\label{Th-SynthesExistenceSW}
    Let $d\in\{1,2\}$, $\lambda,\omega\in\mathbb R$ and $\gamma>0$. Denote by $\mu_0$ the lowest eigenvalue of $-A$. In each of the following situation, there exists a positive solution of \cref{Eq-StationaryAGP} in $\mathrm{D}\left(\sqrt{|A|}\right)$,
    \begin{itemize}
        \item $\lambda>0$ and $\omega>-\mu_0$,
        \item $\lambda = 0$ and $\omega=\mu_0$,
        \item $\lambda<0$ and $\omega<-\mu_0$.
    \end{itemize}
    In every other cases, there is no non-trivial non-negative solution in $\mathrm{D}\left(\sqrt{|A|}\right)$.

\begin{cases}\complexI\p_t u+ H u +\lambda \left|u\right|^{2\gamma}u=0\\ u(0)=u_0,\end{cases} \label{Eq-GP}
\end{equation}
by a white noise potential. In the study of Bose-Einstein condensation, \cref{Eq-GP} appears as a mean-field limit of the many body Schrödinger equation, the nonlinear term appearing as a consequence of particle interactions \cite{BEC_Pitaevskii}. Adding a random potential in \cref{Eq-GP} can be seen as a way of modeling spatial inhomogeneities (see for example \cite{PhysRevLett.104.193901,PhysRevLett.109.243902} and references therein), thus one can see \cref{Eq-AGP} as a toy model of Bose-Einstein condensation in inhomogeneous environment with zero correlation length. In this context, at least formally, \cref{Eq:AGP} has two conserved quantities physically relevant, the mass
$$\mathcal{M}(u)=\frac{1}{2}\int_{\R^d}|u|^2\d x$$
and the energy
$$\mathcal{E}(u)=\frac{1}{2}\langle-(H+\xi)u,u\rangle - \frac{\lambda}{2\gamma+2}\int_{\R^d}|u|^{2\gamma+2}\d x.$$
Standing waves of \cref{Eq:AGP} then appear naturally as solutions of minimal energy at fixed mass. Formally, if $\phi$ minimize the energy at fixed mass, by a Lagrange multiplier argument, there exists $\omega\in\R$ such that
$$-(H+\xi)\phi-\lambda|\phi|^{2\gamma}\phi+\omega \phi = 0$$
and $u(t,x)=\e^{i\omega t}\phi(x)$ is solution of \cref{Eq:AGP}.\\

In dimension 2, it is known from \cite{Mackowiak_2025,MackowiakStrichartzConfAnderson} that \cref{Eq:AGP} should be interpreted in a renormalized sense. This renormalization has to be taken into account in the study of standing waves. The construction of the renormalized operator has been conducted in \cite{MackowiakStrichartzConfAnderson} and will be recall in \cref{Sec-AndersonOp2d}. The renormalized operator is defined through an exponential transform inspired by the one introduced in \cite{Haire_Labbe} to study the continuous parabolic Anderson model, and later used in \cite{debussche_weber_T2,Tzvetkov_2023,tzvetkov2020dimensional} to solve the Anderson nonlinear Schrödinger equation on the torus $\T^2$ and in \cite{debussche2017solution,debussche2023global} to solve the same equation but on the full space $\R^2$. Let $Y=(-H)^{-1}\xi$, it is almost surely of regularity $1-$, so one can define $\rho=\e^Y$ and search for a formula for $(H+\xi)u$ when $u$ is of the form $u=\rho v$. A formal computation gives
$$(H+\xi)u=\rho\left(Hv+2\nabla Y\cdot \nabla v + xY\cdot xv + |\nabla Y|^2v\right).$$
Unfortunately, $\nabla Y$ is of regularity $0-$, thus the product $|\nabla Y|^2$ barely fails to exist. In \cite{Mackowiak_2025}, the author proved that the wick product $\wick{|\nabla Y|^2}$ can be defined as the almost sure limit, in suitable distribution spaces of regularity $0-$, of
$$|\nabla Y_N|^2-\E[|\nabla Y_N|^2],$$
where $Y_N$ is a suitable regularisation of $Y_N$.  Thus, the renormalized operator can be formally defined as
$$(H+\xi)u=\rho\left(Hv+2\nabla Y\cdot \nabla v + xY\cdot xv + \wick{|\nabla Y|^2}v\right).$$
The rigorous construction of $H+\xi$ rely on the associated quadratic form.

Even in dimension 1, the construction of $H+\xi$ is not straightforward as $\xi$ is not a function. On compact interval with no potential, $\p_x^2+\xi$ has first been constructed in \cite{FukushimaNakao} using its quadratic form and has been studied in \cite{DumazLabbeAndersonLoc1D} using a twisted Sturm-Liouville approach, allowing to construct $\p_x^2+\xi$ on unbounded interval too. In \cref{Sec-AndersonOp1d}, we propose a construction of $H+\xi$ as a self-adjoint unbounded operator through its quadratic form. \cref{Prop-IPP1d} shows $\mathrm{D}(H+\xi)=\mathrm{D}(\p_x^2+\xi)\cap \mathrm{D}(|x|^2)$, where $\p_x^2+\xi$ is the unbounded self-adjoint operator on $\L^2(\R)$ constructed in \cite{DumazLabbeAndersonLoc1D} and $\mathrm{D}(|x|^2)=\{u\in\L^2(\R),xu\in\L^2(\R)\}$.\\

We are then interested in standing waves for
\begin{equation}{~}
    \begin{cases}\complexI\p_t u+ Au+\lambda \left|u\right|^{2\gamma}u=0\\ u(0)=u_0,\end{cases}

\begin{cases}\complexI\p_t u+ H u + \xi u+\lambda \left|u\right|^{2\gamma}u=0\\ u(0)=u_0,\end{cases} \label{Eq:AGP}
\end{equation}
of the form $u(t,x)=\e^{i\omega t}\phi(x)$, where $H=\Delta-|x|^2$ is the Hermite operator, $\xi$ is a real spatial white noise, $\lambda\in\R$ and $\gamma>0$. Such a solution is called a standing wave of \cref{Eq:AGP} of frequency $\omega\in\R$ and profile $\phi$. The study of standing waves is motivated by the modeling of Bose-Einstein condensation.
In fact, \cref{Eq:AGP} can be seen as a perturbation of the Gross-Pitaevskii equation
\begin{equation}{~}
    \begin{cases}\complexI\p_t u+ H u +\lambda \left|u\right|^{2\gamma}u=0\\ u(0)=u_0,\end{cases}

\begin{theo}\label{Th-EnergyGS}

    Let $d\in\{1,2\}$, $\lambda,\omega\in\R$, $\gamma>0$ and $A$ be the self-adjoint operator constructed in \cref{Sec-AndersonOp1d,Sec-AndersonOp2d} with $\xi$ a spatial white noise, $\D^{1,2}=\mathrm{D}(\sqrt{|A|})$ and $\D^{-1,2}$ its dual. For $u\in\D^{1,2}$, we define the mass as
    $$\mathcal{M}(u)=\frac{1}{2}\int_{\R^d}|u|^2\d x$$
    and the energy as
    $$\mathcal{E}(u)=\frac{1}{2}\langle -Au,u\rangle_{\D^{-1,2},\D^{1,2}}-\frac{\lambda}{2\gamma+2}\int_{\R^d}|u|^{2\gamma+2}\d x.$$
    Then, it holds
    \begin{enumerate}
        \item For $\lambda\leqslant0$, for any $m>0$, there exists $\phi_m\in\D^{1,2}$ such that $\phi_m>0$ and
        $$\argmin_{\substack{u\in\D^{1,2}\\\mathcal{M}(u)=m}}\mathcal{E}(u) = \left\{\e^{\complexI\theta}\phi_m,\; \theta\in\R\right\}.$$
        \item For $\lambda>0$ and $\gamma<\frac{2}{d}$, for any $m>0$, there exists $\phi_m\in\displaystyle\argmin_{u\in\D^{1,2},\;\mathcal{M}(u)=m}\mathcal{E}(u)$ such that $\phi_m>0$.
        \item For $\lambda>0$ and $\gamma=\frac{2}{d}$, there exists $m_{*,\lambda}(\Xi)>0$ such that for all $m\in(0,m_{*,\lambda}(\Xi))$, there exists $\phi_m\in\displaystyle\argmin_{u\in\D^{1,2},\;\mathcal{M}(u)=m}\mathcal{E}(u)$ such that $\phi_m>0$. Moreover, there exists $m_{*,\lambda}(\Xi)\leqslant m^*_\lambda(\Xi)\leqslant m^*_\lambda$ such that
        $$\forall m>m^*_\lambda(\Xi), \inf_{\substack{u\in\D^{1,2}\\\mathcal{M}(u)=m}}\mathcal{E}(u)=-\infty,$$
        where $m^*_\lambda$ is the critical mass of the classical nonlinear Schrödinger equation (i.e.\ without any potential).
        \item For $\lambda>0$ and $\gamma>\frac{2}{d}$, for any $m>0$,
        $$\inf_{\substack{u\in\D^{1,2}\\\mathcal{M}(u)=m}}\mathcal{E}(u)=-\infty.$$
    \end{enumerate}

