\begin{theorem}[The method of moment from \cite{bollobas2001random}, Theorem 1.23]\label{moment}
Let $\left\{(\Omega_i,\mathbb{P}_i)\right\}_{i\in \mathbb{N}}$ be a sequence of probability spaces.
For $m\in \mathbb{N}$, let $N_{1,i},\cdots,N_{m,i}:\Omega_i\to \mathbb{N}_0$ be random variables for all $i\in \mathbb{N}$
and suppose there exist $\lambda_1,\cdots,\lambda_m\in(0,\infty)$ such that:
$$\lim_{i\to \infty}\mathbb{E}[(N_{1,i})_{k_1}\cdots(N_{m,i})_{k_m}]=\lambda_1^{k_1}\cdots\lambda_m^{k_m}$$
for all $k_1\cdots k_m\in \mathbb{N}$. Then
$$\lim_{i\to \infty}\mathbb{P}[(N_{1,i})=n_1,\cdots,(N_{m,i})=n_m]=\prod^k_{j=1}\frac{\lambda_j^{n_j} e^{-\lambda_j}}{n_j!}.$$
That is, $(N_{1,i},\cdots,N_{m,i}):\Omega_i \to \mathbb{N}^m_0$ converges jointly in distribution to a vector which is independently Poisson distributed with means
$\lambda_1,\cdots,\lambda_m$.

\end{theorem}

\section{Some claims}\label{claim}
Before the proof of the main theorem, we have to make some claims which will simplify the situation.
We consider the stratum whose zero order is $O(1)$, and it inplies $|\kappa|=O(g)$.

\begin{clm} \label{clm1}
A closed saddle connection is non-separable.
\end{clm}
\begin{proof}
For a closed connection $\gamma$ of $(X,\omega)$, if it's separable, then it cuts $X$ into $X_1$ and $X_2$, and $\partial X_1=\partial X_2=\gamma$.
By Stokes theorem, $\int_\gamma \omega=0$, but since $\gamma$ is a saddle connection its length is $|\int_\gamma \omega|$, which is a contradiction.
\end{proof}

\begin{clm} \label{clm2}
The probability measure of the subset of $\H_g(\kappa)$ which has a closed saddle connection involving $q$ cylinders of multiplicity $p$
goes to $0$ as $g\to \infty$, where $p>1$ or $q>0$.
\end{clm}

\begin{proof}
From Theorem \ref{Siegel of closed},
the Siegel-Veech constant of configuration with closed saddle connection involving $q$ cylinders of multiplicity $p$ is $O(\frac{1}{g^{2p+q-2}})$ .
Denote by $\mathcal{C}^{p,q}_g(\kappa)[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]$ the subset of $\H_g(\kappa)$ consisting of surfaces which have a closed saddle connection
of length in the interval $[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]$
involving $q$ cylinders of multiplicity $p$.
Then for $p\geq 2$ or $q\geq 1$, we have
 $$\V(\mathcal{C}^{p,q}_g(\kappa)[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}])\leq |\kappa|O(\frac{1}{g^{q}})\frac{b^2-a^2}{g}\V(\H_g(\kappa))=O(\frac{1}{g})\V(\H_g(\kappa)$$
where the coefficient $|\kappa|$ is the selection of the zero that the closed saddle connection goes through.
So when $g\to \infty$ its probability measure goes to $0$
and it suffices to consider the closed saddle connection with multiplicity $1$ and no cylinders around it.
\end{proof}

\begin{clm} \label{clm3}
The angle of a closed saddle connection at the zero which it connects is odd multiples of $\pi$.
\end{clm}

\begin{proof}
For an Abelian differential $(X,\omega)$, if it has a closed saddle connection $\gamma$ connecting a zero $p$, if its angle at $p$ is $2k\pi$,
then the holonomy of $\gamma$ and $-\gamma$ have same direction. But
$$\int_\gamma \omega=-\int_{-\gamma} \omega.$$
It implies $\int_\gamma \omega=0$, which is a contradiction.
\end{proof}

\begin{clm} \label{clm4}
If a closed saddle connection has angle $\pi$ at one side, it must have a cylinder at the side.
\end{clm}

\begin{proof}
Let $\gamma$ be a colsed saddle connection on $(X,\omega)$ which has angle $\pi$ at one side. Since $(X,\omega)$ is oriented, its normal bundle of $\gamma$ at the side is oriented.
Consider the exponential map from the bundle to $(X,\omega)$, which is well-defined on $\gamma \times [0,s]$ for some $s$ small sufficiently and the image is a cylinder.

\end{proof}

Combine claim $2,3$ and $4$, we have that for a closed saddle connection $\gamma$ on $(X,\omega)$ in principal stratum,
since its total angle around the zero is $4\pi$ and has to be divided into odd multiples of $\pi$,
it must have angle $\pi$ at one side of $\gamma$ and $3\pi$ at another.
So there is a cyliner with $\gamma$ as a boundary. As for the other boundary, if it consists of some open saddle connections,
then these saddle connections have angles $\pi$. That means there are at least two non-homologous saddle connections have angle $\pi$,
which occurs on a set of measure zero.
So we only need to consider the cylinder bounded by curves homologous to $\gamma$.
Then from Claim \ref{clm2} the probability measure of $\mathcal{C}_g(1^{2g-2})[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]$ has limit
 $$\lim_{g\to \infty}\frac{\V(\mathcal{C}_g(1^{2g-2})[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}])}{\V(\H_g(1^{2g-2})}=O(\frac{1}{g})\to 0.$$

So we need to consider the question on stratum with higher order zeros. Next we will consider $\H_g(2^{g-1})$ firstly.

\section{Surgery}\label{surgery}
This section we will introduce some surgeries that, when combined, can collapse a closed saddle connection.
\subsection{Open up a higher-order zero}
Masur, Rafi and Randecker introduce a surgery in \cite{masur2024lengths} which is a variation of a surgery from \cite{eskin2003moduli}.
This surgery can collapse a saddle connection which is not closed and has multiplicity $1$.

Let $(X,\omega)$ be an Abelian differential, $\sigma$ be an open saddle connection with endpoints $v_1$ and $v_2$,
whose orders are $n_1$ and $n_2$ respectively.
Since the total angle at $v_1$ is $2(n_1+1)\pi$,
one can extend $\sigma$ from $v_1$ along its direction to $v_3$ and denote by $\sigma'$ the geodesic segment $v_1v_3$
such that $\ell_\omega(\sigma)=\ell_\omega(\sigma')$
and the angle between $\sigma$ and $\sigma'$ is
$2k_1\pi$ and $2k_2\pi$, where $k_1+k_2=n_1+1$.
If $\sigma'$ does not go through other zeros of $(X,\omega)$,
which means $\sigma'$ is a ray from $v_1$, the following surgery can be carried out.

Cut along $\sigma+\sigma'$ and denote the two copies of $\sigma+\sigma'$ and $v_1$ by $\sigma^{\pm}+(\sigma')^{\pm}$ and $v_1^{\pm}$.
Then glue $\sigma^+$ and $(\sigma')^+$, $\sigma^-$ and $(\sigma')^-$.
This surgery reduces the order of $v_1$ and constructs a new Abelian differential $(X',\omega')$.
On $(X',\omega')$, the total angles at $v_1^+$, $v_1^-$ and $v_2$ are $2k_1\pi$, $2k_2\pi$, and $2(n_2+2)\pi$ respectively,
that is, the orders of $v_1^+$, $v_1^-$ and $v_2$ are $k_1-1$, $k_2-1$, and $n_2+1$.
Particularly, if $v_1$ is a simple zero, the surgery collapses the saddle connection. We call this surgery \emph{collapsing surgery}.
\begin{figure}[h]
    \centering
    \includegraphics[width=0.6\linewidth]{collapsing}
    \caption{The collapsing surgery}
    \label{collapsing}
\end{figure}

In our situation, we need to use the inverse of collapsing surgery which we call \emph{opening surgery}.
From Claim \ref{clm2}, it suffices to consider the subset $\mathcal{C}^{1,0}_g(2^{g-1})[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]$.
For an Abelian differential $(X,\omega)$ in $\mathcal{C}^{1,0}_g(2^{g-1})[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]$,
choose a closed saddle connection $\gamma$ with length in $[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]$ at a double zero $p$.
We can reverse the surgery introduced above as follows.

Since the total angle at $p$ is $6\pi$, from Claim \ref{clm3} and Claim \ref{clm4}, the angles on both side of $\gamma$ at $p$ must be $3\pi$.
Fix the orientation of $\gamma$, choose two rays $\sigma^+$ and $\sigma^-$ from $p$ such that
$\ell_\omega(\sigma^+)=\ell_\omega(\sigma^-)=\ell_\omega(\gamma)$,
and the angle between $\gamma$ and $\sigma^+$
is $\pi$, the angle between $\gamma$ and $\sigma^-$ is $-\pi$, where the sign is consistent with the orientation of the surface.
Denote by $q^+$ and $q^-$ the other endpoints of $\sigma^+$ and $\sigma^-$.
Then cut along $\sigma^+ + \sigma^-$ and glue the two copies of $\sigma^+$ and $\sigma^-$.
After the surgery, the two copies of $p$ become a regular and a simple zero, $q^+$ and $q^-$ are glued to become a new simple zero $q$.

\begin{figure}[h]
   \centering
   \includegraphics[width=0.6\linewidth]{openinglocally}
  \caption{The opening surgery}
  \label{opening}
\end{figure}

This surgery is the inverse of the collapsing surgery above: the double zero is replaced by two simple zeros,
and it can be carried out except for the surgery locus goes through some zero,
that is, it has two non-homologous saddle connections with angle $2\pi$.
From period mapping, such subset has measure zero. We call the subset on which the surgery can be carried out \emph{permissible set},
and denote it by $\mathcal{C}^{1,0}_{g,Per}(2^{g-1})[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]$, from the discussion above we have
\begin{proposition}
$\mathcal{C}^{1,0}_{g,Per}(2^{g-1})[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]$ is a full measure subset of $\mathcal{C}^{1,0}_g(2^{g-1})[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]$.
\end{proposition}

By the surgery of collapsing this map is one-to-one and denote the opening surgery by $F_1$.
\begin{figure}[h]
    \centering
    \includegraphics[width=0.6\linewidth]{opening}
    \caption{The resulting surface}
    \label{opening surface}
\end{figure}

\begin{remark}
We can also choose $\sigma^{\pm}$ to make the angle between $\gamma$ and them are $2\pi$,
it suffices to make sure the opening operation can obtain an abelian differential and a smooth loop homotopic to $\gamma$.
\end{remark}
\subsection{Move zero along closed curve and pinch}
We have constructed the mapping $F_1$,
which takes a closed saddle connection $\gamma$ at a double zero to two saddle connections $\gamma_1$ and $\gamma_2$
sharing same endpoints $p$ and $q$ which are both simple zeros.
Moreover, the angles between the two saddle connections at the two simple zeros (on both sides) are all $2\pi$,
and if we denote by $\ell_\omega(\gamma)=L$, we have $\ell_{\omega'}(\gamma_1)=L$, $\ell_{\omega'}(\gamma_2)=2L$, where $(X',\omega')=F_1(X,\omega)$.
If we want to collapse $\gamma_1$ and $\gamma_2$ simultaneously, we have to make a surgery to move $q$ along $\gamma_2$ to adjust the length of $\gamma_2$.

\subsubsection{Move the zero locally by period mapping.}

First we need to choose a special basis of $H^1(X,\Sigma;\mathbb{C})$, where $\Sigma=\{p,q,p_2,\cdots,p_{g-1}\}$
is the zero set of $(X',\omega')=F_1(X,\omega)$.
From Claim \ref{clm1}, the homology class $[\gamma]=[\gamma_1+\gamma_2]$ is non-separable.
So we can choose a basis of $H^1(X,\mathbb{C})$
$$(\alpha_1,\beta_1,\cdots,\alpha_g,\beta_g),$$
where $\alpha_1=[\gamma]$.

Next we choose the relative homology class
$$(pq,pp_2\cdots,pp_{g-1}),$$
where $[\gamma_1]=pq$, and $pp_i$ is freely homotopic to $0$ in $H^1(X,\mathbb{C})$.
Together they compose a basis of $H^1(X,\Sigma;\mathbb{C})$,
and their holonomy gives a period mapping around $(X',\omega')$ locally.

Let $\omega'(\alpha_1)=(x_1,y_1)$, $\omega'(\beta_1)=(x_2,y_2)$, $\omega'(\alpha_2)=(x_3,y_3)$
$\omega'(\beta_2)=(x_4,y_4)$.
Now choose a relative cohomology class $\upsilon \in H^1(X,\Sigma;\mathbb{C})$ ,
which can be considered as a tangent vector in $T_{\H_g(1,1,\cdots,2)}(X',\omega')$ such that
\begin{equation}\label{moving}
\upsilon(\alpha_1)=(-x_1,-y_1), \upsilon(\alpha_2)=(-\frac{y_2x_1}{y_4},\frac{x_2y_1}{x_4})
\end{equation}
and on the other basis we assign zero to $\upsilon$.

\begin{remark}\label{explain}
Here $\upsilon(\alpha_1)$ is chosen to guarantee that along the curve $(X',\omega')+t\upsilon$ in moduli space the zero moves along $\gamma$,
and $\upsilon(\alpha_2)$ is chosen to guarantee the resulting surface has area $1$, since the area of $(X,\omega)$ can be written by
$$\int_X|\omega|^2=\frac{i}{2}\sum_i(\int_{\alpha_i}\omega \int_{\beta_i}\overline{\omega}-\int_{\beta_i}\omega \int_{\alpha_i}\overline{\omega}).$$
\end{remark}

Consider the curve in $\H_g(1,1,2,,\cdots,2)$: $(X',\omega')+t\upsilon, t\in [0,1]$.
If $\gamma_1$ and $\gamma_2$ don't degenerate along the curve, then we obtain
$$F_2:\mathcal{C}^{1,0}_{g,Per}(2^{g-1})[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]  \to \H_g(1^2,2^{g-1})$$
such that $F_2[(X,\omega)]=F_1[(X,\omega)]+\upsilon$.
From the construction above the resulting surface has two saddle connections
in the relative homology class $\gamma_1$ and $\gamma_2$ with equal length.

\subsubsection{$F_2$ is well-defined almost everywhere}
We have defined a mapping if the saddle connections $\gamma_1$ and $\gamma_2$ are preserved along the curve $(X',\omega')+t\upsilon, t\in [0,1]$
 on $\mathcal{C}^{1,0}_{g,Per}(2^{g-1})$.
Next we will see the mapping can be defined for almost every translation surface in $\mathcal{C}^{1,0}_{g,Per}(2^{g-1})$.

