\begin{teo} \label{thm:main}
	Let $\Omega\subset\mathbb{G}$ be an open and bounded set
	with smooth enough boundary $\partial \Omega$, and let $p\in (0,1)$. Then, there exists $\Lambda > 0$
	such that
	\begin{itemize}
		\item[A)]problem \eqref{EqProblem}$_{\lambda}$ does not admit weak solutions
		for every $\lambda>\Lambda$;
		\item[B)] problem \eqref{EqProblem}$_{\lambda}$ admits at least one weak solution for every $\lambda \in (0,\Lambda]$;
		\item[C)] problem \eqref{EqProblem}$_{\lambda}$ admits at least two weak solutions for every $0<\lambda<\Lambda$.
	\end{itemize}
\end{teo}

Let $\mathbb{G}$ be a Carnot group and let $\Omega \subset \mathbb{G}$ be an bounded and connected
open set with smooth enough boundary $\partial \Omega$. Let $q \in (0,1)$, let $2^{\star}_{Q}:=\tfrac{2Q}{Q-2}$ be the critical Sobolev exponent related to the Sobolev inequality in $\mathbb{G}$, and let $\lambda >0$. We consider the following Dirichlet boundary value problem
\begin{equation}\tag{{$\mathrm{P}$}}\label{EqProblem}
 \left\{\begin{array}{rll}
 -\Delta_{\mathbb{G}}u& = \lambda\, u^{q} + u^{2^{\star}_{Q}-1} & \textrm{ in } \Omega,\\
 u&>0 & \textrm{ in } \Omega,\\
 u&=0 & \textrm{ on } \partial \Omega.
 \end{array}\right.
\end{equation}

We stress that $-\Delta_{\mathbb{G}}$ denotes here the sub-laplacian on $\mathbb{G}$ which is a second-order differential operator with non-negative characteristic form that can be explicitly expressed as a sum of squares of vector fields satisfying the H\"{o}rmander condition, see e.g. \cite{Hormander}. We refer to Section~\ref{sec:Prel} for more details, including the Folland-Stein Sobolev spaces we will work with.\\

Along the paper it will sometimes be useful to denote the above problem as \eqref{EqProblem}$_\lambda$
to make it clear the choice of the parameter.
We immediately state the main result of this paper. In what follows,
we refer to Definition \ref{def:weak_sub_super_sol} for the precise definition of \emph{weak solution}
of \eqref{EqProblem}$_{\lambda}$.

\begin{equation}\tag{{$\mathrm{P}$}}\label{EqProblem}
	\left\{\begin{array}{rll}
		-\Delta_{\mathbb{G}}u& = \lambda\, u^{q} + u^{2^{\star}_{Q}-1} & \textrm{ in } \Omega,\\
		u&>0 & \textrm{ in } \Omega,\\
		u&=0 & \textrm{ on } \partial \Omega.
	\end{array}\right.
\end{equation}

\begin{defin}\label{def:weak_sub_super_sol}
	Let $\Omega\subseteq\mathbb{G}$ be an open, bounded and connected set.
We say that a function $u \in S^{1}_{0}(\Omega)$ is a weak subsolution (resp. supersolution) of \eqref{EqProblem}$_\lambda$ if it satisfies the following properties:
	\begin{itemize}
		\item[(i)] $u>0$ in $\Omega$.
		\item[(ii)] For every $0\leq \varphi \in C^{\infty}_{0}(\Omega)$, it holds that
		\begin{equation}\label{eq:weak_sub_super_sol}
			\int_{\Omega}\langle \nabla_{\mathbb{G}}u, \nabla_{\mathbb{G}}\varphi \rangle_{\mathfrak{g}_{1}} \leq (\textrm{resp. } \geq)  \int_{\Omega}\left(\lambda u^{q}  + \ u^{2^{\star}_{Q}}\right)\varphi.
		\end{equation}
	\end{itemize}
	Finally, we say that $u \in S^{1}_{0}(\Omega)$ is a weak solution of \eqref{EqProblem}$_\lambda$ if it is both a weak subsolution and a weak supersolution of \eqref{EqProblem}$_\lambda$ without the non-negativity condition on $\varphi$.
\end{defin}

The above theorem is the natural generalization to Carnot groups of classical results of \cite{ABC}. We refer e.g. to \cite{BESS,CCP,CoPe,BV} for further generalizations.

We are now ready to properly set the definition of weak sub/supersolution of \eqref{EqProblem}$_\lambda$.
\begin{defin}\label{def:weak_sub_super_sol}
 Let $\Omega\subseteq\mathbb{G}$ be an open, bounded and connected set.
We say that a function $u \in S^{1}_{0}(\Omega)$ is a weak subsolution (resp. supersolution) of \eqref{EqProblem}$_\lambda$ if it satisfies the following properties:
 \begin{itemize}
 \item[(i)] $u>0$ in $\Omega$.
 \item[(ii)] For every $0\leq \varphi \in C^{\infty}_{0}(\Omega)$, it holds that
 \begin{equation}\label{eq:weak_sub_super_sol}
 \int_{\Omega}\langle \nabla_{\mathbb{G}}u, \nabla_{\mathbb{G}}\varphi \rangle_{\mathfrak{g}_{1}} \leq (\textrm{resp. } \geq) \int_{\Omega}\left(\lambda u^{q} + \ u^{2^{\star}_{Q}}\right)\varphi.
 \end{equation}
 \end{itemize}
 Finally, we say that $u \in S^{1}_{0}(\Omega)$ is a weak solution of \eqref{EqProblem}$_\lambda$ if it is both a weak subsolution and a weak supersolution of \eqref{EqProblem}$_\lambda$ without the non-negativity condition on $\varphi$.
\end{defin}

\begin{lemma}\label{lem:Perron}
 Let $\underline{u}, \overline{u}\in S^{1}_{0}(\Omega)$ be a weak subsolution and a weak supersolution, respectively, of problem \eqref{EqProblem}$_\lambda$.
 We assume that
 \begin{itemize}
 \item[a)] $\underline{u}(g) \leq \overline{u}(g)$ for a.e.\,$g\in \Omega$;
 \item[b)] for every open set $\Oo\Subset\Omega$ there exists
 $C = C(\Oo,\underline{u}) > 0$ such that
 $$\text{$\underline{u}\geq C$ a.e.\,in $\Oo$}.$$
 \end{itemize}
 Then, there exists a weak solution $u \in S^{1}_{0}(\Omega)$ of \eqref{EqProblem}$_{\lambda}$ such that
 $$\text{$\underline{u}(g) \leq u(g) \leq \overline{u}(g)$ for a.e. $g \in \Omega$}.$$
\end{lemma}

\noindent We split the proof of I) in two lemmas.
\begin{lemma}\label{lem:lambda0}
 Let $\Lambda$ as defined in \eqref{eq:DefinitionLambda}. Then $\Lambda >0$.
 \begin{proof}
 We will show that there exists a sufficiently small $\lambda >0$ such that
 \eqref{EqProblem}$_{\lambda}$ has a solution. To this aim, we will use Lemma \ref{lem:Perron} exhibiting both a super and a subsolution.
 Looking for a supersolution, we consider the following auxiliary torsion problem
 \begin{equation}
 \left\{\begin{array}{rl}
 -\Delta_{\mathbb{G}} V = 1 & \textrm{in } \Omega,\\
 V = 0 & \textrm{on } \partial \Omega,
 \end{array}\right.
 \end{equation}
 \noindent whose unique solution is provided by Lax-Milgram Theorem. Moreover, by a classical Stampacchia iteration method, it holds that $V\in L^{\infty}(\Omega)$. Observe further that for any positive constant $C$,
 the function $C\cdot x^p-x$, with $p>1$, has negative value for $x>0$ sufficiently small.
 Therefore, for every $C,C'\in \R^+$
 there exists $\lambda^*>0$ such that
 for every $\lambda<\lambda^*$,
 \[
 \exists \, m_\lambda\in \R^+:\quad \lambda\cdot C'\cdot m_\lambda^q+C\cdot m_\lambda^p-m_\lambda\leq0.
 \]
 We fix $\lambda<\lambda^*$ and set $C=\|V\|_{L^{\infty}(\Omega)}^p$, $C'=\|V\|_{L^{\infty}(\Omega)}^q$.
 We define $\overline u_1:=m_\lambda V$, which weakly verifies
 \[
 \left\{\begin{array}{rll}
 -\Delta_{\G}\overline u_1&=m_\lambda \geq \lambda\overline u_1^q+\overline u_1^p &\ \m{in }\Omega\\
 \overline u_1 &>0 &\ \m{in }\Omega\\
 \overline u_1 &=0 &\ \m{on }\de\Omega,
 \end{array}\right.
 \]
 therefore it is a weak supersolution of \eqref{EqProblem}$_\lambda$.\\
 Regarding the weak subsolution to \eqref{EqProblem}, we choose the unique solution $\underline u_\lambda$ to \eqref{eq:Sublinear_Problem}.
 We can now conclude the proof by appealing Lemma \ref{lem:Perron}. Indeed, by Lemma \ref{lem:Weak_Comparison_Model} with $w=\overline{u}_1$ and $v=\underline{u}_{\lambda}$, we get that
 $\overline{u}_1 \geq \underline{u}_{\lambda}$, which is condition a) of Lemma \ref{lem:Perron}. Regarding b) of Lemma \ref{lem:Perron} it is enough to recall \cite[Corollary 2.3]{BiGaVe}. This closes the proof.
 \end{proof}
\end{lemma}

We now set $\overline{u}= u_{\lambda'}$ and $\underline{u} = w_{\lambda}$,
 and we apply Lemma \ref{lem:Perron}: this immediately yields that problem \eqref{EqProblem}$_{\lambda}$ admits a weak solution $u_{\lambda}$ for every $\lambda \in (0,\Lambda)$. Moreover, recalling the definition of $I_{\lambda}$ in \eqref{eq:Def_Ilambda}, such a solution satisfies that
 $$I_{\lambda}(u_\lambda) = \min\{u\in S_0^1(\Omega):\,w_{\lambda}\leq u\leq u_{\lambda'}\}
 \leq I_{\lambda}(w_{\lambda}).$$
 In particular,
 by Theorem \ref{thm:Sublinear_Problem} we have
 \begin{equation} \label{eq:Ilambdaulambdaneg}
 I_{\lambda}(u_{\lambda}) \leq I_{\lambda}(w_{\lambda}) \leq J_{\lambda}(w_{\lambda}) <0.
 \end{equation}
 It remains to consider the case $\lambda = \Lambda$. The proof
 is rather standard and pretty similar to that of \cite[Lemma 3.5]{BiGaVe}. We report it here for the sake of completeness.
 To begin with, we choose a monotone increasing sequence $\{\lambda_k\}_k\subseteq(0,\Lambda)$ such that
 $\lambda_{k} \to \Lambda$ as $k\to+\infty$. Now, for each $k \in \mathbb{N}$, we set
 $$u_k := u_{\lambda_k}\in S^{1}_{0}(\Omega),$$
 \noindent where $u_{\lambda_k}$ is the weak solution of problem \eqref{EqProblem}$_{\lambda_k}$ constructed
 as above by means of Lemma \ref{lem:Perron}. Thanks to
 \eqref{eq:Ilambdaulambdaneg}, for every $k\geq 1$ we have
 \begin{equation} \label{eq:Ilambdakneg}
 I_{\lambda_k,}(u_{k})
 = \dfrac{1}{2} \int_{\Omega}|\nabla_{\mathbb{G}}u_k|^2 - \dfrac{\lambda_k}{q+1}\int_{\Omega}|u_k|^{q+1} - \dfrac{1}{2^{\star}_{Q}}\int_{\Omega}|u_k|^{2^{\star}_{Q}} <0.
 \end{equation}
 Moreover, by using $\varphi = u_k$ in \eqref{eq:weak_sub_super_sol}, and recalling that $u_k$
 solves \eqref{EqProblem}$_{\lambda_k}$, we get
 \begin{equation} \label{eq:testwithukzero}
 \int_{\Omega}|\nabla_{\mathbb{G}}u_k|^2 -\lambda_k\int_\Omega u_k^{q+1}
 -\int_\Omega u_k^{2^{\star}_{Q}} = 0.
 \end{equation}
 Combining \eqref{eq:Ilambdakneg} with \eqref{eq:testwithukzero},
 we notice that the sequence $\{u_k\}_k$ is bounded in~$S^{1}_{0}(\Omega)$.
 Therefore, we can find a function
 $$u_{\Lambda}\in S^{1}_{0}(\Omega)$$
 such that
 (up to a subsequence and as $k\to+\infty$)
 \begin{itemize}
 \item[a)] $u_k\to u_{\Lambda}$ weakly in $S^{1}_{0}(\Omega)$ and strongly
 in $L^p(\Omega)$ for $1\leq p <2^{\star}_{Q}$;
 \item[b)] $u_k\to u_{\Lambda}$ a.e.\,in $\Omega$.
 \end{itemize}
 We now observe that, being $\{\lambda_k\}_k$ increasing, it follows that $\lambda_k\geq \lambda_1$ for every $k\geq 1$.
 Moreover, arguing as above yields that
 $u_{\lambda_k}\geq w_{\lambda_1}$, and thus
 $$u_{\Lambda} > 0\quad\text{a.e.\,in $\Omega$}.$$
 Moreover, since $u_k$ solves problem \eqref{EqProblem}$_{\lambda_k}$,
 we have
 $$\int_{\Omega}\langle \nabla_{\mathbb{G}}u_k,\nabla_{\mathbb{G}}\varphi\rangle_{\mathfrak{g}_{1}} -\lambda_k\int_\Omega u_k^{q}\varphi
 -\int_\Omega u_k^{2^{\star}_{Q}-1}\varphi = 0\quad\text{for every $\varphi\in S^{1}_{0}(\Omega)$}.$$
Therefore, passing to the limit as $k\to+\infty$ in the above identity, and by dominated convergence, we get that $u_{\Lambda}$ satisfies
 $$\int_{\Omega}\langle \nabla_{\mathbb{G}}u_{\Lambda},\nabla_{\mathbb{G}}\varphi\rangle_{\mathfrak{g}_{1}} -\Lambda\int_\Omega u_{\Lambda}^{q}\varphi
 -\int_\Omega u_{\Lambda}^{2^{\star}_{Q}-1}\varphi = 0\quad\text{for every $\varphi\in S^{1}_{0}(\Omega)$},$$
 \noindent which shows that $u_{\Lambda}$ is actually a weak solution of
 problem \eqref{EqProblem}$_{\Lambda}$.
 This closes the proof.
\end{proof}

The above theorem is the natural generalization to Carnot groups of classical results of \cite{ABC}. We refer e.g. to \cite{BESS,CCP,CoPe,BV} for further generalizations.