\end{theo}

\label{Eq-AGP}
\end{equation}
in dimension $d\in\{1,2\}$, where $A=H+\xi$. As $A$ is independent of time, standing waves $u(t,x)=\e^{\complexI\omega t}\phi(x)$ solves \cref{Eq-AGP} if and only if $\p

In this paper, we are interested in particular solutions of the Anderson-Gross-Pitaevskii
\begin{equation}{~}
    \begin{cases}\mathrm{i}\p_t u+ H u + \xi u+\lambda \left|u\right|^{2\gamma}u=0\\ u(0)=u_0,\end{cases} \label{Eq:AGP}
\end{equation}
of the form $u(t,x)=\mathrm{e}^{i\omega t}\phi(x)$, where $H=\Delta-|x|^2$ is the Hermite operator, $\xi$ is a real spatial white noise, $\lambda\in\mathbb R$ and $\gamma>0$. Such a solution is called a standing wave of \cref{Eq:AGP} of frequency $\omega\in\mathbb R$ and profile $\phi$. The study of standing waves is motivated by the modeling of Bose-Einstein condensation.
In fact, \cref{Eq:AGP} can be seen as a perturbation of the Gross-Pitaevskii equation
\begin{equation}{~}
    \begin{cases}\mathrm{i}\p_t u+ H u +\lambda \left|u\right|^{2\gamma}u=0\\ u(0)=u_0,\end{cases} \label{Eq-GP}
\end{equation}
by a white noise potential. In the study of Bose-Einstein condensation, \cref{Eq-GP} appears as a mean-field limit of the many body Schrödinger equation, the nonlinear term appearing as a consequence of particle interactions \cite{BEC_Pitaevskii}. Adding a random potential in \cref{Eq-GP} can be seen as a way of modeling spatial inhomogeneities (see for example \cite{PhysRevLett.104.193901,PhysRevLett.109.243902} and references therein), thus one can see \cref{Eq-AGP} as a toy model of Bose-Einstein condensation in inhomogeneous environment with zero correlation length. In this context, at least formally, \cref{Eq:AGP} has two conserved quantities physically relevant, the mass
$$\mathcal{M}(u)=\frac{1}{2}\int_{\mathbb R^d}|u|^2\mathrm{d} x$$
and the energy
$$\mathcal{E}(u)=\frac{1}{2}\langle-(H+\xi)u,u\rangle - \frac{\lambda}{2\gamma+2}\int_{\mathbb R^d}|u|^{2\gamma+2}\mathrm{d} x.$$
Standing waves of \cref{Eq:AGP} then appear naturally as solutions of minimal energy at fixed mass. Formally, if $\phi$ minimize the energy at fixed mass, by a Lagrange multiplier argument, there exists $\omega\in\mathbb R$ such that
$$-(H+\xi)\phi-\lambda|\phi|^{2\gamma}\phi+\omega \phi = 0$$
and $u(t,x)=\mathrm{e}^{i\omega t}\phi(x)$ is solution of \cref{Eq:AGP}.\\

In dimension 2, it is known from \cite{Mackowiak_2025,MackowiakStrichartzConfAnderson} that \cref{Eq:AGP} should be interpreted in a renormalized sense. This renormalization has to be taken into account in the study of standing waves. The construction of the renormalized operator has been conducted in \cite{MackowiakStrichartzConfAnderson} and will be recall in \cref{Sec-AndersonOp2d}. The renormalized operator is defined through an exponential transform inspired by the one introduced in \cite{Haire_Labbe} to study the continuous parabolic Anderson model, and later used in \cite{debussche_weber_T2,Tzvetkov_2023,tzvetkov2020dimensional} to solve the Anderson nonlinear Schrödinger equation on the torus $\mathbb T^2$ and in \cite{debussche2017solution,debussche2023global} to solve the same equation but on the full space $\mathbb R^2$. Let $Y=(-H)^{-1}\xi$, it is almost surely of regularity $1-$, so one can define $\rho=\mathrm{e}^Y$ and search for a formula for $(H+\xi)u$ when $u$ is of the form $u=\rho v$. A formal computation gives
$$(H+\xi)u=\rho\left(Hv+2\nabla Y\cdot \nabla v + xY\cdot xv + |\nabla Y|^2v\right).$$
Unfortunately, $\nabla Y$ is of regularity $0-$, thus the product $|\nabla Y|^2$ barely fails to exist. In \cite{Mackowiak_2025}, the author proved that the wick product $\wick{|\nabla Y|^2}$ can be defined as the almost sure limit, in suitable distribution spaces of regularity $0-$, of
$$|\nabla Y_N|^2-\mathbb E[|\nabla Y_N|^2],$$
where $Y_N$ is a suitable regularisation of $Y_N$.  Thus, the renormalized operator can be formally defined as
$$(H+\xi)u=\rho\left(Hv+2\nabla Y\cdot \nabla v + xY\cdot xv + \wick{|\nabla Y|^2}v\right).$$
The rigorous construction of $H+\xi$ rely on the associated quadratic form.

Even in dimension 1, the construction of $H+\xi$ is not straightforward as $\xi$ is not a function. On compact interval with no potential, $\p_x^2+\xi$ has first been constructed in \cite{FukushimaNakao} using its quadratic form and has been studied in \cite{DumazLabbeAndersonLoc1D} using a twisted Sturm-Liouville approach, allowing to construct $\p_x^2+\xi$ on unbounded interval too. In \cref{Sec-AndersonOp1d}, we propose a construction of $H+\xi$ as a self-adjoint unbounded operator through its quadratic form. \cref{Prop-IPP1d} shows $\mathrm{D}(H+\xi)=\mathrm{D}(\p_x^2+\xi)\cap \mathrm{D}(|x|^2)$, where $\p_x^2+\xi$ is the unbounded self-adjoint operator on $\mathrm L^2(\mathbb R)$ constructed in \cite{DumazLabbeAndersonLoc1D} and $\mathrm{D}(|x|^2)=\{u\in\mathrm L^2(\mathbb R),xu\in\mathrm L^2(\mathbb R)\}$.\\

We are then interested in standing waves for
\begin{equation}{~}
    \begin{cases}\mathrm{i}\p_t u+ Au+\lambda \left|u\right|^{2\gamma}u=0\\ u(0)=u_0,\end{cases} \label{Eq-AGP}
\end{equation}
in dimension $d\in\{1,2\}$, where $A=H+\xi$. As $A$ is independent of time, standing waves $u(t,x)=\mathrm{e}^{\mathrm{i}\omega t}\phi(x)$ solves \cref{Eq-AGP} if and only if $\phi$ is solution of
\begin{equation}
    -A\phi-\lambda|\phi|^{2\gamma}\phi+\omega \phi = 0. \label{Eq-StationaryAGP}
\end{equation}
This paper aim to construct standing waves in presence of a white noise potential and present some of their properties.

When $\xi=0$ and $H=\Delta$, standing waves are well-known object. See for example \cite{NehariExistence,BerestyckiLions,StraussExistenceSolitary} for existence, \cite{BerestyckiLions} for non-existence, \cite{GidasSymmetry,KwongUniqueness} for uniqueness and \cite{BerestyckiInstabilité,CazenaveLionsStability,CazenaveStabilityInstability,WeinsteinSharpInterpolation,GrillakisStability,GrillakisStabilityII} for results on stability. Let us stress out many of these results relies on the particular structure of \cref{Eq-StationaryAGP} in this case, which is not preserved in presence of potentials. When $\xi=0$ and $H=\Delta-V$ with $V$ a regular and potentially unbounded potential, we refer the reader to \cite{KavianSelfSimilar,FukuizumiStabilityInstability,HiroseStructure,HiroseUniqueness,FukuizumiExponentialDecay,MaedaSymmetry,SelemBifurcation,GuoSeringerMassConcentration,PelinovskyGSEnergyCriticalGP} and references therein for many results on existence, uniqueness, localization and stability.

Nonlinear elliptic equations in presence of white noise potential on compact manifold have been studied in \cite{EulryVariational} using mountain pass methods and in \cite{ZhangConstraintMinAnderson} using constraint minimization. In both \cite{EulryVariational,ZhangConstraintMinAnderson}, the Anderson operator $\Delta+\xi$ is constructed using paracontrolled calculus. We will present here the construction of solution of \cref{Eq-StationaryAGP} through constraint minimization, namely energy minimization at fixed mass and action minimization. This allow to construct a positive solution of \cref{Eq-StationaryAGP} whenever such a solution exists.