From above we know the moving surgery can be realized except for some of $\gamma_1$ or $\gamma_2$ degenerate along $(X',\omega')+t\upsilon, t\in [0,1]$.
Suppose $T$ is the first time when the geodesic in $[\gamma_1]$ can be represented by $pp_i-p_ip_j-\cdots-p_kq$,
since the saddle connection is smooth on $(X',\omega')+t\upsilon,t\in [0,T]$, the corner of $pp_i-p_ip_j-\cdots-p_kq$ must be $\pi$.
But the holonomy of $pp_i$ is not changed on $(X',\omega')+t\upsilon,t\in [0,T]$
This implies there exists some $pp_i$ such that its holonomy on $(X',\omega')$ has the same direction with $p_0p_1$.
And by the relation of holonomy under $F_1$, this implies $p_0p_i$ has the same direction with the closed curve on $(X,\omega)$.
Under period mapping, on each local chart this is a measure-zero set.
So all such $(X,\omega)$ is a subset of measure zero in $ \H_g(2^{g-1})$, also in $\mathcal{C}^{1,0}_{g,Per}(2^{g-1})[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]$.

\subsubsection{Pinching}
On the subset of $\mathcal{C}^{1,0}_{g,Per}(2^{g-1})[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]$ where $F_2$ is well-defined,
we can cut along $\gamma_1+\gamma_2$ and denote the two copies of $p$ and $q$ by $q^+$, $q^-$, $p^+$, $p^-$,
the two copies of $\gamma_1$ and $\gamma_2$ by $\gamma^+_1$, $\gamma^-_1$, $\gamma^+_2$, $\gamma^-_2$.
Notice that the total angles at $q^+$, $q^-$, $p^+$, $p^-$ are all $2\pi$,
which means $\gamma^\pm_1$ have same direction with $\gamma^\pm_2$,
then glue $\gamma^+_1$ with $\gamma^+_2$, $\gamma^-_1$ with $\gamma^-_2$, which make $q^+$, $q^-$, $p^+$, $p^-$ become regular points.
Since $\gamma_1+\gamma_2$ is non-separable, this surgery reduces the genus by one, and obtains a new abelian differential
on which we mark two points $p^+$ and $p^-$ so as to find its inverse.
Considering the choice of which zero is selected after applying the inverse map, we finally obtain:
$$F_3: \mathcal{C}^{1,0}_{g,Per}(2^{g-1})[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]\to M\times \H_{g-1}(2^{g-2},0,0)\times A_{[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]},$$
where $M$ is the combinatorial data of choosing one from $g-1$ zeros.

\begin{figure}[h]
    \centering
    \includegraphics[width=0.6\linewidth]{movingandcut}
    \caption{Moving and cut}
    \label{moving and cut}
\end{figure}

\begin{figure}[h]
    \centering
    \includegraphics[width=0.6\linewidth]{pinching}
    \caption{Pinching}
    \label{pinching}
\end{figure}

\subsubsection{$F_3$ is measure-preserving.}
This surgery can be inverse except for some directions which has saddle connections, which is a measure zero subset $A_0(X,\omega)$
in $A_{[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]}$ for every surface in $\H_{g-1}(2^{g-2},0,0)$:
choose a vector $\kappa$ in $A_{[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]}$,
consider the two rays from $p^+$ and $p^-$ whose holonomy is $\kappa$, if the rays exist, then cut the surface along the two rays
to obtain two closed loci and glue them.
Let
$$\tilde{\H}=\{(X,\omega, \tau): (X,\omega)\in \H_{g-1}(2^{g-2},0,0), \tau \in A_0(X,\omega)\},$$
which is a measure zero subset of $\H_{g-1}(2^{g-2},0,0)\times A_{[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]}$.
So we have
$$F_3( \mathcal{C}^{1,0}_{g,Per}(2^{g-1}))\subset [\H_{g-1}(2^{g-2},0,0)\times (A_{[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]})]\setminus \tilde{\H}.$$

Moreover, under local chart determined by the basis we choose,
$$F_3(z_1,\cdots,z_n)=(z_1,z_i+\upsilon(z_i)),$$
whose Jacobian has deteminant one and thus is measure-preserving.

\section{Collapse closed saddle connections simultaneously}
From the construction above we can collapse one closed saddle connection on some $(X,\omega)\in \mathcal{C}^{1,0}_{g}(2^{g-1}))$ and obtain a new surface
$(X',\omega')\in \H_{g-1}(2^{g-2},0,0)$. if we want to collapse any $k$ closed saddle connections,
we have to ensure the locus of surgery for these saddle connections are disjoint.
For a given interval $[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]$,
define the \emph{exception set} $\mathcal{C}^{exc}_{g,Per}(2^{g-1})[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]$ to be the subset of
$\mathcal{C}^{1,0}_{g,Per}(2^{g-1})[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]$
such that on every $(X,\omega)\in \mathcal{C}^{exc}_{g,Per}(2^{g-1})[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]$ there are two closed saddle connections with length in
$[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]$ whose surgery loci intersect.
First we will show that the measure of exception set goes to zero as genus goes to infinity.

\subsection{The measure of exception set}

For $(X,\omega)\in \mathcal{C}^{exc}_{g,Per}(2^{g-1})[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]$, there are three situations:
\begin{itemize}
  \item[1]  There exist two closed saddle connections on $(X,\omega)$ with lengths in $[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]$
            that do not share a zero. And the loci of opening surgery intersect or the two closed saddle connections intersect.
  \item[2]  There exist two closed saddle connections on $(X,\omega)$ with lengths in $[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]$ sharing one zero.
\end{itemize}

Note that in the first situation, we can find a curve connecting the two zeros with length no more than $\frac{2b}{\sqrt{g}}$.

Fix $B\in \mathbb{R}^+$, define $\mathcal{C}_g^{1}(\frac{B}{\sqrt{g}})$ be the set of $(X,\omega)\in \mathcal{C}^{1,0}_{g,Per}(2^{g-1})[0,\frac{B}{\sqrt{g}}]$
such that there exist two closed saddle connections on $(X,\omega)$
that do not share a zero, moreover the lengths of the two closed saddle connections and the distance between the two zeros are less than $\frac{B}{\sqrt{g}}$.
Define $\mathcal{C}_g^{2}(\frac{B}{\sqrt{g}})$ be the set
of $(X,\omega)\in \mathcal{C}^{1,0}_{g,Per}(2^{g-1})[0,\frac{B}{\sqrt{g}}]$ such that there exists two closed saddle connections with lengths less than $\frac{B}{\sqrt{g}}$ sharing one zero.
Obviously for $B\geq 2b$, we have
$$ \mathcal{C}^{exc}_{g,Per}(2^{g-1})[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]\subset \mathcal{C}_g^{1}(\frac{B}{\sqrt{g}}) \cup \mathcal{C}_g^{2}(\frac{B}{\sqrt{g}}).$$

We need to compute the measure of $\mathcal{C}_g^{1}(\frac{B}{\sqrt{g}})$ and $\mathcal{C}_g^{2}(\frac{B}{\sqrt{g}})$ when $g\to \infty$.
First consider $\mathcal{C}_g^{2}(\frac{B}{\sqrt{g}})$, we have
\begin{proposition}\label{chain}
$$\lim_{g\to \infty}\frac{\V(\mathcal{C}_g^{2}(\frac{B}{\sqrt{g}}))}{\V(\H_g(2^{g-1}))} = 0.$$
\end{proposition}

\begin{proof}
For $(X,\omega) \in \mathcal{C}_g^{2}(\frac{B}{\sqrt{g}})$,
let $p$ be a zero on $(X,\omega)$ and $\gamma_1, \gamma_2$ be the closed saddle connections with lengths less than $\frac{B}{\sqrt{g}}$ at $p$.

As above we can collapse $\gamma_1$ and get a new translation surface in $\H_{g-1}(2^{g-2},0,0)$.
After the surgery $\gamma_2$ will become a segment connecting two marked regular points which are from the pinching surgery.
So the image of mapping $F_3$ on $\mathcal{C}_g^{2}(\frac{B}{\sqrt{g}})$ is in
$\H_{g-1,\frac{B}{\sqrt{g}}}(2^{g-2},0,0)\times D(\frac{B}{\sqrt{g}})$,
where $\H_{g-1,\frac{B}{\sqrt{g}}}(2^{g-2},0,0)$ is the subset that there exists a segment connnecting the two marked points of length less than
$\frac{B}{\sqrt{g}}$ and $D(\frac{B}{\sqrt{g}})$ is the disk of radius $\frac{B}{\sqrt{g}}$.
From \cite[Theorem 1.2]{aggarwal2019large}, the Siegel-Veech constant of saddle connections connecting two fix zeros of order $m_1$ and $m_2$ is
$$c=(m_1+1)(m_2+1)(1+O(\frac{1}{g})).$$
So we have
$$\V(\mathcal{C}_g^{2}(\frac{B}{\sqrt{g}}))\leq (g-1)\frac{B^2}{g}\V(\H_{g-1,\frac{B}{\sqrt{g}}}(2^{g-2},0,0)) \leq c(g-1)\frac{B^4}{g^2}\V(\H_{g-1}(2^{g-2},0,0)),$$
where the coefficient $g-1$ is the choice of $p$.
Then by \ref{Volume}, we have
$$\lim_{g\to \infty}\frac{\V(\mathcal{C}_g^{2}(\frac{B}{\sqrt{g}}))}{\V(\H_g(2^{g-1}))}\leq \lim_{g\to \infty}3c\frac{B^4}{g}=0$$
\end{proof}

For $\mathcal{C}_g^{1}(\frac{B}{\sqrt{g}})$ we also have
\begin{proposition}\label{intersect}
$$\lim_{g\to \infty}\frac{\V(\mathcal{C}_g^{1}(\frac{B}{\sqrt{g}}))}{\V(\H_g(2^{g-1}))} = 0.$$
\end{proposition}

\begin{proof}
For $(X,\omega) \in \mathcal{C}_g^{1}(\frac{B}{\sqrt{g}})$, let $p_1, p_2$ be the two zeros on $(X,\omega)$ with distance less than $\frac{B}{\sqrt{g}}$
and $\gamma_1, \gamma_2$ be the two closed saddle connections connecting $p_1, p_2$.
For every $k\in \mathbb{N}^+$, denote by $\mathcal{C}_g^{1,k}(\frac{B}{\sqrt{g}})$ the subset of $\mathcal{C}_g^{1}(\frac{B}{\sqrt{g}})$
that the geodesic between $p_1$ and $p_2$ is a concatenation of $k$ open saddle connections.
Their lengths are less than $\frac{B}{\sqrt{g}}$.
Choose the shortest saddle connection, using the surgery in \cite{masur2024lengths} we can collapse it and get a new translation surface in
$\H_g(3,1,2^{g-3})$. This is because the locus, that is the extension of the shortest saddle connection can not intersect the whole geodesic.
Repeat the surgery until $p_1$ and $p_2$ are collapsed to one zero and we get a map
$$F:\mathcal{C}_g^{1,k}(\frac{B}{\sqrt{g}})\to \mathcal{C}_g^{2}(2+k,1^k,2^{g-2-k})(\frac{B}{\sqrt{g}})\times D^k(\frac{B}{\sqrt{g}}),$$
where $\mathcal{C}_g^{2}(2+k,1^k,2^{g-2-k})(\frac{B}{\sqrt{g}})$ is the subset of $\H_g(2+k,1^k,2^{g-2-k})$ such that there are
two closed saddle connections $\gamma_1, \gamma_2$ connecting the zero of order $2+k$ with lengths less than $\frac{B}{\sqrt{g}}$ .

Similarly one can collapse $\gamma_1$ and the other will become a saddle connection connecting two zeros.
Note that the order is larger than $2$, although the collapsing surgery can also be realized, we need to require the angles
on the two sides of the closed saddle connections collapsed, which decide the order of zeros after collapsing.
We will explain this situation in detail in Section \ref{general case}.

Let $\mathcal{C}_g^{2,b',b''}(2+k,1^k,2^{g-2-k})(\frac{B}{\sqrt{g}})$ be the subset of
$\mathcal{C}_g^{2}(2+k,1^k,2^{g-2-k})(\frac{B}{\sqrt{g}})$ such that
the two angles on both sides of $\gamma_1$ are $(2b'+1)\pi$ and $(2b''+1)\pi$, where $b'+b''=2+k$.
Then after collapsing we get a map
$$F':\mathcal{C}_g^{2,b',b''}(2+k,1^k,2^{g-2-k})(\frac{B}{\sqrt{g}}) \to \H_{g-1,\frac{B}{\sqrt{g}}}(b'-1,b''-1,1^k,2^{g-2-k})\times D(\frac{B}{\sqrt{g}}).$$

Similar to Proposition \ref{chain}, we have
$$\V(\mathcal{C}_g^{2,b',b''}(2+k,1^k,2^{g-2-k})(\frac{B}{\sqrt{g}}))\leq b'b''\frac{B^4}{g^2}\V(\H_{g-1}(b'-1,b''-1,1^k,2^{g-2-k})).$$

Sum all $k$ and $(b',b'')$ we have
$$\V(\mathcal{C}_g^{1}(\frac{B}{\sqrt{g}}))\leq (g-1)(g-2)\sum_k \sum_{(b',b'')}b'b''\frac{B^4}{g^2}\V(\H_{g-1}(b'-1,b''-1,1^k,2^{g-2-k}))(\frac{B^2}{g})^k,$$
where the coefficient $(g-1)(g-2)$ is the choice of $p_1$ and $p_2$.
Then when $g\to \infty$ we have
$$\frac{\V(\mathcal{C}_g^{1}(\frac{B}{\sqrt{g}}))}{\V(\H_g(2^{g-1}))}\leq
\sum_k \sum_{(b',b'')}b'b''B^4\frac{\V(\H_{g-1}(b'-1,b''-1,1^k,2^{g-2-k}))}{\V(\H_g(2^{g-1}))}(\frac{B^2}{g})^k=O(\frac{1}{g}).$$
\end{proof}

Combine the two proposition we have
\begin{corollary}
$$\frac{\V(\mathcal{C}^{exc}_{g,Per}(2^{g-1})[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}])}{\V(\H_g(2^{g-1}))}=O(\frac{1}{g}).$$
\end{corollary}

\subsection{Collapsing simultaneously}
Now fix a positive integer $K$ and $n_1,\cdots,n_k$ a partition of $K$.
We want to compute the limit of the expectation
$$(L_{g,[a_1,b_1]})_{n_1}\cdots (L_{g,[a_k,b_k]})_{n_k},$$
for which we will collapse $K$ closed saddle connections in order.