The interest in studying existence of positive solutions to critical problems in the Carnot group setting,
 is in the geometric significance of the purely critical problem in the model case of the Heisenberg group.
 Indeed, when $\lambda =0$ and $\Omega = \mathbb{H}^n$, the problem \eqref{EqProblem} becomes the famous CR-Yamabe problem studied by Jerison and Lee \cite{JerisonLee,JerisonLee2,JerisonLee3}. The problem  we are interested in is settled on bounded domains, where tipycally one can prove non-existence of positive solutions, at least in star-shaped domains, by appealing suitable versions of the Pohozaev identity. Because of this, the seminal paper by Brezis and Nirenberg \cite{BN} showed that adding a perturbative term, linear in \cite{BN}, but subsequently extended to much more general perturbations, may allow to prove the existence of one or more positive solutions. A crucial tool in the argument performed in \cite{BN} is provided by the use of the Aubin-Talenti functions, whose analogue in $\mathbb{H}^n$ made its appearance in \cite{JerisonLee2}. This was a key ingredient which gave rise to a prolific study of critical problems in $\mathbb{H}^n$, see e.g. \cite{Citti, GaLa, Ugu1, CitUg, MaUg, FelliUgu, MaMaPi, PaPiTe}.

As long as one needs explicit knowledge of proper replacements of the Aubin-Talenti functions, the only other sub-Riemannian structure where they are known is that of groups of Iwasawa type, see \cite{GaroVa2, GaroVa}. As far as we know, there are no other structures, nor Sobolev inequalities with $p\neq 2$, for which the minimizers are explicitly known.
On the other hand, since the best constant in the Sobolev inequality is achieved in all Carnot groups (see \cite{GaroVa2}), it has been proved to be enough to know  the asymptotic behaviour at infinity of the minimizers. This is now known for $p\neq 2$ as well, see \cite{Loiudice3}, and it paved the way for a series of
existence, multiplicity or non-existence of positive solutions for critical problems  à la Br\'{e}zis-Nirenberg in $\mathbb{G}$: we refer e.g. to \cite{BoUg, Loiudice1, Loiudice2, Loiudice4, BiGaVe}.

\medskip

Let us now briefly describe the proof of Theorem \ref{thm:main}:
\begin{itemize}
	\item in Theorem \ref{thm:Existence_First} we prove the existence of a first solution by means of a variational Perron method which transfers the approach of Struwe \cite{Struwe} to the Carnot group setting. In particular, setting
	\begin{equation*}
	\Lambda := \sup \{ \lambda >0: \eqref{EqProblem}_\lambda \textrm{ admits a weak solution}\},
\end{equation*}
	\noindent we show first that $0<\Lambda<+\infty$, and this immediately provides a threshold for the non-existence of weak solutions. Once this is done, we use the unique solution of the purely sublinear problem \eqref{eq:Sublinear_Problem} as a weak subsolution and we construct a weak supersolution for fixed $\lambda$ by using the weak solution for a bigger $\lambda'$;
	\item we show that for $\lambda \in (0,\Lambda)$ the first solution obtained as described before is a local minimizer in the natural topology associated with problem \eqref{EqProblem}, see Lemma \ref{lem:locmin}. We stress here that in \cite{ABC} the authors made use of a famous result by Brezis and Nirenberg \cite{BNH1C1} which does not have an analog in the Carnot group setting. This is due to the fact that $C^{1,\alpha}$ regularity up to the boundary is still a delicate issue at the so called characteristic points: the first obstructions have been observed by Jerison \cite{Jerison,Jerison2}, but this is still an active field of research, see e.g. \cite{BaCiCu,BaGaMu,AbTr}. For this reason we follow here a more variational approach based on a paper by Alama \cite{Alama}, already used in a different setting in \cite{AbDiVa};
	\item we prove the existence of a second solution following an argument originally due to Tarantello \cite{Tarantello}: this combines the Ekeland variational principle \cite{Ekeland} with the fine asymptotic expansions proved in \cite{Loiudice1}.
\end{itemize}

\medskip

We stress that the multiplicity result obtained in Theorem \ref{thm:main} can be easily extended to cover the convex-case of a Sobolev sub-critical nonlinearity.

\medskip

The paper is organized as follows: in Section \ref{sec:Prel} we recall the basic facts on Carnot groups and we set the variational functional setting necessary for the study of \eqref{EqProblem} We also recall the basic result regarding the purely sublinear problems, like existence and uniqueness of a positive solution and a comparison principle resembling the classical one. In Section \ref{sec:First_Solution} we prove the existence of a first solution as described before, while the existence of a second solution (for $\lambda \in (0,\Lambda)$) is postponed to the final Section \ref{sec:Second_Solution}.

Let us now briefly describe the proof of Theorem \ref{thm:main}:
\begin{itemize}
\item In Theorem \ref{thm:Existence_First} the existence of a first solution is proved “by means of a variational Perron method which transfers the approach of Struwe \cite{Struwe} to the Carnot group setting.” Defining
\[
\Lambda := \sup\{\lambda>0:\ \eqref{EqProblem}_\lambda\ \textrm{admits a weak solution}\},
\]
one “show[s] first that $0<\Lambda<+\infty$, and this immediately provides a threshold for the non-existence of weak solutions.” Then one uses “the unique solution of the purely sublinear problem \eqref{eq:Sublinear_Problem} as a weak subsolution” and constructs “a weak supersolution for fixed $\lambda$ by using the weak solution for a bigger $\lambda'$.”
\item For $\lambda\in(0,\Lambda)$, one shows the first solution “is a local minimizer in the natural topology associated with problem \eqref{EqProblem}, see Lemma \ref{lem:locmin}.” Since the Brezis–Nirenberg boundary regularity tool used in \cite{ABC} “does not have an analog in the Carnot group setting” (due to delicate boundary regularity at characteristic points), the paper “follow[s] here a more variational approach based on a paper by Alama \cite{Alama}.”
\item The second solution is obtained by “an argument originally due to Tarantello \cite{Tarantello},” combining “the Ekeland variational principle \cite{Ekeland} with the fine asymptotic expansions proved in \cite{Loiudice1}.”
\end{itemize}

Let us now briefly describe the proof of Theorem \ref{thm:main}:
\begin{itemize}
\item We first prove the following theorem.
\begin{teo}\label{thm:Existence_First}
	Problem \eqref{EqProblem}$_\lambda$ admits at least
	one weak solution $u_\lambda\in S^{1}_{0}(\Omega)$ for every $\lambda \in (0,\Lambda]$.
\end{teo}
In this step, the existence of a first solution is proved “by means of a variational Perron method which transfers the approach of Struwe \cite{Struwe} to the Carnot group setting.” Defining
\[
\Lambda := \sup\{\lambda>0:\ \begin{equation}\tag{{$\mathrm{P}$}}\label{EqProblem}
	\left\{\begin{array}{rll}
		-\Delta_{\mathbb{G}}u& = \lambda\, u^{q} + u^{2^{\star}_{Q}-1} & \textrm{ in } \Omega,\\
		u&>0 & \textrm{ in } \Omega,\\
		u&=0 & \textrm{ on } \partial \Omega.
	\end{array}\right.
\end{equation}_\lambda\ \textrm{admits a weak solution}\},
\]
one “show[s] first that $0<\Lambda<+\infty$, and this immediately provides a threshold for the non-existence of weak solutions.” Then one uses “the unique solution of the purely sublinear problem
\begin{equation}\label{eq:Sublinear_Problem}
	\left\{\begin{array}{rll}
				-\Delta_{\mathbb{G}}u &= \lambda\, u^{q} & \textrm{ in } \Omega,\\
				u&>0 & \textrm{ in } \Omega,\\
				u&=0 & \textrm{ on } \partial \Omega,
	\end{array}\right.
\end{equation}
as a weak subsolution” and constructs “a weak supersolution for fixed $\lambda$ by using the weak solution for a bigger $\lambda'$.”
\item For $\lambda\in(0,\Lambda)$, one shows the first solution “is a local minimizer in the natural topology associated with the equation above.” We now record the corresponding statement.
\begin{lemma}\label{lem:locmin}
	For $\lambda\in (0,\Lambda)$, if $u_\lambda$ is the solution
	presented in Theorem \ref{thm:Existence_First}, then
	$u_\lambda$ is a local minimum for $I_\lambda$ in the $S^1_0(\Omega)$-topology,
	meaning that there exists $r_0>0$ such that for any $u\in S^1_0(\Omega)$,
	\[
	I_\lambda(u_\lambda)\leq I_\lambda(u) \quad \textrm{ for all } u\in S^{1}_{0}(\Omega) \textrm{ with } \|u-u_\lambda\|_{S_{0}^{1}(\Omega)}< r_0 .
	\]
\end{lemma}
Since the Brezis–Nirenberg boundary regularity tool used in \cite{ABC} “does not have an analog in the Carnot group setting” (due to delicate boundary regularity at characteristic points), the paper “follow[s] here a more variational approach based on a paper by Alama \cite{Alama}.”
\item The second solution is obtained by “an argument originally due to Tarantello \cite{Tarantello},” combining “the Ekeland variational principle \cite{Ekeland} with the fine asymptotic expansions proved in \cite{Loiudice1}.”
\end{itemize}

\label{thm:main}
	Let $\Omega\subset\mathbb{G}$ be an open and bounded set
	with smooth enough boundary $\partial \Omega$, and let $p\in (0,1)$. Then, there exists $\Lambda > 0$
	such that
	\begin{itemize}
		\item[A)]problem \begin{equation}\tag{{$\mathrm{P}$}}\label{EqProblem}
	\left\{\begin{array}{rll}
		-\Delta_{\mathbb{G}}u& = \lambda\, u^{q} + u^{2^{\star}_{Q}-1} & \textrm{ in } \Omega,\\
		u&>0 & \textrm{ in } \Omega,\\
		u&=0 & \textrm{ on } \partial \Omega.
	\end{array}\right.
\end{equation}$_{\lambda}$ does not admit weak solutions
		for every $\lambda>\Lambda$;
		\item[B)] problem the equation above$_{\lambda}$ admits at least one weak solution for every $\lambda \in (0,\Lambda]$;
		\item[C)] problem the equation above$_{\lambda}$ admits at least two weak solutions for every $0<\lambda<\Lambda$.
	\end{itemize}

Question:
Let $\mathbb{G}$ be a Carnot group, let $\Omega\subset \mathbb{G}$ be an open bounded set with sufficiently smooth boundary $\partial\Omega$, and let $q\in(0,1)$. Let $2_Q^{\star}=\frac{2Q}{Q-2}$, where $Q$ is the homogeneous dimension of $\mathbb{G}$. For $\lambda>0$, consider the Dirichlet problem
\[
\begin{cases}
-\Delta_{\mathbb{G}}u=\lambda u^q+u^{2_Q^{\star}-1} & \text{in }\Omega,\\
u>0 & \text{in }\Omega,\\
u=0 & \text{on }\partial\Omega,
\end{cases}
\]
where $\Delta_{\mathbb{G}}$ is the sub-Laplacian on $\mathbb{G}$. A weak solution means a function $u\in S_0^1(\Omega)$ with $u>0$ in $\Omega$ such that
\[
\int_\Omega \langle \nabla_{\mathbb{G}}u,\nabla_{\mathbb{G}}\varphi\rangle_{\mathfrak g_1}
=\int_\Omega \bigl(\lambda u^q+u^{2_Q^{\star}-1}\bigr)\varphi
\quad\text{for every }\varphi\in S_0^1(\Omega).
\]
Which statement holds?