In dimension 2, it is known from \cite{Mackowiak_2025,MackowiakStrichartzConfAnderson} that \cref{Eq:AGP} should be interpreted in a renormalized sense. This renormalization has to be taken into account in the study of standing waves. The construction of the renormalized operator has been conducted in \cite{MackowiakStrichartzConfAnderson} and will be recall in \cref{Sec-AndersonOp2d}. The renormalized operator is defined through an exponential transform inspired by the one introduced in \cite{Haire_Labbe} to study the continuous parabolic Anderson model, and later used in \cite{debussche_weber_T2,Tzvetkov_2023,tzvetkov2020dimensional} to solve the Anderson nonlinear Schrödinger equation on the torus $\mathbb T^2$ and in \cite{debussche2017solution,debussche2023global} to solve the same equation but on the full space $\mathbb R^2$. Let $Y=(-H)^{-1}\xi$, it is almost surely of regularity $1-$, so one can define $\rho=\mathrm{e}^Y$ and search for a formula for $(H+\xi)u$ when $u$ is of the form $u=\rho v$. A formal computation gives
$$(H+\xi)u=\rho\left(Hv+2\nabla Y\cdot \nabla v + xY\cdot xv + |\nabla Y|^2v\right).$$
Unfortunately, $\nabla Y$ is of regularity $0-$, thus the product $|\nabla Y|^2$ barely fails to exist. In \cite{Mackowiak_2025}, the author proved that the wick product $\wick{|\nabla Y|^2}$ can be defined as the almost sure limit, in suitable distribution spaces of regularity $0-$, of
$$|\nabla Y_N|^2-\mathbb E[|\nabla Y_N|^2],$$
where $Y_N$ is a suitable regularisation of $Y_N$. Thus, the renormalized operator can be formally defined as
$$(H+\xi)u=\rho\left(Hv+2\nabla Y\cdot \nabla v + xY\cdot xv + \wick{|\nabla Y|^2}v\right).$$
The rigorous construction of $H+\xi$ rely on the associated quadratic form.

Even in dimension 1, the construction of $H+\xi$ is not straightforward as $\xi$ is not a function. On compact interval with no potential, $\p_x^2+\xi$ has first been constructed in \cite{FukushimaNakao} using its quadratic form and has been studied in \cite{DumazLabbeAndersonLoc1D} using a twisted Sturm-Liouville approach, allowing to construct $\p_x^2+\xi$ on unbounded interval too. In \cref{Sec-AndersonOp1d}, we propose a construction of $H+\xi$ as a self-adjoint unbounded operator through its quadratic form. \cref{Prop-IPP1d} shows $\mathrm{D}(H+\xi)=\mathrm{D}(\p_x^2+\xi)\cap \mathrm{D}(|x|^2)$, where $\p_x^2+\xi$ is the unbounded self-adjoint operator on $\mathrm L^2(\mathbb R)$ constructed in \cite{DumazLabbeAndersonLoc1D} and $\mathrm{D}(|x|^2)=\{u\in\mathrm L^2(\mathbb R),xu\in\mathrm L^2(\mathbb R)\}$.\\

We are then interested in standing waves for
\begin{equation}{~}
 \begin{cases}\mathrm{i}\p_t u+ Au+\lambda \left|u\right|^{2\gamma}u=0\\ u(0)=u_0,\end{cases} \label{Eq-AGP}
\end{equation}
in dimension $d\in\{1,2\}$, where $A=H+\xi$. As $A$ is independent of time, standing waves $u(t,x)=\mathrm{e}^{\mathrm{i}\omega t}\phi(x)$ solves \cref{Eq-AGP} if and only if $\phi$ is solution of
\begin{equation}
 -A\phi-\lambda|\phi|^{2\gamma}\phi+\omega \phi = 0. \label{Eq-StationaryAGP}
\end{equation}
This paper aim to construct standing waves in presence of a white noise potential and present some of their properties.

When $\xi=0$ and $H=\Delta$, standing waves are well-known object. See for example \cite{NehariExistence,BerestyckiLions,StraussExistenceSolitary} for existence, \cite{BerestyckiLions} for non-existence, \cite{GidasSymmetry,KwongUniqueness} for uniqueness and \cite{BerestyckiInstabilité,CazenaveLionsStability,CazenaveStabilityInstability,WeinsteinSharpInterpolation,GrillakisStability,GrillakisStabilityII} for results on stability. Let us stress out many of these results relies on the particular structure of \cref{Eq-StationaryAGP} in this case, which is not preserved in presence of potentials. When $\xi=0$ and $H=\Delta-V$ with $V$ a regular and potentially unbounded potential, we refer the reader to \cite{KavianSelfSimilar,FukuizumiStabilityInstability,HiroseStructure,HiroseUniqueness,FukuizumiExponentialDecay,MaedaSymmetry,SelemBifurcation,GuoSeringerMassConcentration,PelinovskyGSEnergyCriticalGP} and references therein for many results on existence, uniqueness, localization and stability.

Nonlinear elliptic equations in presence of white noise potential on compact manifold have been studied in \cite{EulryVariational} using mountain pass methods and in \cite{ZhangConstraintMinAnderson} using constraint minimization. In both \cite{EulryVariational,ZhangConstraintMinAnderson}, the Anderson operator $\Delta+\xi$ is constructed using paracontrolled calculus. We will present here the construction of solution of \cref{Eq-StationaryAGP} through constraint minimization, namely energy minimization at fixed mass and action minimization. This allow to construct a positive solution of \cref{Eq-StationaryAGP} whenever such a solution exists.

Nonlinear elliptic equations in presence of white noise potential on compact manifold have been studied in \cite{EulryVariational} using mountain pass methods and in \cite{ZhangConstraintMinAnderson} using constraint minimization. In both \cite{EulryVariational,ZhangConstraintMinAnderson}, the Anderson operator $\Delta+\xi$ is constructed using paracontrolled calculus. We will present here the construction of solution of \cref{Eq-StationaryAGP} through constraint minimization, namely energy minimization at fixed mass and action minimization. This allow to construct a positive solution of \cref{Eq-StationaryAGP} whenever such a solution exists.

In order to do so, we use constrained minimization methods. In \cref{Sec-EnergyGS}, we study the minimization of energy at fixed mass and prove the following result.

Let $d\in\{1,2\}$, $\lambda,\omega\in\R$, $\gamma>0$ and $A$ the self-adjoint operator constructed in \cref{Sec-AndersonOp1d,Sec-AndersonOp2d} with $\xi$ a spatial white noise. Let $\D^{1,2}=\mathrm{D}(\sqrt{|A|})$ and $\D^{-1,2}$ its dual. Denote by $\mu_0$ the smallest eigenvalue of $-A$. Let $\omega>-\mu_0$ and define $$\mathcal{S}_\omega = \mathcal{E} + \omega \mathcal{M}$$ and $$\mathcal{N}_\omega =\left\{u\in\D^{1,2}\backslash\{0\},\; \langle \nabla\mathcal{S}_\omega u,u \rangle_{\D^{-1,2},\D^{1,2}}=0\right\}.$$
 Then, there exists $\psi_\omega\in\displaystyle\argmin_{\mathcal{N}_\omega}\mathcal{S}_\omega$ such that $\psi_\omega>0$.