Define $\hat{\mathcal{C}}^{1,0}_{g}(2^{g-1})=(X,\omega,\Gamma_1,\cdots,\Gamma_k)$,
where $(X,\omega)\in \mathcal{C}^{1,0}_{g}(2^{g-1})$,
and $\Gamma_i$ is an ordered list of $n_i$ closed saddle connections $(\gamma_{1,i},\cdots,\gamma_{n_i,i})$
with lengths in $[\frac{a_i}{\sqrt{g}},\frac{b_i}{\sqrt{g}}]$.
Suppose $a_1\leq b_1 \leq a_2\leq b_2 \cdots \leq a_k \leq b_k$, let $B=b_k$.
Then we have
$$\bigcup_i \mathcal{C}^{exc}_{g,Per}(2^{g-1})[\frac{a_i}{\sqrt{g}},\frac{b_i}{\sqrt{g}}]\subset \mathcal{C}_g^{1}(\frac{B}{\sqrt{g}})\cup \mathcal{C}_g^{2}(\frac{B}{\sqrt{g}}).$$

Let $\mathcal{C}'_{g,Per}=\mathcal{C}^{1,0}_{g,Per}(2^{g-1})\setminus (\mathcal{C}_g^{1}(\frac{B}{\sqrt{g}})\cup \mathcal{C}_g^{2}(\frac{B}{\sqrt{g}}))$,
we will adjust the surgery from last section on $\mathcal{C}'_{g,Per}$ to collapse the closed saddle connections in
$\Gamma_1,\cdots,\Gamma_k$ in order.
The opening and pinching operations are same to the situation of one curve. The only modification is the equation \ref{moving}:
by remark \ref{explain}, we need to choose the moving vector $\upsilon$ to ensure the area unchanged,
hence we can choose a basis containing $\gamma'_1,\gamma_1\cdots,\gamma'_K,\gamma_K$,
where $\gamma'_i$ is a saddle connection from opening $\gamma_i$.
Then the moving vector $\upsilon$ can be constructed like equation \ref{moving}: after defining the value on $\gamma_i$,
we can choose another relative homotopy class and give a value of $\upsilon$ on it to ensure the area unchanged.
This means we open and move the closed saddle connections simultaneously and then pinch them.

Finally  we obtain a mapping
$$\hat{F}: \hat{\mathcal{C}}'_{g,Per}(2^{g-1}) \to M_K\cdot \tilde{\H}_{g-K}(2^{g-1-K},0^{2K}) \times \prod^k_{i=1} (A_{[\frac{a_i}{\sqrt{g}},\frac{b_i}{\sqrt{g}}]})^{n_i},$$
where $\hat{\mathcal{C}}'_{g,Per}(2^{g-1})$ is defined as $\hat{\mathcal{C}}^{1,0}_{g}(2^{g-1})$
and $M_K$ is a combinatorial data consisting of choosing $K$ ordered zeros from $g-1$ zeros to label. As the discussion above,
we have
\begin{equation*}
\begin{aligned}
&\int_{\hat{\mathcal{C}}'_{g,Per}(2^{g-1})}(L_{g,[a_1,b_1]})_{n_1}\cdots (L_{g,[a_k,b_k]})_{n_k}d\mu_{MV}\\&=\V(\hat{\mathcal{C}}'_{g}(2^{g-1})))
\\&=|M_K|\prod^k_{i=1}[\frac{\pi(b^2-a^2)}{g}]^{n_i}\V(\H_{g-k}(2^{g-1-K}))\\
&\to \prod^k_{i=1}[\pi(b_i^2-a_i^2)]^{n_i}\V(\H_{g-K}(2^{g-1-K})), g\to \infty,
\end{aligned}
\end{equation*}
where the limit is because $|M_K|=(g-1)\cdots (g-K)$ and
$$\lim_{g\to \infty}\frac{(g-1)\cdots (g-K)}{g^{n_1+\cdots+n_k}}=\frac{(g-1)\cdots (g-K)}{g^K}=1.$$

Let $L'_{g,[a_i,b_i]}$ be the restriction of $L_{g,[a_i,b_i]}$ on $\mathcal{C}'_{g,Per}(2^{g-1})$.
Then the limit of factorial moment of $(L'_{g,[a_i,b_i]})_i$ is
$$\lim_{g\to \infty}\mathbb{E}[(L'_{g,[a_1,b_1]})_{n_1}\cdots (L'_{g,[a_k,b_k]})_{n_k}]=\lim_{g\to \infty}\frac{\V(\H_{g-K}(2^{g-1-K}))}{\V(\mathcal{C}'_{g,Per}(2^{g-1}))}\prod^k_{i=1}[\pi(b_i^2-a_i^2)]^{n_i}.$$
From Proposition \ref{chain} and \ref{intersect}, we have
$$\lim_{g\to \infty}\frac{\V(\mathcal{C}'_{g,Per}(2^{g-1}))}{\V(\H_g(2^{g-1}))}=1.$$

From Theorem \ref{Volume}, we have
$$\lim_{g\to \infty}\frac{\V(\H_{g-K}(2^{g-1-K}))}{\V(\H_{g}(2^{g-1}))}=3^K.$$
So we have
$$\lim_{g\to \infty}\mathbb{E}[(L'_{g,[a_1,b_1]})_{n_1}\cdots (L'_{g,[a_k,b_k]})_{n_k}]=\prod^k_{i=1}[3\pi(b_i^2-a_i^2)]^{n_i}.$$
By theorem \ref{moment}, $(L'_{g,[a_1,b_1]},\cdots,L'_{g,[a_k,b_k]})$ converges jointly in distribution to a vector of random variables with Poisson distributions of means $\lambda_{[a_i,b_i]}$, where
$$\lambda_{[a_i,b_i]}=3\pi(b_i^2-a_i^2).$$

From the property of Poisson distribution, we have
$$\mathbb{P}(L'_{g,[a_1,b_1]}=n_1,\cdots,L'_{g,[a_k,b_k]}=n_k)=\prod_{i=1}^k\frac{\lambda_{[a_i,b_i]}^{n_i}e^{-\lambda_{[a_i,b_i]}}}{n_i!}.$$
On the other hand
$$\mathbb{P}(L'_{g,[a_1,b_1]}=n_1,\cdots,L'_{g,[a_k,b_k]}=n_k)=\frac{\V(\{(X,\omega):L'_{g,[a_i,b_i]}(X,\omega)=n_i,i=1,\cdots,k\})}{\V(\mathcal{C}'_{g,Per}(2^{g-1}))}.$$
Note that
\begin{equation*}
\begin{aligned}
&\V(\{(X,\omega):L_{g,[a_i,b_i]}(X,\omega)=n_i,i=1,\cdots,k\})-
\V(\mathcal{C}_g^{1}(\frac{B}{\sqrt{g}})\cup \mathcal{C}_g^{2}(\frac{B}{\sqrt{g}}))\\ &\leq \V(\{(X,\omega):L'_{g,[a_i,b_i]}(X,\omega)=n_i,i=1,\cdots,k\})\\&\leq
\V(\{(X,\omega):L_{g,[a_i,b_i]}(X,\omega)=n_i,i=1,\cdots,k\}).
\end{aligned}
\end{equation*}
From Proposition \ref{chain} and \ref{intersect}, we have
$$\lim_{g\to \infty}\frac{\V(\mathcal{C}'_{g,Per}(2^{g-1}))}{\V(\H_g(2^{g-1}))}=1,\\ \lim_{g\to \infty}\frac{\V(\mathcal{C}_g^{1}(\frac{B}{\sqrt{g}})\cup \mathcal{C}_g^{2}(\frac{B}{\sqrt{g}}))}{\V(\H_g(2^{g-1}))}=0.$$
So
\begin{equation*}
\begin{aligned}
&\mathbb{P}(L_{g,[a_1,b_1]}=n_1,\cdots,L_{g,[a_k,b_k]}=n_k)-
\frac{\V(\mathcal{C}_g^{1}(\frac{B}{\sqrt{g}})\cup \mathcal{C}_g^{2}(\frac{B}{\sqrt{g}}))}{\V(\H_g(2^{g-1}))}\\ &\leq \mathbb{P}(L'_{g,[a_1,b_1]}=n_1,\cdots,L'_{g,[a_k,b_k]}=n_k)\frac{\V(\mathcal{C}'_{g,Per}(2^{g-1}))}{\V(\H_g(2^{g-1}))}\\&\leq
\mathbb{P}(L_{g,[a_1,b_1]}=n_1,\cdots,L_{g,[a_k,b_k]}=n_k).
\end{aligned}
\end{equation*}

Let $g \to \infty$ we have
$$\mathbb{P}(L_{g,[a_1,b_1]}=n_1,\cdots,L_{g,[a_k,b_k]}=n_k)=\mathbb{P}(L'_{g,[a_1,b_1]}=n_1,\cdots,L'_{g,[a_k,b_k]}=n_k).$$
So $(L_{g,[a_1,b_1]},\cdots,L_{g,[a_k,b_k]})$ also converges jointly in distribution to a vector of random variables
with Poisson distributions of means $\lambda_{[a_i,b_i]}$, where
$$\lambda_{[a_i,b_i]}=3\pi(b_i^2-a_i^2).$$

Since we don't need to deal with simple zeros in the collapsing surgery,
samilarly to the proof of theorem \ref{closed}, we can prove Corollary \ref{general}.

\section{General stratum}\label{general case}
This section we consider the stratum $\H_g(m^{O(g)},1^{2g-2-mO(g)})$, where $m\geq 3$.

If a stratum has zeros of order more than $3$, the angles at both sides are not certain.
But we can consider some fixed configuration of closed saddle connection.
For a configuration $(J,b'_k,b''_k,a'_i,a''_i)$ of closed saddle connection.
We have known it suffices to consider multiplicity $1$ without cylinders, then the configuration become $(b',b'')$, where $b'+b''=m$,
which means the angles of the closed saddle connection are $(2b'+1)\pi$ on one side and $(2b''+1)\pi$ on the other side.
Now we choose all such configuration at each $m$-order zeros.

For $k$ such closed saddle connections, we can use the surgery in Section \ref{surgery} to collapse them.
Note that in this situation the opening surgery will replace the zero of order $m$ to two zeros of order $1$ and $m-1$,
and the angles at both sides of the zero of $m-1$ order are $(2b')\pi$ and $(2b'')\pi$. Then after moving and pinching,
the simple zero become two regular points and the zero of $m-1$ order become two zeros of order $b'-1$ and $b''-1$.
Finally we obtain a mapping
$$\hat{F}: \hat{\mathcal{C}}'_{g,Per}(m^{O(g)},1^{O'(g)})\to \tilde{\H}_{g-K}((b'-1)^K,(b''-1)^K,m^{O(g)-K},1^{O'(g)})\times \prod^k_{i=1} (A_{[\frac{a_i}{\sqrt{g}},\frac{b_i}{\sqrt{g}}]})^{n_i},$$
where $O'(g)=2g-2-mO(g)$.
Again we have
$$\lim_{g\to \infty}\mathbb{E}[(L'_{g,[a_1,b_1]})_{n_1}\cdots (L'_{g,[a_k,b_k]})_{n_k}]=\prod^k_{i=1}(\frac{(m+1)\pi(b_i^2-a_i^2)}{b'b''})^{n_i}.$$

And similar to the proof of theorem \ref{closed} we have
\begin{theorem}
For the stratum $\H(m^{O(g)},1^{2g-2-mO(g)})$, a configuration $(b',b'')$ and $a,b\in \mathbb{R}$,
let the random variable
$$L_{\mathcal{C},g,[a,b]}:\H(m^{O(g)},1^{2g-2-mO(g)}) \to \mathbb{N}_0$$
be the number of closed saddle connections on $(X,\omega)$ satisfying the configuration with lengths in $[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]$.
Then for disjoint intervals $[a_1,b_1],\cdots,[a_k,b_k]$, when $g\to \infty$, the random variable sequence
$$(L_{\mathcal{C},g,[a_1,b_1]},\cdots,L_{\mathcal{C},g,[a_k,b_k]}):\H(m^{O(g)},1^{2g-2-mO(g)}) \to \mathbb{N}^k_0$$
converges jointly in distribution to a vector of random variables with Poisson distributions of means $\lambda_{[a_i,b_i]}$, where
$$\lambda_{[a_i,b_i]}=\frac{(m+1)\pi(b_i^2-a_i^2)}{b'b''}.$$

\end{theorem}

In 2024, Masur, Rafi and Randecker study the distribution of open saddle connections on a random translation surface in \cite{masur2024lengths}.
In this paper we study the distribution of closed saddle connections.

A translation surface can be denoted by $(X,\omega)$ where $X$ is a closed Riemann surface of genus $g$ and $\omega$ is an Abelian differential on $X$.
$(X,\omega)$ has a flat structure defined by the conical metric $|\omega|$, with singularities at the zeros of $\omega$.
If an $|\omega|$-geodesic of $(X,\omega)$ connects two zeros (or one zero to itself) of $\omega$ and has no other zeros in its interior,
we call it an open (or closed) saddle connection.

Let $\H_g$ be the moduli space of unit-area translation surfaces of genus $g$.
$\H_g$ has a natural stratification according to the zero order of Abelian differentials.
Particularly, the stratum consisting of all abelian differentials in $\H_g$ which have only simple zeros is called principal stratum,
denoted by $\H_g(1^{2g-2})$. For convenience, if a stratum consists of Abelian differentials on which the number of zeros of order $d_i$ is $\kappa_i$
we denote it by
$$\H_g(d_1^{\kappa_1},\cdots,d_k^{\kappa_k})$$
where $\sum^{k}_{i=1}\kappa_id_i=2g-2$.

Together with a finite natural measure which is introduced by Masur and Veech denoted by $\mu_{MV}$,
every stratum $\H_g(\kappa)$ of $\H_g$ becomes a probability space
with probability measure $\frac{\mu_{MV}}{\operatorname{Vol}{\H_g(\kappa)}}$, where $\operatorname{Vol}{\H_g(\kappa)}$ is the measure of total stratum,
which we call the Masur-Veech volume of $\H_g(\kappa)$.

Given $(X,\omega)\in \H_g(1^{2g-2})$ and $a,b\in \mathbb{R}^+$,
let $N_{g,[a,b]}(X,\omega)$ be the number of open saddle connections with lengths in the interval $[\frac{a}{g},\frac{b}{g}]$ on $(X,\omega)$.
Then $N_{g,[a,b]}:\H_g(1^{2g-2})\to \mathbb{N}_0$ becomes a random variable.
In \cite{masur2024lengths}, Masur, Rafi and Randecker proved :

Together with a finite natural measure which is introduced by Masur and Veech denoted by $\mu_{MV}$,
every stratum $\H_g(\kappa)$ of $\H_g$ becomes a probability space
with probability measure $\frac{\mu_{MV}}{\V{\H_g(\kappa)}}$, where $\V{\H_g(\kappa)}$ is the measure of total stratum,
which we call the Masur-Veech volume of $\H_g(\kappa)$.