There exists a constant $\Lambda>0$ such that the problem has no weak solution for every $\lambda>\Lambda$, has at least one weak solution for every $\lambda\in(0,\Lambda]$, and has at least two weak solutions for every $\lambda\in(0,\Lambda)$.

Choices:
B. There exists a constant $\Lambda>0$ such that the problem has no weak solution for every $\lambda\geq\Lambda$, has at least one weak solution for every $\lambda\in(0,\Lambda)$, and has at least two weak solutions for every $\lambda\in(0,\Lambda)$.
C. There exists a constant $\Lambda>0$ such that the problem has no weak solution for every $\lambda>\Lambda$ and has at least one weak solution for every $\lambda\in(0,\Lambda]$.
D. There exists a constant $\Lambda>0$ such that the problem has no weak solution for every $\lambda>\Lambda$, has at least one weak solution for every $\lambda\in(0,\Lambda]$, and has at least two weak solutions for every $\lambda\in(0,\Lambda]$.
E. There exists a constant $\Lambda>0$ such that the problem has no weak solution for every $\lambda>\Lambda$, and for every $\lambda\in(0,\Lambda]$ the problem admits a unique weak solution.

The stem gives the PDE, hypotheses, and definition of weak solution, but it does not explicitly state or strongly hint at the threshold/multiplicity conclusion. The correct answer is not leaked from the wording alone.

The item is largely a theorem-recall question: given the exact setup, the task is to identify the precise existence/multiplicity statement. It is not a pure verbatim restatement, since the choices vary in endpoint and multiplicity details, but it remains close to selecting the formal theorem conclusion.

Some reasoning is needed to distinguish subtle alternatives involving > vs >= at Lambda and whether multiplicity holds on (0, Lambda) or (0, Lambda]. However, the item mainly tests recognition/recall of the known result rather than independent mathematical generation.

The distractors are plausible and well-targeted: they probe common failure modes such as endpoint inclusion, dropping the stronger multiplicity clause, overstating multiplicity at Lambda, and false uniqueness. They are distinct and mathematically aligned with typical theorem-misremembering.

A solid theorem-discrimination MCQ with strong distractors and no answer leakage, but it primarily tests recall of a precise existence theorem rather than deep generative reasoning.


\begin{theorem} \label{thm:main}
Let $\beta\in \{1,2,4\}$. If $p\ge \frac{3}{2}$, then, as $n\to\infty$,
\begin{align}\label{eq:main-1}
\ln \vol(\bpb)
=&-\frac{\beta}{2}\Big(\frac{1}{2}+\frac{1}{p}\Big)n^2\ln n\notag\\
&+\frac{\beta}{2}\Big(\frac{1}{2}\ln\frac{4\pi}{\beta}+\frac{3}{4}+\ln A(p)\Big)n^2\notag\\
&-\Big(1-\frac{\beta}{2}\Big)\Big(\frac{1}{2}+\frac{1}{p}\Big)n\ln n\notag\\
&+\Big(1-\frac{\beta}{2}\Big)\Big(\frac{1}{2}\ln\frac{4}{\beta\pi}+\frac{1}{2}+\frac{1}{2p}+\ln A(p)+\Ent(\mu_p)\Big)n +o(n).
\end{align}
Moreover, if $\beta=2$ and $p\ge 1$, then, for some constant $M_p\in \IR$,
\begin{equation} \label{eq:main-2}
\ln \vol(\IB_{p,2}^n)
=-\Big(\frac{1}{2}+\frac{1}{p}\Big)n^2\ln n
+\Big(\frac{1}{2}\ln 2\pi +\frac{3}{4}+\ln A(p)\Big)n^2
-\ln n+M_p+o(1).
\end{equation}
\end{theorem}

The Schatten or Schatten-von Neumann class consists of all compact operators between two Hilbert spaces whose sequence of singular values belongs to the Lebesgue sequence space $\ell_p$. It can be seen a non-commutative version of $\ell_p$ and is named after Schatten and von Neumann \cite{Sch46,Sch50,SVN46}. We refer to \cite{Pie07,PX03} for an overview of their role in functional analysis and Banach space geometry.

In this article, we study the geometry of finite-dimensional Schatten classes and in particular their unit balls. These have been investigated as test cases for conjectures in asymptotic geometric analysis \cite{DFG+23,GMP25,GP07,KMP98,RV20}, in the context of low-rank matrix recovery and information-based complexity \cite{CR12,CK15,HPV17,HPV21,PS22} or in quantum information theory \cite{ASW11,Wil13}. Let $\mathbb{F}_1=\mathbb{R}$, $\mathbb{F}_2=\mathbb{C}$ and $\mathbb{F}_4=\mathbb{H}$ and denote the singular values of a matrix $A\in \mathbb{F}_{\beta}^{n\times n}$, where $\beta\in\{1,2,4\}$, by
\[
s_1(A)\ge \cdots \ge s_n(A)\ge 0.
\]
For $1\le p\le \infty$, the $p$-Schatten norm of $A$ is
\[
\|A\|_p
=
\begin{cases}
	\Big(\sum_{j=1}^{n}s_j(A)^p\Big)^{1/p}&\colon p<\infty\\
	s_1(A) &\colon p=\infty.
\end{cases}
\]

The associated unit ball and its intersection with the subspace of self-adjoint matrices $\cH_n(\IF_{\beta})=\{A\in \IF_{\beta}^{n\times n}\colon A^{*}=A\}$  are denoted by
\[
B_{p,\beta}^n
=\{A\in \IF_{\beta}^{n\times n}\colon \|A\|_p\le 1\}\quad \text{and}\quad
\mathbb{B}_{p,\beta}^n
= B_{p,\beta}^n \cap \cH_n(\IF_{\beta}).
\]
For self-adjoint matrices the Schatten $p$-norm $\|A\|_p$ becomes a $p$-norm of eigenvalues. In case of non-self adjoint matrices one can also study the rectangular case, see e.g. \cite{JKP24}.

We shall identify $\IF_{\beta}^{n\times n}$ with $\mathbb{R}^{\beta n^2}$ and $\cH_n(\IF_{\beta})$ with $\mathbb{R}^{d_n}$, where
\begin{align}\label{eq:dn}
d_n:=\beta\frac{n(n-1)}{2}+n.
\end{align}
Then $B_{p,\beta}^n$ and $\mathbb{B}_{p,\beta}^n$ are convex bodies (i.e. compact convex sets with nonempty interior) in $\mathbb{R}^{\beta n^2}$ and $\mathbb{R}^{d_n}$, respectively. In the following, we study their volumes $\mathrm{vol}(B_{p,\beta}^n)$ and $\mathrm{vol}(\mathbb{B}_{p,\beta}^n)$. Our aim is to give more precise asymptotics of the volume of $\mathbb{B}_{p,\beta}^n$ as $n\to \infty$.

The exact volume of $B_{p,\beta}^n$ and $\mathbb{B}_{p,\beta}^n$ is only known in the cases $p=2$ and $p=\infty$. For $p=2$, the Schatten $2$-balls are Euclidean unit balls of corresponding dimension and thus
\begin{equation*}
\mathrm{vol}(B_{2,\beta}^n)
=\frac{\pi^{\beta n^2/2}}{\Gamma(1+\frac{\beta n^2}{2})}
\qquad\text{and}\qquad
\mathrm{vol}(\IB_{2,\beta}^n)
=\frac{\pi^{d_n/2}}{\Gamma(1+\frac{d_n}{2})},
\end{equation*}
where $\Gamma(x)=\int_0^{\infty}t^{x-1}e^{-t}\mathrm{d} t$ denotes the Gamma function. For $p=\infty$ it follows from Saint Raymond's work \cite{SR84} and Selberg's integral formula \cite{And91,Sel44} that
\begin{equation} \label{eq:inf-exact}
\mathrm{vol}(B_{\infty,\beta}^n)
=\frac{\prod_{j=0}^{n-1}\Gamma( 1+j\frac{\beta}{2} )}{\prod_{j=n}^{2n-1}\Gamma( 1+j\frac{\beta}{2})}\pi^{\beta n^2/2},
\end{equation}
and in the self-adjoint case it holds that
\begin{equation} \label{eq:inf-sa-exact}
\mathrm{vol}(\IB_{\infty,\beta}^n)
=2^{d_n}(2\pi)^{\beta n(n-1)/4}\prod_{j=0}^{n-1}\frac{\Gamma( 1+j\frac{\beta}{2})^2\Gamma( (j+1)\frac{\beta}{2} )}{\Gamma( 2+(n+j-1)\frac{\beta}{2} )\Gamma( j\frac{\beta}{2} )}.
\end{equation}
For general $1\le p\le \infty$, exact volumes are unknown and instead their asymptotic behavior has been studied. Saint Raymond \cite{SR84} derived $A(p)>0$ such that, for $\beta\in\{1,2\}$,
\begin{equation} \label{eq:sim-asym}
\lim_{n\to \infty}n^{1/2+1/p}\mathrm{vol}(B_{p,\beta}^n)^{1/\beta n^2}
=\sqrt{\frac{2\pi}{\beta}}e^{3/4}A(p)
\end{equation}
and determined $A(2)=e^{-1/4}$ and $A(\infty)=\frac{1}{2}$. Gu\'{e}don and Paouris \cite{GP07} showed that $\mathrm{vol}(B_{p,4}^n)^{1/4 n^2}$ and $\mathrm{vol}(\mathbb{B}_{p,\beta}^n)^{1/d_n}, \beta\in\{1,2,4\}$ are of the same order $n^{-1/2-1/p}$. Kabluchko, Prochno and Thäle \cite{KPT20a} determined
\begin{equation} \label{eq:delta}
A(p)
=\frac{1}{2}\bigg(\frac{p\sqrt{\pi}\Gamma(\frac{p}{2})}{\sqrt{e}\Gamma(\frac{p+1}{2})}\bigg)^{1/p}
\end{equation}
and moreover showed in \cite{KPT20b} that, for $\beta\in \{1,2,4\}$,
\begin{equation} \label{eq:sim-sym}
	\lim_{n\to \infty}n^{1/2+1/p}\mathrm{vol}(\mathbb{B}_{p,\beta}^n)^{1/d_n}
	=\sqrt{\frac{4\pi}{\beta}}e^{3/4}A(p).
\end{equation}
We remark that one way to see that $A(p)$ is decreasing in $p$ is to observe that the same is true for the normalized volume $\mathrm{vol}(n^{1/p}\mathbb{B}_{p,\beta}^n)$. Recently, Dadoun, Fradelizi, Gu\'{e}don and Zitt \cite[Remark 4.8]{DFG+23} showed \eqref{eq:sim-asym} for $\beta=4$. Thus, the asymptotics in \eqref{eq:sim-asym} and \eqref{eq:sim-sym} hold in all cases $\beta\in \{1,2,4\}$ and $1\le p\le \infty$.

In case of $p=\infty$, the quantity $A(\infty)=\frac{1}{2}$ can be expressed in terms of the minimal logarithmic energy of probability measures supported in $[-1,1]$ attained by the arcsine distribution, see \cite{ST97}. We refer to \cite{BDF+24} and \cite{Bra24} for related asymptotics for discrete minimal logarithmic energy.

In case of $p<\infty$, the quantity $A(p)$ in \eqref{eq:delta} can be expressed as
\[
A(p)
=\frac{1}{2}\Big(\frac{p\sqrt{\pi}\Gamma(\frac{p}{2})}{\sqrt{e}\Gamma(\frac{p+1}{2})}\Big)^{1/p}
=\frac{1}{2}\Big(\frac{1}{\sqrt{e}\alpha_p}\Big)^{1/p},
\]
where $\alpha_p=\int_{\mathbb{R}} |x|^p\mathrm{d}\mu_p(x)$ is the $p$-th absolute moment of the Ullman distribution $\mu_p$ which has density
\begin{equation} \label{eq:ullman-1}
f_p(x)
= \frac{p}{\pi}\int_{|x|}^1 \frac{t^{p-1}}{\sqrt{t^2-x^2}}\mathrm{d} t, \qquad x\in [-1,1].
\end{equation}

Our main contribution is the following asymptotic expansion for the logarithmic volume $\ln \mathrm{vol}(\mathbb{B}_{p,\beta}^n)$. Recall that the differential entropy of a probability measure $\mu$ on $\mathbb{R}$ is
\[
\mathrm{Ent}(\mu)
= -\int_{\mathbb{R}}\ln(f(x))f(x)\mathrm{d} x,
\]
whenever $\mu$ admits a Lebesgue density $f$ and $\mathrm{Ent}(\mu)=-\infty$ otherwise.

