\label{main1}
Let $X$ be a 1-nodal quadric threefold  and $\pi \colon \widetilde{X} \longrightarrow X$ be its blow-up.

Then there exist a stability condition $\sigma_{\mathcal{K}u (X)}=(Z_\mathcal{A}, \mathcal{A})$ on $\mathcal{K}u (X)$,
and a weak stability condition $\sigma_{\widetilde{\mathcal{D}}'} = (Z_{\widetilde{\mathcal{A}}},\widetilde{\mathcal{A}})$ on the categorical resolution $\widetilde{\mathcal{D}}'$ of $\mathcal{K}u(X)$,
such that they are related as follows.
\begin{enumerate}
    \item[(1)] $Z_{\widetilde{\mathcal{A}}} = Z_\mathcal{A} \circ \pi_\ast$;
    \item[(2)] $\mathbf{R}\pi_\ast \colon \widetilde{\mathcal{A}}\longrightarrow \mathcal{A}$ is an exact functor.
\end{enumerate}

Bridgeland stability conditions on triangulated categories, introduced in \cite{Bri07},
have become a central tool in the study of derived categories.
They are now known to exist on curves \cite{Mac07} and on
surfaces \cite{AB13}.
In dimension three, the tilt-stability approach of Bayer--Macr\`i--Toda
\cite{BMT14} and its refinements provides a general framework for constructing
Bridgeland stability conditions,
and has led to existence results in a number of cases (see for instance \cite{BMS16}).

On the other hand, Kuznetsov studied the semiorthogonal decompositions of Fano threefolds and introduced Kuznetsov components
\cite{Kuz04,Kuz05,Kuz09},
which have played a central role in the study of their derived categories and geometry.
It is therefore natural to investigate stability conditions on Kuznetsov components
of Fano threefolds.
In the smooth case this has been carried out in
\cite{BLMS} for all smooth Fano threefolds of Picard rank one.
This leads to the question of what happens in the presence of mild singularities.

In this article, we extend our previous result in \cite{Cho25} to singular threefolds via Kuznetsov's categorical resolution \cite{Kuz08b},
and we first address the simplest such case, namely a quadric threefold with a single ordinary double point.

Our main result is the following:

\begin{abstract}
Let $X \subset \mathbb{P}^4$ be a quadric threefold with a single ordinary double point, and let $\mathcal{K}u(X)$ be its Kuznetsov component.
In this paper, we construct a weak stability condition $\sigma_{\widetilde{\mathcal{D}}'}$ on its categorical resolution $\widetilde{\mathcal{D}}' \subset \mathrm{D^b}(\widetilde{X})$,
which is compatible with the Verdier localization $\mathbf{R}\pi_\ast$ and descends to a Bridgeland stability condition on $\mathcal{K}u(X)$.
This can be viewed as a three-dimensional analogue of our previous result in \cite{Cho25}.

We describe the geometry of the blow-up $\pi \colon \widetilde{X} \longrightarrow X$ and obtain two semiorthogonal decompositions of $\mathrm{D^b}(\widetilde{X})$,
arising from the projective bundle structure of $\widetilde{X}$ and from Kuznetsov's categorical resolution.
Comparing them, we isolate an admissible subcategory $\widetilde{\mathcal{D}}' \subseteq \mathrm{D^b}(\widetilde{X})$ resolving $\mathcal{K}u(X)$ and show that it admits a full Ext-exceptional collection,
from which we construct $\sigma_{\widetilde{\mathcal{D}}'}$.
\end{abstract}

In this article, we extend our previous result in \cite{Cho25} to singular threefolds via Kuznetsov's categorical resolution \cite{Kuz08b},
and we first address the simplest such case, namely a quadric threefold with a single ordinary double point.

In order to construct stability conditions on the derived category of a singular threefold,
it is natural to first work on an appropriate ``resolution''.
For a nodal quadric threefold $X$, Kuznetsov's theory of categorical resolutions \cite{Kuz08b}
together with the construction of Kuznetsov--Shinder \cite{KS24} produce an admissible subcategory
$\widetilde{\mathcal{D}}\subset\mathrm{D^b}(\widetilde{X})$,
equipped with a functor
$\mathbf{R}\pi_\ast \colon \widetilde{\mathcal{D}} \longrightarrow \mathrm{D^b}(X)$ which is a Verdier localization,
with kernel generated by a single spherical object.

\begin{definition}\label{stability}
A weak stability condition (resp. Bridgeland stability condition) on $\mathcal{C}$ is a pair $\sigma = (Z,\mathcal{H})$ consisting of a group homomorphism (called the central charge of $\sigma$) $Z \colon \Lambda \longrightarrow \mathbb{C}$ and a heart $\mathcal{H}$ of a bounded t-structure on $\mathcal{C}$, such that the following conditions hold:
\begin{enumerate}
 \item[(a)] The composition $Z \circ v \colon {\rm K}(\mathcal{H})={\rm K}(\mathcal{C}) \longrightarrow \Lambda \longrightarrow \mathbb{C}$ is a weak stability function (resp. stability function) on $\mathcal{H}$. This gives a notion of slope: for any $E \in \mathcal{H}$, we set
 \begin{equation*}
 \mu_\sigma(E)=\mu_Z(E):=
 \begin{cases}
 \frac{-\Re Z(E)}{\Im Z(E)}, & \text{if } \Im Z(E)>0, \\
 +\infty, & \text{if } \Im Z(E)=0.
 \end{cases}
 \end{equation*}
 We say that an object $0\neq E \in \mathcal{H}$ is $\sigma$-semistable (resp. $\sigma$-stable) if for every nonzero proper subobject $F \subset E$, we have $\mu_\sigma (F) \leq \mu_\sigma (E)$ (resp. $\mu_\sigma (F) < \mu_\sigma (E)$).

\begin{theorem}[{\normalfont\cite[Theorem 5.8]{KS24}}]\label{KS24}
Let $X$ be a variety of dimension $n \geq 2$ over an algebraically closed
field $k$ of characteristic not equal to $2$ with an ordinary double point
$x_0$ and no other singularities. Let
\[
 \pi : \widetilde{X} = \operatorname{Bl}_{x_0}(X) \longrightarrow X
\]
be the blowup of the singular point, let
$\epsilon \colon E \hookrightarrow \widetilde{X}$ be the embedding of the
exceptional divisor over $x_0$, and let $S$ be a spinor bundle on $E$.
Then the subcategory
$\widetilde{\mathcal{D}}:= \{F \in \mathrm{D^b}(\widetilde{X})|\epsilon^\ast F \in \langle S,\mathcal{O}_{E}\rangle\}$
is admissible in $\mathrm{D^b}(\widetilde{X})$. Moreover,
\begin{enumerate}
 \item the induced functor
 $\mathbf{R}\pi_{*} : \widetilde{\mathcal{D}} \to \mathrm{D^b}(X)$
 is a crepant Verdier localization;
 \item the kernel $\ker(\pi_\ast|_{\widetilde{{\mathcal{D}}}})$ is generated by a single spherical object $K \in \widetilde{\mathcal{D}}$;
 \item when $\dim(X)$ is even, $K=\epsilon_\ast S$ is $2$-spherical,
 and when $\dim(X)$ is odd, $K$ is $3$-spherical and fits into the distinguished triangle $K \longrightarrow \epsilon_\ast S \longrightarrow \epsilon_\ast S'[2]$, where $S'$ is another spinor bundle on $E$.
\end{enumerate}
\end{theorem}

Assume that $\mathcal{N}$ is generated by a single object $A \in\mathcal{C}$ and that $\mathcal{Q}\cap\mathcal{N}$ is a Serre subcategory of $\mathcal{Q}$ (i.e. closed under subobjects, quotients, and extensions).
Then the following are equivalent:
\begin{enumerate}
\item $\mathcal{Q}$ descends to the heart $\mathcal{Q}/(\mathcal{Q}\cap\mathcal{N})$ of a bounded $t$-structure on $\mathcal{C}/\mathcal{N}$.
\item There exists an object $B\in\mathcal{Q}$ such that
$\mathcal{N} = \langle B\rangle$.
\end{enumerate}
In this case, $q\colon \mathcal{Q}\longrightarrow \mathcal{Q}/(\mathcal{Q}\cap\mathcal{N})$ is exact.
\end{lemma}

\begin{proposition}
There exist a weak stability condition $\sigma_{\widetilde{\mathcal{D}}'} = (Z_{\widetilde{\mathcal{A}}},\widetilde{\mathcal{A}})$ on $\widetilde{\mathcal{D}}'$, satisfying the support property with respect to the quotient lattice $\mathrm{K_{num}}(\widetilde{\mathcal{D}}')/\ker (\pi_\ast|_{\widetilde{\mathcal{D}}'})$,
and a Bridgeland stability condition $\sigma_{\mathcal{K}u(X)} = (Z_{\mathcal{A}},\mathcal{A})$ on $\mathcal{K}u(X)$,
such that
\begin{enumerate}
 \item $\mathbf{R}\pi_\ast\colon \widetilde{\mathcal{D}}' \longrightarrow \mathcal{K}u(X)$ is compatible with the central charges, i.e. $Z_{\widetilde{\mathcal{A}}} = Z_{\mathcal{A}} \circ \pi_\ast$.
 \item $\mathbf{R}\pi_\ast\colon \widetilde{\mathcal{A}} \longrightarrow \mathcal{A}$ is an exact functor.

In order to construct stability conditions on the derived category of a singular threefold,
it is natural to first work on an appropriate ``resolution''.
For a nodal quadric threefold $X$, Kuznetsov's theory of categorical resolutions \cite{Kuz08b}
together with the construction of Kuznetsov--Shinder \cite{KS24} produce an admissible subcategory
$\widetilde{\mathcal{D}}\subset\mathrm{D^b}(\widetilde{X})$,
equipped with a functor
$\mathbf{R}\pi_\ast \colon \widetilde{\mathcal{D}} \longrightarrow \mathrm{D^b}(X)$ which is a Verdier localization,
with kernel generated by a single spherical object.

The category $\widetilde{\mathcal{D}}$ admits an explicit description thanks to the projective bundle structure of $\widetilde{X}$,
and it will serve as the starting point of our construction.

First, we exploit two natural semiorthogonal decompositions of $\mathrm{D^b}(\widetilde{X})$.
The first arises from Orlov’s projective bundle formula for $\tilde{X}\cong \mathbb{P}_E(\mathcal{O}_E \oplus \mathcal{O}_E(-1,-1))$,
which yields a semiorthogonal decomposition of $\mathrm{D^b}(\widetilde{X})$ given by eight exceptional line bundles.
The second decomposition comes from Kuznetsov’s categorical resolution of the nodal quadric.

The key point is that, by performing a sequence of mutations relating these two decompositions,
we exhibit a full Ext-exceptional collection of length $3$ in the categorical resolution
$\widetilde{\mathcal{D}}' \subseteq \widetilde{\mathcal{D}}$ of $\mathcal{K}u(X)$,
which greatly simplifies the construction of stability conditions.

More precisely, we have:
\begin{theorem}
The categorical resolution  $\widetilde{\mathcal{D}}'$ of $\mathcal{K}u(X)$ admits a full Ext-exceptional collection of length $3$.
\end{theorem}

Their extension closure immediately gives a heart $\mathcal{B}$ of bounded $t$-structure.
Nevertheless, it can be seen that no weak stability condition with heart $\mathcal{B}$ descends to $\mathcal{K}u(X)$,
and we therefore find a suitable tilt to construct a new heart $\widetilde{\mathcal{A}}\subset \widetilde{\mathcal{D}}'$,
so that its pushforward $\mathcal{A}:=\mathbf{R}\pi_\ast(\widetilde{\mathcal{A}})$ is a heart on $\mathcal{K}u(X)$ and we can then define a localization-compatible central charge $Z_{\widetilde{\mathcal{A}}}$
and induce a stability condition  $\sigma_{\mathcal{A}}=(Z_{\mathcal{A}},\mathcal{A})$ on $\mathcal{K}u(X)$.
\subsection{Related work}
Our result fits into recent developments on Bridgeland stability conditions and moduli on Kuznetsov components of Fano varieties.

