthermore, at the end of the introduction of \cite{FuhrmannKellendonkYassawi2024}, it is pointed out that \cite[Thm.~1.2]{FuhrmannKellendonkYassawi2024} together with \cite[Thm.~1.1]{FuhrmannGroeger2020} yields the equivalence of (i) and (iii) in the binary case. This means that, concerning this equivalence, Theorem~\ref{introthm:tameness} extends this observation to the full general case in the minimal setting.

\begin{lemma}\cite[Lem.\ 9]{DurandRigo2009}\label{lem:growth_of_letters}\mbox{}
\begin{enumerate}
\item\label{lem:growth_of_letters1} For a nonnegative matrix $A$,
\begin{equation}\label{eq:Mfor}
||A^n||_1 = \bigT (\lambda^n n^d),
\end{equation}
where $\lambda$ is the dominant eigenvalue of $A$ and $d\in\N$.
\item Let $\varphi\colon\mathcal{A}\to\mathcal{A}^{*}$ be a substitution, and let $m\geq 1$ be such that $\varphi^m$ is in normal primitive form.
 For each nonerasing letter $a\in\mathcal{A}$ there exist unique algebraic integer $\lambda(a)\geq 0$, integer $d(a)\geq 0$, and $c(a)>0$ such that
\begin{equation}\label{eq:growthtype}
\abs{\varphi^{mn}(a)} \sim c(a)\cdot(mn)^{d(a)}\cdot\lambda(a)^{mn}.
\end{equation}
 If $a\in\A$ is erasing we put $\lambda(a)=0$, $d(a)=1$, $c(a)=1$; in this case $\abs{\varphi^n(a)}=0$ for all $n$ large enough.
\end{enumerate}
\end{lemma}

\begin{proposition}\label{prop:sepnumberproperties}
 Let $\varphi$ be a primitive substitution of length $k\geq 2$ and let $\lambda_s$ be its discrepancy rate.
 The following are true:
\begin{enumerate}
\item\label{prop:sepnumberproperties1} $\lambda_s = 0$ or $1\leq \lambda_s \leq k$,
\item\label{prop:sepnumberproperties2} $\lambda_s = 0$ if and only if $X_{\varphi}$ is finite,
\item\label{prop:sepnumberproperties3} $\lambda_s =k$ if and only if $X_{\varphi}$ does not have discrete spectrum if and only if it is not mean equicontinuous.
\end{enumerate}
\end{proposition}
\begin{proof}
 The first claim follows from the fact that $\lambda_s$ is the dominant eigenvalue of a nonnegative integer matrix.

\begin{lemma}\label{lem:proportion_of_ourmetric_to_Besicovitch}
Let $\varphi\colon \A \to \A^*$ be a primitive substitution of length $k\geq 2$ and height 1. Then
\begin{enumerate}
\item\label{lem:proportion_of_ourmetric_to_Besicovitch1} $D_{\mg}(x,y)=0$ if and only if $D_{\mg}(\varphi(x),\varphi(y))=0$ for any $x,y\in X_{\varphi}$,
\item\label{lem:proportion_of_ourmetric_to_Besicovitch2} for any $x,y\in X_{\varphi}$ with $D_{\fp}(x,y)>0$ and $n$ big enough we have
\begin{equation}\label{eq:proportion_of_ourmetric_to_Besicovitch}
\frac{D_{\mg}(\varphi^n(x),\varphi^n(y))}{D_{\fp}(\varphi^n(x),\varphi^n(y))}\geq
\frac{D_{\mg}(x,y)}{D_{\fp}(x,y)}.
\end{equation}
\end{enumerate}
\end{lemma}
\begin{proof}
 Property \eqref{lem:proportion_of_ourmetric_to_Besicovitch1} is clear, since for any distinct pair of letters $\alpha={a\choose b}$ of not maximal growth (i.e.\ lying in $\A_s\setminus \mg$), all (pairs of) letters appearing in $\varphi_s(\alpha)$ are again of not maximal growth and for any distinct pair of letters $\alpha={a\choose b}$ of maximal growth, $\varphi(\alpha)$ contains at least one appearance of a (pair of) letters of maximal growth.