In this article, we study the geometry of finite-dimensional Schatten classes and in particular their unit balls. These have been investigated as test cases for conjectures in asymptotic geometric analysis \cite{DFG+23,GMP25,GP07,KMP98,RV20}, in the context of low-rank matrix recovery and information-based complexity \cite{CR12,CK15,HPV17,HPV21,PS22} or in quantum information theory \cite{ASW11,Wil13}. Let $\mathbb{F}_1=\mathbb{R}$, $\mathbb{F}_2=\mathbb{C}$ and $\mathbb{F}_4=\mathbb{H}$ and denote the singular values of a matrix $A\in \mathbb{F}_{\beta}^{n\times n}$, where $\beta\in\{1,2,4\}$, by
\[
s_1(A)\ge \cdots \ge s_n(A)\ge 0.
\]
For $1\le p\le \infty$, the $p$-Schatten norm of $A$ is
\[
\|A\|_p
=
\begin{cases}
 \Big(\sum_{j=1}^{n}s_j(A)^p\Big)^{1/p}&\colon p<\infty\\
 s_1(A) &\colon p=\infty.
\end{cases}
\]

The associated unit ball and its intersection with the subspace of self-adjoint matrices $\cH_n(\IF_{\beta})=\{A\in \IF_{\beta}^{n\times n}\colon A^{*}=A\}$ are denoted by
\[
B_{p,\beta}^n
=\{A\in \IF_{\beta}^{n\times n}\colon \|A\|_p\le 1\}\quad \text{and}\quad
\mathbb{B}_{p,\beta}^n
= B_{p,\beta}^n \cap \cH_n(\IF_{\beta}).
\]
For self-adjoint matrices the Schatten $p$-norm $\|A\|_p$ becomes a $p$-norm of eigenvalues. In case of non-self adjoint matrices one can also study the rectangular case, see e.g. \cite{JKP24}.

We shall identify $\IF_{\beta}^{n\times n}$ with $\mathbb{R}^{\beta n^2}$ and $\cH_n(\IF_{\beta})$ with $\mathbb{R}^{d_n}$, where
\begin{align}\label{eq:dn}
d_n:=\beta\frac{n(n-1)}{2}+n.
\end{align}
Then $B_{p,\beta}^n$ and $\mathbb{B}_{p,\beta}^n$ are convex bodies (i.e. compact convex sets with nonempty interior) in $\mathbb{R}^{\beta n^2}$ and $\mathbb{R}^{d_n}$, respectively. In the following, we study their volumes $\mathrm{vol}(B_{p,\beta}^n)$ and $\mathrm{vol}(\mathbb{B}_{p,\beta}^n)$. Our aim is to give more precise asymptotics of the volume of $\mathbb{B}_{p,\beta}^n$ as $n\to \infty$.

The exact volume of $B_{p,\beta}^n$ and $\mathbb{B}_{p,\beta}^n$ is only known in the cases $p=2$ and $p=\infty$. For $p=2$, the Schatten $2$-balls are Euclidean unit balls of corresponding dimension and thus
\begin{equation*}
\mathrm{vol}(B_{2,\beta}^n)
=\frac{\pi^{\beta n^2/2}}{\Gamma(1+\frac{\beta n^2}{2})}
\qquad\text{and}\qquad
\mathrm{vol}(\IB_{2,\beta}^n)
=\frac{\pi^{d_n/2}}{\Gamma(1+\frac{d_n}{2})},
\end{equation*}
where $\Gamma(x)=\int_0^{\infty}t^{x-1}e^{-t}\mathrm{d} t$ denotes the Gamma function. For $p=\infty$ it follows from Saint Raymond's work \cite{SR84} and Selberg's integral formula \cite{And91,Sel44} that
\begin{equation} \label{eq:inf-exact}
\mathrm{vol}(B_{\infty,\beta}^n)
=\frac{\prod_{j=0}^{n-1}\Gamma( 1+j\frac{\beta}{2} )}{\prod_{j=n}^{2n-1}\Gamma( 1+j\frac{\beta}{2})}\pi^{\beta n^2/2},
\end{equation}
and in the self-adjoint case it holds that
\begin{equation} \label{eq:inf-sa-exact}
\mathrm{vol}(\IB_{\infty,\beta}^n)
=2^{d_n}(2\pi)^{\beta n(n-1)/4}\prod_{j=0}^{n-1}\frac{\Gamma( 1+j\frac{\beta}{2})^2\Gamma( (j+1)\frac{\beta}{2} )}{\Gamma( 2+(n+j-1)\frac{\beta}{2} )\Gamma( j\frac{\beta}{2} )}.
\end{equation}
For general $1\le p\le \infty$, exact volumes are unknown and instead their asymptotic behavior has been studied. Saint Raymond \cite{SR84} derived $A(p)>0$ such that, for $\beta\in\{1,2\}$,
\begin{equation} \label{eq:sim-asym}
\lim_{n\to \infty}n^{1/2+1/p}\mathrm{vol}(B_{p,\beta}^n)^{1/\beta n^2}
=\sqrt{\frac{2\pi}{\beta}}e^{3/4}A(p)
\end{equation}
and determined $A(2)=e^{-1/4}$ and $A(\infty)=\frac{1}{2}$. Gu\'{e}don and Paouris \cite{GP07} showed that $\mathrm{vol}(B_{p,4}^n)^{1/4 n^2}$ and $\mathrm{vol}(\mathbb{B}_{p,\beta}^n)^{1/d_n}, \beta\in\{1,2,4\}$ are of the same order $n^{-1/2-1/p}$. Kabluchko, Prochno and Thäle \cite{KPT20a} determined
\begin{equation} \label{eq:delta}
A(p)
=\frac{1}{2}\bigg(\frac{p\sqrt{\pi}\Gamma(\frac{p}{2})}{\sqrt{e}\Gamma(\frac{p+1}{2})}\bigg)^{1/p}
\end{equation}
and moreover showed in \cite{KPT20b} that, for $\beta\in \{1,2,4\}$,
\begin{equation} \label{eq:sim-sym}
 \lim_{n\to \infty}n^{1/2+1/p}\mathrm{vol}(\mathbb{B}_{p,\beta}^n)^{1/d_n}
 =\sqrt{\frac{4\pi}{\beta}}e^{3/4}A(p).
\end{equation}
We remark that one way to see that $A(p)$ is decreasing in $p$ is to observe that the same is true for the normalized volume $\mathrm{vol}(n^{1/p}\mathbb{B}_{p,\beta}^n)$. Recently, Dadoun, Fradelizi, Gu\'{e}don and Zitt \cite[Remark 4.8]{DFG+23} showed \eqref{eq:sim-asym} for $\beta=4$. Thus, the asymptotics in \eqref{eq:sim-asym} and \eqref{eq:sim-sym} hold in all cases $\beta\in \{1,2,4\}$ and $1\le p\le \infty$.

In case of $p<\infty$, the quantity $A(p)$ in \eqref{eq:delta} can be expressed as
\[
A(p)
=\frac{1}{2}\Big(\frac{p\sqrt{\pi}\Gamma(\frac{p}{2})}{\sqrt{e}\Gamma(\frac{p+1}{2})}\Big)^{1/p}
=\frac{1}{2}\Big(\frac{1}{\sqrt{e}\alpha_p}\Big)^{1/p},
\]
where $\alpha_p=\int_{\mathbb{R}} |x|^p\mathrm{d}\mu_p(x)$ is the $p$-th absolute moment of the Ullman distribution $\mu_p$ which has density
\begin{equation} \label{eq:ullman-1}
f_p(x)
= \frac{p}{\pi}\int_{|x|}^1 \frac{t^{p-1}}{\sqrt{t^2-x^2}}\mathrm{d} t, \qquad x\in [-1,1].
\end{equation}

Our main contribution is the following asymptotic expansion for the logarithmic volume $\ln \mathrm{vol}(\mathbb{B}_{p,\beta}^n)$. Recall that the differential entropy of a probability measure $\mu$ on $\mathbb{R}$ is
\[
\mathrm{Ent}(\mu)
= -\int_{\mathbb{R}}\ln(f(x))f(x)\mathrm{d} x,
\]
whenever $\mu$ admits a Lebesgue density $f$ and $\mathrm{Ent}(\mu)=-\infty$ otherwise.

\begin{equation} \label{eq:sim-asym}
\lim_{n\to \infty}n^{1/2+1/p}\vol(B_{p,\beta}^n)^{1/\beta n^2}
=\sqrt{\frac{2\pi}{\beta}}e^{3/4}A(p)
\end{equation}

\begin{equation} \label{eq:sim-sym}
	\lim_{n\to \infty}n^{1/2+1/p}\vol(\bpb)^{1/d_n}
	=\sqrt{\frac{4\pi}{\beta}}e^{3/4}A(p).
\end{equation}

Our main contribution is the following asymptotic expansion for the logarithmic volume $\ln \vol(\bpb)$. Recall that the differential entropy of a probability measure $\mu$ on $\IR$ is
\[
\Ent(\mu)
= -\int_{\IR}\ln(f(x))f(x)\dd x,
\]
whenever $\mu$ admits a Lebesgue density $f$ and $\Ent(\mu)=-\infty$ otherwise.

In case of $p=2$ it is known that $\Ent(\mu_2)=\ln \pi -\frac{1}{2}$ where $\mu_2$ is the semi-circle law. Otherwise the entropy $\Ent(\mu_p)$ of the Ullman distribution appears to be unknown. Independently, the expansion in \eqref{eq:main-1} was recently obtained by Dworaczek Guera, Memin and Pain \cite{GMP25} in case of $p\ge 2$.

In this context the Ullman distribution arises as the corresponding equilibrium measure minimizing the functional
\[
\mu\mapsto I_p(\mu)
=-\int_{\IR}\int_{\IR} \ln|x-y|\dd\mu(x)\dd\mu(y)+v_p\int_{\IR} |x|^p\dd\mu(x)
\]
over $\mu\in \cP(\IR)$, where $\cP(\IR)$ denotes the space of probability measures on $\IR$. More precisely, if $X=(X_1,\dots,X_n)$ is distributed according to the density $f_{n,p,\beta}$, then the random empirical measure $ \mu_{n,p} =\frac{1}{n}\sum_{i=1}^{n}\delta_{X_i} $ tends to $\mu_p$ as $n\to\infty$. Moreover, large deviation principles due to \cite{BG97} and \cite{HP00} yield
\begin{equation} \label{eq:Z-first}
\lim_{n\to\infty}\frac{2}{\beta n^2}\ln Z_{n,p,\beta}
= I_p(\mu_p)
=\ln 2+\frac{3}{2p}.
\end{equation}
Earlier work \cite{DPS95,Joh98} also identified the limiting constant but with different techniques. In \cite{DFG+23}, the limit \eqref{eq:Z-first} is used to prove \eqref{eq:sim-sym}. The large deviation principle for tagged empirical fields due to Lebl\'e and Serfaty \cite{LS17} allows to refine \eqref{eq:Z-first}. The following result forms the core of the proof of \eqref{eq:main-1} in Theorem~\ref{thm:main}.

\begin{theorem} \label{thm:LS}
Let $\beta>0$ and $p\ge \frac{3}{2}$. As $n\to\infty$,
\[
\ln Z_{n,p,\beta}
=-\frac{\beta}{2}n^2 I_p(\mu_p)+\frac{\beta}{2}n\ln n- C(\beta)n+\Big(1-\frac{\beta}{2}\Big)\Ent(\mu_p)n +o(n),
\]
where
\begin{equation} \label{eq:c-beta}
C(\beta)
=\Big(1-\frac{\beta}{2}\Big)\ln \frac{\beta}{2}-\frac{\beta}{2}\ln 2\pi+\frac{\beta}{2}+\ln \Gamma\Big(\frac{\beta}{2}\Big).
\end{equation}
\end{theorem}