\begin{itemize}
 \item Assume $\gamma>\frac{2}{d}$, let $v\in\W^{1,2}\backslash\{0\}$ and $u=\rho v$, then for any $\alpha>0$, \cref{Eq-EnergyVside} implies
 \begin{align*}
 \mathcal{E}(u_\alpha) &= - \frac{\lambda\alpha^{d\gamma}}{2\gamma+2}\int_{\R^d}|v(x)|^{2\gamma+2}\rho(x/\alpha)^{2\gamma+2}\d x &(I)\\
 &+ \frac{\alpha^2}{2}\int_{\R^d}\rho(x/\alpha)^2|\nabla v(x)|^2\d x &(II)\\
 &+ \frac{1}{2\alpha^2}\int_{\R^d} (1-Y(x/\alpha))\rho(x/\alpha)^2|xv(x)|^2\d x &(III)\\
 &-\frac{1}{2}\langle Z,|u_\alpha|^2\rangle. &(IV)
 \end{align*}
 \cref{Lem-EstimScalTransf,Lem-ProductRule} imply $(IV)=O(\alpha^s)$ as $\alpha$ goes to infinity, for any $s\in(0,2)$, as $Z\in\W^{0-,\infty}$. By dominated convergence, $(III)=O(\alpha^{-2})$ as $\alpha$ goes to infinity. Finally, $(II)=O(\alpha^2)$ and
 $$(I)\sim - \frac{\lambda\alpha^{d\gamma}\rho(0)^{2\gamma+2}}{2\gamma+2}\int_{\R^d}|v(x)|^{2\gamma+2}\d x.$$
 Chose $v = \sqrt{2m}\rho(0)^{-1}h_0$ ($m>0$), then $\mathcal{M}(u_\alpha)$ converges to $m$ whereas $\mathcal{E}(u_\alpha)$ goes to $-\infty$, because $d\gamma>2$. Thus, \cref{Lem-AsympEnergy} implies
 $$\forall m>0,\; \inf_{\Atop{u\in\D^{1,2}}{\mathcal{M}(u)=m}}\mathcal{E}(u)=-\infty.$$
 \item Assume $\gamma=\frac{2}{d}$ and let $Q$ be the unique positive and radial solution of
 $$\Delta Q + \lambda Q^{2\gamma+1} = Q.$$
 Recall $\mathcal{M}(Q)=m_\lambda^*=\frac{1}{2}\left(\frac{(\gamma+1)J}{\lambda}\right)^{\frac{1}{\gamma}}$. For $c>0$, let $u=c\rho Q$. For any $\alpha>0$, it holds
 \begin{align*}
 \mathcal{E}(u_\alpha) &= \frac{\alpha^2}{2}\int_{\R^d}\rho(x/\alpha)^2|c\nabla Q(x)|^2\d x - \frac{\lambda\alpha^{2}}{2\gamma+2}\int_{\R^d}|cQ(x)|^{2\gamma+2}\rho(x/\alpha)^{2\gamma+2}\d x &(I)\\
 &+ \frac{c^2}{2\alpha^2}\int_{\R^d} (1-Y(x/\alpha))\rho(x/\alpha)^2|xQ(x)|^2\d x -\frac{1}{2}\langle Z,|u_\alpha|^2\rangle. &(II)
 \end{align*}
 As previously $(II)=O(\alpha^s)$ as $\alpha$ goes to infinity, for any $s\in(0,2)$, and for $c\rho(0)\neq 1$,
 $$(I)\sim\frac{\alpha^2\lambda(c\rho(0))^2}{2\gamma+2}\left(1-(c\rho(0))^{2\gamma}\right)\int_{\R^d}|Q(x)|^{2\gamma+2}\d x$$
 as $\alpha$ goes to infinity, using \cref{Eq-PohozaevQ}. Thus, for $c\rho(0)>1$, \cref{Lem-AsympEnergy} implies
 $$\inf_{\Atop{u\in\D^{1,2}}{\mathcal{M}(u)=c\rho(0)m^*_\lambda}}\mathcal{E}(u)=-\infty,$$
 that is
 $$\forall m>m^*_\lambda,\; \inf_{\Atop{u\in\D^{1,2}}{\mathcal{M}(u)=m}}\mathcal{E}(u)=-\infty.$$
 Let $m'>m>0$ and $\phi\in\D^{1,2}$ such that $\mathcal{M}(\phi)=m$. Let $\psi=\sqrt{\frac{m'}{m}}\phi$. Then,
 \begin{align*}
 \mathcal{E}(\phi) &= \frac{m}{m'}\left[\frac{1}{2}\langle -A\psi,\psi\rangle - \frac{\lambda}{2\gamma+2}\left(\frac{m}{m'}\right)^{\gamma}\int_{\R^d}|\psi|^{2\gamma+2}\d x\right]\\
 &=\frac{m}{m'}\mathcal{E}(\psi) + \frac{m\lambda}{m'(2\gamma+2)}\left[1 - \left(\frac{m}{m'}\right)^{\gamma}\right]\int_{\R^d}|\phi|^{2\gamma+2}\d x\\
 &\geqslant \frac{m}{m'}\mathcal{E}(\psi).
 \end{align*}
 Thus,
 $$m'\inf_{\Atop{u\in\D^{1,2}}{\mathcal{M}(u)=m}}\mathcal{E}(u) \geqslant m \inf_{\Atop{u\in\D^{1,2}}{\mathcal{M}(u)=m}'}\mathcal{E}(u).$$
 In particular, if $\displaystyle\inf_{u\in\D^{1,2},\; \mathcal{M}(u)=m}\mathcal{E}(u)=-\infty$, then $\displaystyle\inf_{u\in\D^{1,2},\; \mathcal{M}(u)=m'}\mathcal{E}(u)=-\infty$. Similarly, if $\displaystyle\inf_{u\in\D^{1,2},\; \mathcal{M}(u)=m'}\mathcal{E}(u)>-\infty$, then $\displaystyle\inf_{u\in\D^{1,2},\; \mathcal{M}(u)=m}\mathcal{E}(u)>-\infty$. Hence, we can define
 \begin{align*}
 m^*_\lambda(\Xi) &= \inf\left\{m>0,\; \inf_{\Atop{u\in\D^{1,2}}{\mathcal{M}(u)=m}}\mathcal{E}(u)=-\infty\right\}\\
 &= \sup\left\{m>0,\; \inf_{\Atop{u\in\D^{1,2}}{\mathcal{M}(u)=m}}\mathcal{E}(u)>-\infty\right\} \in [0,m^*_\lambda]
 \end{align*}
 To conclude, we only have to prove a lower bound for $m^*_\lambda(\Xi)$.

Our aim is now to prove \cref{Th-ACtionGS}. Showing directly that the minimization problem
\begin{equation}
 \inf_{\mathcal{N}_{\omega,\lambda}}\mathcal{S}_{\omega,\lambda}\label{Eq-MinAction}
\end{equation}
has non-negative solutions may be a bit technical as it is not straightforward that
$$u\in\argmin_{\mathcal{N}_{\omega,\lambda}}\mathcal{S}_{\omega,\lambda}\Rightarrow |u|\in\argmin_{\mathcal{N}_{\omega,\lambda}}\mathcal{S}_{\omega,\lambda}.$$
Thus, we will introduce two other, but equivalent, minimization problems which are easier to study. Remark if $u$ solves \cref{Eq-StationaryAGP} for some $\lambda>0$, then $\alpha u$ solves \cref{Eq-StationaryAGP} for $\lambda'=\frac{\lambda}{|\alpha|^{2\gamma}}$. Thus, the parameter $\lambda$ can be seen as a Lagrange multiplier, and solving \cref{Eq-StationaryAGP} for some $\lambda>0$ is equivalent to solving \cref{Eq-StationaryAGP} for all $\lambda>0$. Following an idea of \cite{ROSE1988207}, we define for $u\in\D^{1,2}$
\begin{equation}
 \mathcal{P}_\omega(u)=\langle (-A+\omega) u, u\rangle,\label{Eq-POmega}
\end{equation}
\begin{equation}
 \mathcal{Q}(u) = \int_\R |u|^{2(\gamma+1)}\d x\label{Eq-Q(u)}
\end{equation}
and if $u\neq0,$
\begin{equation}
 \mathcal{J}_\omega(u) = \frac{\mathcal{P}_\omega(u)}{\mathcal{Q}(u)^{1/(\gamma+1)}}.\label{Eq-JOmega}
\end{equation}
Let us stress out $\mathcal{J}_\omega(\alpha u)=\mathcal{J}_\omega(u)$ for any $\alpha\in\C^*$, thus if $u$ minimizes $\mathcal{J}_\omega$, $\alpha u$ minimizes $\mathcal{J}_\omega$ too. Using standard argument of variational calculus, one can show minimizers of both
\begin{equation}
 \inf_{\Atop{u\in\D^{1,2}}{\mathcal{Q}(u)=q}}\mathcal{P}_\omega(u)\label{Eq-MinP/Q=q}
\end{equation}
and
\begin{equation}
 \inf_{\D^{1,2}\backslash\{0\}}\mathcal{J}_\omega\label{Eq-MinJ}
\end{equation}
are solutions of \cref{Eq-StationaryAGP} for some $\lambda>0$. For $\lambda>0$, one can rewrite
$$\mathcal{S}_{\omega,\lambda}(u)=\frac{1}{2}\mathcal{P}_\omega(u)-\frac{\lambda}{2(\gamma+1)}\mathcal{Q}(u)$$
and
\begin{equation}
 \mathcal{N}_{\omega,\lambda} = \left\{u\in\D^{1,2}\backslash\{0\},\; \mathcal{J}_\omega(u) = \lambda \mathcal{Q}(u)^{\gamma/(\gamma+1)} \right\}.\label{Eq-NehariJQ}
\end{equation}
As remarked in \cite{ROSE1988207}, minimizers of \cref{Eq-MinP/Q=q,Eq-MinJ} are essentially the same, that is up to a multiplicative factor, due to the the homogeneity of $\mathcal{J}_\omega$. The following proposition shows minimization problems \cref{Eq-MinAction,Eq-MinP/Q=q,Eq-MinJ} are equivalent, up to a multiplicative factor (see also \cite{LiuRotatingActionGS}).

In order to do so, we use constrained minimization methods. In \cref{Sec-EnergyGS}, we study the minimization of energy at fixed mass and prove the following result.