Their method is based on the relationship between Poisson distribution and its factorial moment \cite{bollobas2001random},
which was used first in the work of Mirzakhani–Petri \cite{mirzakhani2019lengths} to study the distribution of closed hyperbolic geodesics on random hyperbolic surfaces in Teichm\"uller space with respect to Weil-Peterson measure.

\begin{theorem}\label{closed}
For the stratum $\H(2^{g-1})$ and disjoint intervals $[a_1,b_1],\cdots,[a_k,b_k]$, when $g\to \infty$, the random variable sequence
$$(L_{g,[a_1,b_1]},\cdots,L_{g,[a_k,b_k]}):\H(2^{g-1}) \to \mathbb{N}^k_0$$
converges jointly in distribution to a vector of random variables with Poisson distributions of means $\lambda_{[a_i,b_i]}$, where
$$\lambda_{[a_i,b_i]}=3\pi(b_i^2-a_i^2).$$
\end{theorem}
\begin{remark}
We will expalin the question is trivial for the principal stratum in Section \ref{claim}:
the probability measure of $\mathcal{C}_g(1^{2g-2})[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]$ is zero when $g\to \infty$.
For principal stratum, the length interval to be considered should be $[a,b]$.
\end{remark}

Recall that a random variable $X:\Omega \to \mathbb{N}_0$ is Poisson distributed with mean $\lambda$ if
$$\mathbb{P}(X=k)=\frac{\lambda^k e^{-\lambda}}{k!}.$$
The following theorem from \cite{bollobas2001random} gives the relationship between factorial moment and distribution for Poisson distribution:
\begin{theorem}[The method of moment from \cite{bollobas2001random}, Theorem 1.23]\label{moment}
Let $\left\{(\Omega_i,\mathbb{P}_i)\right\}_{i\in \mathbb{N}}$ be a sequence of probability spaces.
For $m\in \mathbb{N}$, let $N_{1,i},\cdots,N_{m,i}:\Omega_i\to \mathbb{N}_0$ be random variables for all $i\in \mathbb{N}$
and suppose there exist $\lambda_1,\cdots,\lambda_m\in(0,\infty)$ such that:
$$\lim_{i\to \infty}\mathbb{E}[(N_{1,i})_{k_1}\cdots(N_{m,i})_{k_m}]=\lambda_1^{k_1}\cdots\lambda_m^{k_m}$$
for all $k_1\cdots k_m\in \mathbb{N}$. Then
$$\lim_{i\to \infty}\mathbb{P}[(N_{1,i})=n_1,\cdots,(N_{m,i})=n_m]=\prod^k_{j=1}\frac{\lambda_j^{n_j} e^{-\lambda_j}}{n_j!}.$$
That is, $(N_{1,i},\cdots,N_{m,i}):\Omega_i \to \mathbb{N}^m_0$ converges jointly in distribution to a vector which is independently Poisson distributed with means
$\lambda_1,\cdots,\lambda_m$.

From Theorem \ref{Volume}, we have
$$\lim_{g\to \infty}\frac{\V(\H_{g-K}(2^{g-1-K}))}{\V(\H_{g}(2^{g-1}))}=3^K.$$
So we have
$$\lim_{g\to \infty}\mathbb{E}[(L'_{g,[a_1,b_1]})_{n_1}\cdots (L'_{g,[a_k,b_k]})_{n_k}]=\prod^k_{i=1}[3\pi(b_i^2-a_i^2)]^{n_i}.$$
By theorem \ref{moment}, $(L'_{g,[a_1,b_1]},\cdots,L'_{g,[a_k,b_k]})$ converges jointly in distribution to a vector of random variables with Poisson distributions of means $\lambda_{[a_i,b_i]}$, where
$$\lambda_{[a_i,b_i]}=3\pi(b_i^2-a_i^2).$$

From the property of Poisson distribution, we have
$$\mathbb{P}(L'_{g,[a_1,b_1]}=n_1,\cdots,L'_{g,[a_k,b_k]}=n_k)=\prod_{i=1}^k\frac{\lambda_{[a_i,b_i]}^{n_i}e^{-\lambda_{[a_i,b_i]}}}{n_i!}.$$
On the other hand
$$\mathbb{P}(L'_{g,[a_1,b_1]}=n_1,\cdots,L'_{g,[a_k,b_k]}=n_k)=\frac{\V(\{(X,\omega):L'_{g,[a_i,b_i]}(X,\omega)=n_i,i=1,\cdots,k\})}{\V(\mathcal{C}'_{g,Per}(2^{g-1}))}.$$
Note that
\begin{equation*}
\begin{aligned}
&\V(\{(X,\omega):L_{g,[a_i,b_i]}(X,\omega)=n_i,i=1,\cdots,k\})-
\V(\mathcal{C}_g^{1}(\frac{B}{\sqrt{g}})\cup \mathcal{C}_g^{2}(\frac{B}{\sqrt{g}}))\\ &\leq \V(\{(X,\omega):L'_{g,[a_i,b_i]}(X,\omega)=n_i,i=1,\cdots,k\})\\&\leq
\V(\{(X,\omega):L_{g,[a_i,b_i]}(X,\omega)=n_i,i=1,\cdots,k\}).
\end{aligned}
\end{equation*}
From Proposition \ref{chain} and \ref{intersect}, we have
$$\lim_{g\to \infty}\frac{\V(\mathcal{C}'_{g,Per}(2^{g-1}))}{\V(\H_g(2^{g-1}))}=1,\\ \lim_{g\to \infty}\frac{\V(\mathcal{C}_g^{1}(\frac{B}{\sqrt{g}})\cup \mathcal{C}_g^{2}(\frac{B}{\sqrt{g}}))}{\V(\H_g(2^{g-1}))}=0.$$
So
\begin{equation*}
\begin{aligned}
&\mathbb{P}(L_{g,[a_1,b_1]}=n_1,\cdots,L_{g,[a_k,b_k]}=n_k)-
\frac{\V(\mathcal{C}_g^{1}(\frac{B}{\sqrt{g}})\cup \mathcal{C}_g^{2}(\frac{B}{\sqrt{g}}))}{\V(\H_g(2^{g-1}))}\\ &\leq \mathbb{P}(L'_{g,[a_1,b_1]}=n_1,\cdots,L'_{g,[a_k,b_k]}=n_k)\frac{\V(\mathcal{C}'_{g,Per}(2^{g-1}))}{\V(\H_g(2^{g-1}))}\\&\leq
\mathbb{P}(L_{g,[a_1,b_1]}=n_1,\cdots,L_{g,[a_k,b_k]}=n_k).
\end{aligned}
\end{equation*}

Let $g \to \infty$ we have
$$\mathbb{P}(L_{g,[a_1,b_1]}=n_1,\cdots,L_{g,[a_k,b_k]}=n_k)=\mathbb{P}(L'_{g,[a_1,b_1]}=n_1,\cdots,L'_{g,[a_k,b_k]}=n_k).$$
So $(L_{g,[a_1,b_1]},\cdots,L_{g,[a_k,b_k]})$ also converges jointly in distribution to a vector of random variables
with Poisson distributions of means $\lambda_{[a_i,b_i]}$, where
$$\lambda_{[a_i,b_i]}=3\pi(b_i^2-a_i^2).$$

And similar to the proof of theorem \ref{closed} we have
\begin{theorem}
For the stratum $\H(m^{O(g)},1^{2g-2-mO(g)})$, a configuration $(b',b'')$ and $a,b\in \mathbb{R}$,
let the random variable
$$L_{\mathcal{C},g,[a,b]}:\H(m^{O(g)},1^{2g-2-mO(g)}) \to \mathbb{N}_0$$
be the number of closed saddle connections on $(X,\omega)$ satisfying the configuration with lengths in $[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]$.
Then for disjoint intervals $[a_1,b_1],\cdots,[a_k,b_k]$, when $g\to \infty$, the random variable sequence
$$(L_{\mathcal{C},g,[a_1,b_1]},\cdots,L_{\mathcal{C},g,[a_k,b_k]}):\H(m^{O(g)},1^{2g-2-mO(g)}) \to \mathbb{N}^k_0$$
converges jointly in distribution to a vector of random variables with Poisson distributions of means $\lambda_{[a_i,b_i]}$, where
$$\lambda_{[a_i,b_i]}=\frac{(m+1)\pi(b_i^2-a_i^2)}{b'b''}.$$

Their method is based on the relationship between Poisson distribution and its factorial moment \cite{bollobas2001random},
which was used first in the work of Mirzakhani–Petri \cite{mirzakhani2019lengths} to study the distribution of closed hyperbolic geodesics on random hyperbolic surfaces in Teichm\"uller space with respect to Weil-Peterson measure.

As for translation surface, to use the method,
\cite{masur2024lengths} introduces a surgery to collapse the open saddle connections to obtain a new general translation surface in a new stratum.
This surgery does not work for closed saddle connections and they proposed the distribution question in the situation of closed saddle connections,
which inspires this work.
This paper gives a new surgery to collapse a closed saddle connection, which ensures the same method can be used for closed saddle connections.

Let $L_{g,[a,b]}(X,\omega)$ be the number of closed saddle connections on $(X,\omega)$ with lengths in $[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]$.
Denote by $\mathcal{C}_g(\kappa)[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]$ the subset of $\H_g(\kappa)$ consisting of translation
surfaces on which there exist closed saddle connections
with lengths in $[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]$.
The main result of this paper is the following theorem

\begin{theorem}\label{closed}
For the stratum $\H(2^{g-1})$ and disjoint intervals $[a_1,b_1],\cdots,[a_k,b_k]$, when $g\to \infty$, the random variable sequence
$$(L_{g,[a_1,b_1]},\cdots,L_{g,[a_k,b_k]}):\H(2^{g-1}) \to \mathbb{N}^k_0$$
converges jointly in distribution to a vector of random variables with Poisson distributions of means $\lambda_{[a_i,b_i]}$, where
$$\lambda_{[a_i,b_i]}=3\pi(b_i^2-a_i^2).$$
\end{theorem}
\begin{remark}
We will expalin the question is trivial for the principal stratum in Section \ref{claim}:
the probability measure of $\mathcal{C}_g(1^{2g-2})[\frac{a}{\sqrt{g}},\frac{b}{\sqrt{g}}]$ is zero when $g\to \infty$.
For principal stratum, the length interval to be considered should be $[a,b]$.
\end{remark}

\begin{corollary}\label{general}
Let $\H_g(2^{O(g)},1^{2g-2-2O(g)})$ be a stratum with $O(g)$ double zeros, then the result in Theorem \ref{closed} is also true.
\end{corollary}

Their method is based on the relationship between Poisson distribution and its factorial moment \cite{bollobas2001random}, which was used first in the work of Mirzakhani--Petri \cite{mirzakhani2019lengths}. For translation surfaces, \cite{masur2024lengths} introduces a surgery to collapse the open saddle connections to obtain a new general translation surface in a new stratum. "This surgery does not work for closed saddle connections"; the paper "gives a new surgery to collapse a closed saddle connection, which ensures the same method can be used for closed saddle connections."

\label{open}
Let $[a_1,b_1],\cdots,[a_k,b_k]$ be $k$ disjoint intervals. Then as $g\to \infty$, the vector of random variables
$$(N_{g,[a_1,b_1]},\cdots,N_{g,[a_k,b_k]}):\H_g(1^{2g-2})\to \mathbb{N}_0^k$$
converges jointly in distribution to a vector of random variables with Poisson distributions of means $\lambda_{[a_i,b_i]}$, where
$$\lambda_{[a_i,b_i]}=8\pi(b_i^2-a_i^2)$$
for $i=1,\cdots,k$. That is,
$$\mathbb{P}(N_{g,[a_1,b_1]}=n_1,\cdots,N_{g,[a_k,b_k]}=n_k)=\prod_{i=1}^k\frac{\lambda_{[a_i,b_i]}^{n_i}e^{-\lambda_{[a_i,b_i]}}}{n_i!}$$,

Question: Let H_g(1^{2g-2}) denote the principal stratum of unit-area translation surfaces of genus g, consisting of Abelian differentials with 2g-2 simple zeros, equipped with the normalized Masur–Veech probability measure. For (X,ω) in H_g(1^{2g-2}) and an interval [a,b] with a,b>0, let N_{g,[a,b]}(X,ω) be the number of open saddle connections on (X,ω), meaning |ω|-geodesic segments joining two distinct zeros of ω and containing no zero in their interior, whose lengths lie in [a/g,b/g]. If [a_1,b_1],...,[a_k,b_k] are k pairwise disjoint intervals, then as g→∞, which limiting statement holds for the random vector (N_{g,[a_1,b_1]},...,N_{g,[a_k,b_k]})?

As g→∞, (N_{g,[a_1,b_1]},...,N_{g,[a_k,b_k]}) converges jointly in distribution to a vector of independent Poisson random variables with means λ_[a_i,b_i] = 8π(b_i^2-a_i^2) for i=1,...,k. Equivalently, for every fixed n_1,...,n_k in N_0, lim_{g→∞} P(N_{g,[a_1,b_1]}=n_1,...,N_{g,[a_k,b_k]}=n_k) = ∏_{i=1}^k [λ_[a_i,b_i]^{n_i} e^{-λ_[a_i,b_i]} / n_i!].

Choices:
B. As g→∞, (N_{g,[a_1,b_1]},...,N_{g,[a_k,b_k]}) converges jointly in distribution to a vector of independent Poisson random variables with means λ_[a_i,b_i] = 8π(b_i-a_i) for i=1,...,k. Equivalently, for every fixed n_1,...,n_k in N_0, lim_{g→∞} P(N_{g,[a_1,b_1]}=n_1,...,N_{g,[a_k,b_k]}=n_k) = ∏_{i=1}^k [λ_[a_i,b_i]^{n_i} e^{-λ_[a_i,b_i]} / n_i!].
C. As g→∞, each individual random variable N_{g,[a_i,b_i]} converges in distribution to a Poisson random variable with mean λ_[a_i,b_i] = 8π(b_i^2-a_i^2) for i=1,...,k.
D. As g→∞, (N_{g,[a_1,b_1]},...,N_{g,[a_k,b_k]}) converges jointly in distribution to a vector of Poisson random variables with means λ_[a_i,b_i] = 8π(b_i^2-a_i^2) for i=1,...,k, but in general the limiting coordinates need not be independent. Equivalently, for every fixed n_1,...,n_k in N_0, lim_{g→∞} P(N_{g,[a_1,b_1]}=n_1,...,N_{g,[a_k,b_k]}=n_k) exists and has Poisson marginals with those means.
E. As g→∞, (N_{g,[a_1,b_1]},...,N_{g,[a_k,b_k]}) converges jointly in distribution to a vector of independent Poisson random variables with means λ_[a_i,b_i] = 3π(b_i^2-a_i^2) for i=1,...,k. Equivalently, for every fixed n_1,...,n_k in N_0, lim_{g→∞} P(N_{g,[a_1,b_1]}=n_1,...,N_{g,[a_k,b_k]}=n_k) = ∏_{i=1}^k [λ_[a_i,b_i]^{n_i} e^{-λ_[a_i,b_i]} / n_i!].

quadratic area-scaling in the Poisson mean

joint convergence/independence of counts across disjoint intervals

factorial-moment method yields independent joint Poisson law, not merely Poisson marginals

mean constant specific to open saddle connections in the principal stratum

The stem defines the objects and asks for the limiting law, but it does not explicitly state the correct conclusion or uniquely signal option A. No direct answer leakage is present.