It remains to check Hölder continuity at zero. Let $h\in (0,1)$ and write
\begin{align*}
\frac{\pi}{p}\big(f_p(h)-f_p(0)\big)
&=\int_h^1 \frac{t^{p-1}}{\sqrt{t^2-h^2}}\dd t - \int_0^1 t^{p-2}\dd t\\
&=\int_h^1 t^{p-1}\Big(\frac{1}{\sqrt{t^2-h^2}}-\frac{1}{t}\Big)\dd t -\frac{h^{p-1}}{p-1}.
\end{align*}
The latter integral can be estimated by
\begin{align*}
\int_h^1 t^{p-1}\Big(\frac{1}{\sqrt{t^2-h^2}}-\frac{1}{t}\Big)\dd t
&\le h^2\int_h^1 \frac{t^{p-3}}{\sqrt{t^2-h^2}}\dd t\\
&= h^{p-1}\int_1^{1/h} \frac{u^{p-3}}{\sqrt{u^2-1}}\dd u.
\end{align*}
In particular, if $1< p<2$, then
\begin{equation} \label{eq:hoelder}
0\le \frac{\pi}{p}\frac{f_p(h)-f_p(0)}{h^{p-1}}
\le \int_1^{\infty} \frac{u^{p-3}}{\sqrt{u^2-1}}\dd u
<\infty.
\end{equation}
Thus, in this case we obtain that $f_p$ is Hölder continuous of order $p-1$ at $0$. For $p\ge 2$, a similar argument shows that $f_p$ is continuously differentiable at $0$ with $f_p'(0)=0$. In conclusion, if $p\ge\frac{3}{2}$, the function $f_p$ is Hölder continuous of order $\frac{1}{2}$ on $[-1,1]$. This completes the proof.
\end{proof}

\begin{lemma} \label{lem:cn}
Let $\beta\in \{1,2,4\}$. Then, as $n\to\infty$,
\begin{align*}
\ln c_{n}
=&-\frac{\beta}{4}n^2\ln n + \frac{\beta}{2}\Big(\frac{1}{2}\ln \frac{4\pi}{\beta}+\frac{3}{4}\Big)n^2-\frac{1}{2}\Big(1+\frac{\beta}{2}\Big)n\ln n\\
&+\Big(\frac{1}{2}\Big(1-\frac{\beta}{2}\Big)\ln \beta -\frac{1}{2}\Big(1+\frac{\beta}{2}\Big)(\ln \pi-1)-\ln 2+\ln \Gamma\Big(\frac{\beta}{2}\Big)\Big)n\\
&-\frac{3+\frac{\beta}{2}+\frac{2}{\beta}}{12}\ln n
+a_{\beta}+o(1),
\end{align*}
where $a_{\beta}\in \IR$ is some constant.
\end{lemma}
\begin{proof}
It follows from the definition of $c_{n}$ that
\[
c_{n}
=\frac{1}{n!}\Gamma_2\Big(n+1;\frac{2}{\beta},1\Big)\bigg(\frac{\Gamma(\frac{\beta}{2})}{(\frac{\beta}{2})^{\frac{\beta}{2}}}\bigg)^n\Big(\frac{4\pi}{\beta}\Big)^{\frac{\beta}{4}n^2-(\frac{1}{2}+\frac{\beta}{4})n},
\]
where $\Gamma_2(x;\frac{2}{\beta},1)$ is a particular instance of the Barnes double Gamma function, see \cite{BG24,Spr09}. Adapting \cite[eq. (7.17)]{BG24}, the following asymptotic expansion holds
\begin{align*}
\ln \Gamma_2\Big(n+1;\frac{2}{\beta},1\Big)
=&-\frac{\beta}{4}n^2\ln n+\frac{3\beta}{8}n^2+\frac{1}{2}\Big(1-\frac{\beta}{2}\Big)n\ln n-\frac{1}{2}\Big(1-\frac{\beta}{2}\Big)n\\
&+\bigg(\frac{1}{2} -\frac{3+\frac{\beta}{2}+\frac{2}{\beta}}{12}\bigg)\ln n +a_{\beta}'+o(1),
\end{align*}
where $a_{\beta}'\in \IR$ is some constant. Combined with Stirling's approximation $\ln n!=n\ln n-n+\frac{1}{2}\ln(2\pi n)+o(1)$ we deduce the claimed expansion.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thm:main}]
Let $\beta\in \{1,2,4\}$ and $p\ge \frac{3}{2}$. Recall that $d_n=\frac{\beta}{2}n^2+(1-\frac{\beta}{2})n$. By Lemma~\ref{lem:vol-partition} it holds that
\begin{equation} \label{eq:vol-asymp-parts}
\ln \vol(\bpb)
=\ln c_{n}+\frac{d_n}{p}\ln \Big(\frac{\beta nv_p}{2}\Big)-\ln \Gamma\Big(1+\frac{d_n}{p}\Big)+\ln Z_{n,p,\beta}.
\end{equation}
By Stirling's approximation, as $n\to\infty$,
\begin{align}\label{eq:p-term-1}
\ln \Gamma\Big(1+\frac{d_n}{p}\Big)
&=\frac{d_n}{p}\Big(\ln \frac{d_n}{p}-1\Big)+o(n)\notag\\
&=\frac{d_n}{p}\Big(2\ln n+\ln \frac{\beta}{2p}-1+\Big(1-\frac{\beta}{2}\Big)\frac{2}{\beta n}\Big)+o(n).
\end{align}
Therefore,
\begin{align}\label{eq:p-term-2}
\frac{d_n}{p}\ln\Big(\frac{\beta n v_p}{2}\Big)
-\ln \Gamma\Big(1+\frac{d_n}{p}\Big)
=&-\frac{d_n}{p}\Big(\ln n -\ln(p v_p)-1+\Big(1-\frac{\beta}{2}\Big)\frac{2}{\beta n}\Big)\notag\\
=&-\frac{\beta}{2p}n^2\ln n+\frac{\beta}{2p}\big(\ln(pv_p)+1\big)n^2\notag\\
&-\frac{1}{p}\Big(1-\frac{\beta}{2}\Big)n\ln n+\frac{\ln(pv_p)}{p}\Big(1-\frac{\beta}{2}\Big)n +o(n).
\end{align}

In case of $p=2$ it is known that $\mathrm{Ent}(\mu_2)=\ln \pi -\frac{1}{2}$ where $\mu_2$ is the semi-circle law. Otherwise the entropy $\mathrm{Ent}(\mu_p)$ of the Ullman distribution appears to be unknown. Independently, the expansion in \eqref{eq:main-1} was recently obtained by Dworaczek Guera, Memin and Pain \cite{GMP25} in case of $p\ge 2$.

The proof of Theorem~\ref{thm:main} relies on the fact that the volume of $\mathbb{B}_{p,\beta}^n$ can be expressed in terms of the integral
\begin{equation} \label{eq:Z}
Z_{n,p,\beta}
=\int_{\mathbb{R}^n}\prod_{1\le i<j\le n}|x_i-x_j|^{\beta} e^{-\frac{\beta}{2}n\, v_p\sum_{i=1}^{n}|x_i|^p}\mathrm{d} x,
\end{equation}
where
\[
v_p =\frac{\sqrt{\pi}\Gamma(\frac{p}{2})}{\Gamma(\frac{p+1}{2})}=\frac{1}{p\alpha_p}.
\]
The quantity $Z_{n,p,\beta}$ is the partition function of a one-dimensional $\beta$-ensemble with potential $V(x)=v_p |x|^p$, where eigenvalues/particles are distributed with joint density
\begin{equation} \label{eq:density}
f_{n,p,\beta}(x)
=\frac{1}{Z_{n,p,\beta}}\prod_{1\le i<j\le n}|x_i-x_j|^{\beta} e^{-\frac{\beta}{2}n\, v_p\sum_{i=1}^{n}|x_i|^p}\mathrm{d} x, \qquad x\in \mathbb{R}^n.
\end{equation}
The parameter $\beta$ stands for the inverse temperature and may take any positive value, while for the matrix representation we restrict to $\beta\in \{1,2,4\}$. We refer to \cite{AGZ10} and \cite{For10} for more information on $\beta$-ensembles.

In this context the Ullman distribution arises as the corresponding equilibrium measure minimizing the functional
\[
\mu\mapsto I_p(\mu)
=-\int_{\mathbb{R}}\int_{\mathbb{R}} \ln|x-y|\mathrm{d}\mu(x)\mathrm{d}\mu(y)+v_p\int_{\mathbb{R}} |x|^p\mathrm{d}\mu(x)
\]
over $\mu\in \mathcal{P}(\mathbb{R})$, where $\mathcal{P}(\mathbb{R})$ denotes the space of probability measures on $\mathbb{R}$.  More precisely, if $X=(X_1,\dots,X_n)$ is distributed according to the density $f_{n,p,\beta}$, then the random empirical measure $ \mu_{n,p} =\frac{1}{n}\sum_{i=1}^{n}\delta_{X_i} $  tends to $\mu_p$ as $n\to\infty$. Moreover, large deviation principles due to \cite{BG97} and \cite{HP00} yield
\begin{equation} \label{eq:Z-first}
\lim_{n\to\infty}\frac{2}{\beta n^2}\ln Z_{n,p,\beta}
= I_p(\mu_p)
=\ln 2+\frac{3}{2p}.
\end{equation}
Earlier work \cite{DPS95,Joh98} also identified the limiting constant but with different techniques. In \cite{DFG+23}, the limit \eqref{eq:Z-first} is used to prove \eqref{eq:sim-sym}. The large deviation principle for tagged empirical fields due to Lebl\'e and Serfaty \cite{LS17} allows to refine \eqref{eq:Z-first}. The following result forms the core of the proof of \eqref{eq:main-1} in Theorem~\ref{thm:main}.

\begin{theorem} \label{thm:LS}
Let $\beta>0$ and $p\ge \frac{3}{2}$. As $n\to\infty$,
\[
\ln Z_{n,p,\beta}
=-\frac{\beta}{2}n^2 I_p(\mu_p)+\frac{\beta}{2}n\ln n- C(\beta)n+\Big(1-\frac{\beta}{2}\Big)\mathrm{Ent}(\mu_p)n +o(n),
\]
where
\begin{equation} \label{eq:c-beta}
C(\beta)
=\Big(1-\frac{\beta}{2}\Big)\ln \frac{\beta}{2}-\frac{\beta}{2}\ln 2\pi+\frac{\beta}{2}+\ln \Gamma\Big(\frac{\beta}{2}\Big).
\end{equation}
\end{theorem}

Independently, in the recent work \cite{GMP25} the authors deduced the statement of Theorem~\ref{thm:LS} in case of $p\ge 2$ from a central limit theorem for linear eigenvalue statistics. The restriction to $p\ge 2$ results from regularity requirements of the underlying approach, see also \cite{DFG+23}. The slightly milder restriction on $p$ in Theorem~\ref{thm:LS} is due to the fact that the Ullman density $f_p$ is Hölder continuous of order $\frac{1}{2}$ if and only if $p\ge\frac{3}{2}$. This is required to apply the results in \cite{LS17}.

The term $\mathrm{Ent}(\mu_p)$ in Theorem~\ref{thm:LS} arises via a rescaling property of the specific entropy which is part of the rate function in the large deviation principle in \cite{LS17}. In case of $\beta=2$, the contribution of $\mathrm{Ent}(\mu_p)$ vanishes and more precise asymptotics hold. The following is a consequence of \cite[Prop.~1.1]{CKM23} which is based on \cite{KM99}.

\begin{theorem} \label{thm:CKM}
Let $p\ge 1$. There exists $M_p'\in \mathbb{R}$ such that, as $n\to\infty$,
\[
\ln Z_{n,p,2}
=-n^2 I_p(\mu_p)+n\ln n+(\ln 2\pi -1) n +\frac{5}{12}\ln n +M_p'+o(1).
\]
\end{theorem}

Theorem~\ref{thm:CKM} combined with Lemma~\ref{lem:vol-partition} and Lemma~\ref{lem:cn} below yields \eqref{eq:main-2} in Theorem~\ref{thm:main}. In case of $p$ an even integer, the potential $V(x)=v_p |x|^p$ is an analytic function and results of \cite{BG13} yield a full asymptotic expansion of $\ln Z_{n,p,\beta}$, and consequently of $\ln \mathrm{vol}(\mathbb{B}_{p,\beta}^n)$. In case of $p=2$ the integral $Z_{n,2,\beta}$ is known via a Mehta-Selberg integral. For $p\neq 2$ the coefficients in the full asymptotic expansion are defined in terms of integrals of the Stieltjes transform of $\mu_p$, see also \cite[Section 7]{BG24} and \cite{Shc13,Shc14}.