In the smooth case, stability conditions on Fano threefolds of Picard rank one and their Kuznetsov components are constructed in \cite{Li19} and \cite{BLMS}, respectively.

On the singular side, Kuznetsov and Shinder introduced the notion of categorical absorption of singularities and constructed, under natural hypotheses,
categorical absorptions for projective varieties with isolated ordinary double points \cite{KS24}.
They also studied the derived categories of 1-nodal prime Fano threefolds in \cite{KS25}.

Independently, in \cite{CGL+} the authors studied kernels of categorical resolutions of nodal singularities and proved that for nodal varieties the kernel is generated by a single spherical object, with applications to nodal cubic fourfolds.

Our weak stability condition on the categorical resolution also fits into the framework of partial compactifications of stability spaces by massless objects developed in \cite{BPPW22}. In their terminology it can be viewed as a lax stability condition lying on a boundary stratum,
with massless subcategory corresponding to the  kernel of the Verdier localization.

Finally, in our previous work \cite{Cho25} we constructed Bridgeland stability conditions on a singular surface and its resolution,
and the present paper can be viewed as a threefold counterpart.

\subsection{Organization of the paper}
In Section 2 we review background on stability conditions, hearts of $t$-structures, and tilting.

Section 3 is devoted to the geometry of the nodal quadric threefold $X$ and its resolution $\widetilde{X}$; we describe the projective bundle structure of $\widetilde{X}$ and derive a semiorthogonal decomposition of $\mathrm{D^b}(\widetilde{X})$ by Orlov's formula.

In Section 4, we introduce Kuznetsov’s categorical resolution $\widetilde{\mathcal{D}}$ of $\mathrm{D^b}(X)$ and isolate the admissible subcategory $\widetilde{\mathcal{D}}'$ which resolves the Kuznetsov component $\mathcal{K}u(X)$. We then perform a sequence of mutations, using the projective bundle decomposition from Section 3, to exhibit a full exceptional triple generating $\widetilde{\mathcal{D}}'$.

In Section 5, we use this exceptional collection to construct a new heart $\widetilde{\mathcal{A}}$ on $\widetilde{\mathcal{D}}'$ and prove that it descends to a heart $\mathcal{A}$ on $\mathcal{K}u(X)$.
We also carry out the construction of the stability condition on $\mathcal{K}u(X)$ and prove the main result.

Throughout the paper, we work over the complex numbers $\mathbb{C}$.

In order to prove Theorem~\ref{main1}, the construction proceeds by working first on an appropriate categorical ``resolution''. For a nodal quadric threefold $X$, one uses Kuznetsov's theory of categorical resolutions together with the construction of Kuznetsov--Shinder to obtain an admissible subcategory $\widetilde{\mathcal{D}}\subset\mathrm{D^b}(\widetilde{X})$ equipped with a functor $\mathbf{R}\pi_\ast\colon \widetilde{\mathcal{D}}\to\mathrm{D^b}(X)$ which is a Verdier localization, with kernel generated by a single spherical object.

The category $\widetilde{\mathcal{D}}$ is described explicitly using the projective bundle structure of $\widetilde{X}$, and the construction starts by exploiting two semiorthogonal decompositions of $\mathrm{D^b}(\widetilde{X})$: one from Orlov’s projective bundle formula (giving eight exceptional line bundles) and one from Kuznetsov’s categorical resolution. By performing a sequence of mutations relating these decompositions, one obtains a full Ext-exceptional collection of length $3$ in the categorical resolution $\widetilde{\mathcal{D}}'\subseteq\widetilde{\mathcal{D}}$ of $\mathcal{K}u(X)$, which simplifies the stability construction.

The extension closure of this exceptional collection gives a heart $\mathcal{B}$, but “no weak stability condition with heart $\mathcal{B}$ descends to $\mathcal{K}u(X)$”, so one performs “a suitable tilt” to produce a new heart $\widetilde{\mathcal{A}}\subset \widetilde{\mathcal{D}}'$. This is arranged so that its pushforward $\mathcal{A}:=\mathbf{R}\pi_\ast(\widetilde{\mathcal{A}})$ is a heart on $\mathcal{K}u(X)$ and $\mathbf{R}\pi_\ast\colon \widetilde{\mathcal{A}}\to\mathcal{A}$ is exact. Finally, one defines a “localization-compatible” central charge $Z_{\widetilde{\mathcal{A}}}$ and induces a stability condition $\sigma_{\mathcal{A}}=(Z_{\mathcal{A}},\mathcal{A})$ on $\mathcal{K}u(X)$ with $Z_{\widetilde{\mathcal{A}}}=Z_{\mathcal{A}}\circ\pi_\ast$.

In order to prove the main theorem, the construction proceeds by working first on an appropriate categorical ``resolution''. For a nodal quadric threefold $X$, one uses Kuznetsov's theory of categorical resolutions together with the construction of Kuznetsov--Shinder to obtain an admissible subcategory $\widetilde{\mathcal{D}}\subset\mathrm{D^b}(\widetilde{X})$ equipped with a functor $\mathbf{R}\pi_\ast\colon \widetilde{\mathcal{D}}\to\mathrm{D^b}(X)$ which is a Verdier localization, with kernel generated by a single spherical object.

The category $\widetilde{\mathcal{D}}$ is described explicitly using the projective bundle structure of $\widetilde{X}$, and the construction starts by exploiting two semiorthogonal decompositions of $\mathrm{D^b}(\widetilde{X})$: one from Orlov’s projective bundle formula (giving eight exceptional line bundles) and one from Kuznetsov’s categorical resolution. By performing a sequence of mutations relating these decompositions, one obtains a full Ext-exceptional collection of length $3$ in the categorical resolution $\widetilde{\mathcal{D}}'\subseteq\widetilde{\mathcal{D}}$ of $\mathcal{K}u(X)$, which simplifies the stability construction.

The extension closure of this exceptional collection gives a heart $\mathcal{B}$, but “no weak stability condition with heart $\mathcal{B}$ descends to $\mathcal{K}u(X)$”, so one performs “a suitable tilt” to produce a new heart $\widetilde{\mathcal{A}}\subset \widetilde{\mathcal{D}}'$. This is arranged so that its pushforward $\mathcal{A}:=\mathbf{R}\pi_\ast(\widetilde{\mathcal{A}})$ is a heart on $\mathcal{K}u(X)$ and $\mathbf{R}\pi_\ast\colon \widetilde{\mathcal{A}}\to\mathcal{A}$ is exact. Finally, one defines a “localization-compatible” central charge $Z_{\widetilde{\mathcal{A}}}$ and induces a stability condition $\sigma_{\mathcal{A}}=(Z_{\mathcal{A}},\mathcal{A})$ on $\mathcal{K}u(X)$ with $Z_{\widetilde{\mathcal{A}}}=Z_{\mathcal{A}}\circ\pi_\ast$.

Question: Let \(X\) be a 1-nodal quadric threefold, i.e. a quadric threefold with exactly one ordinary double point, and let \(\pi\colon \widetilde{X}\to X\) be the blow-up of the node. Let \(\mathcal{K}u(X)\) be the Kuznetsov component of \(X\), and let \(\widetilde{\mathcal{D}}'\) be a categorical resolution of \(\mathcal{K}u(X)\) inside \(\mathrm{D}^b(\widetilde{X})\). A Bridgeland stability condition or weak stability condition on a triangulated category is a pair \((Z,\mathcal{H})\) consisting of a central charge \(Z\) and the heart \(\mathcal{H}\) of a bounded \(t\)-structure. Which existence statement holds under these assumptions?

There exist a Bridgeland stability condition \(\sigma_{\mathcal{K}u(X)}=(Z_{\mathcal{A}},\mathcal{A})\) on \(\mathcal{K}u(X)\) and a weak stability condition \(\sigma_{\widetilde{\mathcal{D}}'}=(Z_{\widetilde{\mathcal{A}}},\widetilde{\mathcal{A}})\) on the categorical resolution \(\widetilde{\mathcal{D}}'\) such that \(Z_{\widetilde{\mathcal{A}}}=Z_{\mathcal{A}}\circ \pi_*\) and \(\mathbf{R}\pi_*\colon \widetilde{\mathcal{A}}\to \mathcal{A}\) is an exact functor.

Choices:
B. There exist Bridgeland stability conditions \(\sigma_{\mathcal{K}u(X)}=(Z_{\mathcal{A}},\mathcal{A})\) on \(\mathcal{K}u(X)\) and \(\sigma_{\widetilde{\mathcal{D}}'}=(Z_{\widetilde{\mathcal{A}}},\widetilde{\mathcal{A}})\) on the categorical resolution \(\widetilde{\mathcal{D}}'\) such that \(Z_{\widetilde{\mathcal{A}}}=Z_{\mathcal{A}}\circ \pi_*\) and \(\mathbf{R}\pi_*\colon \widetilde{\mathcal{A}}\to \mathcal{A}\) is an exact functor.
C. There exist a Bridgeland stability condition \(\sigma_{\mathcal{K}u(X)}=(Z_{\mathcal{A}},\mathcal{A})\) on \(\mathcal{K}u(X)\) and a weak stability condition \(\sigma_{\widetilde{\mathcal{D}}'}=(Z_{\widetilde{\mathcal{A}}},\widetilde{\mathcal{A}})\) on the categorical resolution \(\widetilde{\mathcal{D}}'\) such that \(Z_{\widetilde{\mathcal{A}}}=Z_{\mathcal{A}}\circ \pi_*\).
D. There exist a Bridgeland stability condition \(\sigma_{\mathcal{K}u(X)}=(Z_{\mathcal{A}},\mathcal{A})\) on \(\mathcal{K}u(X)\) and a weak stability condition \(\sigma_{\widetilde{\mathcal{D}}'}=(Z_{\widetilde{\mathcal{A}}},\widetilde{\mathcal{A}})\) on the categorical resolution \(\widetilde{\mathcal{D}}'\) such that \(Z_{\widetilde{\mathcal{A}}}=Z_{\mathcal{A}}\circ \pi_*\) and \(\mathbf{R}\pi_*\colon \widetilde{\mathcal{A}}\to \mathcal{A}\) is a faithful functor.
E. There exist a weak stability condition \(\sigma_{\mathcal{K}u(X)}=(Z_{\mathcal{A}},\mathcal{A})\) on \(\mathcal{K}u(X)\) and a Bridgeland stability condition \(\sigma_{\widetilde{\mathcal{D}}'}=(Z_{\widetilde{\mathcal{A}}},\widetilde{\mathcal{A}})\) on the categorical resolution \(\widetilde{\mathcal{D}}'\) such that \(Z_{\widetilde{\mathcal{A}}}=Z_{\mathcal{A}}\circ \pi_*\) and \(\mathbf{R}\pi_*\colon \widetilde{\mathcal{A}}\to \mathcal{A}\) is an exact functor.

weak-vs-Bridgeland status on the resolution

exactness of \(\mathbf{R}\pi_*\) replaced by faithfulness

assignment of Bridgeland vs weak stability to the two categories

The stem does not explicitly reveal the correct option. It only sets up the geometric/categorical context and asks which existence statement is valid; the exact conclusion must be identified from the choices.

This is essentially a theorem-recall item: the task is to recognize the precise stated existence theorem among slight variants, rather than derive a new conclusion from the assumptions.

Some reasoning/attention is needed to distinguish subtle modifications (Bridgeland vs weak stability, exactness vs faithfulness, omitted conditions), but the item mainly tests recognition of the exact theorem statement rather than genuine generative mathematical reasoning.

The distractors are close, plausible, and mathematically meaningful. They target realistic confusion points: strengthening weak to Bridgeland, dropping exactness, replacing exactness by faithfulness, and swapping which category gets weak vs Bridgeland stability.