\begin{proposition}\label{prop:simplekernelstructure}
 Let $\varphi\colon\A\to\A^*$ be a primitive substitution of length $k\geq 2$ and height 1. Let $x$ be a fixed point of $\varphi$.
 We have
 \begin{equation}\label{eq:growthdm}
 \d_m(k,x) = \bigT (m^{d_{s}}\lambda_s^m),
 \end{equation}
 where $(\lambda_s,d_{s})$ is the discrepancy type of $\varphi$.
 Furthermore, assuming $x$ is not periodic, the following are equivalent:
 \begin{enumerate}
 \item\label{prop:simplekernelstructure1} $x$ has simple $k$-kernel structure,
 \item\label{prop:simplekernelstructure2} $\lambda_s=1$,
 \item\label{prop:simplekernelstructure3} for each nonconstant $y\in \Ker_k(x)$ and $m\geq 1$, $y$ appears at most once in $y$ as $k^m$-$\AP$,
 \item\label{prop:simplekernelstructure4} for each $\mathcal{A}'\in \C(\varphi)$ with $\abs{\A'}\geq 2$ and each $m\geq 1$ there exists at most one $0\leq i<k^m$ such that $\varphi^m_i(\mathcal{A}')=\mathcal{A}'$.
\end{enumerate}
 \end{proposition}
 \begin{proof}

\begin{remark}\label{rem:graph}
 In \cite{CovenQuasYassawi}, the authors associate with each primitive substitution $\varphi\colon\A\to\A^*$ of length $k\geq 2$ a certain directed graph $\mathcal G_\varphi$.
 If $\varphi$ has height 1, the graph $\mathcal G_\varphi$ is defined as follows: its vertices are given by $\C(\varphi)$ and there is an oriented edge from $\mathcal{B}$ to $\mathcal{C}$ if $\varphi_j(\mathcal{B})=\mathcal{C}$ for some $0\leq j<k$. If $\varphi$ does not have height 1, then $\mathcal G_\varphi$ is defined as the graph of its pure base.
 It is straightforward to note that Condition \eqref{prop:simplekernelstructure4} in Proposition \ref{prop:kernelandnullness} corresponds to the fact that $\mathcal G_{\varphi}$ does not contain two distinct cycles which share a common vertex.
 In \cite[Thm.\ 2.2]{FuhrmannKellendonkYassawi2024} it is shown that this condition precisely characterises systems $X_{\varphi}$ that are tame.
 \end{remark}

\begin{theorem}\label{thm:tameness}
Let $\varphi\colon \A\to\A^*$ be a primitive substitution of length $k\geq 2$ and assume that $X_{\varphi}$ is infinite. The following are equivalent:
\begin{enumerate}
\item\label{thm:tameness1} $\ac(X_{\varphi})=1$.
\item\label{thm:tameness2} $X_{\varphi}$ is null,
\item\label{thm:tameness3} $X_{\varphi}$ is tame.
\end{enumerate}
\end{theorem}
\begin{proof}

In particular, Theorem~\ref{introthm:tameness} yields that a system $X_{\varphi}$ is null iff it is tame (recall that nullness always implies tameness); to the best of our knowledge, this is a new result even in case of a binary alphabet. Furthermore, at the end of the introduction of \cite{FuhrmannKellendonkYassawi2024}, it is pointed out that \cite[Thm.~1.2]{FuhrmannKellendonkYassawi2024} together with \cite[Thm.~1.1]{FuhrmannGroeger2020} yields the equivalence of (i) and (iii) in the binary case. This means that, concerning this equivalence, Theorem~\ref{introthm:tameness} extends this observation to the full general case in the minimal setting.

Theorem~\ref{introthm:tameness} offers a natural interpretation of tameness and nullness in the context of minimal automatic systems. As shown in \cite[Cor.~1.3]{FuhrmannGroeger2020}, amorphic complexity of such systems is either $0$ or lies in the interval $[1, \infty]$. Furthermore, it is straightforward to demonstrate that amorphic complexity can attain a dense set of values in $[1, \infty)$ in the class of minimal automatic systems over a fixed alphabet of size at least two (see Proposition~\ref{prop:denseness}). Beyond the trivial case where amorphic complexity is $0$ (corresponding to periodic behaviour), this reveals a complexity gap between $0$ and $1$, with tame and null systems occupying exactly the minimal nontrivial complexity level of $1$.