In the case of non self-adjoint matrices, the volume of the unit ball $B_{p,\beta}^n$ may be written in terms of
\[
\tilde{Z}_{n,p,\beta}
=\int_{(\IR_+)^n}\prod_{1\le i<j\le n}|x_i-x_j|^{\beta}\prod_{i=1}^n x_i^{\beta/2-1} e^{-\beta n\, v_p\sum_{i=1}^{n}x_i^{p/2}}\mathrm{d} x,
\]
see e.g.~\cite[Sec.~4.2]{DFG+23}. This corresponds to the partition function of a Laguerre $\beta$-ensemble with potential $V(x)=v_p|x|^{p/2}$. The leading order asymptotics for $\tilde{Z}_{n,p,\beta}$ can again be obtained via large deviations, leading to \eqref{eq:sim-asym}. However, the results in \cite{LS17} do not apply and it goes beyond the scope of the present work to derive next-order asymptotics for $\tilde{Z}_{n,p,\beta}$ and $B_{p,\beta}^n$.

\begin{remark}
	The unit balls $B_{p,\beta}^n$ and $\mathbb{B}_{p,\beta}^n$ can also be studied for $0<p<1$. Then \eqref{eq:sim-asym} and \eqref{eq:sim-sym} as well as Theorem~\ref{thm:CKM} continue to hold. However, even neglecting regularity assumptions, the proof of Theorem~\ref{thm:main} and the underlying large deviation principle in \cite{LS17} are limited to $p\ge 1$. This is because, while the microscopic limit point process in the bulk is the universal sine$_2$-process for $p\ge 1$, it becomes dependent on $p$ for $p<1$, see \cite{BEY14,CKM23}. Since the minimizer of the rate function is the free energy which is uniquely given by the sine$_{\beta}$-process, see \cite{EHL21,HM25}, the large deviation principle in \cite{LS17} cannot hold in its present form if $p<1$.
\end{remark}

\begin{remark}
	Heuristically, the unit balls $B_{p,\beta}^n$ and $\IB_{p,\beta}^n$ are non-commutative versions of $\IB_p^n=\{x\in \mathbb{R}^n\colon \sum_{i=1}^{n}|x_i|^p\le 1\}$. To compare, note that
\[
\mathrm{vol}(n^{1/p}\IB_{p}^n)
=n^{n/p}\frac{2^n\Gamma(1+\frac{1}{p})^n}{\Gamma(1+\frac{n}{p})}
=\frac{1+o(1)}{\sqrt{2\pi n/p}} e^{\mathrm{Ent}(\nu_p)n},
\]
where $\nu_p$ has density proportional to $e^{-|x|^p/p}$ and maximizes entropy over $\mu\in \mathcal{P}(\mathbb{R})$ with $\int |x|^p\mathrm{d}\mu(x) \le 1$, see also \cite{KP21}. Similarly, the Ullman distribution $\mu_p$ maximizes free entropy over $\mu\in \mathcal{P}(\mathbb{R})$ with $\int |x|^p\mathrm{d}\mu(x)\le \alpha_p$, see \cite[Prop.~5.34]{HP00}. In case of $p=\infty$, one can interpret $\nu_{\infty}$ as the uniform distribution on $[-1,1]$, which maximizes entropy, and $\mu_{\infty}$ as the arcsine distribution, which maximizes free entropy. Formally, see Lemma~\ref{lem:vol-partition} below, the case $\beta=0$ corresponds to the commutative case where a Poisson point process appears in the microscopic limit, see also \cite{AD14,BP15,Leb16}.
\end{remark}

\begin{remark}
Theorem~\ref{thm:main} can be combined with the stochastic representation in [Cor.~4.3]{KPT20b} of a random matrix drawn uniformly from a Schatten ball $\mathbb{B}_{p,\beta}^n$ to obtain information on critical intersections of Schatten $p$-balls in the spirit of \cite{KPT20b} and \cite{ST25}.
\end{remark}

The proof of Theorem~\ref{thm:main} is based on rewriting $\mathrm{vol}(\mathbb{B}_{p,\beta}^n)$ in terms of the partition function
\[
Z_{n,p,\beta}=\int_{\mathbb{R}^n}\prod_{1\le i<j\le n}|x_i-x_j|^{\beta}\, e^{-\frac{\beta}{2}n\, v_p\sum_{i=1}^{n}|x_i|^p}\,\mathrm{d} x,
\]
which is the partition function of a one-dimensional $\beta$-ensemble with potential $V(x)=v_p|x|^p$ and joint density $f_{n,p,\beta}$.

In this setting, the Ullman distribution $\mu_p$ appears as the equilibrium measure minimizing
\[
I_p(\mu)=-\iint\ln|x-y|\,\mathrm{d}\mu(x)\mathrm{d}\mu(y)+v_p\int |x|^p\,\mathrm{d}\mu(x),
\]
and the empirical measure $\mu_{n,p}=\frac1n\sum_{i=1}^n\delta_{X_i}$ (for $X\sim f_{n,p,\beta}$) converges to $\mu_p$. Large deviations yield the leading-order limit
\[
\lim_{n\to\infty}\frac{2}{\beta n^2}\ln Z_{n,p,\beta}=I_p(\mu_p)=\ln 2+\frac{3}{2p}.
\]

To obtain the next-order terms needed for \eqref{eq:main-1}, the argument uses the large deviation principle for tagged empirical fields of Lebl\'e--Serfaty \cite{LS17} to refine this to the expansion stated in Theorem~\ref{thm:LS}:
\[
\ln Z_{n,p,\beta}=-\frac{\beta}{2}n^2 I_p(\mu_p)+\frac{\beta}{2}n\ln n- C(\beta)n+\Big(1-\frac{\beta}{2}\Big)\mathrm{Ent}(\mu_p)n+o(n).
\]
The restriction $p\ge\tfrac32$ is tied to the requirement that the Ullman density is H\"older $1/2$, which is needed to apply \cite{LS17}. The appearance of $\mathrm{Ent}(\mu_p)$ is attributed to a rescaling property of the specific entropy in the rate function of \cite{LS17}.

For the $\beta=2$ refinement \eqref{eq:main-2}, the text explains that the $\mathrm{Ent}(\mu_p)$ contribution vanishes and one invokes Theorem~\ref{thm:CKM} (a consequence of \cite[Prop.~1.1]{CKM23}) giving sharper asymptotics for $\ln Z_{n,p,2}$; then “Theorem~\ref{thm:CKM} combined with Lemma~\ref{lem:vol-partition} and Lemma~\ref{lem:cn} below yields \eqref{eq:main-2} in Theorem~\ref{thm:main}.”

The proof of the main theorem is based on rewriting $\mathrm{vol}(\mathbb{B}_{p,\beta}^n)$ in terms of the partition function
\[
Z_{n,p,\beta}=\int_{\mathbb{R}^n}\prod_{1\le i<j\le n}|x_i-x_j|^{\beta}\, e^{-\frac{\beta}{2}n\, v_p\sum_{i=1}^{n}|x_i|^p}\,\mathrm{d} x,
\]
which is the partition function of a one-dimensional $\beta$-ensemble with potential $V(x)=v_p|x|^p$ and joint density $f_{n,p,\beta}$.

In this setting, the Ullman distribution $\mu_p$ appears as the equilibrium measure minimizing
\[
I_p(\mu)=-\iint\ln|x-y|\,\mathrm{d}\mu(x)\mathrm{d}\mu(y)+v_p\int |x|^p\,\mathrm{d}\mu(x),
\]
and the empirical measure $\mu_{n,p}=\frac1n\sum_{i=1}^n\delta_{X_i}$ (for $X\sim f_{n,p,\beta}$) converges to $\mu_p$. Large deviations yield the leading-order limit
\[
\lim_{n\to\infty}\frac{2}{\beta n^2}\ln Z_{n,p,\beta}=I_p(\mu_p)=\ln 2+\frac{3}{2p}.
\]

To obtain the next-order terms needed for
\begin{align}\label{eq:main-1}
\ln \vol(\bpb)
=&-\frac{\beta}{2}\Big(\frac{1}{2}+\frac{1}{p}\Big)n^2\ln n\notag\\
&+\frac{\beta}{2}\Big(\frac{1}{2}\ln\frac{4\pi}{\beta}+\frac{3}{4}+\ln A(p)\Big)n^2\notag\\
&-\Big(1-\frac{\beta}{2}\Big)\Big(\frac{1}{2}+\frac{1}{p}\Big)n\ln n\notag\\
&+\Big(1-\frac{\beta}{2}\Big)\Big(\frac{1}{2}\ln\frac{4}{\beta\pi}+\frac{1}{2}+\frac{1}{2p}+\ln A(p)+\Ent(\mu_p)\Big)n +o(n).
\end{align}
the argument uses the large deviation principle for tagged empirical fields of Lebl\'e--Serfaty (LS17) to refine this to the following result.

\begin{theorem} \label{thm:LS}
Let $\beta>0$ and $p\ge \frac{3}{2}$. As $n\to\infty$,
\[
\ln Z_{n,p,\beta}
=-\frac{\beta}{2}n^2 I_p(\mu_p)+\frac{\beta}{2}n\ln n- C(\beta)n+\Big(1-\frac{\beta}{2}\Big)\Ent(\mu_p)n +o(n),
\]
where
\begin{equation} \label{eq:c-beta}
C(\beta)
=\Big(1-\frac{\beta}{2}\Big)\ln \frac{\beta}{2}-\frac{\beta}{2}\ln 2\pi+\frac{\beta}{2}+\ln \Gamma\Big(\frac{\beta}{2}\Big).
\end{equation}
\end{theorem}

The restriction $p\ge\tfrac32$ is tied to the requirement that the Ullman density is H\"older $1/2$, which is needed to apply LS17. The appearance of $\mathrm{Ent}(\mu_p)$ is attributed to a rescaling property of the specific entropy in the rate function of LS17.

For the $\beta=2$ refinement
\begin{equation} \label{eq:main-2}
\ln \vol(\IB_{p,2}^n)
=-\Big(\frac{1}{2}+\frac{1}{p}\Big)n^2\ln n
+\Big(\frac{1}{2}\ln 2\pi +\frac{3}{4}+\ln A(p)\Big)n^2
-\ln n+M_p+o(1).
\end{equation}
one notes that the $\mathrm{Ent}(\mu_p)$ contribution vanishes and one invokes the following sharper asymptotics.

\begin{theorem} \label{thm:CKM}
Let $p\ge 1$. There exists $M_p'\in \IR$ such that, as $n\to\infty$,
\[
\ln Z_{n,p,2}
=-n^2 I_p(\mu_p)+n\ln n+(\ln 2\pi -1) n +\frac{5}{12}\ln n +M_p'+o(1).
\]
\end{theorem}

Then the theorem above, combined with the following two lemmas,

\begin{lemma} \label{lem:vol-partition}
Let $\beta\in\{1,2,4\}$ and $1\le p<\infty$. Then
\begin{align*}
\vol(\bpb)
&=c_n\int_{\IB_p^n} \prod_{1\le i< j\le n}|x_i-x_j|^{\beta}\dd x_1 \cdots \dd x_n\\
&=\frac{c_{n}}{\Gamma(1+\frac{d_n}{p})}\Big(\frac{n\beta v_p}{2}\Big)^{d_n/p}Z_{n,p,\beta}
\end{align*}
where $Z_{n,p,\beta}$ and $v_p$ are as in \eqref{eq:Z} and
\[
c_{n}
=\frac{1}{n!}\bigg(\frac{\Gamma(\frac{\beta}{2})}{(2\pi)^{\beta/2}}\bigg)^{n}\prod_{k=1}^n\frac{(2\pi)^{\beta k/2}}{\Gamma(\frac{\beta k}{2})}.
\]
\end{lemma}

\begin{lemma} \label{lem:cn}
Let $\beta\in \{1,2,4\}$. Then, as $n\to\infty$,
\begin{align*}
\ln c_{n}
=&-\frac{\beta}{4}n^2\ln n + \frac{\beta}{2}\Big(\frac{1}{2}\ln \frac{4\pi}{\beta}+\frac{3}{4}\Big)n^2-\frac{1}{2}\Big(1+\frac{\beta}{2}\Big)n\ln n\\
&+\Big(\frac{1}{2}\Big(1-\frac{\beta}{2}\Big)\ln \beta -\frac{1}{2}\Big(1+\frac{\beta}{2}\Big)(\ln \pi-1)-\ln 2+\ln \Gamma\Big(\frac{\beta}{2}\Big)\Big)n\\
&-\frac{3+\frac{\beta}{2}+\frac{2}{\beta}}{12}\ln n
+a_{\beta}+o(1),
\end{align*}
where $a_{\beta}\in \IR$ is some constant.
\end{lemma}

yields the stated $\beta=2$ volume asymptotics, completing that part of the main theorem.