The item is close to a theorem-recall question: it essentially asks for the exact asymptotic statement of a known result. The alternatives do introduce meaningful variations, but the prompt is still a fairly direct request for the theorem’s conclusion.

Some reasoning is required to distinguish the full joint independent Poisson limit from weaker marginal convergence, incorrect dependence claims, and wrong mean formulas/constants. However, solving it is driven more by precise recall/recognition than by substantial generative mathematical reasoning.

The distractors are strong and mathematically plausible: one uses the wrong scaling in the mean, one gives a weaker true statement, one removes independence, and one changes the constant. These align well with realistic confusion points.

A solid MCQ with no answer leakage and high-quality distractors, but it functions mainly as theorem recognition rather than a deeply generative reasoning task.


\begin{thm}
\label{mnthm}
Let $H^\omega$ be as in \eqref{model} and assume Hypothesis~\ref{hypo}.
Then, for each $f \in C^1_{d,0}\big[-\|V\|_\infty,\infty\big)$, we have the following convergence of random variables:
\begin{equation}
\label{clt-cn}
\frac{1}{|\Lambda_L|^{\frac{1}{2}}}
\bigg(
  \operatorname{Tr}\!\big( f(H^\omega_{\Lambda_L,X}) \big)
  - \mathbb{E}\!\big[\operatorname{Tr}\!\big( f(H^\omega_{\Lambda_L,X}) \big)\big]
\bigg)
\xrightarrow[L\to\infty]{\;\;\text{distribution}\;\;}
\mathcal{N}\!\big(0,\sigma^2_{f,X}\big),
\end{equation}
where $X \in \{D,N\}$ and $\mathcal{N}\!\big(0,\sigma^2_{f,X}\big)$ denotes the normal distribution with mean $0$ and variance $\sigma^2_{f,X}$, as defined in \eqref{lim-vr}. \\~\\
Moreover, for any $f \in C^1_{d,0}\big[-\|V\|_\infty,\infty\big)$, the limiting variance $\sigma^2_{f,X}$ is finite and independent of the boundary condition $X \in \{D,N\}$. In particular,
$\sigma^2_f := \sigma^2_{f,N} = \sigma^2_{f,D} < \infty$,
and its exact expression is given by
\begin{align}
\label{extlmv}
\sigma^2_f
&= \mathbb{E}\!\Bigg[\omega_{\vec{1}_d}\,
  \mathbb{E}\!\bigg(
    \int_0^1
    \operatorname{Tr}\!\Big( u_{\vec{1}_d}(x)\,
    f'(H^\omega)_{(\omega_{\vec{1}_d}\to t\omega_{\vec{1}_d})} \Big)\,dt
  \;\Big|\; \mathcal{F}^d_{\vec{1}_d}  \bigg) \nonumber\\
&\qquad
- \mathbb{E}\!\bigg(
    \int_0^1 \omega_{\vec{1}_d}\,
    \operatorname{Tr}\!\Big( u_{\vec{1}_d}(x)\,
    f'(H^\omega)_{(\omega_{\vec{1}_d}\to t\omega_{\vec{1}_d})} \Big)\,dt
    \;\Big|\; \mathcal{F}^d_{\vec{1}_{d-1,0}}
  \bigg)
\Bigg]^2.
\end{align}
Finally, if $f \in C^1_{d,0}\big[-\|V\|_\infty, \infty\big)$ is strictly monotone and $u \ge 0$ or $u \le 0$ with $\|u\|_2 \neq 0$, then $\sigma_f^2 > 0$.
\end{thm}

In this work, we consider the random Schr\"{o}dinger operator \( H^\omega \) on the Hilbert space \( L^2(\mathbb{R}^d) \) in the presence of a magnetic field, defined by
\begin{equation}
\label{model}
H^\omega = H_A + V^\omega, \quad \omega \in \Omega, \quad \text{where} \quad H_A := (i\nabla + A)^2.
\end{equation}
Here, \( A \in \big(L^2_{\mathrm{loc}}(\mathbb{R}^d)\big)^d \) denotes the vector potential of a constant magnetic field. The operator \( H_A\), known as the \emph{free magnetic Laplacian}, is a positive, unbounded, self-adjoint operator on \( L^2(\mathbb{R}^d) \), defined via an associated quadratic form. A detailed description of the operator \( H_A \) and its domain is provided in Remark~\ref{fl-d-lp}.
We consider the alloy-type random potential $V^\omega$ defined as the bounded multiplication operator
\begin{equation}
\label{pntal}
\big(V^\omega \varphi\big)(x) = \sum_{n \in \mathbb{Z}^d} \omega_n u(x - n)\varphi(x), \quad \forall \, \varphi \in L^2(\mathbb{R}^d),
\end{equation}
where the single site potential \( u \in L^\infty(\mathbb{R}^d) \) is real-valued and compactly supported. The coefficients \( \{\omega_n\}_{n \in \mathbb{Z}^d} \) are independent and identically distributed (i.i.d.), bounded, real-valued random variables with a common distribution \( \mu \), referred to as the \emph{single site distribution} (SSD).
The underlying probability space is the product measure space \( (\Omega, \mathcal{B}_\Omega, \mathbb{P}) := \big(\mathbb{R}^{\mathbb{Z}^d}, \mathcal{B}_{\mathbb{R}^{\mathbb{Z}^d}}, \mathbb{P} \big) \), where \( \mathbb{P} = \bigotimes_{n \in \mathbb{Z}^d} \mu \) is constructed via Kolmogorov’s extension theorem. We denote elements by \( \omega = (\omega_n)_{n \in \mathbb{Z}^d} \in \Omega \).
It follows from \cite{WF} and \cite{TLMS} that \( H^\omega \) is self-adjoint on \( L^2(\mathbb{R}^d) \) for almost every \( \omega \in \Omega \). Since \( V^\omega \) is a bounded operator, we have $D(H^\omega) = D(H_A),$ for a.e. $\omega \in \Omega$.
Moreover, the mapping \( \omega \mapsto H^\omega \) is measurable with respect to
the product $\sigma$-algebra \( \mathcal{B}_\Omega \), so that \( \{H^\omega\}_{\omega \in \Omega} \) forms a measurable family of random operators. For further details, we refer to \cite{Wernr} and \cite{CL}.
 \\
Let $O\subset \mathbb{R}^d$ be a bounded open set. We denote
$H^\omega_{O, N}$ and $H^\omega_{O, D}$ as the finite-volume restriction of $H^\omega$ to the Hilbert space $L^2(O)$ with the Neumann and Dirichlet boundary conditions, respectively.   The finite-volume restriction of $H^\omega$ is defined by
\begin{equation}
\label{NDrst}
H^\omega_{O, X}=H^X_{A,O}+V^\omega_O~~\text{on}~~L^2(O),~\omega\in\Omega~~\text{and}~~X=D,N.
\end{equation}
In the above, $H^X_{A,O},~X=D, N$ denotes the finite-volume restriction of the free magnetic Laplacian $H_A$ to the Hilbert space $L^2(O)$ with Dirichlet and Neumann boundary conditions.
The operators $H^D_{A,O}$ and $H^N_{A,O}$ are unbounded, positive, self-adjoint on $L^2(O)$, a detailed description is given in Lemma \ref{dir-neu-lp}. The bounded operator $V^\omega_O$ is the restriction of $V^\omega$ to the Hilbert space $L^2(O)$, defined by $\big(V^\omega_O\varphi\big)(x)=\big(V^\omega \varphi\big)(x)~\forall~x\in O$ where $\varphi\in L^2(O)$.   Also $\{H^\omega_{O,X}\}_{\omega\in\Omega}$ forms a measurable collection of random operators,  \big(see \cite{KMcre},  \cite{TLMS} and \cite{CL}\big).
Since $V^\omega_O$ is a bounded operator, therefore $H^\omega_{O,X}$ is an unbounded self-adjoint operator on $L^2(O)$ with $D\big( H^\omega_{O,X} \big)=D\big( H^X_{A,O} \big)$ for a.e. $\omega$,  here $X=D,N$.\\
Before describing our result, we formally make assumptions on the vector potential $A$, single site potential $u$,  and the single site distribution (SSD) $\mu$.
\begin{hyp}\indent
\label{hypo}
\begin{enumerate}
\item The single site distribution (SSD) $\mu$ is compactly supported.
\item The single site potential $u\in L^\infty(\mathbb{R}^d)$ is real-valued and compactly supported and denote $u_n(x)=u(x-n)$, $n\in\mathbb{Z}^d$.
\item  The vector potential $A(x)=\big(A_1(x),A_2(x),\cdots,A_d(x)\big):\mathbb{R}^d\to\mathbb{R}^d$ is Borel-measurable and it is given by
$$A(x)=\frac{1}{2}Bx,~x=(x_1,x_2,\cdots,x_d)\in\mathbb{R}^d.$$
Here $B=\big[ B_{i,j} \big]_{1\leq i,j\leq d}$ is a real skew-symmetric $d\times d$ matrix
representing the constant magnetic field.
\end{enumerate}
\end{hyp}
\begin{rem}
\label{bnd-f-pnt}
\noindent Under Hypothesis~\ref{hypo}, there exists a deterministic constant $\|V\|_\infty$ such that $\|V^\omega_O\| \leq \|V\|_\infty~\text{for a.e.}~\omega$ and every open set $O \subseteq \mathbb{R}^d$.
\end{rem}
\begin{rem}
Under Hypothesis~\ref{hypo}, it is clear that each component $A_k$ of the vector potential $A$ can be written as
$A_k(x_1,x_2,\cdots, x_d)=\frac{1}{2}\displaystyle\sum_{j=1}^dx_jB_{k,j}$, for $k=1,2,\cdots, d$.  Now $A_k\in L^2_{\mathrm{loc}}(\mathbb{R}^d)$ is immediate.
\end{rem}
\noindent Denote $m=(m_1,m_2,\cdots,m_d)\in\mathbb{Z}^d$. We define a family of the unitary operators $\{U_m\}_{m\in\mathbb{Z}^d}$ on $L^2(\mathbb{R}^d)$ as
\begin{equation}
\label{unop}
\big(U_m\varphi\big)(x)=e^{i\Psi_m(x)}\varphi(x-m)~~\forall~~\varphi\in L^2(\mathbb{R}^d),~\text{here}~i=\sqrt{-1},
\end{equation}
here $\Psi_m(x)=\displaystyle\frac{1}{2}\sum_{j,k=1}^d(m_j-x_j)B_{k,j}m_k,~x=(x_1,x_2,\cdots,x_d)\in\mathbb{R}^d$.  We also define an ergodic family $\{T_m\}_{m\in\mathbb{Z}^d}$ of measure-preserving transformations on the product probability space $\big(\Omega, \mathcal{B}_\Omega, \mathbb{P} \big)$ as
\begin{equation}
\label{mpt}
\big(T_m\omega\big)_n=\omega_{n-m},~~\omega=(\omega_n)_{n\in\mathbb{Z}^d}\in\Omega=\mathbb{R}^{\mathbb{Z}^d}
\end{equation}
$\{H^\omega\}_{\omega\in\Omega}$ is an ergodic family of random self-adjoint operators (unbounded) on $L^2(\mathbb{R}^d)$,  in the sense that $U_mH^\omega U_m^*=H^{T_m\omega}~~\forall~~\omega\in\Omega$ and $m\in\mathbb{Z}^d$.  We refer to \cite{Wernr} for details about ergodic operators (self-adjoint); see also \cite{CL}.   Since $H^\omega$ is an ergodic operator (self-adjoint), its spectral components are non-random subsets of the real line for a.e $\omega$, and the discrete spectrum of $H^\omega$ is empty for a.e. $\omega$, details can be found in \cite{Wernr},  \cite{CL} and \cite{ U}.  The spectrum of the finite-volume restriction $H^\omega_{O, X}$ \big(of $H^\omega$\big) is always purely discrete for a.e. $\omega$, $X=D,N,$  see \cite{TLMS} and \cite{RS4} for details.\\
First, we set a few notations to define the integrated density of states (IDS) of the ergodic operator $H^\omega$.  Let $\Lambda_L\subseteq\mathbb{R}^d$ be the cube (open) of side length $L$ centered at the origin i.e.
\begin{equation}
\label{cbL}
\Lambda_L=\bigg\{x=(x_1,x_2,\cdots, x_d)\in\mathbb{R}^d:|x_i|< \frac{L}{2}  \bigg\},~~L\in\mathbb{N}.
\end{equation}
Now the integrated density of states (IDS) of $H^\omega$ can be described by the limit
\begin{equation}
\label{ids}
\lim_{L\to\infty}\frac{1}{\big|\Lambda_L  \big|}\operatorname{Tr}\!\big( f\big(H^\omega_{\Lambda_L,X}  \big) \big)=
\mathbb{E}\left[\operatorname{Tr}\!\left(\chi_{\Lambda_1}f\big( H^\omega\big)\chi_{\Lambda_1} \right)\right]~~a.e.~\omega,~~\forall~f\in C_c(\mathbb{R}).
\end{equation}
In the above, $C_c(\mathbb{R})$ denotes the set of all continuous functions defined on $\mathbb{R}$ with compact support. The existence of the limit in $(\ref{ids})$ is independent of the choice of boundary condition $X=D,N$. The distribution function $\mathcal{N}(\cdot)$ defined by $\mathcal{N}(x)=\mathbb{E}\left[\operatorname{Tr}\!\left(\chi_{\Lambda_1} E_{ H^\omega}(-\infty,x]\chi_{\Lambda_1}\right) \right],~x\in\mathbb{R}$ is known as the integrated density of states (IDS) of $H^\omega$.  \\
The existence of the limit $(\ref{ids})$ is a well-studied topic in the literature.
A proof of the existence of IDS in the sense of vague convergence (a.e. $\omega$) of random measures can be found in \cite{TLMS1},  the generalisation of \cite{PF}, which deals with the case when the magnetic field is absent. Use of the functional-analytic argument is given in \cite{MH}; it is first presented in \cite{KM} without a magnetic field. A different approach using Feynman-Kac(-It\^{o}) functional-integral representation of Schr\"{o}dinger semigroups can be found in \cite{U, BHL} and it goes back to \cite{pasl,  Nas} for $A=0$.    For the uniqueness of the limit $(\ref{ids})$,  \big(its independence of the boundary condition $X=D,N$\big), we refer to \cite{TLMS1,  DIM,  Na} for non-zero magnetic field and \cite{Wernr,  CL} for zero magnetic field. More details about IDS and, in general, random Schr\"{o}dinger operator is well documented in \cite{KB, His, CFKS, ves}.\\
Let us regard the limit in \((\ref{ids})\) as the analogue of the law of large numbers (LLN) for the trace functional $\operatorname{Tr}\!\big(f(H^\omega_{\Lambda_L,X})\big)$.
Our main goal is to establish an analogue of the central limit theorem (CLT) for \((\ref{ids})\) and to demonstrate its independence from the choice of boundary conditions \(X = D, N\). More explicitly, we aim to study the fluctuations of the trace functional $\operatorname{Tr}\!\big(f(H^\omega_{\Lambda_L,X})\big)$ around its mean as \(L \to \infty\).
\begin{rem}
\label{lbs}
In view of Remark~\ref{bnd-f-pnt}, we have
$\sigma(H^\omega_{O,X}) \subseteq \big[-\|V\|_\infty,\infty\big)$,  and
$\sigma(H^\omega) \subseteq \big[-\|V\|_\infty,\infty\big)$ a.e. $\omega$,
where $O \subset \mathbb{R}^d$ and $X \in \{D,N\}$.
\end{rem}
\noindent We now define the class of test functions $C^1_{d,0}[-\|V\|_\infty,\infty)$, for which we will establish an analogue of the central limit theorem for the limit (\ref{ids}).
\begin{defin}
\label{defcl}
We say $f\in C^1_{d, 0}\big[-\|V\|_\infty,\infty\big)$ if both $f$ and $f'$ are real-valued continuous functions on $\big[-\|V\|_\infty,\infty\big)$ and $|f(x)|=O(x^{-{m_1}})$, $|f'(x)|=O(x^{-{m_2}})$ as $x\to\infty$ for some $m_1>d+1$,  $m_2>d+1$.
\end{defin}
\noindent Now for any real-valued Borel measurable function $f$ on $\mathbb{R}$ we define the random variable $Y_{f,X,L}(\omega)$ as
\begin{equation}
\label{rv}
Y_{f,X,L}(\omega)=\bigg(\operatorname{Tr}\!\big( f\big(H^\omega_{\Lambda_L,X}  \big) \big)-\mathbb{E}
\big[\operatorname{Tr}\!\big( f\big(H^\omega_{L,X}  \big) \big) \big]\bigg),~~X=D,N.
\end{equation}
\noindent We denote $\sigma^2_{f,X}$as the limiting variance of the random variable $Y_{f,X,L}$, as $L\to\infty$ and it is given by
\begin{align}
\label{lim-vr}
\sigma^2_{f,X}&=\lim_{L\to\infty}\frac{1}{|\Lambda_L|} \mathrm{Var}\big(Y_{f,X,L}  \big),~~~~X=D,N\nonumber\\
&=\lim_{L\to\infty}\frac{1}{|\Lambda_L|}\mathbb{E}\bigg[\operatorname{Tr}\!\big( f\big(H^\omega_{\Lambda_L,X}  \big) \big)-\mathbb{E}\big[\operatorname{Tr}\!\big( f\big(H^\omega_{\Lambda_L,X}  \big) \big) \big]\bigg]^2.
\end{align}
\begin{rem}
We will show that for \( f \in C^1_{d, 0}\big[-\|V\|_\infty,\infty\big) \), the limit~\eqref{lim-vr} exists, and the limiting variance \(\sigma^2_{f,X}\) is independent of the choice of boundary conditions \(X = D, N\).
\end{rem}
\noindent To express the limiting variance $\sigma^2_{f,X}$ explicitly, we introduce a family of subsets of $\mathbb{Z}^d$ together with the corresponding $\sigma$-algebras. For $n = (n_1, n_2,\ldots, n_d) \in \mathbb{Z}^d$, define
\[
\begin{aligned}
A_0^1 &= \{\, n \in \mathbb{Z}^d : n_1 \le 0 \,\}, \\
A_1^1 &= \{\, n \in \mathbb{Z}^d : n_1 \le 1 \,\}, \\
A_{1,0}^2 &= A_0^1 \,\cup\, \{\, n \in \mathbb{Z}^d : n_1 \le 1,\, n_2 \le 0 \,\}, \\
A_{1,1}^2 &= A_0^1 \,\cup\, \{\, n \in \mathbb{Z}^d : n_1 \le 1,\, n_2 \le 1 \,\}, \\
A_{1,1,0}^3 &= A_{1,0}^2 \,\cup\, \{\, n \in \mathbb{Z}^d : n_1 \le 1,\, n_2 \le 1,\, n_3 \le 0 \,\}, \\
A_{1,1,1}^3 &= A_{1,0}^2 \,\cup\, \{\, n \in \mathbb{Z}^d : n_1 \le 1,\, n_2 \le 1,\, n_3 \le 1 \,\}.
\end{aligned}
\]
Now successively we can define
\begin{equation*}
\label{sgmalg}
\begin{split}
& A^d_{\underbrace{\scriptstyle 1,1, \cdots,1,0}_{\scriptstyle d}}=A^{d-1}_{\underbrace{\scriptstyle 1,1,\cdots,1,0}_{\scriptstyle d-1}}
\cup \big\{n \in\mathbb{Z}^d: n_1\leq 1, n_2\leq 1\cdots,n_{d-1}\leq 1, n_d\leq 0\big\},\\
& A^d_{\underbrace{\scriptstyle 1,1,\cdots,1,1}_{\scriptstyle d}}=A^{d-1}_{\underbrace{\scriptstyle 1,1,\cdots,1,0}_{\scriptstyle d-1}}
\cup \big\{n \in\mathbb{Z}^d: n_1\leq 1, n_2\leq 1,\cdots,n_{d-1}\leq 1, n_d\leq 1\big\}.
\end{split}
\end{equation*}
Now we define the $\sigma$-algebras associated with the above two subsets of $\mathbb{Z}^d$ as
\begin{align}
\label{smag}
\begin{split}
 \mathcal{F}^d_{\vec{1}_d}&=
\sigma\bigg(\omega_n:n\in A^d_{\underbrace{\scriptstyle 1,1,\cdots,1,1}_{\scriptstyle d}}  \bigg),~~~~\vec{1}_d= (\underbrace { 1,1,\cdots,1,1,1}_{\scriptstyle d})\in\mathbb{Z}^d,\\
\mathcal{F}^d_{\vec{1}_{d-1,0}}&=
\sigma\bigg(\omega_n:n\in A^d_{\underbrace{\scriptstyle 1,1,\cdots,1,0}_{\scriptstyle d}}  \bigg),~~\vec{1}_{d-1,0}= ( \underbrace { 1,1,\cdots,1,1}_{\scriptstyle d-1},0)\in\mathbb{Z}^d.
\end{split}
\end{align}
For each \( k \in \mathbb{Z}^d \) and \( t \in [0,1] \), we define the modified random operator
\[
H^\omega\big|_{(\omega_k \to t\omega_k)}
= H_A + t\omega_k\,u(\cdot - k)
+ \sum_{\substack{n \in \mathbb{Z}^d \\ n \ne k}} \omega_n\,u(\cdot - n).
\]
Then, using the spectral theorem for self-adjoint operators, we define, for any measurable function \( g : \mathbb{R} \to \mathbb{C} \),
\begin{equation}
\label{mdfmdl}
g(H^\omega)_{(\omega_k \to t\omega_k)} :=
g\!\left(H^\omega\big|_{(\omega_k \to t\omega_k)}\right).
\end{equation}
\noindent Now we are ready to state the main result of this work.