To establish Theorem \ref{introthm:tameness} we need to determine amorphic complexity for arbitrary minimal automatic systems.
For this, to each primitive substitution $\varphi$ of constant length we associate its \emph{discrepancy substitution} $\varphi_s$ and define the \emph{discrepancy rate} $\lambda_s$ of $\varphi$ as the dominant eigenvalue of the incidence matrix of $\varphi_s$ (see Section \ref{sec:discrepancysubstitution}).
The discrepancy substitution $\varphi_s$ is straightforward to compute; however, in general, it is not primitive, which makes its analysis considerably more delicate.
For this reason, we introduce suitable dynamically defined pseudometrics that allow for a refined analysis of the discrepancy structure.
With this we derive a closed formula for the amorphic complexity of any minimal automatic systems, showing, in particular, that it always exists.

\begin{introtheorem}\label{introthm:ac_of_automatic_systems} Let $\varphi$ be a primitive substitution of length $k\geqslant 2$.  The amorphic complexity of $X_{\varphi}$ is given by
\[\ac(X_{\varphi})=\frac{\log k}{\log k - \log \lambda_s},\]
where $\lambda_s$ is the discrepancy rate of $\varphi$.\footnote{Assuming the usual conventions when dividing by 0 or $\infty$ and $-\log 0 = \infty$.}
\end{introtheorem}

We note that in case of a binary alphabet, Theorem~\ref{introthm:ac_of_automatic_systems} coincides with \cite[Thm.~1.1]{FuhrmannGroeger2020}. Moreover, for general alphabets, \cite{FuhrmannGroeger2020} also provides a potential algorithm to compute concrete lower and upper bounds for the lower and upper amorphic complexity of $X_\varphi$, whereas \cite{BaakeGaehlerGohlke2025} gives improved lower and upper bounds; see also Remark~\ref{rem:Baakeresult}.

The paper is organized as follows. In Section~\ref{sec:setup} we collect the necessary preliminaries. In Section~\ref{sec:discrepancytools} we introduce the discrepancy substitution and new dynamically generated pseudometrics, which serve as the main technical tools throughout the paper. In Section~\ref{sec:complexityofsubstitution} we prove Theorem~\ref{introthm:ac_of_automatic_systems} concerning the formula for the amorphic complexity. Finally, in Section~\ref{sec:tamenessandnullness} we establish Theorem~\ref{introthm:tameness}.

To establish Theorem~\ref{introthm:tameness} the authors first “need to determine amorphic complexity for arbitrary minimal automatic systems.” For this, “to each primitive substitution $\varphi$ of constant length we associate its \emph{discrepancy substitution} $\varphi_s$ and define the \emph{discrepancy rate} $\lambda_s$ of $\varphi$ as the dominant eigenvalue of the incidence matrix of $\varphi_s$.” Although “the discrepancy substitution $\varphi_s$ is straightforward to compute,” it “is not primitive,” so its analysis is “considerably more delicate.” To handle this they “introduce suitable dynamically defined pseudometrics that allow for a refined analysis of the discrepancy structure.” With these tools they “derive a closed formula for the amorphic complexity of any minimal automatic systems, showing, in particular, that it always exists,” namely Theorem~\ref{introthm:ac_of_automatic_systems} with
\[\ac(X_{\varphi})=\frac{\log k}{\log k - \log \lambda_s}.\]
Finally, “in Section~\ref{sec:tamenessandnullness} we establish Theorem~\ref{introthm:tameness},” i.e., the equivalence between “$\ac(X_{\varphi})\in\{0,1\}$,” “$X_{\varphi}$ is null,” and “$X_{\varphi}$ is tame,” using the amorphic-complexity determination above.

Question: Let \(\varphi:\mathcal A\to\mathcal A^*\) be a primitive substitution of constant length \(k\ge 2\) on a finite alphabet \(\mathcal A\), and let \(X_{\varphi}\subseteq \mathcal A^{\mathbb Z}\) be the associated substitution subshift consisting of all bi-infinite sequences whose every finite block appears in some iterate \(\varphi^n(a)\). Writing \(\operatorname{ac}(X_{\varphi})\) for the amorphic complexity of \(X_{\varphi}\), which statement is equivalent to both of the properties \("X_{\varphi}\) is null" and \("X_{\varphi}\) is tame"?

\(\operatorname{ac}(X_{\varphi})\in\{0,1\}\), with \(\operatorname{ac}(X_{\varphi})=0\) exactly when \(X_{\varphi}\) is finite.

Choices:
B. \(\operatorname{ac}(X_{\varphi})=1\), with \(\operatorname{ac}(X_{\varphi})=0\) impossible for primitive substitutions of constant length.
C. \(\operatorname{ac}(X_{\varphi})\le 1\).
D. \(\operatorname{ac}(X_{\varphi})\in\{0,1\}\), with \(\operatorname{ac}(X_{\varphi})=1\) exactly when \(X_{\varphi}\) is finite.
E. \(\operatorname{ac}(X_{\varphi})\) is finite.

dropped exact discrete range and finiteness characterization

reversal of the finite/ac=0 correspondence

discrete dichotomy replaced by mere boundedness