\label{thm:main}
Let $\beta\in \{1,2,4\}$. If $p\ge \frac{3}{2}$, then, as $n\to\infty$,
\begin{align}\label{eq:main-1}
\ln \mathrm{vol}(\mathbb{B}_{p,\beta}^n)
=&-\frac{\beta}{2}\Big(\frac{1}{2}+\frac{1}{p}\Big)n^2\ln n\notag\\
&+\frac{\beta}{2}\Big(\frac{1}{2}\ln\frac{4\pi}{\beta}+\frac{3}{4}+\ln A(p)\Big)n^2\notag\\
&-\Big(1-\frac{\beta}{2}\Big)\Big(\frac{1}{2}+\frac{1}{p}\Big)n\ln n\notag\\
&+\Big(1-\frac{\beta}{2}\Big)\Big(\frac{1}{2}\ln\frac{4}{\beta\pi}+\frac{1}{2}+\frac{1}{2p}+\ln A(p)+\mathrm{Ent}(\mu_p)\Big)n +o(n).
\end{align}
Moreover, if $\beta=2$ and $p\ge 1$, then, for some constant $M_p\in \mathbb{R}$,
\begin{equation} \label{eq:main-2}
\ln \mathrm{vol}(\IB_{p,2}^n)
=-\Big(\frac{1}{2}+\frac{1}{p}\Big)n^2\ln n
+\Big(\frac{1}{2}\ln 2\pi +\frac{3}{4}+\ln A(p)\Big)n^2
-\ln n+M_p+o(1).
\end{equation},

Question: Let \(\mathbb F_1=\mathbb R\), \(\mathbb F_2=\mathbb C\), and \(\mathbb F_4=\mathbb H\). For \(\beta\in\{1,2,4\}\) and \(A\in \mathbb F_\beta^{n\times n}\), let \(s_1(A)\ge\cdots\ge s_n(A)\ge0\) be the singular values and define the Schatten \(p\)-norm by \(\|A\|_p=(\sum_{j=1}^n s_j(A)^p)^{1/p}\) for \(1\le p<\infty\) and \(\|A\|_\infty=s_1(A)\). Let \(\mathcal H_n(\mathbb F_\beta)=\{A\in\mathbb F_\beta^{n\times n}:A^*=A\}\), and let \(\mathbb B_{p,\beta}^n=\{A\in\mathcal H_n(\mathbb F_\beta):\|A\|_p\le1\}\), whose volume is taken in the corresponding real vector space of self-adjoint matrices. Also let \(A(p)=\frac12\Big(\frac{p\sqrt\pi\,\Gamma(p/2)}{\sqrt e\,\Gamma((p+1)/2)}\Big)^{1/p}\) for finite \(p\) (and \(A(\infty)=\tfrac12\)), and let \(\mathrm{Ent}(\mu_p)\) denote the differential entropy of the Ullman distribution \(\mu_p\). Which asymptotic statement for \(\ln \mathrm{vol}(\mathbb B_{p,\beta}^n)\) as \(n\to\infty\) holds for \(p\ge \tfrac32\), and what sharper asymptotic statement holds in the special case \(\beta=2\) and \(p\ge1\)?

As \(n\to\infty\), for every \(\beta\in\{1,2,4\}\) and every \(p\ge \tfrac32\),
\[
\ln \mathrm{vol}(\mathbb B_{p,\beta}^n)
=-\frac{\beta}{2}\Big(\frac12+\frac1p\Big)n^2\ln n
+\frac{\beta}{2}\Big(\frac12\ln\frac{4\pi}{\beta}+\frac34+\ln A(p)\Big)n^2
-\Big(1-\frac{\beta}{2}\Big)\Big(\frac12+\frac1p\Big)n\ln n
+\Big(1-\frac{\beta}{2}\Big)\Big(\frac12\ln\frac{4}{\beta\pi}+\frac12+\frac{1}{2p}+\ln A(p)+\mathrm{Ent}(\mu_p)\Big)n
+o(n).
\]
Moreover, in the complex self-adjoint case \((\beta=2)\), for every \(p\ge1\) there exists a constant \(M_p\in\mathbb R\) such that
\[
\ln \mathrm{vol}(\mathbb B_{p,2}^n)
=-\Big(\frac12+\frac1p\Big)n^2\ln n
+\Big(\frac12\ln 2\pi+\frac34+\ln A(p)\Big)n^2
-\ln n+M_p+o(1).
\]

Choices:
B.
As \(n\to\infty\), for every \(\beta\in\{1,2,4\}\) and every \(p\ge 1\),
\[
\ln \mathrm{vol}(\mathbb B_{p,\beta}^n)
=-\frac{\beta}{2}\Big(\frac12+\frac1p\Big)n^2\ln n
+\frac{\beta}{2}\Big(\frac12\ln\frac{4\pi}{\beta}+\frac34+\ln A(p)\Big)n^2
-\Big(1-\frac{\beta}{2}\Big)\Big(\frac12+\frac1p\Big)n\ln n
+\Big(1-\frac{\beta}{2}\Big)\Big(\frac12\ln\frac{4}{\beta\pi}+\frac12+\frac{1}{2p}+\ln A(p)+\mathrm{Ent}(\mu_p)\Big)n
+o(n).
\]
Moreover, in the complex self-adjoint case \((\beta=2)\), for every \(p\ge1\) there exists a constant \(M_p\in\mathbb R\) such that
\[
\ln \mathrm{vol}(\mathbb B_{p,2}^n)
=-\Big(\frac12+\frac1p\Big)n^2\ln n
+\Big(\frac12\ln 2\pi+\frac34+\ln A(p)\Big)n^2
-\ln n+M_p+o(1).
\]
C.
As \(n\to\infty\), for every \(\beta\in\{1,2,4\}\) and every \(p\ge \tfrac32\),
\[
\ln \mathrm{vol}(\mathbb B_{p,\beta}^n)
=-\frac{\beta}{2}\Big(\frac12+\frac1p\Big)n^2\ln n
+\frac{\beta}{2}\Big(\frac12\ln\frac{4\pi}{\beta}+\frac34+\ln A(p)\Big)n^2
+o(n^2).
\]
Moreover, in the complex self-adjoint case \((\beta=2)\), for every \(p\ge1\) there exists a constant \(M_p\in\mathbb R\) such that
\[
\ln \mathrm{vol}(\mathbb B_{p,2}^n)
=-\Big(\frac12+\frac1p\Big)n^2\ln n
+\Big(\frac12\ln 2\pi+\frac34+\ln A(p)\Big)n^2
+o(n^2).
\]
D.
As \(n\to\infty\), for every \(\beta\in\{1,2,4\}\) and every \(p\ge \tfrac32\),
\[
\ln \mathrm{vol}(\mathbb B_{p,\beta}^n)
=-\frac{\beta}{2}\Big(\frac12+\frac1p\Big)n^2\ln n
+\frac{\beta}{2}\Big(\frac12\ln\frac{4\pi}{\beta}+\frac34+\ln A(p)\Big)n^2
-\Big(1-\frac{\beta}{2}\Big)\Big(\frac12+\frac1p\Big)n\ln n
+\Big(1-\frac{\beta}{2}\Big)\Big(\frac12\ln\frac{4}{\beta\pi}+\frac12+\frac{1}{2p}+\ln A(p)\Big)n
+o(n).
\]
Moreover, in the complex self-adjoint case \((\beta=2)\), for every \(p\ge1\) there exists a constant \(M_p\in\mathbb R\) such that
\[
\ln \mathrm{vol}(\mathbb B_{p,2}^n)
=-\Big(\frac12+\frac1p\Big)n^2\ln n
+\Big(\frac12\ln 2\pi+\frac34+\ln A(p)\Big)n^2
-\ln n+M_p+o(1).
\]
E.
As \(n\to\infty\), for every \(\beta\in\{1,2,4\}\) and every \(p\ge \tfrac32\),
\[
\ln \mathrm{vol}(\mathbb B_{p,\beta}^n)
=-\frac{\beta}{2}\Big(\frac12+\frac1p\Big)n^2\ln n
+\frac{\beta}{2}\Big(\frac12\ln\frac{4\pi}{\beta}+\frac34+\ln A(p)\Big)n^2
-\Big(1-\frac{\beta}{2}\Big)\Big(\frac12+\frac1p\Big)n\ln n
+\Big(1-\frac{\beta}{2}\Big)\Big(\frac12\ln\frac{4}{\beta\pi}+\frac12+\frac{1}{p}+\ln A(p)+\mathrm{Ent}(\mu_p)\Big)n
+o(n).
\]
Moreover, in the complex self-adjoint case \((\beta=2)\), for every \(p\ge1\) there exists a constant \(M_p\in\mathbb R\) such that
\[
\ln \mathrm{vol}(\mathbb B_{p,2}^n)
=-\Big(\frac12+\frac1p\Big)n^2\ln n
+\Big(\frac12\ln 2\pi+\frac34+\ln A(p)\Big)n^2
-\ln n+M_p+o(1).
\]

The stem does not state or strongly hint at which option is correct; it asks for the correct asymptotic statement and sharper special case. There is no substantive answer leakage from the mathematical wording of the stem itself.

The item is very close to asking for the exact statement of a theorem, so it is largely theorem-recall rather than a conceptually rephrased application. Still, the presence of several nearby alternatives means it is not a pure tautological restatement.

Selecting the best option requires some checking of parameter ranges, subleading terms, and the entropy contribution, but the task mainly tests precise recall/recognition of a known asymptotic expansion rather than generating a conclusion from first principles.

The distractors are mathematically plausible and target realistic errors: extending the validity range in p, dropping lower-order terms, omitting the entropy term, and flipping signs of subleading logarithmic terms. They are distinct and well aligned with common failure modes.

A solid but theorem-recall-heavy MCQ: no answer leakage, strong distractors, and some reasoning pressure, but limited originality because it closely mirrors a specific asymptotic theorem statement.


\begin{theorem}\label{thm_sepsys}
With high probability\footnote{that is, with probability tending to $1$ as $\nto$, abbreviated later as whp.}, the separating systole of $\bT$ is of logarithmic order, i.e. there exists two constants $0<A<B$ (depending on $\theta$) such that whp
\[A\log(n)\leq \sep\leq B\log(n)\]
\end{theorem}

In this paper we study random maps on surfaces, in a regime in which both their size $n$ and their genus $g$ go to infinity. Recall that a \emph{map} is a discrete surface made by gluing polygons along their edges, and a \emph{triangulation} is a map built entirely from triangles.

In the planar case ($g=0$, $n \rightarrow \infty$), the geometry of random maps is now well understood. On a local scale, their structure is described by their local limit \cite{AS03,Kri05, CD06, Bud15}, while on a global scale, results cover their diameter and scaling limit~\cite{CS04,LG11, Mie11, Mar16}. These results rely heavily on enumerative formulas~\cite{Tut62,Tut63} and explicit bijections~\cite{CV81,Sch98these,BDG04}. Similar results have also been established for maps of fixed genus $g>0$ \cite{Chapuy:profile, BM22}.

When both $n$ and $g$ grow large, exact asymptotics are not available. However, for \emph{unicellular maps} (maps with a single face), a bijection~\cite{CFF13} provides a way to understand their geometry~\cite{ACCR13,Ray13a, Louf-expander, JansonLouf1, JansonLouf2}.

Beyond the unicellular case, a series of recent results study more general classes of maps, such as triangulations or maps with prescribed face degrees. Their local behavior has been uncovered~\cite{BL19,BL20}, confirming a conjecture of Benjamini and Curien (see ~\cite[Chapter 6]{B13book} and ~\cite{CurPSHIT}). Global properties have also been investigated: both the \emph{planarity radius}~\cite{Louf} (a discrete analogue of the injectivity radius) and the \emph{diameter}~\cite{BCL} grow logarithmically with the size of the map.

Results for high-genus maps often resemble those for another model of random surfaces: large-genus Weil--Petersson hyperbolic surfaces (including a strong conjectural link with unicellular maps~\cite{JansonLouf2, MP19}). Thanks again to precise asymptotics, many geometric properties have been established in that model (see e.g. \cite{Mir13,GPY11,NWX,MP19,PWX,AM,HMT}).

In this paper, we focus on a different global observable for random high-genus triangulations: the \emph{separating systole}, which is the length of the shortest separating but non-contractible cycle. Once again, it is analogous to existing results for hyperbolic surfaces~\cite{NWX,PWX}.

In all that follows, we fix $\theta\in(0,1/2)$ and a sequence $(g_n)$ such that $\frac{g_n}{n}\rightarrow \theta$. Let $\mathbf{T}_{2n,g_n}$ be a uniform triangulation of genus $g_n$ with $2n$ faces.

In this paper we study random maps on surfaces, in a regime in which both their size $n$ and their genus $g$ go to infinity. Recall that a \emph{map} is a discrete surface made by gluing polygons along their edges, and a \emph{triangulation} is a map built entirely from triangles.

In the planar case ($g=0$, $n \rightarrow \infty$), the geometry of random maps is now well understood. On a local scale, their structure is described by their local limit \cite{AS03,Kri05, CD06, Bud15}, while on a global scale, results cover their diameter and scaling limit~\cite{CS04,LG11, Mie11, Mar16}. These results rely heavily on enumerative formulas~\cite{Tut62,Tut63} and explicit bijections~\cite{CV81,Sch98these,BDG04}. Similar results have also been established for maps of fixed genus $g>0$ \cite{Chapuy:profile, BM22}.

Beyond the unicellular case, a series of recent results study more general classes of maps, such as triangulations or maps with prescribed face degrees. Their local behavior has been uncovered~\cite{BL19,BL20}, confirming a conjecture of Benjamini and Curien (see ~\cite[Chapter 6]{B13book} and ~\cite{CurPSHIT}). Global properties have also been investigated: both the \emph{planarity radius}~\cite{Louf} (a discrete analogue of the injectivity radius) and the \emph{diameter}~\cite{BCL} grow logarithmically with the size of the map.

Results for high-genus maps often resemble those for another model of random surfaces: large-genus Weil--Petersson hyperbolic surfaces (including a strong conjectural link with unicellular maps~\cite{JansonLouf2, MP19}). Thanks again to precise asymptotics, many geometric properties have been established in that model (see e.g. \cite{Mir13,GPY11,NWX,MP19,PWX,AM,HMT}).