, \\
A_1^1 &= \{\, n \in \mathbb{Z}^d : n_1 \le 1 \,\}, \\
A_{1,0}^2 &= A_0^1 \,\cup\, \{\, n \in \mathbb{Z}^d : n_1 \le 1,\, n_2 \le 0 \,\}, \\
A_{1,1}^2 &= A_0^1 \,\cup\, \{\, n \in \mathbb{Z}^d : n_1 \le 1,\, n_2 \le 1 \,\}, \\
A_{1,1,0}^3 &= A_{1,0}^2 \,\cup\, \{\, n \in \mathbb{Z}^d : n_1 \le 1,\, n_2 \le 1,\, n_3 \le 0 \,\}, \\
A_{1,1,1}^3 &= A_{1,0}^2 \,\cup\, \{\, n \in \mathbb{Z}^d : n_1 \le 1,\, n_2 \le 1,\, n_3 \le 1 \,\}.
\end{aligned}
\]
Now successively we can define
\begin{equation*}
\label{sgmalg}
\begin{split}
& A^d_{\underbrace{\scriptstyle 1,1, \cdots,1,0}_{\scriptstyle d}}=A^{d-1}_{\underbrace{\scriptstyle 1,1,\cdots,1,0}_{\scriptstyle d-1}}
\cup \big\{n \in\mathbb{Z}^d: n_1\leq 1, n_2\leq 1\cdots,n_{d-1}\leq 1, n_d\leq 0\big\},\\
& A^d_{\underbrace{\scriptstyle 1,1,\cdots,1,1}_{\scriptstyle d}}=A^{d-1}_{\underbrace{\scriptstyle 1,1,\cdots,1,0}_{\scriptstyle d-1}}
\cup \big\{n \in\mathbb{Z}^d: n_1\leq 1, n_2\leq 1,\cdots,n_{d-1}\leq 1, n_d\leq 1\big\}.
\end{split}
\end{equation*}
Now we define the $\sigma$-algebras associated with the above two subsets of $\mathbb{Z}^d$ as
\begin{align}
\label{smag}
\begin{split}
 \mathcal{F}^d_{\vec{1}_d}&=
\sigma\bigg(\omega_n:n\in A^d_{\underbrace{\scriptstyle 1,1,\cdots,1,1}_{\scriptstyle d}} \bigg),~~~~\vec{1}_d= (\underbrace { 1,1,\cdots,1,1,1}_{\scriptstyle d})\in\mathbb{Z}^d,\\
\mathcal{F}^d_{\vec{1}_{d-1,0}}&=
\sigma\bigg(\omega_n:n\in A^d_{\underbrace{\scriptstyle 1,1,\cdots,1,0}_{\scriptstyle d}} \bigg),~~\vec{1}_{d-1,0}= ( \underbrace { 1,1,\cdots,1,1}_{\scriptstyle d-1},0)\in\mathbb{Z}^d.
\end{split}
\end{align}
For each \( k \in \mathbb{Z}^d \) and \( t \in [0,1] \), we define the modified random operator
\[
H^\omega\big|_{(\omega_k \to t\omega_k)}
= H_A + t\omega_k\,u(\cdot - k)
+ \sum_{\substack{n \in \mathbb{Z}^d \\ n \ne k}} \omega_n\,u(\cdot - n).
\]
Then, using the spectral theorem for self-adjoint operators, we define, for any measurable function \( g : \mathbb{R} \to \mathbb{C} \),
\begin{equation}
\label{mdfmdl}
g(H^\omega)_{(\omega_k \to t\omega_k)} :=
g\!\left(H^\omega\big|_{(\omega_k \to t\omega_k)}\right).
\end{equation}
\noindent Now we are ready to state the main result of this work.

\begin{lem}
\label{dir-neu-lp}
Let $O\subseteq \mathbb{R}^d$ be an open set and consider the sesquilinear forms $h^N_{A, O}(\cdot, \cdot)$ and $h^D_{A, O}(\cdot, \cdot)$ as defined in (\ref{ses-neu}) and (\ref{ses-dir}),  respectively.
\begin{enumerate}
\item[(i)] These two sesquilinear forms are positive, symmetric, and closed. Therefore, they each uniquely define a positive self-adjoint operator denoted $H^N_{A, O}$ and $H^D_{A, O}$, respectively.
\item[(ii)] The restricted Neumann magnetic Laplacian $H^N_{A, O}$ is given by
 \begin{equation}
\label{neu-lapla}
h^N_{A,O}(\psi, u)=\big\langle \psi, H^N_{A, O}u\big\rangle,~\forall~\psi\in W^{1,2}_A(O),~u\in
D(H^N_{A,O}).
\end{equation}
Here the domain of the operator $H^N_{A,O}$ is described by
\begin{align}
\label{neu-dom}
D(H^N_{A,O})&=\bigg\{u\in W^{1,2}_A(O): \text{there~exists}~\tilde{u}\in L^2(O),~\text{such that}\nonumber\\
&\qquad \qquad \qquad h^N_{A, O}(\psi,u)=\langle \psi, \tilde{u}\rangle~\forall~\psi\in W^{1,2}_A(O)  \bigg\},
\end{align}
and for $u\in D(H^N_{A, O})$ we write $H^N_{A, O}u=\tilde{u}$.
\item[(iii)] The restricted Dirichlet magnetic Laplacian $H^D_{A, O}$ is given by
 \begin{equation}
\label{dir-lapla}
h^D_{A,O}(\psi, u)=\big\langle \psi, H^D_{A, O}u\big\rangle,~\forall~\psi\in W^{1,2}_{A,cl}(O),~u\in
D(H^D_{A,O}).
\end{equation}
Here the domain of the operator $H^D_{A,O}$ is described by
\begin{align}
\label{dir-dom}
D(H^D_{A,O})&=\bigg\{u\in W^{1,2}_{A,cl}(O): \text{there~exists}~\tilde{u}\in L^2(O),~\text{such that}\nonumber\\
&\qquad \qquad \qquad h^D_{A, O}(\psi,u)=\langle \psi, \tilde{u}\rangle~\forall~\psi\in W^{1,2}_{A,cl}(O)  \bigg\},
\end{align}
and for $u\in D(H^D_{A, O})$ we write $H^D_{A, O}u=\tilde{u}$.
\end{enumerate}
\end{lem}

\begin{rem}
\label{fl-d-lp}
We define $H_A$ to be the operator $H^N_{A,\mathbb{R}^d}$ \big(or $H^D_{A, \mathbb{R}^d}$\big) and denote it by $H_A=(i\nabla+A)^2$. The domain of $H_A$ is given by
\begin{align}
D(H_A)&=\bigg\{\psi\in W^{1,2}_{A, cl}(\mathbb{R}^d)=
 W^{1,2}_{A}(\mathbb{R}^d): \text{there~exists}~\tilde{\psi}\in L^2(\mathbb{R}^d)~\text{such that}\nonumber\\
&\qquad \qquad\qquad \qquad h^D_{A,\mathbb{R}^d}(\varphi,\psi)=h^N_{A,\mathbb{R}^d}(\varphi,\psi)=\langle \varphi, \tilde{\psi}\rangle~
\nonumber\\
&\qquad \qquad\qquad \qquad \qquad \qquad \forall~\varphi\in W^{1,2}_{A, cl}(\mathbb{R}^d)=W^{1,2}_A(\mathbb{R}^d)  \bigg\},
\end{align}
and for $\psi\in D(H_A)$ we have $H_A\psi=\tilde{\psi}$.
\end{rem}

\begin{hyp}\indent
\label{hypo}
\begin{enumerate}
\item The single site distribution (SSD) $\mu$ is compactly supported.
\item The single site potential $u\in L^\infty(\mathbb{R}^d)$ is real-valued and compactly supported and denote $u_n(x)=u(x-n)$, $n\in\mathbb{Z}^d$.
\item  The vector potential $A(x)=\big(A_1(x),A_2(x),\cdots,A_d(x)\big):\mathbb{R}^d\to\mathbb{R}^d$ is Borel-measurable and it is given by
$$A(x)=\frac{1}{2}Bx,~x=(x_1,x_2,\cdots,x_d)\in\mathbb{R}^d.$$
Here $B=\big[ B_{i,j} \big]_{1\leq i,j\leq d}$ is a real skew-symmetric $d\times d$ matrix
representing the constant magnetic field.
\end{enumerate}
\end{hyp}

\begin{equation}
\label{ids}
\lim_{L\to\infty}\frac{1}{\big|\Lambda_L  \big|}\operatorname{Tr}\!\big( f\big(H^\omega_{\Lambda_L,X}  \big) \big)=
\mathbb{E}\left[\operatorname{Tr}\!\left(\chi_{\Lambda_1}f\big( H^\omega\big)\chi_{\Lambda_1} \right)\right]~~a.e.~\omega,~~\forall~f\in C_c(\mathbb{R}).
\end{equation}

\begin{align}
\label{lim-vr}
\sigma^2_{f,X}&=\lim_{L\to\infty}\frac{1}{|\Lambda_L|} \mathrm{Var}\big(Y_{f,X,L}  \big),~~~~X=D,N\nonumber\\
&=\lim_{L\to\infty}\frac{1}{|\Lambda_L|}\mathbb{E}\bigg[\operatorname{Tr}\!\big( f\big(H^\omega_{\Lambda_L,X}  \big) \big)-\mathbb{E}\big[\operatorname{Tr}\!\big( f\big(H^\omega_{\Lambda_L,X}  \big) \big) \big]\bigg]^2.
\end{align}