A solid theorem-recognition MCQ with strong distractors and no direct answer leakage, but it is largely tautological and only moderately tests reasoning rather than generative understanding.


\begin{thm}\label{mainthm}
    Every pseudo-Anosov homeomorphism has an invariant train track whose transition matrix is irreducible.
\end{thm}

\par In slightly more detail,
given a boundaryless surface with negative Euler characteristic
and a train track folding sequence (equivalently, splitting sequence) for a pseudo-Anosov $f\colon S \to S$, one can construct the veering triangulation of the mapping torus of $f$, whose
flow graph encodes the transition matrix for this train track. Using the work of \cite{AT}, we show that the only obstructions to irreducibility are branches which are dual to infinitesimal cycles of the flow graph and characterize the structure of veering tracks near these branches. This local picture allows us to modify the track by contracting the obstructing branches, and recover a support map for the modified train track with irreducible transition matrix.

\subsection{Connections to literature}\label{discuss}

\par Work of Penner-Papadopoulos \cite[Theorem 4.1]{PP} is sometimes cited as producing $f$-invariant generic train tracks with irreducible transition matrix for a pseudo-Anosov $f$, however it was pointed out to us by Chris Leininger that this construction seems to only produce train tracks which are $f^n$-invariant with irreducible transition matrix for some $n$ possibly greater than one.
\par In the setting of \cite[Theorem 4.1]{PP},
the support map $\sigma$ produced for the $f$-invariant train track $\tau$ is not required to map switches to switches, rather the transition matrix is defined by choosing test points in the interiors of branches and counting the number of times the image of each branch hits each test point. Since switches don't have to map to switches, a branch $b$ of $\tau$ could e.g. be mapped by $\sigma \circ f$ over itself and a small piece of an adjacent branch. Then the transition matrix for $f$ will record that $b$ only maps over itself. However, for some $n$, $(\sigma \circ f)^n(b)$ will be mapped over all of the adjacent branch, so the transition matrix for $(\sigma \circ f)^n$ will record that $b$ maps over itself and the adjacent branch. Hence, it's possible that $f^n(\tau) \prec \tau$ has irreducible transition matrix while $f(\tau) \prec \tau$ does not.

\par Returning to discussion of the present work, we note that much of the theory of train tracks assumes that the tracks considered are \textit{\textbf{generic}}, i.e. trivalent, since this simplification can often be made without loss of generality. In particular, the train tracks associated to veering triangulations are always generic. However, if the flow graph for a pseudo-Anosov $f$ is not irreducible, \Cref{mainthm} necessarily produces train tracks which are not generic.
This motivates the following question:

\begin{qn}
    For all $f\colon S \to S$ pseudo-Anosov, does there exist a \textit{generic} invariant train track for $f$ with irreducible transition matrix?
\end{qn}

\par \textbf{Acknowledgements.} I would like to thank Sam Taylor for his generous guidance and feedback. I would also like to thank Chi Cheuk Tsang for helpful conversations and comments on a draft of this paper.

\section{Background}\label{background}

\par A \textit{\textbf{strongly connected component}} of a directed graph $G$ is a maximal subset $C$ of the vertices of $G$ such that for any two vertices in $C$, there is a path in $G$ from the first vertex to the second.
If $G$ has only one strongly connected component, it is called \textit{\textbf{strongly connected}}.
A nonnegative square matrix $A$ is called \textit{\textbf{irreducible}} if for every pair of indices $i$, $j$ there is an $n>0$ such that the $(i,j)$-th entry of $A^n$ is positive. $A$ is irreducible if and only if it is the adjacency matrix of a strongly connected graph.
\par
    Given an orientable finite-type surface $S$ without boundary and negative Euler characteristic, a homeomorphism $f\colon S \to S$ is called \textit{\textbf{pseudo-Anosov}} if there is a pair of transverse measured singular foliations $(\Lambda^u,\mu_u)$ and $(\Lambda^s,\mu_s)$, called the \textit{\textbf{unstable}} and \textit{\textbf{stable}} foliations, and a constant $\lambda > 1$, called the \textit{\textbf{stretch factor}}, such that $f(\Lambda^u)=\Lambda^u$, $f(\Lambda^s)=\Lambda^s$ and $f(\mu^u) = \lambda \mu^u$, $f(\mu^s) = \lambda^{-1} \mu^s$. No nontrivial power of a pseudo-Anosov map fixes the homotopy class of any essential closed curve in $S$. A pseudo-Anosov map is a diffeomorphism except at the singular points of $\Lambda^u$.

    \par The \textit{\textbf{mapping torus of $f$}} is the 3-manifold
    $$ M_f = \faktor{(S \times [0,1])}{ (f(p),0) \sim (p,1)} $$ which fibers over the circle with fiber $S$ and monodromy $f$.

    \par If all of the singularities of the foliations are at the punctures of $S$, we say $f$ is \textit{\textbf{fully punctured}}. We will study general pseudo-Anosovs by deleting the singularities of the foliations to obtain a new surface $S^\circ$ on which the restriction of $f$ is fully punctured.
For full background on pseudo-Anosov homeomorphisms, see Fathi--Laudenbach--Poénaru \cite{FLP}.

\subsection{Train tracks}\label{tts}
\par A \textit{\textbf{train track}} $\tau$ is a closed 1-complex embedded in a surface $S$ with a ``smoothing" at each vertex so that $\tau$ has a well-defined tangent space at each vertex, and $S \setminus \tau$ contains no nullgons, unpunctured monogons or unpunctured bigons. See Penner-Harer \cite{PH}. The vertices of $\tau$ are called \textit{\textbf{switches}} and the edges are called \textit{\textbf{branches}}. If the switches of $\tau$ all have degree three, $\tau$ is called \textit{\textbf{generic}}.

\par Given two train tracks $\tau_1$ and $\tau_2$ on $S$, $\tau_2$ is said to \textit{\textbf{carry}} $\tau_1$ (or $\tau_1$ \textit{\textbf{is carried by}} $\tau_2$), denoted $\tau_1 \prec \tau_2$, if there is a $C^1$ map $\sigma\colon S \to S$ called the \textit{\textbf{support map}}
such that
    \begin{enumerate}
        \item $\sigma$ is homotopic to the identity,
        \item $\sigma(\tau_1) \subseteq \tau_2$,
        \item for all points $x$ in $\tau_1$,
    $d_x \sigma_{|\tau_1}\colon T_x \tau_1 \to T_{h(x)}\tau_2$ is an isomorphism of tangent spaces,
    \end{enumerate}
Note that carrying is transitive: if $\tau_1 \prec \tau_2$ and $\tau_2 \prec \tau_3$, then
$\tau_1 \prec \tau_3$.
We also write $\tau_1 \prec_\sigma \tau_2$ to indicate that $\sigma$ is the support map for $\tau_1 \prec \tau_2$. A carrying map $\sigma$ for $\tau_1 \prec \tau_2$  is called \textit{\textbf{combinatorial}} if for every switch $v$ in $\tau_1$, $\sigma(v)$ is a switch in $\tau_2$.
    \par
    Given $\tau_1$ with branches $\{b^1_j\}_{j=1\ldots n}$ and $\tau_2$ with branches $\{b^2_i\}_{i=1\ldots m}$ such that $\tau_1$ is carried by $\tau_2$ with combinatorial support map $\sigma$,
    the \textit{\textbf{transition matrix}} is the $m \times n$ matrix $T = [t_{i,j}]$ where
    $t_{i,j}$ is the number of times the image of $b^1_j$ under $\sigma$ passes over $b^2_i$.
\par
    A \textit{\textbf{train path}} is a $C^1$ immersion $p\colon [0,1] \to \tau$ so that the endpoints $\{0,1\} = \partial[0,1]$ are mapped to switches. We will sometimes conflate $p$ with its image in $\tau$. For $t \in (0,1)$ such that $p(t)$ is a switch, the sub-paths $p_0 = p_{|[0,t]}\colon [0,t] \to \tau$, $p_1 = p_{|[t,1]}\colon [t,1] \to \tau$ are also train paths (after reparametrizing). Similarly, if $p_0$ and $p_1$ are train paths such that $p_0(1)=p_1(0)$ and the differentials $d_1 p_0(\partial_t)$ and $d_0 p_1(\partial_t)$ are either both positive or both negative. The concatenation $(p_0 p_1)\colon [0,2] \to \tau$ is a train path after reparametrizing, hence any train path can be written as $(p_0\ldots p_{n-1})$, where $p_i\colon\left[\frac{i}{n},\frac{i+1}{n}\right] \to \tau$ has image consisting of a single branch. If $p$ is injective we say the train path is \textit{\textbf{embedded}}.
\par
    A \textit{\textbf{fold}} is a particular combinatorial support map $\phi$ for $\tau_1 \prec \tau_2$ which induces a bijection between the branches of $\tau_1$ and $\tau_2$
    except on a particular connected subgraph where it is defined according to either of the two pictures in \Cref{fig:fold}. The three branches of $\tau_1$ which are mapped over the small branch $e'$ in the domain of \Cref{fig:fold} are said to \textit{\textbf{fold to}} $e'$.
    If there is a $\tau_1 \prec_\phi \tau_2$
    we say $\tau_1$ \textit{\textbf{folds to}} $\tau_2$, denoted $\tau_1 \leftharpoonup \tau_2$. If $\tau_1$ folds to $\tau_2$, $\tau_1$ is carried by $\tau_2$. If $\tau_1 \prec \tau_2$ with support map given by a sequence of folds we also say $\tau_1$ folds to $\tau_2$.
    \begin{figure}[h]
    \centering
    \includegraphics[width=0.66\textwidth]{foldgood.pdf}
    \caption{The two possible pictures of a fold. The fold on the left is a left fold and the one on the right is a right fold. Colors indicate the image of each branch on the top after folding.}
    \label{fig:fold}
\end{figure}
    \par
    A train track $\tau$ is an \textit{\textbf{invariant}} track for a map $f: S \to S$ with support map $\sigma$ if $f(\tau)$ is carried by $\tau$ with support map $\sigma$.
    If the transition matrix for $f(\tau) \prec_\sigma \tau$ is irreducible, we also say $\tau$ is \textit{\textbf{irreducible}}.
By smoothly isotoping $\tau$ if necessary, we may assume it is disjoint from the singularities of $\Lambda^u$. This will be convenient later when we remove the singularities of $\Lambda^u$. The following theorem due to Agol is the starting point of the construction of the veering triangulation of the mapping torus of a pseudo-Anosov $f$:

\begin{thm}[\protect{\cite[Theorem 3.5]{A}}]\label{ttfs} Given $f\colon S \to S$ pseudo-Anosov, there exists a generic invariant train track $\tau$ for $f$ such that $f(\tau)$ folds to $\tau$, i.e. there exist a finite sequence of train tracks
$$f(\tau) = \tau_0 \leftharpoonup \tau_1 \leftharpoonup \ldots \leftharpoonup \tau_n = \tau,$$
where $\tau_i$ is carried by $\tau_{i+1}$ with support map given by a single fold.
\end{thm}

\label{intro}
Given a pseudo-Anosov homeomorphism $f$, an invariant train track provides a combinatorial way of recording how curves and other objects behave under iteration of $f$.
The existence of invariant train tracks with irreducible transition matrix
allows the application of Perron-Frobenius theory to the study of the properties of these maps.
For example, when an invariant train track is irreducible it can be used to build a Markov partition for the map.
The existence of irreducible invariant train tracks is often taken for granted
(See e.g. \cite{FR}, \cite{F}, \cite{J}, \cite{EF}, \cite{M}, \cite{Pe})
but to our knowledge no complete proof is currently in the literature. See \Cref{discuss} for more details.