\begin{theo}\label{Th-EnergyGS}

    Let $d\in\{1,2\}$, $\lambda,\omega\in\mathbb R$, $\gamma>0$ and $A$ be the self-adjoint operator constructed in \cref{Sec-AndersonOp1d,Sec-AndersonOp2d} with $\xi$ a spatial white noise, $\mathcal D^{1,2}=\mathrm{D}(\sqrt{|A|})$ and $\mathcal D^{-1,2}$ its dual. For $u\in\mathcal D^{1,2}$, we define the mass as
    $$\mathcal{M}(u)=\frac{1}{2}\int_{\mathbb R^d}|u|^2\mathrm{d} x$$
    and the energy as
    $$\mathcal{E}(u)=\frac{1}{2}\langle -Au,u\rangle_{\mathcal D^{-1,2},\mathcal D^{1,2}}-\frac{\lambda}{2\gamma+2}\int_{\mathbb R^d}|u|^{2\gamma+2}\mathrm{d} x.$$
    Then, it holds
    \begin{enumerate}
        \item For $\lambda\leqslant0$, for any $m>0$, there exists $\phi_m\in\mathcal D^{1,2}$ such that $\phi_m>0$ and
        $$\argmin_{\substack{u\in\mathcal D^{1,2}\\\mathcal{M}(u)=m}}\mathcal{E}(u) = \left\{\mathrm{e}^{\mathrm{i}\theta}\phi_m,\; \theta\in\mathbb R\right\}.$$
        \item For $\lambda>0$ and $\gamma<\frac{2}{d}$, for any $m>0$, there exists $\phi_m\in\displaystyle\argmin_{u\in\mathcal D^{1,2},\;\mathcal{M}(u)=m}\mathcal{E}(u)$ such that $\phi_m>0$.
        \item For $\lambda>0$ and $\gamma=\frac{2}{d}$, there exists $m_{*,\lambda}(\Xi)>0$ such that for all $m\in(0,m_{*,\lambda}(\Xi))$, there exists $\phi_m\in\displaystyle\argmin_{u\in\mathcal D^{1,2},\;\mathcal{M}(u)=m}\mathcal{E}(u)$ such that $\phi_m>0$. Moreover, there exists $m_{*,\lambda}(\Xi)\leqslant m^*_\lambda(\Xi)\leqslant m^*_\lambda$ such that
        $$\forall m>m^*_\lambda(\Xi), \inf_{\substack{u\in\mathcal D^{1,2}\\\mathcal{M}(u)=m}}\mathcal{E}(u)=-\infty,$$
        where $m^*_\lambda$ is the critical mass of the classical nonlinear Schrödinger equation (i.e.\ without any potential).
        \item For $\lambda>0$ and $\gamma>\frac{2}{d}$, for any $m>0$,
        $$\inf_{\substack{u\in\mathcal D^{1,2}\\\mathcal{M}(u)=m}}\mathcal{E}(u)=-\infty.$$
    \end{enumerate}

\end{theo}

It is classical that the set of minimal energy states of fixed mass is stable in $\mathcal D^{1,2}$ for \cref{Eq-AGP}. In particular, \cref{Th-EnergyGS} implies the orbital stability of the ground-state $\phi_m$ in the defocusing case ($\lambda<0$). \cref{Prop-ExistenceEnergyGS,Prop-FrequencyEnergyGS} show that, for $\lambda\leqslant 0$ or $\gamma<\frac{2}{d}$, energy minimization at fixed mass provide positive solutions of \cref{Eq-StationaryAGP} for any possible value of $\omega$. We also prove the small mass asymptotic of $\phi_m$ and its frequency $\omega_m$.\\

In \cref{Sec-ActionGS}, we investigate the minimization of action $\mathcal{S}_\omega = \mathcal{E} + \omega \mathcal{M}$. In the focusing case ($\lambda> 0$), one easily verifies the action is not lowerbounded. We thus introduce the Nehari manifold and show the following result.

\begin{theo}\label{Th-ACtionGS}

    Let $d\in\{1,2\}$, $\lambda,\omega\in\mathbb R$, $\gamma>0$ and $A$ the self-adjoint operator constructed in \cref{Sec-AndersonOp1d,Sec-AndersonOp2d} with $\xi$ a spatial white noise. Let $\mathcal D^{1,2}=\mathrm{D}(\sqrt{|A|})$ and $\mathcal D^{-1,2}$ its dual. Denote by $\mu_0$ the smallest eigenvalue of $-A$. Let $\omega>-\mu_0$ and define $$\mathcal{S}_\omega = \mathcal{E} + \omega \mathcal{M}$$ and $$\mathcal{N}_\omega =\left\{u\in\mathcal D^{1,2}\backslash\{0\},\; \langle \nabla\mathcal{S}_\omega u,u \rangle_{\mathcal D^{-1,2},\mathcal D^{1,2}}=0\right\}.$$
    Then, there exists $\psi_\omega\in\displaystyle\argmin_{\mathcal{N}_\omega}\mathcal{S}_\omega$ such that $\psi_\omega>0$.

\end{theo}

Let us stress out an important property of solution of \cref{Eq-StationaryAGP} compared to the noiseless case. In presence of white noise, standing waves have a limited Sobolev regularity.
\begin{theo}\label{Th-MaxRegSW}
    Let $d\in\{1,2\}$, $\lambda,\omega\in\mathbb R$ and $\gamma>0$. Almost surely any solution $u$ of \cref{Eq-StationaryAGP} belongs to $\mathcal W^{2-\frac{d}{2}-,q}$ for all $q\in[2,+\infty)$, where $\mathcal W^{s,q}$ spaces are defined in \cref{Sec-ConfiningSobolev}. Moreover, almost surely any solution $u$ of \cref{Eq-StationaryAGP} which belongs to $\mathrm{H}^{2-\frac{d}{2},q}(U)$ for some open set $U\subset\mathbb R^d$ and $q\in[2,+\infty]$ vanishes everywhere in $U$.
\end{theo}

In order to prove Theorem~\ref{Th-SynthesExistenceSW}, the text says they "use constrained minimization methods".

- In \cref{Sec-EnergyGS}, they "study the minimization of energy at fixed mass" (Theorem~\ref{Th-EnergyGS}). It is then stated that \cref{Prop-ExistenceEnergyGS,Prop-FrequencyEnergyGS} "show that, for $\lambda\leqslant 0$ or $\gamma<\frac{2}{d}$, energy minimization at fixed mass provide positive solutions of \cref{Eq-StationaryAGP} for any possible value of $\omega$".

- In \cref{Sec-ActionGS}, they instead "investigate the minimization of action $\mathcal{S}_\omega=\mathcal{E}+\omega\mathcal{M}$". In the focusing case ($\lambda>0$), they note that "one easily verifies the action is not lowerbounded", so they "introduce the Nehari manifold" and obtain an action minimizer on it (Theorem~\ref{Th-ACtionGS}) for $\omega>-\mu_0$.

- They also "stress out" that, in the presence of white noise, standing waves have "limited Sobolev regularity" (Theorem~\ref{Th-MaxRegSW}), giving the maximal regularity and a unique-continuation-type statement (if a solution lies in $\mathrm{H}^{2-\frac{d}{2},q}(U)$ on some open set $U$, then it "vanishes everywhere in $U$").

In order to prove the main theorem, the text says they "use constrained minimization methods".

Next, they "study the minimization of energy at fixed mass" and use the following result.

\begin{theo}\label{Th-EnergyGS}

    Let $d\in\{1,2\}$, $\lambda,\omega\in\R$, $\gamma>0$ and $A$ be the self-adjoint operator constructed in \cref{Sec-AndersonOp1d,Sec-AndersonOp2d} with $\xi$ a spatial white noise, $\D^{1,2}=\mathrm{D}(\sqrt{|A|})$ and $\D^{-1,2}$ its dual. For $u\in\D^{1,2}$, we define the mass as
    $$\mathcal{M}(u)=\frac{1}{2}\int_{\R^d}|u|^2\d x$$
    and the energy as
    $$\mathcal{E}(u)=\frac{1}{2}\langle -Au,u\rangle_{\D^{-1,2},\D^{1,2}}-\frac{\lambda}{2\gamma+2}\int_{\R^d}|u|^{2\gamma+2}\d x.$$
    Then, it holds
    \begin{enumerate}
        \item For $\lambda\leqslant0$, for any $m>0$, there exists $\phi_m\in\D^{1,2}$ such that $\phi_m>0$ and
        $$\argmin_{\substack{u\in\D^{1,2}\\\mathcal{M}(u)=m}}\mathcal{E}(u) = \left\{\e^{\complexI\theta}\phi_m,\; \theta\in\R\right\}.$$
        \item For $\lambda>0$ and $\gamma<\frac{2}{d}$, for any $m>0$, there exists $\phi_m\in\displaystyle\argmin_{u\in\D^{1,2},\;\mathcal{M}(u)=m}\mathcal{E}(u)$ such that $\phi_m>0$.
        \item For $\lambda>0$ and $\gamma=\frac{2}{d}$, there exists $m_{*,\lambda}(\Xi)>0$ such that for all $m\in(0,m_{*,\lambda}(\Xi))$, there exists $\phi_m\in\displaystyle\argmin_{u\in\D^{1,2},\;\mathcal{M}(u)=m}\mathcal{E}(u)$ such that $\phi_m>0$. Moreover, there exists $m_{*,\lambda}(\Xi)\leqslant m^*_\lambda(\Xi)\leqslant m^*_\lambda$ such that
        $$\forall m>m^*_\lambda(\Xi), \inf_{\substack{u\in\D^{1,2}\\\mathcal{M}(u)=m}}\mathcal{E}(u)=-\infty,$$
        where $m^*_\lambda$ is the critical mass of the classical nonlinear Schrdinger equation (i.e.\ without any potential).
        \item For $\lambda>0$ and $\gamma>\frac{2}{d}$, for any $m>0$,
        $$\inf_{\substack{u\in\D^{1,2}\\\mathcal{M}(u)=m}}\mathcal{E}(u)=-\infty.$$
    \end{enumerate}

\end{theo}

It is then stated that \cref{Prop-ExistenceEnergyGS,Prop-FrequencyEnergyGS} show that, for $\lambda\leqslant 0$ or $\gamma<\frac{2}{d}$, energy minimization at fixed mass provides positive solutions of
\begin{equation}
    -A\phi-\lambda|\phi|^{2\gamma}\phi+\omega \phi = 0. \label{Eq-StationaryAGP}
\end{equation}
for any possible value of $\omega$.