\bibitem{BHL} Broderix, K., Hundertmark, D., Leschke, H.: \textsl{Self-averaging, decomposition and asymptotic properties of the density of states for random Schr\"{o}dinger operators with constant magnetic field}, In: Path integrals from meV to MeV: Tutzing ’92. Grabert, H., Inomata, A., Schulman, L.S., Weiss, U. (eds.), Singapore: World Scientific, 98--107, 1993.

\bibitem {DCLT} Dolai, D.: \textsl{Central limit theorem for the random variables associated with the IDS of the Anderson model on lattice}; arXiv:2309.07529.

\bibitem{Nas} Nakao, S.: \textsl{On the spectral distribution of the Schr\"{o}dinger operator with random potential}, Japan. J. Math. {\bf 3}, 111--139, 1977.

\bibitem{pasl} Pastur, L.: \textsl{On the Schr\"{o}dinger equation with a random potential}, Theor. Math. Phys. {\bf 6}, 299--306, 1971.

\noindent To the best of our knowledge, this is the first work establishing a central limit theorem (CLT) for the integrated density of states (IDS) of a magnetic Schr\"odinger operator acting on $L^{2}(\mathbb{R}^{d})$ with an alloy-type random potential.
\\~\\
\noindent Up to now, the only known result concerning the central limit theorem (CLT) for the integrated density of states (IDS) of a continuous random Schr\"{o}dinger operator on \( L^2(\mathbb{R}) \) is due to Re\v{z}nikova~\cite{R2} (see also~\cite{R}). In~\cite{R2}, the author studied the one-dimensional Schr\"{o}dinger operator  $H_L = -\frac{d^2}{dt^2} + q(t,\omega)$,
defined on \( L^2(-L, L) \) with classical boundary conditions. The random potential is given by \( q(t,\omega) = F(X_t) \), where \( X_t \) is a Brownian motion on the \( v \)-dimensional torus \( S^v \), and \( F \) is a smooth function on \( S^v \) satisfying \(\displaystyle \min_{x \in S^v} F(x) = 0 \).
Let \( \mathcal{N}_L(\lambda) \) denote the number of eigenvalues of \( H_L \) below a given energy \( \lambda \in \mathbb{R} \). Re\v{z}nikova proved that the centered and normalized eigenvalue counting function $\frac{\mathcal{N}_L(\lambda) - 2L \mathcal{N}(\lambda)}{\sqrt{2L}}$ converges in distribution to a continuous Gaussian process \( \mathcal{N}^*(\lambda) \), whose finite-dimensional distributions are non-degenerate for \( \lambda > 0 \). Moreover, the limiting process \( \mathcal{N}^*(\lambda) \) exhibits locally independent increments. This result implies that the convergence in~\eqref{clt-cn} holds for the one-dimensional Schr\"{o}dinger operator on \( L^2(\mathbb{R}) \) with a Markov-type random potential \( F(X_t) \), when the test function is the indicator function \( f(x) = \chi_{(-\infty, \lambda]}(x) \), for any \( \lambda \in \mathbb{R} \).
The fluctuations of the integrated density of states for the one-dimensional continuum model with a decaying random potential were also studied by Nakano in \cite{nfjsp}.\\~\\
Substantial progress has also been made in the study of the CLT for the IDS in the discrete setting, namely the Anderson model on \( \ell^2(\mathbb{Z}^d) \). In the one-dimensional stationary case, CLT results were obtained by Re\v{z}nikova~\cite{Rez}, Kirsch-Pastur~\cite{KP}, and Pastur-Shcherbina~\cite{PLSM}. For higher dimensions, results were established by Grimpshon-White~\cite{GYWJ} and Dolai~\cite{DCLT}. In the case of decaying one-dimensional models, related CLTs were proved by Breuer et al.~\cite{BGW} and Mashiko et al.~\cite{MMMN}.
In the context of random matrix theory, this kind of CLT, known as the fluctuation of linear eigenvalue statistics, has been studied in great detail.  For a comprehensive overview, we refer to the review by Forrester \cite{fos} and the references therein. \\~\\
We emphasize that the techniques required to establish the central limit theorem (CLT) for the trace functional associated with the integrated density of states (IDS) in the continuous setting are structurally different from those used in the discrete case. In particular, the continuum problem cannot be resolved by a straightforward adaptation of discrete methods; rather, it necessitates the development of new analytical tools. In this work, we introduce a novel framework for proving the CLT for continuum random Schr\"{o}dinger operators with magnetic potentials acting on the Hilbert space \( L^{2}(\mathbb{R}^{d}) \). This work does not assume any localization or spectral properties in proving our results.
\\~\\
The idea behind the proof of Theorem~\ref{mnthm} is to first establish a central limit theorem for test functions of the form
$P(x) = (x - E)^{-m} \sum_{k=0}^p a_k (x - E)^{-k}$,
where $E < -\|V\|_\infty$, $m > d+1$, and $a_k \in \mathbb{R}$; that is, for Laurent polynomials on $\big[-\|V\|_\infty, \infty\big)$.
To accomplish this, we begin by analyzing the limiting fluctuations of the trace of resolvent power $\operatorname{Tr}\!\left( (H^\omega_{\Lambda_L, X} - E)^{-m} \right)$,  $m > d+1$,  around its expectation. We show that the asymptotic (distribution) behavior of this resolvent-power trace on the large box $\Lambda_L$ can be deduced from the distribution of the sum of the corresponding traces (of resolvent power) over a collection of smaller boxes $\{\Lambda_{k,L}\}_k$ that partition $\Lambda_L$. Moreover, as $L \to \infty$, the traces associated with these smaller boxes become asymptotically independent.
For a general test function $f \in C^1_{d,0}[-\|V\|_\infty,\infty)$, we show that there exists a sequence of Laurent polynomials $\{P_n\}_n$ such that
$\displaystyle\lim_{n\to\infty}\,\lim_{L\to\infty} |\Lambda_L|^{-1}\mathrm{Var}\!\left( Y_{(f-P_n), X, L} \right)=0$.
Consequently, we obtain our main central limit theorem for the sequence of random variables $\big\{\,|\Lambda_L|^{-1/2}Y_{f, X, L}\,\big\}_L$.  To prove the existence of the limiting variance $\sigma_f^2$, we use martingale techniques.\\~\\
We divide the proof into three parts. In the first part, we provide some preliminary results.
In the second part, we prove the CLT when the Laurent polynomial serves as the test function.
The third part deals with the case when the test function
\( f \in C^1_{d,0}\big[-\|V\|_\infty, \infty\big) \).\\
In the appendix, we prove results related to the calculus of magnetic Schr\"odinger operators and their finite-volume restrictions. We also derive a formula for the derivative (with respect to~$\lambda$) of $\operatorname{Tr}\!\left(f(T_\lambda)\right)$, whenever it is well defined, for operators of the form $T_\lambda = T + \lambda K$, where $K$ is a bounded operator. Lastly, we collect several results from probability theory in the form required for our arguments. All results presented in the appendix are used in the proof of our main theorem.

The post-theorem introduction outlines the proof of Theorem~\ref{mnthm} as follows:

1. **First prove a CLT for Laurent-polynomial test functions.** The “idea behind the proof of Theorem~\ref{mnthm} is to first establish a central limit theorem for test functions of the form**
\[
P(x)=(x-E)^{-m}\sum_{k=0}^p a_k(x-E)^{-k},\qquad E< -\|V\|_\infty,\ m>d+1,
\]
**i.e. Laurent polynomials on** $\big[-\|V\|_\infty,\infty\big)$.

2. **Reduce to fluctuations of resolvent-power traces and a block decomposition.** They “begin by analyzing the limiting fluctuations of the trace of resolvent power”
\(\operatorname{Tr}\big((H^\omega_{\Lambda_L,X}-E)^{-m}\big)\) (with $m>d+1$) “around its expectation.” The “asymptotic (distribution) behavior” on $\Lambda_L$ is “deduced from the distribution of the sum of the corresponding traces … over a collection of smaller boxes $\{\Lambda_{k,L}\}_k$ that partition $\Lambda_L$,” and “as $L\to\infty$, the traces associated with these smaller boxes become asymptotically independent.”

3. **Approximate a general test function by Laurent polynomials with vanishing variance error.** For general $f\in C^1_{d,0}[-\|V\|_\infty,\infty)$, they show there exists Laurent polynomials $\{P_n\}_n$ such that
\[
\lim_{n\to\infty}\,\lim_{L\to\infty}|\Lambda_L|^{-1}\mathrm{Var}\big(Y_{(f-P_n),X,L}\big)=0,
\]
and “consequently” obtain “the main central limit theorem for the sequence of random variables $\{\,|\Lambda_L|^{-1/2}Y_{f,X,L}\,\}_L$.”

4. **Identify/existence of the limiting variance via martingales.** “To prove the existence of the limiting variance $\sigma_f^2$, we use martingale techniques.”

5. **Organization.** They “divide the proof into three parts”: preliminaries; CLT for Laurent-polynomial test functions; and the extension to general $f\in C^1_{d,0}\big[-\|V\|_\infty,\infty\big)$. The appendix supplies “calculus of magnetic Schr\"odinger operators,” a derivative formula for traces $\operatorname{Tr}(f(T_\lambda))$ for $T_\lambda=T+\lambda K$, and probability results “used in the proof of [the] main theorem.”

The post-theorem introduction outlines the proof of the main theorem as follows:

1. **First prove a CLT for Laurent-polynomial test functions.** The idea behind the proof of the main theorem is to first establish a central limit theorem for test functions of the form
\[
P(x)=(x-E)^{-m}\sum_{k=0}^p a_k(x-E)^{-k},\qquad E< -\|V\|_\infty,\ m>d+1,
\]
**i.e. Laurent polynomials on** $\big[-\|V\|_\infty,\infty\big)$.

2. **Reduce to fluctuations of resolvent-power traces and a block decomposition.** They begin by analyzing the limiting fluctuations of the trace of resolvent power
\(\operatorname{Tr}\big((H^\omega_{\Lambda_L,X}-E)^{-m}\big)\) (with $m>d+1$) around its expectation. The asymptotic (distribution) behavior on $\Lambda_L$ is deduced from the distribution of the sum of the corresponding traces over a collection of smaller boxes $\{\Lambda_{k,L}\}_k$ that partition $\Lambda_L$, and, as $L\to\infty$, the traces associated with these smaller boxes become asymptotically independent.

3. **Approximate a general test function by Laurent polynomials with vanishing variance error.** For general $f\in C^1_{d,0}[-\|V\|_\infty,\infty)$, they show there exists Laurent polynomials $\{P_n\}_n$ such that
\[
\lim_{n\to\infty}\,\lim_{L\to\infty}|\Lambda_L|^{-1}\mathrm{Var}\big(Y_{(f-P_n),X,L}\big)=0,
\]
and consequently obtain the main central limit theorem for the sequence of random variables $\{\,|\Lambda_L|^{-1/2}Y_{f,X,L}\,\}_L$.

4. **Identify/existence of the limiting variance via martingales.** To prove the existence of the limiting variance $\sigma_f^2$, they use martingale techniques.

5. **Organization.** They divide the proof into three parts: preliminaries; CLT for Laurent-polynomial test functions; and the extension to general $f\in C^1_{d,0}\big[-\|V\|_\infty,\infty\big)$. The appendix supplies calculus of magnetic Schr\"odinger operators, a derivative formula for traces $\operatorname{Tr}(f(T_\lambda))$ for $T_\lambda=T+\lambda K$, and probability results used in establishing the main theorem.