\par The veering triangulation associated to $f$ provides a unified way to study both the topology of its mapping torus and the dynamics of its invariant train tracks.
One important datum attached to a veering triangulation is its \textit{flow graph}, which encodes a Markov partition of a pseudo-Anosov map or flow. Flow graphs have been used by Landry--Minsky--Taylor used to count periodic points of pseudo-Anosov maps, and to define polynomial invariants of pseudo-Anosov flows without perfect fits \cite{LMT1}.
When the flow graph associated to the veering triangulation of a pseudo-Anosov mapping torus is strongly connected, the triangulation can be used to obtain invariant train tracks with
irreducible transition matrices.
\par For general veering triangulations, Agol--Tsang have characterized the strongly connected components of the flow graph, which consist of one ``reduced component" from which there are paths to every vertex, and some number of ``infinitesimal cycles" which do not have any paths back to the rest of the graph \cite{AT}.
By studying the interaction between these infinitesimal components and the invariant train tracks arising from the triangulation, we are able to produce new train tracks where the obstructions to irreducibility coming from the infinitesimal components have been bypassed. This allows us to prove the following:

\par In slightly more detail,
given a boundaryless surface with negative Euler characteristic
and a train track folding sequence (equivalently, splitting sequence) for a pseudo-Anosov $f\colon S \to S$, one can construct the veering triangulation of the mapping torus of $f$, whose
flow graph encodes the transition matrix for this train track. Using the work of \cite{AT}, we show that the only obstructions to irreducibility are branches which are dual to infinitesimal cycles of the flow graph and characterize the structure of veering tracks near these branches. This local picture allows us to modify the track by contracting the obstructing branches, and recover a support map for the modified train track with irreducible transition matrix.

\begin{qn}
 For all $f\colon S \to S$ pseudo-Anosov, does there exist a \textit{generic} invariant train track for $f$ with irreducible transition matrix?
\end{qn}

\par Given two train tracks $\tau_1$ and $\tau_2$ on $S$, $\tau_2$ is said to \textit{\textbf{carry}} $\tau_1$ (or $\tau_1$ \textit{\textbf{is carried by}} $\tau_2$), denoted $\tau_1 \prec \tau_2$, if there is a $C^1$ map $\sigma\colon S \to S$ called the \textit{\textbf{support map}}
such that
 \begin{enumerate}
 \item $\sigma$ is homotopic to the identity,
 \item $\sigma(\tau_1) \subseteq \tau_2$,
 \item for all points $x$ in $\tau_1$,
 $d_x \sigma_{|\tau_1}\colon T_x \tau_1 \to T_{h(x)}\tau_2$ is an isomorphism of tangent spaces,
 \end{enumerate}
Note that carrying is transitive: if $\tau_1 \prec \tau_2$ and $\tau_2 \prec \tau_3$, then
$\tau_1 \prec \tau_3$.
We also write $\tau_1 \prec_\sigma \tau_2$ to indicate that $\sigma$ is the support map for $\tau_1 \prec \tau_2$. A carrying map $\sigma$ for $\tau_1 \prec \tau_2$ is called \textit{\textbf{combinatorial}} if for every switch $v$ in $\tau_1$, $\sigma(v)$ is a switch in $\tau_2$.
 \par
 Given $\tau_1$ with branches $\{b^1_j\}_{j=1\ldots n}$ and $\tau_2$ with branches $\{b^2_i\}_{i=1\ldots m}$ such that $\tau_1$ is carried by $\tau_2$ with combinatorial support map $\sigma$,
 the \textit{\textbf{transition matrix}} is the $m \times n$ matrix $T = [t_{i,j}]$ where
 $t_{i,j}$ is the number of times the image of $b^1_j$ under $\sigma$ passes over $b^2_i$.
\par
 A \textit{\textbf{train path}} is a $C^1$ immersion $p\colon [0,1] \to \tau$ so that the endpoints $\{0,1\} = \partial[0,1]$ are mapped to switches. We will sometimes conflate $p$ with its image in $\tau$. For $t \in (0,1)$ such that $p(t)$ is a switch, the sub-paths $p_0 = p_{|[0,t]}\colon [0,t] \to \tau$, $p_1 = p_{|[t,1]}\colon [t,1] \to \tau$ are also train paths (after reparametrizing). Similarly, if $p_0$ and $p_1$ are train paths such that $p_0(1)=p_1(0)$ and the differentials $d_1 p_0(\partial_t)$ and $d_0 p_1(\partial_t)$ are either both positive or both negative. The concatenation $(p_0 p_1)\colon [0,2] \to \tau$ is a train path after reparametrizing, hence any train path can be written as $(p_0\ldots p_{n-1})$, where $p_i\colon\left[\frac{i}{n},\frac{i+1}{n}\right] \to \tau$ has image consisting of a single branch. If $p$ is injective we say the train path is \textit{\textbf{embedded}}.
\par
 A \textit{\textbf{fold}} is a particular combinatorial support map $\phi$ for $\tau_1 \prec \tau_2$ which induces a bijection between the branches of $\tau_1$ and $\tau_2$
 except on a particular connected subgraph where it is defined according to either of the two pictures in \Cref{fig:fold}. The three branches of $\tau_1$ which are mapped over the small branch $e'$ in the domain of \Cref{fig:fold} are said to \textit{\textbf{fold to}} $e'$.
 If there is a $\tau_1 \prec_\phi \tau_2$
 we say $\tau_1$ \textit{\textbf{folds to}} $\tau_2$, denoted $\tau_1 \leftharpoonup \tau_2$. If $\tau_1$ folds to $\tau_2$, $\tau_1$ is carried by $\tau_2$. If $\tau_1 \prec \tau_2$ with support map given by a sequence of folds we also say $\tau_1$ folds to $\tau_2$.
 \begin{figure}[h]
 \centering
 \includegraphics[width=0.66\textwidth]{foldgood.pdf}
 \caption{The two possible pictures of a fold. The fold on the left is a left fold and the one on the right is a right fold. Colors indicate the image of each branch on the top after folding.}
 \label{fig:fold}
\end{figure}
 \par
 A train track $\tau$ is an \textit{\textbf{invariant}} track for a map $f: S \to S$ with support map $\sigma$ if $f(\tau)$ is carried by $\tau$ with support map $\sigma$.
 If the transition matrix for $f(\tau) \prec_\sigma \tau$ is irreducible, we also say $\tau$ is \textit{\textbf{irreducible}}.
By smoothly isotoping $\tau$ if necessary, we may assume it is disjoint from the singularities of $\Lambda^u$. This will be convenient later when we remove the singularities of $\Lambda^u$. The following theorem due to Agol is the starting point of the construction of the veering triangulation of the mapping torus of a pseudo-Anosov $f$:

\par Note that the referenced theorem is stated in terms of \textit{splits} rather than folds, which are the combinatorial inverse of folds.
 Additionally, the referenced theorem is stronger than what we state here because the splitting sequence is shown to be canonical up to ``commuting maximal splits": at each step in the sequence, the train track is split at a branch carrying maximal measure of the invariant lamination, and the splits of maximal branches commute if there is not a unique branch carrying maximal measure.
 This theorem motivates the following definition:
\begin{defn}
 An invariant train track $\tau$ for a pseudo-Anosov map $f$ is called \textit{\textbf{veering}} if it
 is generic and $\tau$ can be obtained from $f(\tau)$ by a sequence of folds, in which case there is a sequence of train tracks
 $$f(\tau) = \tau_0 \leftharpoonup \tau_1 \leftharpoonup \ldots \leftharpoonup \tau_n = \tau$$
 where $\tau_i$ is carried by $\tau_{i+1}$ with support map consisting of a single fold.
\end{defn}
\par In \Cref{clvts} we will see that a veering train track as defined here is sufficient to produce the veering triangulation of the fully punctured mapping torus of $f$. While veering train tracks are not necessarily irreducible, they will be the beginning of our construction of irreducible tracks.
\par
 Given two train tracks $\tau_1$, $\tau_2$ on $S$, a (surface) \textit{\textbf{train track map}} is a map $t: S \to S$ so that $t(\tau_1)$ is contained in $\tau_2$, and the restriction of $t$ to $\tau_1$ is
 such that for any train path $p\colon [0,1] \to \tau_1$, $t \circ p\colon [0,1] \to \tau_2$ is also a train path. In other words, a train track map is like a combinatorial support map except that it does not have to be homotopic to the identity map.
 \par The following lemma will be useful for producing support maps:
\begin{lemma}\label{supplemma}
 Let $t\colon S \to S$ be a train track map taking $\tau_1$ to $\tau_2$, and $g\colon S \to S$ be a diffeomorphism.
 If $t$ is homotopic to $g$, then $g(\tau_1) \prec \tau_2$ with support map given by $t \circ g^{-1}$.
\end{lemma}
\begin{proof}
 Since $t$ and $g$ are homotopic, $t \circ g^{-1}$ is homotopic to the identity. Since $t(\tau_1) \subseteq \tau_2$, $(t \circ g^{-1})(g(\tau_1)) = t(\tau_1) \subseteq \tau_2$. Let $p \in g(\tau_1)$. Since $g$ is a diffeomorphism, the restriction of the differential
 $$d_p g^{-1}_{|g(\tau_1)}\colon T_p g(\tau_1) \to T_{g^{-1}(p)} \tau_1$$
 is an isomorphism. Since $t$ is a train track map, $d_{g^{-1}(p)} t_{| \tau_1}$ is an isomorphism, so the composition
 $$d_p (t \circ g^{-1})_{|g(\tau_1)} = (d_{g^{-1}(p)} t_{| \tau_1}) \circ (d_p g^{-1}_{|g(\tau_1)})\colon T_p g(\tau_1) \to T_{(t \circ g^{-1})(p)} \tau_2$$
 is also an isomorphism.
\end{proof}

Given a boundaryless surface with negative Euler characteristic and a train track folding (splitting) sequence for a pseudo-Anosov $f\colon S\to S$, one constructs the veering triangulation of the mapping torus of $f$, whose flow graph encodes the transition matrix for this train track. Using the work of \cite{AT}, one shows that the only obstructions to irreducibility are branches dual to infinitesimal cycles of the flow graph, and characterizes the local structure of veering tracks near these branches. Using this local picture, one modifies the track by contracting the obstructing branches, and recovers a support map for the modified train track with irreducible transition matrix.

\label{mainthm}
    Every pseudo-Anosov homeomorphism has an invariant train track whose transition matrix is irreducible.,

Question: Let \(f\colon S\to S\) be a pseudo-Anosov homeomorphism. A train track \(\tau\) on \(S\) is called invariant for \(f\) if \(f(\tau)\) is carried by \(\tau\) via a combinatorial support map. If the branches of \(f(\tau)\) are \(\{b^1_j\}\) and the branches of \(\tau\) are \(\{b^2_i\}\), the associated transition matrix is the nonnegative matrix \(T=[t_{i,j}]\), where \(t_{i,j}\) is the number of times the image of \(b^1_j\) under the support map passes over \(b^2_i\). A nonnegative matrix is irreducible if for every \(i,j\) there exists \(n\ge 1\) such that \((T^n)_{i,j}>0\). Under these assumptions, which existence statement holds?

For every pseudo-Anosov homeomorphism \(f\colon S\to S\), there exists an invariant train track \(\tau\) for \(f\) such that the transition matrix associated to \(f(\tau)\prec \tau\) is irreducible.

Choices:
B. For every pseudo-Anosov homeomorphism \(f\colon S\to S\), every invariant train track \(\tau\) for \(f\) has the property that the transition matrix associated to \(f(\tau)\prec \tau\) is irreducible.
C. For every pseudo-Anosov homeomorphism \(f\colon S\to S\), there exists an invariant train track \(\tau\) for \(f\) such that the transition matrix associated to \(f(\tau)\prec \tau\) is nonzero.
D. For every pseudo-Anosov homeomorphism \(f\colon S\to S\), there exists an invariant train track \(\tau\) for \(f\) such that for every pair of branches \(b^2_i,b^2_j\) of \(\tau\), one has \(t_{i,j}>0\) in the transition matrix associated to \(f(\tau)\prec \tau\).
E. For every pseudo-Anosov homeomorphism \(f\colon S\to S\), there exists an invariant train track \(\tau\) for \(f\) such that each strongly connected component of the transition matrix associated to \(f(\tau)\prec \tau\) is irreducible.

existential choice of modified track after contracting obstructing branches

irreducibility requirement dropped to mere existence of a nontrivial transition matrix

irreducible replaced by strictly positive one-step transition matrix

infinitesimal-cycle obstruction bypassed only after contracting branches, not for the original component decomposition

The stem gives definitions of invariant train tracks and irreducibility, but it does not explicitly state the target existence theorem or uniquely point to choice A. The correct answer is not leaked, though the setup clearly narrows the topic.