Later, they instead "investigate the minimization of action $\mathcal{S}_\omega=\mathcal{E}+\omega\mathcal{M}$". In the focusing case ($\lambda>0$), they note that "one easily verifies the action is not lowerbounded", so they "introduce the Nehari manifold" and obtain an action minimizer on it (Theorem~\ref{Th-ACtionGS}) for $\omega>-\mu_0$.

They also "stress out" that, in the presence of white noise, standing waves have "limited Sobolev regularity" (Theorem~\ref{Th-MaxRegSW}), giving the maximal regularity and a unique-continuation-type statement (if a solution lies in $\mathrm{H}^{2-\frac{d}{2},q}(U)$ on some open set $U$, then it "vanishes everywhere in $U$").

\label{Th-SynthesExistenceSW}
    Let $d\in\{1,2\}$, $\lambda,\omega\in\mathbb R$ and $\gamma>0$. Denote by $\mu_0$ the lowest eigenvalue of $-A$. In each of the following situation, there exists a positive solution of
\begin{equation}
    -A\phi-\lambda|\phi|^{2\gamma}\phi+\omega \phi = 0. \label{Eq-StationaryAGP}
\end{equation}
 in $\mathrm{D}\left(\sqrt{|A|}\right)$,
    \begin{itemize}
        \item $\lambda>0$ and $\omega>-\mu_0$,
        \item $\lambda = 0$ and $\omega=\mu_0$,
        \item $\lambda<0$ and $\omega<-\mu_0$.
    \end{itemize}
    In every other cases, there is no non-trivial non-negative solution in $\mathrm{D}\left(\sqrt{|A|}\right)$.,

Question:
Let \(d\in\{1,2\}\), let \(A=H+\xi\) be the Anderson Hamiltonian in dimension \(d\), and let \(\mu_0\) denote the lowest eigenvalue of \(-A\). For \(\lambda,\omega\in\mathbb R\) and \(\gamma>0\), consider the stationary equation
\[
-A\phi-\lambda|\phi|^{2\gamma}\phi+\omega\phi=0.
\]
Which statement about positive or nonnegative solutions \(\phi\in \mathrm D(\sqrt{|A|})\) is valid?

A positive solution \(\phi\in \mathrm D(\sqrt{|A|})\) exists in each of the following three cases: \(\lambda>0\) and \(\omega>-\mu_0\); \(\lambda=0\) and \(\omega=\mu_0\); \(\lambda<0\) and \(\omega<-\mu_0\). In every other case, there is no non-trivial non-negative solution in \(\mathrm D(\sqrt{|A|})\).

Choices:
B. A positive solution \(\phi\in \mathrm D(\sqrt{|A|})\) exists in each of the following three cases: \(\lambda>0\) and \(\omega\ge -\mu_0\); \(\lambda=0\) and \(\omega=\mu_0\); \(\lambda<0\) and \(\omega\le -\mu_0\). In every other case, there is no non-trivial non-negative solution in \(\mathrm D(\sqrt{|A|})\).
C. If \(\lambda>0\) and \(\omega>-\mu_0\), or if \(\lambda=0\) and \(\omega=\mu_0\), or if \(\lambda<0\) and \(\omega<-\mu_0\), then there exists a non-trivial non-negative solution \(\phi\in \mathrm D(\sqrt{|A|})\).
D. A positive solution \(\phi\in \mathrm D(\sqrt{|A|})\) exists whenever \(\lambda>0\) and \(\omega\in\mathbb R\), whenever \(\lambda=0\) and \(\omega=\mu_0\), and whenever \(\lambda<0\) and \(\omega< -\mu_0\). In every other case, there is no non-trivial non-negative solution in \(\mathrm D(\sqrt{|A|})\).
E. A positive solution \(\phi\in \mathrm D(\sqrt{|A|})\) exists in each of the following three cases: \(\lambda>0\) and \(\omega>-\mu_0\); \(\lambda=0\) and \(\omega\ge \mu_0\); \(\lambda<0\) and \(\omega<-\mu_0\). In every other case, there is no non-trivial non-negative solution in \(\mathrm D(\sqrt{|A|})\).


\begin{equation}    \L\big(n,\frac{\alpha}{\ln^2 n}\big)= n^{1- \frac{\pi}{\sqrt{2 \alpha}} +o_p(1)},\label{alpha:polynomial}\end{equation}

\noindent where $o_p(1)$ denotes some quantity which converges to $0$ in probability as $n\to\infty$. This, together with Theorem \ref{t:main}, shows that there is a phase transition at $\lambda_c=\frac{\pi^2}{2}\frac1{\ln^2 n}$. We also note that \eqref{alpha:polynomial} is consistent with \eqref{Ding-2} when $\alpha$ is taken sufficiently large.

This critical value $\lambda_c$ also appears in a related problem concerning the minimum average weight of a cycle in $K_n$.
Ding, Sun and Wilson \cite{ding_sun_wilson} proved that conditionally on the minimum being nonnegative, its value is $\frac{\pi^2}{2 \ln^2 n}  + O(\frac{1}{\ln^3 n})$ and that the length of the corresponding cycle is of order $\ln^3 n$. It would be interesting to investigate further the connection between these two models.

\bigskip
Now, we outline the main ideas of the proofs.
 Heuristically, we  explore the graph in the following way. Let $N=o(n)$ be an integer. We start with an initial set of $N$ vertices, called generation $0$. We examine all adjacent edges of these vertices
  and select the $N$ edges with the  smallest labels. The endpoints of the selected edges form generation $1$, and the associated labels stand for their positions. We then delete the initial set of vertices and all their adjacent edges, and continue the exploration from the vertices of generation $1$. Each incident edge of a vertex at generation $1$ carries a weight. Add this weight to the position of the vertex,
  then select the $N$ edges associated to the $N$ smallest obtained numbers. The endpoints of the selected edges form the vertices of generation $2$ and the associated numbers stand for their positions.  This process can be repeated for roughly $\frac{n}{N}$ steps before the graph is exhausted. The weights of the paths obtained by this strategy resemble an (inhomogeneous)  branching random walk  in which only the $N$ leftmost particles are retained at each step. This model was introduced by Brunet and Derrida \cite{brunet_derrida2} who conjectured that the selection mechanism was shifting the speed of the population by a correction term of order $\frac{1}{\ln^2 N}$, a result later proved by B\'erard and Gou\'er\'e \cite{berard_gouere}.   We refer to Maillard \cite{maillard}
  for more  results on the model in continuous time. In our setting, the minimum of the exploration process at time $t$ is of order
\[
-\ln N + \frac{\pi^2}{2 \ln^2 N} t.
\]
Taking  $N=n^{1-o(1)}$  and $N=e^{\frac{\pi}{\sqrt{2\beta_n}}}$ yields Theorem \ref{t:main} and Theorem \ref{t:main2}, respectively. A slight variation of this strategy is used to establish the lower bound for $\L(n,\lambda)$. The upper bound follows a similar approach to that of \cite{ding,ding_goswami,ding_sun_wilson} by computing suitable truncated first moments.

\bigskip

Finally, we mention the recent preprint by Jorritsma, Maillard and M\"orters \cite{JMM} in which the authors make use of a killed branching random walk to explore growing sparse random graphs. Even if the models are different, our results  show similarity with several features of  their model, see \cite{JMM} and the references therein.

\bigskip

The paper is organized as follows. Sections \ref{s:upper} and \ref{s:lower} give the proofs of the upper bounds and lower bounds for $\L(n,\lambda)$ respectively.  Section \ref{s:rw-brw} collects results on real-valued random walks and  branching random walks that are used in Sections \ref{s:upper} and \ref{s:lower}.