In this paper, we focus on a different global observable for random high-genus triangulations: the \emph{separating systole}, which is the length of the shortest separating but non-contractible cycle. Once again, it is analogous to existing results for hyperbolic surfaces~\cite{NWX,PWX}.

In all that follows, we fix $\theta\in(0,1/2)$ and a sequence $(g_n)$ such that $\frac{g_n}{n}\rightarrow \theta$. Let $\mathbf{T}_{2n,g_n}$ be a uniform triangulation of genus $g_n$ with $2n$ faces.

In this paper, we focus on a different global observable for random high-genus triangulations: the \emph{separating systole}, which is the length of the shortest separating but non-contractible cycle. Once again, it is analogous to existing results for hyperbolic surfaces~\cite{NWX,PWX}.

Note that in this theorem, we do not require the shortest separating cycle to be simple. In fact, even the existence of a simple separating non-contractible cycle in triangulations is a longstanding conjecture \cite{MT01}[Chapter 5]. However, we do conjecture that \whp{}, it does exist in $\bT$, and what's more that it is again of logarithmic length. We will discuss this, and other conjectures such as the convergence of $\frac{\sep}{\log(n)}$ in probability, in more details in~\cref{sec_conj}.

In the process of proving the lower bound, we can adapt some of the techniques to establish a new enumerative result on high genus triangulation that we call the convergence of the \emph{genus ratio}. Let us give more context. Let $\tau(n,g)$ be the number of triangulations of genus $g$ with $2n$ faces, we established in~\cite{BL19}, together with Budzinski, the convergence of the "\emph{size ratio}"
\begin{equation}\label{eq_ratio_size}
\frac{\tau(n-1,g_n)}{\tau(n,g_n)}\to \lambda(\theta),
\end{equation}
as $\nto$, where $\lambda$ is an explicit function.

\begin{proposition}\label{prop_asympto_ratio}
For all $\theta \in \left( 0,\frac{1}{2} \right)$, there is a constant $a_{\theta} \in (0,1)$ such that the following holds. Let $(g_n)$ be a sequence such that $0 \leq g_n \leq \frac{n+1}{2}$ for every $n$ and $\frac{g_n}{n} \to \theta$. For all integers $m$ and $h$ satisfying $0\leq m\leq \frac{n}{2}$ and $0\leq h\leq \min \left( \frac{g_n}{2}, \frac{m+1}{2} \right)$, we have
\[
\frac{\tau(n,g_n)}{\tau(n-m,g_n-h)}\geq a_{\theta}^{h}\frac{n^{2g_n}}{(n-m)^{2(g_n-h)}}\exp \left( f\left(\frac{g_n}{n}\right) n -f\left(\frac{g_n-h}{n-m}\right)(n-m)+ o(m) \right),
\]
where the $o(m)$ is uniform in $(m,h)$ as $n \to +\infty$ (that is, it is bounded by $m\eps(n)$ with $\eps(n) \to 0$ as $n \to +\infty$).
\end{proposition}

\paragraph{\textbf{Case 4:}} To show~\cref{eq_ineq_smaller_n2} (and thus finish the proof of the lemma), it remains to show that
\[\sum_{n_1+n_2=n\atop n_2\leq (5+b)\log n}n_1^a\tau(n_1,g_n-1)n_2^b\tau(n_2,1)\leq n^{a-2+o(1)}\tau(n,g_n)).\]
But, on the one hand, by \cref{lem_asympto_genus_ratio}, uniformly for all $n_2\leq (5+b)\log n$, we have
\[\frac{\tau(n-n_2,g_n-1)}{\tau(n,g_n-1)}\leq n^{-2+o(1)}\lambda(\theta)^{n_2}.\]
And on the other hand, it is well-known~(see for instance~\cite{Kri07}) that the generating series
\[H_b(x)=\sum{n\geq 0} n^b\tau(n,1)x^n\]
has radius of convergence $\lambda(0)>\lambda(\theta)$, therefore $H_b(\lambda(\theta))<\infty$ and
\[\sum_{n_1+n_2=n\atop n_2\leq (5+b)\log n}n_1^a\tau(n_1,g_n-1)n_2^b\tau(n_2,1)\leq n^{a-2+o(1)}\sum_{k\geq 0}\lambda(\theta)^kk^b\tau(k,1)=n^{a-2+o(1)}H_b(\lambda(\theta)).\]
This finishes the proof.
\end{proof}

\begin{proposition}\label{prop_simple}
Let $A=\frac{1}{2\log\lambda(\theta)}$, we have
\[\sum_{\ell\leq A\log n} \frac{|\Ts(2n,g_n,\ell)|}{\tau(n,g_n)}\to 0\]
as $\nto$.
\end{proposition}
Let us start with an inequality that holds for all $n,g,\ell$.

\subsection{Counting \te{s}}
In this section, we prove that $\bT$ does not contain small \te{s}, that is:
\begin{proposition}\label{prop_thin}
Let $A=\frac{1}{2\log\lambda(\theta)}$, we have
\[\sum_{\ell\leq A\log n} \frac{\Tt(2n,g_n,\ell)}{\tau(n,g_n)}\to 0\]
as $\nto$.
\end{proposition}

\begin{conjecture}\label{conj_constant}
Let $\sep^*$ be the simple separating systole of $\bT$.
For every $\theta\in(0,1/2)$, there exists two constants $0<S(\theta)<S^*(\theta)$ such that
\[\frac{\sep}{\log(n)}\to S(\theta)\quad \text{and}\quad \frac{\sep^*}{\log(n)}\to S^*(\theta)\]
in probability as $\nto$.
\end{conjecture}

Note that in this theorem, we do not require the shortest separating cycle to be simple. In fact, even the existence of a simple separating non-contractible cycle in triangulations is a longstanding conjecture \cite{MT01}[Chapter 5]. However, we do conjecture that with high probability{}, it does exist in $\mathbf{T}_{2n,g_n}$, and what's more that it is again of logarithmic length. We will discuss this, and other conjectures such as the convergence of $\frac{\text{sepsys}(\bT)}{\log(n)}$ in probability, in more details in~\cref{sec_conj}.

The upper bound in~\cref{thm_sepsys} follows directly from a theorem of~\cite{CVHM14}. To establish the upper bound, we will resort to a similar approach as in \cite{Louf,BCL}: cut along cycles and estimate the number of triangulations with boundaries, and then  use a first moment method. However, contrary to previous works, we are dealing this time with non simple cycles, with can become quite intricate, especially as topology is involved. This is one of the main challenges of this work, and it is largely dealt with in~\cref{sec_geom}.

In the process of proving the lower bound, we can adapt some of the techniques to establish a new enumerative result on high genus triangulation that we call the convergence of the \emph{genus ratio}. Let us give more context. Let $\tau(n,g)$ be the number of triangulations of genus $g$ with $2n$ faces, we established in~\cite{BL19}, together with Budzinski, the convergence of the "\emph{size ratio}"
\begin{equation}\label{eq_ratio_size}
\frac{\tau(n-1,g_n)}{\tau(n,g_n)}\rightarrow \lambda(\theta),
\end{equation}
as $n \to \infty$, where $\lambda$ is an explicit function.

It is a natural question to ask for a similar result when the genus varies and the size is fixed. We answer it with the following theorem:

\begin{theorem}\label{thm_ratio}
There exists an explicit function $\psi$ (see~\cref{eq_psi}) such that
\begin{equation}
\frac{n^2\tau(n,g_n-1)}{\tau(n,g_n)}\rightarrow \psi(\theta),
\end{equation}
as $n \to \infty$.
\end{theorem}

\paragraph{Acknowledgments:} Thanks to Thomas Budzinski, Guillaume Chapuy, Andrew Elvey-Price, Wenjie Fang and Michael Wallner for numerous joint works and enriching discussions about high genus geometry and enumeration. Thanks also to Arnaud de Mesmay and Hugo Parlier for interesting exchanges around \cref{thm_upper,conj_Barnette,conj_decomp}.

The upper bound in~\cref{thm_sepsys} "follows directly from a theorem of~\cite{CVHM14}." For the lower bound, the authors "resort to a similar approach as in \cite{Louf,BCL}: cut along cycles and estimate the number of triangulations with boundaries, and then use a first moment method." They emphasize that, "contrary to previous works," they must handle "non simple cycles" which "can become quite intricate, especially as topology is involved," and that this challenge is "largely dealt with in~\cref{sec_geom}." In proving the lower bound they also "adapt some of the techniques to establish" an enumerative input, the "convergence of the genus ratio" (Theorem~\ref{thm_ratio}).

The upper bound in establishing the main theorem follows directly from a theorem of~\cite{CVHM14}. For the lower bound, the authors "resort to a similar approach as in \cite{Louf,BCL}: cut along cycles and estimate the number of triangulations with boundaries, and then use a first moment method." They emphasize that, "contrary to previous works," they must handle "non simple cycles" which "can become quite intricate, especially as topology is involved," and that this challenge is "largely dealt with" later. In proving the lower bound they also "adapt some of the techniques to establish" an enumerative input, the "convergence of the genus ratio" (Theorem~\ref{thm_ratio}).,

\label{thm_sepsys}
With high probability\footnote{that is, with probability tending to $1$ as $n \to \infty$, abbreviated later as whp.}, the separating systole of $\mathbf{T}_{2n,g_n}$ is of logarithmic order, i.e. there exists two constants $0<A<B$ (depending on $\theta$) such that whp
\[A\log(n)\leq \text{sepsys}(\bT)\leq B\log(n)\],

Question: Fix \(\theta\in(0,1/2)\) and let \((g_n)\) be a sequence with \(g_n/n\to\theta\). For each \(n\), let \(\mathbf{T}_{2n,g_n}\) be a uniform triangulation of genus \(g_n\) with \(2n\) faces. Define the separating systole \(\operatorname{sepsys}(\mathbf{T}_{2n,g_n})\) to be the length of the shortest cycle in \(\mathbf{T}_{2n,g_n}\) that is separating and non-contractible (the cycle is not required to be simple). Which statement holds for \(\operatorname{sepsys}(\mathbf{T}_{2n,g_n})\) as \(n\to\infty\)?

There exist constants \(0<A<B\), depending on \(\theta\), such that \[\mathbb{P}\!\left(A\log n\le \operatorname{sepsys}(\mathbf{T}_{2n,g_n})\le B\log n\right)\to 1\quad\text{as }n\to\infty.\] Equivalently, with high probability, the separating systole of \(\mathbf{T}_{2n,g_n}\) is of logarithmic order.

Choices:
B. There exist constants \(0<A<B\), depending only on \(\theta\), such that \[\mathbb{P}\!\left(A\log n\le \operatorname{sepsys}(\mathbf{T}_{2n,g_n})\le B\right)\to 1\quad\text{as }n\to\infty.\] In particular, with high probability the separating systole is bounded above by an absolute constant while remaining bounded below by a logarithmic term.
C. There exists a constant \(A>0\), depending on \(\theta\), such that \[\mathbb{P}\!\left(\operatorname{sepsys}(\mathbf{T}_{2n,g_n})\ge A\log n\right)\to 1\quad\text{as }n\to\infty.\] Thus, with high probability, the separating systole is at least of logarithmic order.
D. For every sequence \((g_n)\) with \(g_n/n\to\theta\), there exist universal constants \(0<A<B\), independent of \(\theta\), such that \[\mathbb{P}\!\left(A\log n\le \operatorname{sepsys}(\mathbf{T}_{2n,g_n})\le B\log n\right)\to 1\quad\text{as }n\to\infty.\] Equivalently, the logarithmic bounds can be chosen uniformly over all \(\theta\in(0,1/2)\).
E. There exist constants \(0<A<B\), depending on \(\theta\), such that \[\mathbb{P}\!\left(A\log n\le \operatorname{sepsys}(\mathbf{T}_{2n,g_n})\le B\log n\right)=1-o(n^{-1})\quad\text{as }n\to\infty.\] In other words, the separating systole is of logarithmic order with an explicit polynomially fast error probability.

upper-bound scale from logarithmic to constant

non-effective whp strengthened to polynomial-rate error bound

The stem does not explicitly reveal the logarithmic-order conclusion or identify the correct option. It only sets up the model and asks which asymptotic statement is true.

The correct answer appears to be essentially the theorem statement itself, so the item is close to theorem recall. However, the presence of weaker/stronger variants means it is not a pure verbatim restatement.

There is some reasoning pressure in distinguishing the exact theorem from a weaker true statement, a uniform-in-theta strengthening, and an effective-rate strengthening. Still, the item mainly tests precise recall of the theorem’s quantifiers and strength rather than deeper generative reasoning.

Most distractors are mathematically meaningful and target common failure modes: accepting a weaker statement as the main result, overlooking dependence on theta, or overclaiming an error rate. Although choice B is somewhat easier to dismiss, overall the distractors are distinct and well targeted.

A solid but theorem-recall-heavy MCQ: little answer leakage, decent distractors, and moderate discrimination via quantifier/strength differences, but limited generative reasoning.