The item is close to a theorem-recall question: after introducing the relevant definitions, it asks which existence statement holds. It is not a verbatim restatement, because the alternatives vary in quantifier strength and matrix properties, but it still largely tests recognition of a standard result.

There is some reasoning pressure in separating 'there exists' from 'for every,' and irreducible from merely nonzero or strictly positive. However, for a student who knows the theorem, the answer is mainly direct identification rather than substantial derivation.

The distractors are mathematically plausible and target natural failure modes: overstrengthening to every invariant track (B), weakening to nonzero (C), confusing irreducibility with positivity in one step (D), and using a subtle component-wise condition (E). They are distinct and well aligned with common misunderstandings.

A solid MCQ with strong distractors and little answer leakage, but it leans toward theorem recognition rather than deep generative reasoning.


\label{thm:main}
Let $(\theta_n)$ be i.i.d. random variables on $[0,2\pi)$ satisfying \eqref{eqn:Nunidimensional}, and $(S_n, n \geq 1)$ the ERW defined by \eqref{eqn:defERWC}.
\begin{itemize}
  \item If $\Re(\Phi_1) < 1/2$, then writing $\sigma^2 = \frac{1}{1 - 2 \Re(\Phi_1)} \in (1/3,\infty)$, we have
  \begin{equation}
   \label{eqn:mainSub}
    \lim_{n \to \infty} \frac{S_n}{n^{1/2}} = \mathcal{N}_\mathbb{C}(0,\sigma^2) \quad \text{ in law}.
  \end{equation}
  \item If $\Re(\Phi_1)= 1/2$, then
  \begin{equation}
    \label{eqn:mainCrit}
    \lim_{n \to \infty} \frac{S_n}{(n \log n)^{1/2}} = \mathcal{N}_\mathbb{C}(0, 1)\quad \text{ in law}.
  \end{equation}
  \item If $\Re(\Phi_1)> 1/2$, there exists a $\mathbb{C}$-valued random variable $W$ such that, writing $\sigma^2 =\frac{1}{2 \Re(\Phi_1) - 1} \in(1,\infty)$, we have
  \begin{equation}
    \label{eqn:mainSup}
    \lim_{n \to \infty} S_n\mathrm{e}^{- \Phi_1 \log n} = W \quad \text{a.s. \quad and} \quad \lim_{n \to \infty} \frac{S_n  - \mathrm{e}^{\Phi_1 \log n} W}{n^{1/2}} = \mathcal{N}_\mathbb{C}(0,\sigma^2) \quad \text{ in law}.
  \end{equation}
\end{itemize}

\begin{theorem}
\label{thm:main}
Let $(\theta_n)$ be i.i.d. random variables on $[0,2\pi)$ satisfying \eqref{eqn:Nunidimensional}, and $(S_n, n \geq 1)$ the ERW defined by \eqref{eqn:defERWC}.
\begin{itemize}
  \item If $\Re(\Phi_1) < 1/2$, then writing $\sigma^2 = \frac{1}{1 - 2 \Re(\Phi_1)} \in (1/3,\infty)$, we have
  \begin{equation}
   \label{eqn:mainSub}
    \lim_{n \to \infty} \frac{S_n}{n^{1/2}} = \mathcal{N}_\C(0,\sigma^2) \quad \text{ in law}.
  \end{equation}
  \item If $\Re(\Phi_1)= 1/2$, then
  \begin{equation}
    \label{eqn:mainCrit}
    \lim_{n \to \infty} \frac{S_n}{(n \log n)^{1/2}} = \mathcal{N}_\C(0, 1)\quad \text{ in law}.
  \end{equation}
  \item If $\Re(\Phi_1)> 1/2$, there exists a $\C$-valued random variable $W$ such that, writing $\sigma^2 =\frac{1}{2 \Re(\Phi_1) - 1} \in(1,\infty)$, we have
  \begin{equation}
    \label{eqn:mainSup}
    \lim_{n \to \infty} S_n\erm^{- \Phi_1 \log n} = W \quad \text{a.s. \quad and} \quad \lim_{n \to \infty} \frac{S_n  - \erm^{\Phi_1 \log n} W}{n^{1/2}} = \mathcal{N}_\C(0,\sigma^2) \quad \text{ in law}.
  \end{equation}
\end{itemize}
\end{theorem}

The elephant random walk (ERW) is a well-studied reinforcement process, introduced by Schütz and Trimper \cite{ScT} to investigate the influence of memory on random walk dynamics. In $\mathbb{Z}$, the process is constructed as follows. Given a memory parameter $p \in [0,1]$, the process starts from initial position $S_0 = 0$, then moves to $S_1 = 1$ at the first step\footnote{Alternatively, this step could be randomized, but by construction, flipping the initial path as the same effect as flipping the whole trajectory of the walk along the $x$-axis.}. At each subsequent time $n \geq 2$, the walker chooses uniformly at random one of its $n-1$ past steps. It then repeats this step with probability $p$, or with complementary probability $1-p$ moves one step in the opposite direction. In other words, the ERW is defined by induction by
\begin{equation}
  \label{eqn:defERW1}
  S_0 = 0, \quad S_1 = 1, \quad S_{n} = S_{n-1} + R_{n} (S_{I_n} - S_{I_n - 1}) \text{ for $n \geq 2$,}
\end{equation}
where $(R_n, n \geq 2)$ are i.i.d. Radamacher random variables with parameter $p$ that are
further independent from the sequence $(I_n, n \geq 2)$ of independent random variables with $I_k$ uniformly distributed on $\{1,\ldots,k-1\}$.

Provided that $p>1/2$, this dynamic can be rephrased as follows. At each step, the walker remembers, with probability $2p-1$, one of its past steps chosen uniformly at random, and repeats this step. With complementary probability $2(1-p)$, it performs an independent step equal to $1$ or $-1$. An analogous definition can be taken when $p<1/2$ considering $1-2p$ as the probability of performing the opposite remembered step. The parameter $p$ is usually referred to as the memory parameter of the ERW, but we see from the previous rephrasing the effective memory is more accurately described by $a := 2p-1$. The value $a=0$ (i.e. $p=1/2$) corresponds to the absence of memory, while $|a|$ measures the strength of the dependence on past increments. When $a>0$, the walker tends to repeat the chosen past step (“positive” memory), whereas for $a<0$ it tends to do the opposite (“negative” memory). In both cases, the memory is strong when $|a|$ is close to $1$.

From this perspective, the walk appears to exhibit long-range dependence, as the walk tends for example to reinforce previous directions when $a>0$ -- the ERW is a special case of \textit{step-reinforced random walk}. Consequently, the walker’s trajectory can deviate significantly from a simple random walk, leading to superdiffusive behavior when $a>1/2$ while for $a\in(-1,1/2)$ the process remains diffusive but with different variance compared to the classical case.

A natural generalisation for this process would be to define a version of the elephant random walk in dimension $d$. A $\mathbb{Z}^d$ version of this model was first introduced in \cite{BercuLaulin2019}, while the works \cite{CuL,Qin} further investigate its properties, notably recurrence and transience.
The process considered was the following: at each time $n$, after selecting uniformly at random a previous step, the walker either moves according to this step with probability $p$, or with probability $1-p$ moves towards one of the $2d-1$ other neighbour of $S_n$ chosen uniformly at random. It can be shown in this situation that the ERW undergoes a phase transition when the memory parameter crosses $p_c = \frac{2d+1}{4d}$. If $p \leq p_c$, the ERW is in the diffusive regime, while it drifts at some sub-linear rate if $p > p_c$. This behaviour is therefore very close to the one observed in dimension $1$.

The objective of this note is to show that a larger variety of behaviours may be observed in dimension greater than $1$, by modifying the above described evolution of the process: instead of moving to a uniformly chosen neighbour of $S_n$ with probability $1-p$, we consider a process in which, at each steps, the walker moves according to a random rotation of the previously chosen step. More precisely, let $(R_n, n \geq 2)$ be i.i.d. random rotations of $\mathbb{R}^d$ (i.e. random variables in $\mathcal{O}(\mathbb{R}^d)$) independent of the sequence $(I_n, n \geq 2)$ defined above, we then define the ERW in $\mathbb{R}^d$ as
\begin{equation}
  \label{eqn:defERWd}
    S_0 = (0,\ldots, 0), \quad S_1 = (1,0,\ldots, 0) \quad \text{and} \quad S_{n} = S_{n-1} + R_n(S_{I_{n}} - S_{I_{n}-1}).
\end{equation}
This model clearly generalizes the one considered above, since we no longer restrict the law of $R_n$ to satisfy $\mathbb{P}(R_n = \mathrm{Id}) = p$ and $\mathcal{L}\big(R_n | R_n \neq \mathrm{Id}\big)$ uniformly selected among the non-identity rotations that preserve the canonical base of $\mathbb{R}^d$.

In the present article, we restrict our attention to elephant random walks in the plane. For our purpose, it will be convenient to identify $\mathbb{R}^2$ with the complex plane $\mathbb{C}$ in the usual fashion, and we define the ERW in $\mathbb{C}$ as the following complex-valued stochastic process. Let $(\theta_n, n \geq 1)$ be i.i.d. $[0,2\pi)$-valued random variables, further independent from the sequence $(I_n, n \geq 2)$ defined above, we set
\begin{equation}
  \label{eqn:defERWC}
  S_0 = 0, \quad S_1 = 1 \quad \text{and}\quad S_n= S_{n-1} + (S_{I_n} - S_{I_n-1}) \mathrm{e}^{i \theta_n}.
\end{equation}
Some sample trajectories of $S$ are represented in Figure~\ref{fig:ex} when $\theta_1$ is a.s. a constant.

\begin{figure}[ht]
\centering
  \begin{subfigure}[t]{.3\linewidth}
    \centering\includegraphics[width=.95\linewidth]{erw2d1}
    \caption{$\theta_1= \frac{\pi}{3} - 0.1$ a.s.}
  \end{subfigure}
  \begin{subfigure}[t]{.3\linewidth}
    \centering\includegraphics[width=.95\linewidth]{erw2d3}
    \caption{$\theta_1 = \frac{\pi}{3}$ a.s.}
  \end{subfigure}
  \begin{subfigure}[t]{.3\linewidth}
    \centering\includegraphics[width=.95\linewidth]{erw2d5}
    \caption{$\theta_1 = \frac{\pi}{3} + 0.1$ a.s.}
  \end{subfigure}
\caption{Sample path of $1000$ steps of an elephant random walks in $\mathbb{R}^2$ as defined in \eqref{eqn:defERWC}, in which $\theta_1$ is a.s. a constant.}
\label{fig:ex}
\end{figure}

The main result of this article is a central limit theorem for the the ERW $(S_n, n \geq 1)$. We show that the asymptotic behaviour of $S_n$ depends mainly of
\begin{equation}
  \Phi_k := \mathbb{E}(\mathrm{e}^{i k \theta_1}), \quad \text{for $k \geq 1$},
\end{equation}
the Fourier coefficient of the law of $\theta_1$ and, more specifically,  of $\Phi_1$, that plays a role analogue to the memory parameter $a$ in dimension $1$. Indeed, when $d=1$ and $\theta_1$ takes values in $\{0,\pi\}$ with $\mathbb{P}(\theta_1=0)=p$, one recovers $\Phi_1=2p-1=a$, the usual memory parameter. For $d>1$, $\Phi_1$ describes the average rotation applied to the remembered steps, therefore $|\Phi_1| \in [0,1]$ may roughly be thought of as the strength of that memory, minimal for the random walk $(\Phi_1=0)$, and maximal in the situation represented in Figure~\ref{fig:ex} in which $\Phi_1 \in \mathbb{S}^1$.