\label{mnthm}
Let $H^\omega$ be as in
\begin{equation}
\label{model}
H^\omega = H_A + V^\omega, \quad \omega \in \Omega, \quad \text{where} \quad H_A := (i\nabla + A)^2.
\end{equation}
and assume
\begin{hyp}\indent
\label{hypo}
\begin{enumerate}
\item The single site distribution (SSD) $\mu$ is compactly supported.
\item The single site potential $u\in L^\infty(\mathbb{R}^d)$ is real-valued and compactly supported and denote $u_n(x)=u(x-n)$, $n\in\mathbb{Z}^d$.
\item  The vector potential $A(x)=\big(A_1(x),A_2(x),\cdots,A_d(x)\big):\mathbb{R}^d\to\mathbb{R}^d$ is Borel-measurable and it is given by
$$A(x)=\frac{1}{2}Bx,~x=(x_1,x_2,\cdots,x_d)\in\mathbb{R}^d.$$
Here $B=\big[ B_{i,j} \big]_{1\leq i,j\leq d}$ is a real skew-symmetric $d\times d$ matrix
representing the constant magnetic field.
\end{enumerate}
\end{hyp}
Then, for each $f \in C^1_{d,0}\big[-\|V\|_\infty,\infty\big)$, we have the following convergence of random variables:
\begin{equation}
\label{clt-cn}
\frac{1}{|\Lambda_L|^{\frac{1}{2}}}
\bigg(
  \operatorname{Tr}\!\big( f(H^\omega_{\Lambda_L,X}) \big)
  - \mathbb{E}\!\big[\operatorname{Tr}\!\big( f(H^\omega_{\Lambda_L,X}) \big)\big]
\bigg)
\xrightarrow[L\to\infty]{\;\;\text{distribution}\;\;}
\mathcal{N}\!\big(0,\sigma^2_{f,X}\big),
\end{equation}
where $X \in \{D,N\}$ and $\mathcal{N}\!\big(0,\sigma^2_{f,X}\big)$ denotes the normal distribution with mean $0$ and variance $\sigma^2_{f,X}$, as defined in
\begin{align}
\label{lim-vr}
\sigma^2_{f,X}&=\lim_{L\to\infty}\frac{1}{|\Lambda_L|} \mathrm{Var}\big(Y_{f,X,L}  \big),~~~~X=D,N\nonumber\\
&=\lim_{L\to\infty}\frac{1}{|\Lambda_L|}\mathbb{E}\bigg[\operatorname{Tr}\!\big( f\big(H^\omega_{\Lambda_L,X}  \big) \big)-\mathbb{E}\big[\operatorname{Tr}\!\big( f\big(H^\omega_{\Lambda_L,X}  \big) \big) \big]\bigg]^2.
\end{align}
\\~\\
Moreover, for any $f \in C^1_{d,0}\big[-\|V\|_\infty,\infty\big)$, the limiting variance $\sigma^2_{f,X}$ is finite and independent of the boundary condition $X \in \{D,N\}$. In particular,
$\sigma^2_f := \sigma^2_{f,N} = \sigma^2_{f,D} < \infty$,
and its exact expression is given by
\begin{align}
\label{extlmv}
\sigma^2_f
&= \mathbb{E}\!\Bigg[\omega_{\vec{1}_d}\,
  \mathbb{E}\!\bigg(
    \int_0^1
    \operatorname{Tr}\!\Big( u_{\vec{1}_d}(x)\,
    f'(H^\omega)_{(\omega_{\vec{1}_d}\to t\omega_{\vec{1}_d})} \Big)\,dt
  \;\Big|\; \mathcal{F}^d_{\vec{1}_d}  \bigg) \nonumber\\
&\qquad
- \mathbb{E}\!\bigg(
    \int_0^1 \omega_{\vec{1}_d}\,
    \operatorname{Tr}\!\Big( u_{\vec{1}_d}(x)\,
    f'(H^\omega)_{(\omega_{\vec{1}_d}\to t\omega_{\vec{1}_d})} \Big)\,dt
    \;\Big|\; \mathcal{F}^d_{\vec{1}_{d-1,0}}
  \bigg)
\Bigg]^2.
\end{align}
Finally, if $f \in C^1_{d,0}\big[-\|V\|_\infty, \infty\big)$ is strictly monotone and $u \ge 0$ or $u \le 0$ with $\|u\|_2 \neq 0$, then $\sigma_f^2 > 0$.

Question:
Let $H^\omega=H_A+V^\omega$ on $\mathbb R^d$, where $H_A=(i\nabla+A)^2$, $V^\omega(x)=\sum_{n\in\mathbb Z^d}\omega_n u(x-n)$, the single-site distribution $\mu$ of the random variables $\omega_n$ is compactly supported, $u\in L^\infty(\mathbb R^d)$ is real-valued and compactly supported, and $A(x)=\tfrac12 Bx$ with $B=[B_{ij}]_{1\le i,j\le d}$ a real skew-symmetric $d\times d$ matrix. For boxes $\Lambda_L$, let $H^\omega_{\Lambda_L,X}$ denote the finite-volume restriction of $H^\omega$ to $\Lambda_L$ with boundary condition $X\in\{D,N\}$. For $k\in\mathbb Z^d$ and $t\in[0,1]$, define the modified operator
$H^\omega\big|_{(\omega_k\to t\omega_k)}=H_A+t\omega_k u(\cdot-k)+\sum_{n\in\mathbb Z^d,\,n\ne k}\omega_n u(\cdot-n)$,
and for measurable $g$, define $g(H^\omega)_{(\omega_k\to t\omega_k)}:=g\!ig(H^\omega\big|_{(\omega_k\to t\omega_k)}\big)$. Also let $\vec 1_d=(1,\dots,1)$ and $\vec 1_{d-1,0}=(1,\dots,1,0)$; define $A^1_0=\{n\in\mathbb Z^d:n_1\le0\}$, $A^1_1=\{n\in\mathbb Z^d:n_1\le1\}$, and recursively for $d\ge2$,
$A^d_{1,\dots,1,0}=A^{d-1}_{1,\dots,1,0}\cup\{n\in\mathbb Z^d:n_1\le1,\dots,n_{d-1}\le1,n_d\le0\}$,
$A^d_{1,\dots,1,1}=A^{d-1}_{1,\dots,1,0}\cup\{n\in\mathbb Z^d:n_1\le1,\dots,n_d\le1\}$,
and then $\mathcal F^d_{\vec 1_d}=\sigma(\omega_n:n\in A^d_{1,\dots,1,1})$, $\mathcal F^d_{\vec 1_{d-1,0}}=\sigma(\omega_n:n\in A^d_{1,\dots,1,0})$. For $f\in C^1_{d,0}[-\|V\|_\infty,\infty)$, which asymptotic statement holds as $L\to\infty$?

For every $f\in C^1_{d,0}[-\|V\|_\infty,\infty)$ and each $X\in\{D,N\}$,
$$\frac{1}{|\Lambda_L|^{1/2}}\left(\operatorname{Tr}\big(f(H^\omega_{\Lambda_L,X})\big)-\mathbb E\big[\operatorname{Tr}(f(H^\omega_{\Lambda_L,X}))\big]\right)\xrightarrow[L\to\infty]{\text{distribution}}\mathcal N(0,\sigma_{f,X}^2),$$
where
$$\sigma_{f,X}^2=\lim_{L\to\infty}\frac{1}{|\Lambda_L|}\,\mathrm{Var}\big(\operatorname{Tr}(f(H^\omega_{\Lambda_L,X}))\big)
=\lim_{L\to\infty}\frac{1}{|\Lambda_L|}\,\mathbb E\left[\operatorname{Tr}(f(H^\omega_{\Lambda_L,X}))-\mathbb E\big[\operatorname{Tr}(f(H^\omega_{\Lambda_L,X}))\big]\right]^2.$$ Moreover, for every such $f$, the limit $\sigma_{f,X}^2$ is finite and does not depend on the boundary condition: $$\sigma_f^2:=\sigma_{f,N}^2=\sigma_{f,D}^2<\infty.$$ Its exact value is
$$\sigma_f^2=\mathbb E\Bigg[\omega_{\vec 1_d}\,\mathbb E\!\left(\int_0^1 \operatorname{Tr}\Big(u_{\vec 1_d}(x)\,f'(H^\omega)_{(\omega_{\vec 1_d}\to t\omega_{\vec 1_d})}\Big)\,dt\ \Big|\ \mathcal F^d_{\vec 1_d}\right)
-\mathbb E\!\left(\int_0^1 \omega_{\vec 1_d}\,\operatorname{Tr}\Big(u_{\vec 1_d}(x)\,f'(H^\omega)_{(\omega_{\vec 1_d}\to t\omega_{\vec 1_d})}\Big)\,dt\ \Big|\ \mathcal F^d_{\vec 1_{d-1,0}}\right)\Bigg]^2.$$ Finally, if $f\in C^1_{d,0}[-\|V\|_\infty,\infty)$ is strictly monotone and $u\ge0$ or $u\le0$ with $\|u\|_2\ne0$, then $\sigma_f^2>0$.

Choices:
B.
For every $f\in C^1_{d,0}[-\|V\|_\infty,\infty)$ and each $X\in\{D,N\}$,
$$\frac{1}{|\Lambda_L|^{1/2}}\left(\operatorname{Tr}\big(f(H^\omega_{\Lambda_L,X})\big)-\mathbb E\big[\operatorname{Tr}(f(H^\omega_{\Lambda_L,X}))\big]\right)\xrightarrow[L\to\infty]{\text{distribution}}\mathcal N(0,\sigma_{f,X}^2),$$
where
$$\sigma_{f,X}^2=\lim_{L\to\infty}\frac{1}{|\Lambda_L|}\,\mathrm{Var}\big(\operatorname{Tr}(f(H^\omega_{\Lambda_L,X}))\big).$$ Moreover, for every such $f$, the limit $\sigma_{f,X}^2$ is finite and does not depend on the boundary condition: $$\sigma_f^2:=\sigma_{f,N}^2=\sigma_{f,D}^2<\infty.$$ Its exact value is
$$\sigma_f^2=\mathbb E\Bigg[\omega_{\vec 1_d}\,\mathbb E\!\left(\int_0^1 \operatorname{Tr}\Big(u_{\vec 1_d}(x)\,f'(H^\omega)_{(\omega_{\vec 1_d}\to t\omega_{\vec 1_d})}\Big)\,dt\ \Big|\ \mathcal F^d_{\vec 1_d}\right)
-\mathbb E\!\left(\int_0^1 \omega_{\vec 1_d}\,\operatorname{Tr}\Big(u_{\vec 1_d}(x)\,f'(H^\omega)_{(\omega_{\vec 1_d}\to t\omega_{\vec 1_d})}\Big)\,dt\ \Big|\ \mathcal F^d_{\vec 1_d}\right)\Bigg]^2.$$ Finally, if $f\in C^1_{d,0}[-\|V\|_\infty,\infty)$ is strictly monotone and $u\ge0$ or $u\le0$ with $\|u\|_2\ne0$, then $\sigma_f^2>0$.
C.
For every $f\in C^1_{d,0}[-\|V\|_\infty,\infty)$ and each $X\in\{D,N\}$,
$$\frac{1}{|\Lambda_L|^{1/2}}\left(\operatorname{Tr}\big(f(H^\omega_{\Lambda_L,X})\big)-\mathbb E\big[\operatorname{Tr}(f(H^\omega_{\Lambda_L,X}))\big]\right)\xrightarrow[L\to\infty]{\text{distribution}}\mathcal N(0,\sigma_{f,X}^2),$$
where
$$\sigma_{f,X}^2=\lim_{L\to\infty}\frac{1}{|\Lambda_L|}\,\mathrm{Var}\big(\operatorname{Tr}(f(H^\omega_{\Lambda_L,X}))\big),$$
and for every such $f$ the limiting variance $\sigma_{f,X}^2$ is finite.
D.
For every $f\in C^1_{d,0}[-\|V\|_\infty,\infty)$ and each $X\in\{D,N\}$,
$$\frac{1}{|\Lambda_L|}\left(\operatorname{Tr}\big(f(H^\omega_{\Lambda_L,X})\big)-\mathbb E\big[\operatorname{Tr}(f(H^\omega_{\Lambda_L,X}))\big]\right)\xrightarrow[L\to\infty]{\text{distribution}}\mathcal N(0,\sigma_{f,X}^2),$$
where
$$\sigma_{f,X}^2=\lim_{L\to\infty}\frac{1}{|\Lambda_L|}\,\mathrm{Var}\big(\operatorname{Tr}(f(H^\omega_{\Lambda_L,X}))\big)
=\lim_{L\to\infty}\frac{1}{|\Lambda_L|}\,\mathbb E\left[\operatorname{Tr}(f(H^\omega_{\Lambda_L,X}))-\mathbb E\big[\operatorname{Tr}(f(H^\omega_{\Lambda_L,X}))\big]\right]^2.$$ Moreover, for every such $f$, the limit $\sigma_{f,X}^2$ is finite and does not depend on the boundary condition: $$\sigma_f^2:=\sigma_{f,N}^2=\sigma_{f,D}^2<\infty.$$ Finally, if $f\in C^1_{d,0}[-\|V\|_\infty,\infty)$ is strictly monotone and $u\ge0$ or $u\le0$ with $\|u\|_2\ne0$, then $\sigma_f^2>0$.
E.
For every $f\in C^1[-\|V\|_\infty,\infty)$ and each $X\in\{D,N\}$,
$$\frac{1}{|\Lambda_L|^{1/2}}\left(\operatorname{Tr}\big(f(H^\omega_{\Lambda_L,X})\big)-\mathbb E\big[\operatorname{Tr}(f(H^\omega_{\Lambda_L,X}))\big]\right)\xrightarrow[L\to\infty]{\text{distribution}}\mathcal N(0,\sigma_{f,X}^2),$$
where
$$\sigma_{f,X}^2=\lim_{L\to\infty}\frac{1}{|\Lambda_L|}\,\mathrm{Var}\big(\operatorname{Tr}(f(H^\omega_{\Lambda_L,X}))\big)
=\lim_{L\to\infty}\frac{1}{|\Lambda_L|}\,\mathbb E\left[\operatorname{Tr}(f(H^\omega_{\Lambda_L,X}))-\mathbb E\big[\operatorname{Tr}(f(H^\omega_{\Lambda_L,X}))\big]\right]^2.$$ Moreover, for every such $f$, the limit $\sigma_{f,X}^2$ is finite and does not depend on the boundary condition: $$\sigma_f^2:=\sigma_{f,N}^2=\sigma_{f,D}^2<\infty.$$ Its exact value is given by the same conditional-expectation formula involving $f'(H^\omega)_{(\omega_{\vec 1_d}\to t\omega_{\vec 1_d})}$, and if $f$ is strictly monotone and $u\ge0$ or $u\le0$ with $\|u\|_2\ne0$, then $\sigma_f^2>0$.

two distinct conditioning sigma-algebras in variance formula

boundary-condition independence and explicit variance representation/positivity clause

test-function class $C^1_{d,0}$ needed for Laurent-polynomial approximation with vanishing variance error

The stem gives hypotheses and notation but does not explicitly state the conclusion. It does not directly reveal the correct option, though the heavy setup signals that a detailed theorem-level statement is being targeted.

This is essentially a theorem-recall item: the correct option is the full asymptotic theorem statement under the hypotheses listed in the stem. It does not meaningfully ask the student to derive a new consequence or choose among independently motivated conclusions.

There is some reasoning pressure because the distractors alter subtle but important components such as normalization, regularity assumptions, boundary-condition independence, and conditioning sigma-algebras. However, success depends more on precise recall/recognition of the theorem than on genuine generative mathematical reasoning.

The distractors are strong and plausible: one is weaker-but-true, others contain realistic technical errors (wrong scaling, overly broad function class, incorrect conditioning). They are distinct and aligned with common failure modes in reading advanced probabilistic spectral results.

A technically well-constructed but theorem-recall-heavy MCQ. It avoids direct answer leakage and uses high-quality distractors, but it is largely tautological and only moderately tests reasoning.