\section{Proofs of the upper bounds in Theorems \ref{t:main} and \ref{t:main2}}
\label{s:upper}

Let $(S_1,S_2,\ldots)$ denote a random walk with step distribution $X$ of \eqref{def:X}, starting at $S_0=0$. For any $\ell\ge 1$ and Borel set $A\subset \r^\ell$,
\begin{equation}\label{comp:moment1}
\e\Big[\#\{\gamma \in {\mathcal P}_\ell\colon S(\gamma) \in A\}\Big] \le n^{\ell+1}\p\Big((S_1,\ldots,S_\ell) \in A\Big),
\end{equation}

\begin{proof}[Proof of the upper bound in Theorem \ref{t:main2}] Let $n$ be large and  $a \in (0, 1)$. Consider $\ell= \lceil ne^{- \frac{\pi}{\sqrt{ 2 \beta_n}} (1-a)}\rceil$,  $r=e^{\frac{\pi}{\sqrt{2\beta_n}}(1-\frac{a}2)}$ and  let $m'_\Delta(L)$ be the number of $i\in \lb 0,L\rb$ such that $\inf_{j \in \lb i,L\rb} S_j\le S_i- \Delta$ with $\Delta:=\ln r$.  Let $b_n=\ln n + \ln\ln n$.  By the same argument leading to \eqref{eq:L>ell-bn}, but using Corollary \ref{cor:jump} (ii) in addition, we obtain that  for any $M>2$, $$\p\Big(\L(n, \beta_n) \ge \ell\Big) \le  \sum_{L=\ell}^{2\ell} \widetilde a_L + \frac{2}{M}+ o(1),$$

\noindent where for $L\in \lb \ell, 2\ell \rb$, $$ \widetilde a_L:= \p\Big(\exists \gamma\in {\mathcal P}_L: S_L(\gamma) \le \beta_n L, m'_\Delta(L) \le M \frac{n}{r},\, \min_{0\le i \le j \le L} (S_j(\gamma)- S_i(\gamma)) >-  b_n \Big).$$

By \eqref{comp:moment1} and \eqref{eq:many-to-one}, \begin{align*}    \widetilde a_L & \le n \, e^{\beta_n L}  \p\Big( Y_L\le \beta_n L , \, m_\Delta(L) \le M \frac{n}{r},\,  \min_{0\le i \le j \le L} (Y_j - Y_i ) >- b_n\Big)
\\
& \le n \, e^{\beta_n L}  \p\Big(   \max_{i \in \lb 0, L\rb} Y_i \le 2 \beta_n L, \,  m_\Delta(L) \le M \frac{n}{r}\Big)
 ,
\end{align*} where $m_\Delta(L)$ is as defined in Lemma \ref{main2:upper} and in the last inequality, we have used the fact that $\max_{i \in \lb 0, L\rb} Y_i\le Y_L+b_n \le \beta_n L+ b_n \le 2 \beta_n L$. Choose $d\in (0, \frac{\pi^2}2)$
  such that $\eta:= \frac{2d}{\pi^2 (1-\frac{a}2)^2} - 1>0$. Fix an arbitrary $c\in (0, 1)$. If $\varepsilon>0$ is chosen small enough in Theorem \ref{t:main2}, we have $L\ge \Delta^{2+c}$, $M \frac{n}{r} \le L \Delta^{-2-c}$ and $2 \beta_n L \le L \Delta^{-1-c}$.  Note that $\Delta$ is arbitrarily large when $\varepsilon$ goes to $0$.   Lemma \ref{main2:upper} yields that $$ \widetilde a_L \le n e^{\beta_n L} e^{- d \frac{L}{\Delta^2}} = n e^{- \eta \,\beta_n L}. $$

 Notice that for all large $n$, $\beta_n L \ge \beta_n \ell \ge n^{a'}$ for some positive constant $a'$. Hence   $\sum_{L=\ell}^{2\ell} \widetilde a_L \le n^2  e^{- \eta\, n^{a'}}\to 0$ and we conclude the proof by letting $n \to\infty$ then $M\to\infty$.  \end{proof}

\section{Proofs of the lower bounds}
\label{s:lower}

We consider a branching random walk $\V$ on $\r$. It starts with a certain number of particles at $0$. At each discrete time $k\ge 0$, each particle of the current population splits into an infinite number of children, whose displacements from their parent  are given by a copy of
\begin{equation} \label{eq:BRW-V}
\sum_{i=1}^\infty {\delta}_{\{e T_i -1\}}
\end{equation}

\noindent where $0<T_1<T_2<\ldots$ are the atoms of a Poisson point process of intensity $1$ on $\r_+$. The position of a particle $u$ is denoted by $\V(u)$.
The offspring distribution corresponds to the limiting law  of $\sum_{u=2}^{n} \delta_{X(1,u)}$, so that, when $n$ is large, exploring the graph around a given vertex locally approximates the branching random walk $\V$. To make it precise, we will  couple labels $X(u,v)$ and copies of the Poisson point process $(T_i)_{i\ge 1}$ as follows.

\medskip

Consider $(X_i)_{1\le i \le n-1}$ a family of $(n-1)$ i.i.d. copies of $X= n e \mbox{Exp}(1)-1$.  Its increasing order statistics $ X_{(0)}:=-1 < X_{(1)}< ...< X_{(n-1)}$ verify that the variables $X_{(i)}-X_{(i-1)}$ are independent, distributed as $\frac{n e }{n-i}\text{Exp}(1)$.   One can couple it with a Poisson point process $T_1<T_2<\ldots$ of intensity $1$, by
\[     X_{(i)}-X_{(i-1)} = \frac{n e}{n-i} (T_{i}-T_{i-1}), \qquad i\in \lb 1, n-1\rb,
\]where we set $T_0=0$.  In particular,  for all $i\in \lb 1, n-1\rb$, \begin{equation}\label{coupling}
   e T_i-1 \le  X_{(i)} \le   \frac{n}{n-i} (e T_{i}-1) + \frac{i}{n-i}.
\end{equation}

For each $u\in K_n$, this gives a coupling between $(X(u,v))_{v\neq u}$ and a copy of $(T_i)_{i\ge 1}$.  We further need the following simple observation.  Let $m \in \lb 1, n-1 \rb$. If we choose   $m$  points uniformly from $(X_{(i)})_{1\le i \le n-1}$ and arrange them in increasing order, then they have the same distribution as  the increasing order statistics of $(X_i)_{1\le i \le m}$. In other words, there exists a coupling in the sense of \eqref{coupling}, between $X(u, v)$ when $v$ varies over a subset of $K_n$ with $m$ vertices, and the family of $m$  points  chosen uniformly from $(T_i)_{1\le i \le n-1}$.

Let $N$ be an integer between $1$ and $n-1$.  We denote by $\V^N$ the branching random walk $\V$  in which, at each step, we only keep the $N$ leftmost particles of the next generation. For a particle $u$ in $\V^N$, let   $|u|$ denote its generation in the genealogical tree of $\V^N$, and let $\V^N(u)$ denote its position in $\r$.

\begin{proposition}\label{p:first_moment} Fix $a\in [0, 1]$ and $t>0$.    If $\V^N$ starts with $\lfloor N^a\rfloor$ particles located at $0$, then for any sequence $\ell_N$ satisfying $\frac{\ell_N}{\ln^3 N} \to t$,  \begin{equation}  \frac{1}{\ln N}  \min_{|u|=\ell_N} \V^N(u)   \,\cvL2\, -a+\frac{\pi^2}{2}t  , \qquad N \to\infty. \label{Nbrw-cvL2}\end{equation}
\end{proposition}

We will apply the above proposition to two cases: $a=1$ and $a=0$.
The proof of Proposition \ref{p:first_moment} is postponed to Section \ref{sub:brw}.

In what follows, we construct a coupling between $\V^N$ and an exploration process on $K_n$.
We will color particles of $\V^{N}$ in blue, red or purple. Blue particles will correspond to  vertices of $K_{n}$ and will be the particles of interest.

\subsection{Construction of an exploration process $(Z_j, {\mathcal Q}_j)_{j\ge 0}$ in $K_n$ and its coupling with $\V^N$.}
 \label{s:exploration}

For each $j\ge 0$, we define a  set $Z_j$ of so-called {\it active} vertices of $K_n$ and a set of paths ${\mathcal Q}_j\subset {\mathcal P}_j$  such that each vertex  in $Z_j$ is associated with a unique  path in ${\mathcal Q}_j$ ending at this point. The set $Z_j$ will correspond to the blue particles of $\V^{N}$ at time $j$. We will therefore write $u$ either for the blue particle or its corresponding  vertex in $K_n$.  For each active vertex $u$ at step $j$, we will write $V(u)$  for $S_j(\gamma)$ where $\gamma$ is the path in ${\mathcal Q}_j$ ending at $u$.

Let $G_0$ be a complete subgraph of $K_n$. Let $Z_0\subset G_0$ be a set of $N_0\le N$ active vertices at time $0$. We start  the branching random walk $\V$ with  $N_0$ blue particles,  each particle corresponding to an active vertex in $Z_0$.  We let $\mathcal Q_0$ be the collection $\{\{u\},\, u\in Z_0\}$.  We denote by $\#G_0$ the number of vertices in $G_0$. Let $m:=\#G_0- \#Z_0$.   Each $u\in Z_0$ corresponds to a blue particle in $\V$, still denoted by $u$. We have $\V(u)=V(u)=0$. We  choose $m$ children $v$ of $u$ in $\V$ uniformly among the $n-1$ leftmost ones and color them {\it blue.}  By the coupling described above, for each blue particle $u$, $\V(v)-\V(u)$ is coupled with $X(u, v)$ for $v \in G_0 \backslash Z_0$ (where, as for $u$, the variable $v$ denotes both a particle in $\V$ and a vertex in $G_0$).  Let $i_v$ be the index of the point $T_i$ corresponding to $v$ in the coupling \eqref{coupling}. We have  $$X(u,v) = X_{(i_v)} \le \frac{n}{n-i_v} e T_{i_v} -1 =   \frac{n}{n-i_v} (\V(v)-\V(u))+\frac{i_v}{n-i_v}.$$ We also have $X(u,v) \ge \V(v)-\V(u)$.
We will see that $i_v \le N$ for all $v$ of interest.  We color the remaining children of $u$ in $\V$ {\it red.}  We then retain the $N$ leftmost particles of $\V$ in generation $1$ which is in fact the population of $\V^N$ at generation $1$.  This population may contain red and blue particles.  If a blue particle $v$  is selected in $\V^N$, we select the corresponding edge $(u, v)$ in $G_0$. Note that in this case we indeed have $i_v \le N$. We also remark that several selected edges may point to the same vertex $v$. If this happens, we keep blue the corresponding particle in $\V^N$ which has the minimal position in $\V$,  and color the other particles corresponding to $v$ {\it purple.} We then define $Z_1$ to be the set of blue particles in ${\mathcal V}^N$ at generation $1$, after removing all purple particles. We identify $Z_1$ with a subset of $G_0$: a blue particle of $Z_1$ in $\V^N$ corresponds exactly to one (active) vertex of $G_0$, and the paths $(u, v)_{u\in Z_0, v \in Z_1}$ form  ${\mathcal Q}_1$. Since $i_v\le N$ if $(u,v)\in {\mathcal Q}_1$,  we have $\V^N(v)-\V^N(u)= \V(v)-\V(u)$ and
 \begin{equation}\label{eq:index}
  V(v)-V(u):=X(u,v) \le \frac{n}{n-N} (\V^N(v)-\V^N(u)) + \frac{N}{n-N}.
\end{equation}
 Finally,  $G_1$ is the sub-complete graph formed by  removing $Z_0$ and all incident edges in $G_0$.