To avoid considering a unidimensional system, we always work in this article under the assumption
\begin{equation}
  \label{eqn:Nunidimensional}
  \mathbb{P}(\theta \in \{0,\pi\}) < 1, \quad \text{or equivalently}\quad \Re(\Phi_2) < 1.
\end{equation}
For $\sigma^2 > 0$, we denote by $\mathcal{N}_\mathbb{C}(0,\sigma^2)$ the centred complex normal distribution, such that $N$ has law $\mathcal{N}_\mathbb{C}(0,\sigma^2)$ if $\Re(N)$ and $\Im(N)$, its real and imaginary parts, are independent real-valued centred normal random variables with variance $\sigma^2/2$.

The main result of this article is that the ERW in $\mathbb{C}$ exhibits a superdiffusive behaviour if $\Re(\Phi_1)>1/2$, while it remains diffusive when $\Re(\Phi_1)<1/2$. In addition, we show that when superdiffusive, the path of the ERW scales towards a randomly rotated logarithmic spiral.

The elephant random walk (ERW) is a well-studied reinforcement process, introduced by Schütz and Trimper \cite{ScT} to investigate the influence of memory on random walk dynamics. In $\mathbb{Z}$, the process is constructed as follows. Given a memory parameter $p \in [0,1]$, the process starts from initial position $S_0 = 0$, then moves to $S_1 = 1$ at the first step\footnote{Alternatively, this step could be randomized, but by construction, flipping the initial path as the same effect as flipping the whole trajectory of the walk along the $x$-axis.}. At each subsequent time $n \geq 2$, the walker chooses uniformly at random one of its $n-1$ past steps. It then repeats this step with probability $p$, or with complementary probability $1-p$ moves one step in the opposite direction. In other words, the ERW is defined by induction by
\begin{equation}
 \label{eqn:defERW1}
 S_0 = 0, \quad S_1 = 1, \quad S_{n} = S_{n-1} + R_{n} (S_{I_n} - S_{I_n - 1}) \text{ for $n \geq 2$,}
\end{equation}
where $(R_n, n \geq 2)$ are i.i.d. Radamacher random variables with parameter $p$ that are
further independent from the sequence $(I_n, n \geq 2)$ of independent random variables with $I_k$ uniformly distributed on $\{1,\ldots,k-1\}$.

The objective of this note is to show that a larger variety of behaviours may be observed in dimension greater than $1$, by modifying the above described evolution of the process: instead of moving to a uniformly chosen neighbour of $S_n$ with probability $1-p$, we consider a process in which, at each steps, the walker moves according to a random rotation of the previously chosen step. More precisely, let $(R_n, n \geq 2)$ be i.i.d. random rotations of $\mathbb{R}^d$ (i.e. random variables in $\mathcal{O}(\mathbb{R}^d)$) independent of the sequence $(I_n, n \geq 2)$ defined above, we then define the ERW in $\mathbb{R}^d$ as
\begin{equation}
 \label{eqn:defERWd}
 S_0 = (0,\ldots, 0), \quad S_1 = (1,0,\ldots, 0) \quad \text{and} \quad S_{n} = S_{n-1} + R_n(S_{I_{n}} - S_{I_{n}-1}).
\end{equation}
This model clearly generalizes the one considered above, since we no longer restrict the law of $R_n$ to satisfy $\mathbb{P}(R_n = \mathrm{Id}) = p$ and $\mathcal{L}\big(R_n | R_n \neq \mathrm{Id}\big)$ uniformly selected among the non-identity rotations that preserve the canonical base of $\mathbb{R}^d$.

In the present article, we restrict our attention to elephant random walks in the plane. For our purpose, it will be convenient to identify $\mathbb{R}^2$ with the complex plane $\mathbb{C}$ in the usual fashion, and we define the ERW in $\mathbb{C}$ as the following complex-valued stochastic process. Let $(\theta_n, n \geq 1)$ be i.i.d. $[0,2\pi)$-valued random variables, further independent from the sequence $(I_n, n \geq 2)$ defined above, we set
\begin{equation}
 \label{eqn:defERWC}
 S_0 = 0, \quad S_1 = 1 \quad \text{and}\quad S_n= S_{n-1} + (S_{I_n} - S_{I_n-1}) \mathrm{e}^{i \theta_n}.
\end{equation}
Some sample trajectories of $S$ are represented in Figure~\ref{fig:ex} when $\theta_1$ is a.s. a constant.

The main result of this article is a central limit theorem for the the ERW $(S_n, n \geq 1)$. We show that the asymptotic behaviour of $S_n$ depends mainly of
\begin{equation}
 \Phi_k := \mathbb{E}(\mathrm{e}^{i k \theta_1}), \quad \text{for $k \geq 1$},
\end{equation}
the Fourier coefficient of the law of $\theta_1$ and, more specifically, of $\Phi_1$, that plays a role analogue to the memory parameter $a$ in dimension $1$. Indeed, when $d=1$ and $\theta_1$ takes values in $\{0,\pi\}$ with $\mathbb{P}(\theta_1=0)=p$, one recovers $\Phi_1=2p-1=a$, the usual memory parameter. For $d>1$, $\Phi_1$ describes the average rotation applied to the remembered steps, therefore $|\Phi_1| \in [0,1]$ may roughly be thought of as the strength of that memory, minimal for the random walk $(\Phi_1=0)$, and maximal in the situation represented in Figure~\ref{fig:ex} in which $\Phi_1 \in \mathbb{S}^1$.

To avoid considering a unidimensional system, we always work in this article under the assumption
\begin{equation}
 \label{eqn:Nunidimensional}
 \mathbb{P}(\theta \in \{0,\pi\}) < 1, \quad \text{or equivalently}\quad \Re(\Phi_2) < 1.
\end{equation}
For $\sigma^2 > 0$, we denote by $\mathcal{N}_\mathbb{C}(0,\sigma^2)$ the centred complex normal distribution, such that $N$ has law $\mathcal{N}_\mathbb{C}(0,\sigma^2)$ if $\Re(N)$ and $\Im(N)$, its real and imaginary parts, are independent real-valued centred normal random variables with variance $\sigma^2/2$.

The main result of this article is that the ERW in $\mathbb{C}$ exhibits a superdiffusive behaviour if $\Re(\Phi_1)>1/2$, while it remains diffusive when $\Re(\Phi_1)<1/2$. In addition, we show that when superdiffusive, the path of the ERW scales towards a randomly rotated logarithmic spiral.

\begin{equation}
  \label{eqn:Nunidimensional}
  \IP(\theta \in \{0,\pi\}) < 1, \quad \text{or equivalently}\quad \Re(\Phi_2) < 1.
\end{equation}

\begin{equation}
  \label{eqn:defERWC}
  S_0 = 0, \quad S_1 = 1 \quad \text{and}\quad S_n= S_{n-1} + (S_{I_n} - S_{I_n-1}) \erm^{i \theta_n}.
\end{equation}

\label{fig:ex}
\end{figure}

The main result of this article is a central limit theorem for the the ERW $(S_n, n \geq 1)$. We show that the asymptotic behaviour of $S_n$ depends mainly of
\begin{equat

To avoid considering a unidimensional system, we always work in this article under the assumption
\begin{equation}
 \label{eqn:Nunidimensional}
 \IP(\theta \in \{0,\pi\}) < 1, \quad \text{or equivalently}\quad \Re(\Phi_2) < 1.
\end{equation}
For $\sigma^2 > 0$, we denote by $\mathcal{N}_\C(0,\sigma^2)$ the centred complex normal distribution, such that $N$ has law $\mathcal{N}_\C(0,\sigma^2)$ if $\Re(N)$ and $\Im(N)$, its real and imaginary parts, are independent real-valued centred normal random variables with variance $\sigma^2/2$.

The main result of this article is that the ERW in $\C$ exhibits a superdiffusive behaviour if $\Re(\Phi_1)>1/2$, while it remains diffusive when $\Re(\Phi_1)<1/2$. In addition, we show that when superdiffusive, the path of the ERW scales towards a randomly rotated logarithmic spiral.

In order to provide variety to this article, we propose to prove Theorem~\ref{thm:main} in two different ways. We study the regime $\Re(\Phi_1) \leq 1/2$ in Section~\ref{sec:diffusive} using Lindeberg's central limit theorem. The key observation in this section is that
\begin{equation}
 \label{eqn:defMartingale}
 \left( \frac{S_n}{\prod_{j=1}^n \left( 1 + \frac{\Phi_1}{j} \right)}, n\geq 1 \right) \text{ is a martingale,}
\end{equation}
its asymptotic behaviour can be deduced from studying its quadratic variation. Up to the difficulty of dealing with a complex martingale, the argument there follows closely standard methods in dimension 1 or $d$ \cite{BaB,BercuLaulin2019}.

In this model, remark that $\Phi_1= p - r + i (q-s)$, therefore Theorem~\ref{thm:main} implies the following behaviour for $(S_n, n \geq 1)$, depending on the sign of $p-r-1/2$.
\begin{corollary}
Let $(S_n, n \geq 1) $ be the ERW in $\Z^2$ with parameters $p,q,r,s$ defined above.
\begin{itemize}
 \item If $p < r+1/2$, then $\lim_{n \to \infty} \frac{S_n}{n^{1/2}} = \mathcal{N}(0, \frac{1}{2 - 4 (p-r)} I_2)$ in law.
 \item If $p=r+1/2$, then $\lim_{n \to \infty} \frac{S_n}{(n\log )^{1/2}}= \mathcal{N}(0, \frac{1}{2} I_2)$ in law.
 \item If $p > r+1/2$, then there exists a random vector $W \in \R^2$ such that
 \begin{multline*}
 \lim_{n \to \infty} \frac{\mathcal{R}_{-(q-s)\log n}S_n}{n^{p-r}} = W \quad \text{a.s.} \\ \text{and}\quad \lim_{n \to \infty} \frac{S_n - n^{p-r} \mathcal{R}_{(q-s)\log n}W}{n^{1/2}} = \mathcal{N}(0,\frac{1}{4(p-r)-2}I_2) \quad \text{ in law},
 \end{multline*}
 writing $\mathcal{R}_\theta$ for a rotation of angle $\theta$.
\end{itemize}
\end{corollary}

We denote by $(S_n, n \geq 1)$ an ERW defined by \eqref{eqn:defERWC}, and we assume in this section that
\begin{equation}
 \label{eqn:cdSub}
 \Re(\Phi_1) = \Re\left(\E\left( \erm^{i\theta_1} \right)\right) < \frac{1}{2}.
\end{equation}
We prove in this section that
\[
 \lim_{n \to \infty} \frac{S_n}{n^{1/2}} = \mathcal{N}_\C\left(0,\frac{1}{1- 2 \Re(\Phi_1)}\right) \quad \text{ in distribution.}
\]
The proof of \eqref{eqn:mainCrit} follows a similar path with minor modifications, and will be sketched at the end of the section. Let us mention that \eqref{eqn:mainSup} also could have be obtained using the same techniques as developed here.

We denote by $\Gamma$ Euler's Gamma function defined on $\C \backslash \Z_-$, and observe that we can rewrite
\begin{align*}
 a^{(k)}_n &= \prod_{j=1}^{n-1} \left( 1 + \frac{\Phi_k}{j} \right) = \prod_{j=1}^{n-1}\frac{j+\Phi_k}{j} \frac{\Gamma(j+\Phi_k)}{\Gamma(j)} \frac{\Gamma(j)}{\Gamma(j+\Phi_k)}\\
 &= \prod_{j=1}^{n-1} \frac{\Gamma(j + \Phi_k+ 1)}{\Gamma(j+\Phi_k)} \frac{\Gamma(j)}{\Gamma(j + 1)} = \frac{\Gamma(n+\Phi_k)}{\Gamma(n)\Gamma(\Phi_k)},
\end{align*}
from which we deduce there exists $c_k > 0$ such that
\begin{equation}
 \label{eqn:data}
 a_n^{(k)} \sim C_k n^{\Phi_k}, \text{ and } |a_n^{(k)}|^2 \sim |C_k|^2 n^{2 \Re(\Phi_k)} \text{ as $n \to \infty$.}
\end{equation}
Using that $\Re(\Phi_k) \in (-1,1)$ due to \eqref{eqn:Nunidimensional} we conclude that
\[
 \sum_{j=1}^n \frac{1}{|a_j^{(k)}|^2} \begin{cases}
 = O(1) & \text{ if } \Re(\Phi_k)>1/2\\
 \sim |C_k|^2 \log n & \text{ if } \Re(\Phi_k) = 1/2\\
 \sim \frac{|C_k|^2}{1 - 2 \Re(\Phi_k)} n^{1 - 2 \Re(\Phi_k)} &\text{ if } \Re(\Phi_k) < 1/2
 \end{cases}
 \quad \text{ as $n \to \infty$.}
\]
Solving the recursion for $u$ in all three cases, we conclude the proof.
\end{proof}

\begin{proof}
Let $n \geq 1$, we observe that
\begin{align*}
 \Delta M_k &= M_{k+1} - M_k = \frac{S_n + \Delta S_n}{a_{n+1}} - \frac{S_n}{a_n} = \frac{1}{a_{n+1}}\left( S_n + \Delta S_n - \left(1+ \frac{\Phi_1}{n}\right) S_n \right)\\
 &= \frac{1}{a_{n+1}} \left( \Delta S_n - \frac{\Phi_1}{n}S_n \right).
\end{align*}
Therefore,
\begin{align*}
 \E\left( |\Delta M_k|^2 \middle| \mathcal{F}_k \right)
 &= \frac{1}{|a_{k+1}|^2}\E\left( |\Delta S_k|^2 + \left| \frac{\Phi_1}{k} S_k \right|^2 - 2 \Re\left( \frac{\Phi_1}{k} S_k \overline{\Delta S_k} \right) \middle| \mathcal{F}_k \right)\\
 &= \frac{1}{|a_{k+1}|^2} \left( 1 + \left| \frac{\Phi_1}{k} S_k \right|^2 - 2 \Re\left( \left|\frac{\Phi_1}{k} S_k\right|^2 \right)\right) \text{ a.s.},
\end{align*}
using that $|\Delta S_k| = 1$ and $\E(\Delta S_{k}| \mathcal{F}_k) = \Phi_1\frac{S_k}{k}$ a.s. This yields
\begin{equation*}
 \E\left( |\Delta M_k|^2 \middle| \mathcal{F}_k \right) = \frac{1}{|a_{k+1}|^2}\left( 1 - \left| \frac{\Phi_1}{k} S_k \right|^2\right) \text{ a.s.}
\end{equation*}
As a consequence, we have
\begin{equation}
 \label{eqn:formula}
 \crochet{M,\bar{M}}_n = \sum_{k=1}^{n-1} \frac{1}{|a_{k+1}|^2} - \sum_{k=1}^{n-1} \frac{1}{|a_{k+1}|^2} \left|\Phi_1\frac{S_k}{k}\right|^2 \quad \text{a.s.}
\end{equation}
We now study the asymptotic behaviour of these two sums.
We deduce from \eqref{eqn:data} that
\begin{equation}
 \label{eqn:asymp}
 v_n := \sum_{k=1}^{n-1} \frac{1}{|a_{k+1}|^2} \sim |C_1|^2 \frac{n^{1 - 2 \Re(\Phi_{1})}}{1 - 2 \Re(\Phi_1)},
\end{equation}
and from Lemma~\ref{lem:martingale} that
\[
 \frac{1}{|a_{n+1}|^2 n^2}\E\left( |S_n|^2\right) \sim_{n \to \infty} |C_1|^2 n^{-(1+ 2\Re(\Phi_1))},
\]
as $n \to \infty$. Therefore, there exists $C > 0$ such that for all $n$ large enough, we have
\begin{equation}
 \label{eqn:intermediate}
 \E\left( \sum_{k=1}^{n-1} \frac{1}{|a_{k+1}|^2} \left|\Phi_1\frac{S_k}{k}\right|^2\right) \leq C n^{-2\Re(\Phi_1)} = o(v_n).
\end{equation}
It shows that $\lim_{n \to \infty} \frac{1}{v_n}\sum_{k=1}^{n-1} \frac{1}{|a_{k+1}|^2} \left|\Phi_1\frac{S_k}{k}\right|^2 = 0$ in probability, which completes the proof.
\end{proof}

We now turn to the central limit theorem result, writing
\begin{align*}
 &\phantom{=}\frac{S_n - n^{\Re(\Phi) \erm^{i \Im(\Phi) \log n}} W \exp(-\Phi_1 \log E) }{\sqrt{n}}\\
 &= \frac{Z_1(\tau_n) - W \exp(\Phi_1 \log (N_{\tau_n}/E))}{\sqrt{n}}\\
 &= \erm^{-\tau_n/2}\frac{Z_1(\tau_n) - W \erm^{\Phi_1 \tau_n} + W \erm^{\Phi_1 \tau_n} - W \exp(\Phi_1 \log (N_{\tau_n}/E)}{\sqrt{N_{\tau_n} \erm^{-\tau_n}}}.
\end{align*}
By Corollary~\ref{cor:obvious}, we observe that
\[
 \lim_{t \to \infty} \erm^{-t/2}\frac{Z_1(t) - W \erm^{\Phi_1 t}}{\sqrt{N_t \erm^{-t}}} = \mathcal{N}_\C(0,\frac{1}{2 \Re(\Phi_1) - 1}) \quad \text{in law},
\]
from which we deduce straightforwardly that $\erm^{-\tau_n/2}\frac{Z_1(\tau_n) - W \erm^{\Phi_1 \tau_n} }{\sqrt{N_{\tau_n} \erm^{-\tau_n}}}$ admit the same limit in distribution as $n \to \infty$, using that $\tau_n - \log n$ is tight. In addition, we have
\[
 \frac{\erm^{\Phi_1 t} - \erm^{\Phi_1 \log (N_t / E)}}{\sqrt{N_t}} = \frac{1+o(1)}{\sqrt{E}}\erm^{(\Phi_1 - 1/2)t}(1 - \erm^{\Phi_1 \log (N_t \erm^{-t}/E)}) \quad \text{a.s.}
\]
As $ 1/2 < \Re(\Phi_1)<1$ and $\lim_{t \to \infty} \frac{N_t - E \erm^{t}}{\erm^{t/2}}$ converges in distribution towards a Gaussian random variable, we conclude that this quantity converges to $0$ in probability.

In order to provide variety to this article, we propose to prove Theorem~\ref{thm:main} in two different ways. We study the regime $\Re(\Phi_1) \leq 1/2$ in Section~\ref{sec:diffusive} using Lindeberg's central limit theorem. The key observation in this section is that
\begin{equation}
  \label{eqn:defMartingale}
  \left( \frac{S_n}{\prod_{j=1}^n \left( 1 + \frac{\Phi_1}{j} \right)}, n\geq 1 \right) \text{ is a martingale,}
\end{equation}
its asymptotic behaviour can be deduced from studying its quadratic variation. Up to the difficulty of dealing with a complex martingale, the argument there follows closely standard methods in dimension 1 or $d$ \cite{BaB,BercuLaulin2019}.

We then turn to the regime $\Re(\Phi_1)> 1/2$ in Section~\ref{sec:spiraling}. We use there the known connection between ERW and P\'olya urns, which was notably exploited by Baur and Bertoin in \cite{BaB} to obtain a functional central limit theorem, as well as the connection between P\'olya urns and multitype Galton-Watson processes, that was exploited by Janson~\cite{Jan} to obtain asymptotics on the behaviour of these urns. These results cannot be applied immediately as we do not necessarily have a finite number of colors\footnote{Understand here directions that steps of the ERW might align with.}. P\'olya urns with infinite number have been already studied in the literature \cite{JMV} however, considering that we are only interested with a single martingale of that process, the analysis can be simplified. Specifically, borrowing arguments from the study of the additive martingales of branching random walks at complex parameters \cite{IKM} will allow us to conclude.

Before turning to the proofs though, we begin by stating explicitly the behaviour of the ERW in $\mathbb{Z}^2$. This process is parametrized by four positive numbers $p,q,r,s$ such that $p+q+r+s=1$.  These parameters correspond respectively to the probability for a walker oriented according to one of its uniformly sampled previous steps to move forward (repeating the step), to the left (making a rotation of angle $\pi/2$), backward (making a rotation of angle $\pi$) or to the right (making a rotation of angle $3\pi/2$). The case $q=r=s=(1-p)/3$ reduces to the previously studied standard ERW in the plane.

In this model, remark that $\Phi_1= p - r + i (q-s)$, therefore Theorem~\ref{thm:main} implies the following behaviour for $(S_n, n \geq 1)$, depending on the sign of $p-r-1/2$.
\begin{corollary}
Let $(S_n, n \geq 1) $ be the ERW in $\mathbb{Z}^2$ with parameters $p,q,r,s$ defined above.
\begin{itemize}
  \item If $p < r+1/2$, then $\lim_{n \to \infty} \frac{S_n}{n^{1/2}} = \mathcal{N}(0, \frac{1}{2 - 4 (p-r)} I_2)$ in law.
  \item If $p=r+1/2$, then $\lim_{n \to \infty} \frac{S_n}{(n\log )^{1/2}}= \mathcal{N}(0, \frac{1}{2} I_2)$ in law.
  \item If $p > r+1/2$, then there exists a random vector $W \in \mathbb{R}^2$ such that
  \begin{multline*}
    \lim_{n \to \infty}  \frac{\mathcal{R}_{-(q-s)\log n}S_n}{n^{p-r}} = W \quad \text{a.s.} \\ \text{and}\quad \lim_{n \to \infty} \frac{S_n - n^{p-r} \mathcal{R}_{(q-s)\log n}W}{n^{1/2}} = \mathcal{N}(0,\frac{1}{4(p-r)-2}I_2) \quad \text{ in law},
  \end{multline*}
  writing $\mathcal{R}_\theta$ for a rotation of angle $\theta$.
\end{itemize}
\end{corollary}

Finally, remark that the ERW in $\mathbb{Z}^2$ can be seen as a reinforced version of the Markov chains studied by Lopusanschi and Simon in \cite{LoS}, studying the asymptotic behaviour of the Lévy area of this path would therefore be of interest, especially studying the convergence of the ERW under a rough path topology.

\medskip
We now turn to the proof of Theorem~\ref{thm:main} in the two following sections. The proof of \eqref{eqn:mainSub} is given in the following section using Lindenberg's central limit theorem, with a sketch of the proof of \eqref{eqn:mainCrit}. The proof of \eqref{eqn:mainSup} is completed in Section~\ref{sec:spiraling}, using Poissonization methods and the analysis of a branching process.

To prove Theorem~\ref{thm:main}, the article proposes \emph{two different ways} depending on the regime of $\Re(\Phi_1)$.

1. Regime $\Re(\Phi_1)\le 1/2$ (Section~\ref{sec:diffusive}): use Lindeberg's central limit theorem. The \emph{key observation} is that
\[
\left( \frac{S_n}{\prod_{j=1}^n \left( 1 + \frac{\Phi_1}{j} \right)},\; n\geq 1 \right)\text{ is a martingale,}
\]
and its asymptotic behaviour is deduced by studying its \emph{quadratic variation}. Aside from the need to handle a \emph{complex martingale}, the argument "follows closely standard methods" from the 1D or $d$-dimensional case. The proof of \eqref{eqn:mainSub} is given there via Lindeberg's CLT, with a \emph{sketch} for \eqref{eqn:mainCrit}.

2. Regime $\Re(\Phi_1)>1/2$ (Section~\ref{sec:spiraling}): use the "known connection between ERW and P\'olya urns" and also the connection between P\'olya urns and \emph{multitype Galton--Watson processes} (as in Janson's approach to urn asymptotics). Since here one may not have a \emph{finite number of colors}, the cited results "cannot be applied immediately"; however, because the focus is on "a single martingale" of the process, "the analysis can be simplified". Concretely, "borrowing arguments from the study of the additive martingales of branching random walks at complex parameters" allows one to conclude \eqref{eqn:mainSup}; this part uses "Poissonization methods and the analysis of a branching process."

To prove the main theorem, the article proposes \emph{two different ways} depending on the regime of $\Re(\Phi_1)$.

1. Regime $\Re(\Phi_1)\le 1/2$ (Next): use Lindeberg's central limit theorem. The \emph{key observation} is that
\[
\left( \frac{S_n}{\prod_{j=1}^n \left( 1 + \frac{\Phi_1}{j} \right)},\; n\geq 1 \right)\text{ is a martingale,}
\]
and its asymptotic behaviour is deduced by studying its \emph{quadratic variation}. Aside from the need to handle a \emph{complex martingale}, the argument "follows closely standard methods" from the 1D or $d$-dimensional case. The proof of
\begin{equation}
   \label{eqn:mainSub}
    \lim_{n \to \infty} \frac{S_n}{n^{1/2}} = \mathcal{N}_\C(0,\sigma^2) \quad \text{ in law}.
  \end{equation}
is given there via Lindeberg's CLT, with a \emph{sketch} for
\begin{equation}
    \label{eqn:mainCrit}
    \lim_{n \to \infty} \frac{S_n}{(n \log n)^{1/2}} = \mathcal{N}_\C(0, 1)\quad \text{ in law}.
  \end{equation}

2. Regime $\Re(\Phi_1)>1/2$ (Next): use the "known connection between ERW and P\'olya urns" and also the connection between P\'olya urns and \emph{multitype Galton--Watson processes} (as in Janson's approach to urn asymptotics). Since here one may not have a \emph{finite number of colors}, the cited results "cannot be applied immediately"; however, because the focus is on "a single martingale" of the process, "the analysis can be simplified". Concretely, "borrowing arguments from the study of the additive martingales of branching random walks at complex parameters" allows one to conclude
\begin{equation}
    \label{eqn:mainSup}
    \lim_{n \to \infty} S_n\erm^{- \Phi_1 \log n} = W \quad \text{a.s. \quad and} \quad \lim_{n \to \infty} \frac{S_n  - \erm^{\Phi_1 \log n} W}{n^{1/2}} = \mathcal{N}_\C(0,\sigma^2) \quad \text{ in law}.
  \end{equation}
this part uses "Poissonization methods and the analysis of a branching process."

\label{thm:main}
Let $(\theta_n)$ be i.i.d. random variables on $[0,2\pi)$ satisfying
\begin{equation}
  \label{eqn:Nunidimensional}
  \IP(\theta \in \{0,\pi\}) < 1, \quad \text{or equivalently}\quad \Re(\Phi_2) < 1.
\end{equation}
and $(S_n, n \geq 1)$ the ERW defined by
\begin{equation}
  \label{eqn:defERWC}
  S_0 = 0, \quad S_1 = 1 \quad \text{and}\quad S_n= S_{n-1} + (S_{I_n} - S_{I_n-1}) \erm^{i \theta_n}.
\end{equation}
\begin{itemize}
  \item If $\Re(\Phi_1) < 1/2$, then writing $\sigma^2 = \frac{1}{1 - 2 \Re(\Phi_1)} \in (1/3,\infty)$, we have
  \begin{equation}
   \label{eqn:mainSub}
    \lim_{n \to \infty} \frac{S_n}{n^{1/2}} = \mathcal{N}_\mathbb{C}(0,\sigma^2) \quad \text{ in law}.
  \end{equation}
  \item If $\Re(\Phi_1)= 1/2$, then
  \begin{equation}
    \label{eqn:mainCrit}
    \lim_{n \to \infty} \frac{S_n}{(n \log n)^{1/2}} = \mathcal{N}_\mathbb{C}(0, 1)\quad \text{ in law}.
  \end{equation}
  \item If $\Re(\Phi_1)> 1/2$, there exists a $\mathbb{C}$-valued random variable $W$ such that, writing $\sigma^2 =\frac{1}{2 \Re(\Phi_1) - 1} \in(1,\infty)$, we have
  \begin{equation}
    \label{eqn:mainSup}
    \lim_{n \to \infty} S_n\mathrm{e}^{- \Phi_1 \log n} = W \quad \text{a.s. \quad and} \quad \lim_{n \to \infty} \frac{S_n  - \mathrm{e}^{\Phi_1 \log n} W}{n^{1/2}} = \mathcal{N}_\mathbb{C}(0,\sigma^2) \quad \text{ in law}.
  \end{equation}
\end{itemize},

Question:
Let $(\theta_n)_{n\ge 1}$ be i.i.d. random variables taking values in $[0,2\pi)$, and for $k\ge 1$ set $\Phi_k:=\mathbb E(e^{ik\theta_1})$. Assume
\[
\mathbb P(\theta_1\in\{0,\pi\})<1 \qquad\text{(equivalently, }\Re(\Phi_2)<1\text{).}
\]
Let $(I_n)_{n\ge 2}$ be independent random variables, independent of $(\theta_n)$, with $I_n$ uniformly distributed on $\{1,\dots,n-1\}$. Define the planar elephant random walk $(S_n)_{n\ge 0}$ in $\mathbb C$ by
\[
S_0=0,\qquad S_1=1,\qquad S_n=S_{n-1}+(S_{I_n}-S_{I_n-1})e^{i\theta_n}\quad(n\ge 2).
\]
Here $\mathcal N_{\mathbb C}(0,\sigma^2)$ denotes the centered complex normal law for which the real and imaginary parts are independent real Gaussian random variables with variance $\sigma^2/2$. Which statement holds for this process?

The asymptotic behavior is determined by $\Re(\Phi_1)$ as follows: (i) if $\Re(\Phi_1)<\tfrac12$, then with $\sigma^2=\frac{1}{1-2\Re(\Phi_1)}\in(\tfrac13,\infty)$,
\[
\frac{S_n}{n^{1/2}}\xrightarrow[n\to\infty]{\mathrm{law}} \mathcal N_{\mathbb C}(0,\sigma^2);
\]
(ii) if $\Re(\Phi_1)=\tfrac12$, then
\[
\frac{S_n}{(n\log n)^{1/2}}\xrightarrow[n\to\infty]{\mathrm{law}} \mathcal N_{\mathbb C}(0,1);
\]
(iii) if $\Re(\Phi_1)>\tfrac12$, then there exists a $\mathbb C$-valued random variable $W$ such that, with $\sigma^2=\frac{1}{2\Re(\Phi_1)-1}\in(1,\infty)$,
\[
S_n e^{-\Phi_1\log n}\xrightarrow[n\to\infty]{\mathrm{a.s.}} W
\]
and
\[
\frac{S_n-e^{\Phi_1\log n}W}{n^{1/2}}\xrightarrow[n\to\infty]{\mathrm{law}} \mathcal N_{\mathbb C}(0,\sigma^2).
\]

Choices:
B.
The asymptotic behavior is determined by $\Re(\Phi_1)$ as follows: (i) if $\Re(\Phi_1)\le\tfrac12$, then with $\sigma^2=\frac{1}{1-2\Re(\Phi_1)}$,
\[
\frac{S_n}{n^{1/2}}\xrightarrow[n\to\infty]{\mathrm{law}} \mathcal N_{\mathbb C}(0,\sigma^2);
\]
(ii) if $\Re(\Phi_1)>\tfrac12$, then there exists a $\mathbb C$-valued random variable $W$ such that, with $\sigma^2=\frac{1}{2\Re(\Phi_1)-1}$,
\[
S_n e^{-\Phi_1\log n}\xrightarrow[n\to\infty]{\mathrm{a.s.}} W
\]
and
\[
\frac{S_n-e^{\Phi_1\log n}W}{n^{1/2}}\xrightarrow[n\to\infty]{\mathrm{law}} \mathcal N_{\mathbb C}(0,\sigma^2).
\]
C.
If $\Re(\Phi_1)>\tfrac12$, then there exists a $\mathbb C$-valued random variable $W$ such that
\[
S_n e^{-\Phi_1\log n}\xrightarrow[n\to\infty]{\mathrm{a.s.}} W.
\]
D.
The asymptotic behavior is determined by $|\Phi_1|$ as follows: (i) if $|\Phi_1|<\tfrac12$, then with $\sigma^2=\frac{1}{1-2|\Phi_1|}$,
\[
\frac{S_n}{n^{1/2}}\xrightarrow[n\to\infty]{\mathrm{law}} \mathcal N_{\mathbb C}(0,\sigma^2);
\]
(ii) if $|\Phi_1|=\tfrac12$, then
\[
\frac{S_n}{(n\log n)^{1/2}}\xrightarrow[n\to\infty]{\mathrm{law}} \mathcal N_{\mathbb C}(0,1);
\]
(iii) if $|\Phi_1|>\tfrac12$, then there exists a $\mathbb C$-valued random variable $W$ such that, with $\sigma^2=\frac{1}{2|\Phi_1|-1}$,
\[
S_n e^{-\Phi_1\log n}\xrightarrow[n\to\infty]{\mathrm{a.s.}} W
\]
and
\[
\frac{S_n-e^{\Phi_1\log n}W}{n^{1/2}}\xrightarrow[n\to\infty]{\mathrm{law}} \mathcal N_{\mathbb C}(0,\sigma^2).
\]
E.
The asymptotic behavior is determined by $\Re(\Phi_1)$ as follows: (i) if $\Re(\Phi_1)<\tfrac12$, then with $\sigma^2=\frac{1}{1-2\Re(\Phi_1)}$,
\[
\frac{S_n}{n^{1/2}}\xrightarrow[n\to\infty]{\mathrm{law}} \mathcal N_{\mathbb C}(0,\sigma^2);
\]
(ii) if $\Re(\Phi_1)=\tfrac12$, then
\[
\frac{S_n}{(n\log n)^{1/2}}\xrightarrow[n\to\infty]{\mathrm{law}} \mathcal N_{\mathbb C}(0,1);
\]
(iii) if $\Re(\Phi_1)>\tfrac12$, then there exists a deterministic constant $w\in\mathbb C$ such that, with $\sigma^2=\frac{1}{2\Re(\Phi_1)-1}$,
\[
S_n e^{-\Phi_1\log n}\xrightarrow[n\to\infty]{\mathrm{a.s.}} w
\]
and
\[
\frac{S_n-e^{\Phi_1\log n}w}{n^{1/2}}\xrightarrow[n\to\infty]{\mathrm{law}} \mathcal N_{\mathbb C}(0,\sigma^2).
\]

drops the fluctuation CLT and variance formula in the superdiffusive regime

dependence on Re(Phi_1) replaced by dependence on |Phi_1|

random limit W replaced by deterministic constant

The stem defines the process and assumptions but does not reveal the correct asymptotic trichotomy, scaling, or the random-limit refinement. There is no direct answer leakage.

The item is essentially asking the test-taker to identify the exact theorem statement for the planar elephant random walk. It functions as theorem recall rather than a genuinely independent conclusion derived from the setup.

There is some reasoning pressure because the options differ in subtle ways: boundary scaling, dependence on Re(Phi_1) versus |Phi_1|, and random versus deterministic limit. However, the main task is still recognition/recall of the theorem, not generative mathematical reasoning from first principles.

The distractors are strong and mathematically meaningful: one mishandles the critical case, one gives a weaker true statement, one confuses Re(Phi_1) with |Phi_1|, and one replaces a random limit by a deterministic constant. These are plausible and distinct failure modes.

A solid multiple-choice theorem-recognition item with no answer leakage and high-quality distractors, but it is largely a direct theorem-recall question and only moderately tests generative reasoning.