For the next steps, we proceed by induction on $j$ as long as $Z_j\neq \emptyset$. See Figure \ref{f:coupling}.

 \begin{figure}[ht!]
\centering
 \begin{tikzpicture}[scale=0.9,
    >=Stealth,
    vertex/.style={circle,draw,inner sep=1pt,minimum size=5mm},
    bluev/.style={vertex,fill=blue!30},
    redv/.style={vertex,fill=red!50},
    purplev/.style={vertex,fill=violet!55},
    thin edge/.style={line width=0.05pt},
    dottedbox/.style={dotted,thick},
    small/.style={font=\scriptsize}
]

\node at (0,4) {$G_j$};

\node[bluev]   (u1) at (-2,2) {$u_1$};
\node[bluev]   (u2) at (-0.3,3) {$u_2$};
\node[bluev]   (u3) at (2,2) {$u_3$};

\node[vertex] (v1)  at (-2,0.5) {$v_1$};
\node[vertex] (v2)  at (-1,-0.8) {$v_2$};
\node[vertex] (v3)  at (0,-1.2) {$v_3$};
\node[vertex] (v4)  at (1,-0.8) {$v_4$};
\node[vertex] (v5)  at (2,0.5) {$v_5$};

\foreach \x/\y in {u1/u2,u2/u3,u1/u3,u1/v1,u1/v2,u1/v3,u2/v1,u2/v2,u2/v3,u3/v1,u3/v2,u3/v3, v1/v2, v1/v3, v2/v3, u1/v4, u2/v4, u3/v4, v1/v4, v2/v4, v4/v4, u1/v5, u2/v5, u3/v5, v1/v5, v2/v5, v3/v5, v4/v5}
  \draw[thin edge] (\x) -- (\y);

\node[small] at (0,-2) {$Z_j=\{u_1,u_2,u_3\}$};
\node[small] at (0,-2.5) {$G_j\setminus Z_j=\{v_1,v_2,v_3, v_4, v_5\}$};

\node at (6.8,4) {\small $\V^N$ at time $j$};

\node[bluev] (r1) at (6.3,3) {$u_1$};
\node[bluev] (r2) at (7,3) {$u_2$};
\node[bluev] (r3) at (8.4,3) {$u_3$};
\node[purplev] (r4) at (7.7,3) {$\cdot$};
\node[redv] (r5) at (5.6,3) {$\cdot$};

\draw[dottedbox] ($(r5)+(-0.5,-0.5)$) rectangle ($(r3)+(0.5,0.5)$);

\node[bluev] (a1) at (5.6,0) {$v_1$};
\node[redv]  (a2) at (6.3,0) {$\cdot$};

\node[purplev] (b1) at (7,0) {$v_1$};
\node[bluev]   (b2) at (7.7,0) {$v_2$};

\node[redv] (c1) at (8.4,0) {$\cdot$};

\node[bluev]  (c2) at (9.35,0) {$v_5$};
\node[bluev]  (c3) at (10.05,0) {$v_4$};
\node[redv]  (c4) at (10.75,0) {$\cdot$};

\draw (r1)--(a1);
\draw (r1)--(b2);
\draw [dashed] (r1)--(a2);

\draw (r2)--(b1);

\draw  (r3)--(c2);
\draw  (r3)--(c3);
\draw [dashed] (r5)--(c4);

\draw [dashed](r4)--(c1);

\draw[dottedbox] ($(a1)+(-0.5,-0.5)$) rectangle ($(c1)+(0.5,0.5)$);

\node[small] at (6.8,-0.8) {$\mathcal V^N$ at time $j+1$};
\node[small] at (6.8,-1.2) {by keeping the $N$ leftmost particles};

\node[small] (Zj1) at (6.8,-2) {$Z_{j+1}=\{v_1,v_2\}$};
\node[small] (Zj1) at (6.8,-2.5) {$G_{j+1}=\{v_1, v_2, v_3, v_4, v_5\}$};

\end{tikzpicture}
\caption{\small  Schematic illustration of the coupling between $(Z_j, G_j)$ and $\V^N$.}
\label{f:coupling}
\end{figure}

 Suppose we are at step $j$ and have constructed the graph $G_j$, the set $Z_j\subset G_j$ and the paths in $\mathcal Q_j$.  Each active vertex  $u\in Z_j$ corresponds to a blue particle $u$ of $\V^N$ at time $j$. We randomly color $\#G_j-\#Z_j$ children of $u$ in $\V$ blue, among the $n-1$ leftmost ones, so that each blue child $v$ corresponds to an edge in $\{u\}\times (G_j\setminus Z_j)$.  The remaining children of $u$ in $\V$ are colored {\it red}.  We couple the family $(X(u,v))_{v\in G_j\setminus Z_j}$ with the displacements of the blue particles in $\V$ via the same coupling described above.

 Purple and red particles at time $j$ give birth only to red particles. We then retain the $N$ leftmost particles of the population at generation $j+1$, which form the population of $\V^N$ at generation $j+1$.  It is composed of blue/red particles. If a blue particle $|v|=j+1$ with parent $u$ is selected in $\V^N$, we select the corresponding edge $(u,v)$ in $G_j$.  In the case several selected edges  point to the same vertex $v$ in $G_j \setminus Z_j$, so that multiple blue particles at time $j+1$ correspond to the same vertex $v$, we keep the one with minimal position in $\V$ {\it blue} and color the other particles {\it purple.}  By definition, $V(v):=V(u)+X(u,v)$ is the weight $S_{j+1}(\gamma,v)$ of the path $(\gamma,v)$ in $K_n$ obtained by  adding the edge $e=(u,v)$ to the unique path $\gamma \in {\mathcal Q}_j$ ending at $u$. The blue particles  $v$ in $\V^N$ at time $j+1$ form the set $Z_{j+1}$, and the paths $(\gamma,v)$ the set ${\mathcal Q}_{j+1}$.  Finally, we remove $Z_j$ and all incident edges to obtain the graph $G_{j+1}$.  We note that \eqref{eq:index} also holds for any $v \in Z_{j+1}$ (if it exists). Summing these inequalities, we obtain that \begin{equation}\label{eq:index-2}
V(v) \le \frac{n}{n-N}  \V^N(v)+\frac{N}{n-N}(j+1) , \qquad \forall\, v \in Z_{j+1}.
\end{equation}

\noindent Therefore, we have constructed an exploration process $(Z_j, {\mathcal Q}_j)_{j\ge 0}$, together with its coupling with $\V^N$,  up to the first index $j$ for which $Z_j=\emptyset$.  \hfill {\footnotesize $\Box$}

\medskip
   Let $|u|=j$ be a blue particle in $\V^N$ at time $j$ and $v$ be a child of $u$ in $\V$ among the $n-1$ leftmost children. Conditionally on the whole branching random walk $\V$ and     on the exploration process up to time $j$, the probability for $v$ to be colored  red is
\[ 1-\frac{ \#G_{j+1}}{n-1}= \frac{n-1-\#G_0+\#Z_0 + \#Z_1+\ldots+\#Z_j }{n-1}\le \frac{n-\#G_0+N(j+1)}{n-1}.
\]

Now, we estimate the probability that $v$ is colored purple, given it is not red. In particular, $v$ belongs to $\V^N$ at time $j+1$. To avoid confusion, write temporarily $w$ for the vertex in $K_n$ associated to the particle $v$ in $\V^N$. Whenever $v$ is colored purple,  there must exist some particle $u'\neq u$  in $\V^N$ at time $j$ and some child $v'$ of $u'$ in $\V^N$ such that  $\V^N(v')-\V^N(u')$  is less than $x:=\V^N(v)- \V^N(u')$, and $v'$ is coupled with the vertex $w$. Let  ${\mathcal N}_{u'}$ be the set of children of $u'$ that are not red and  whose displacement (in $\V$) with respect to $u'$ is less than $x$.  By our coupling, we identify ${\mathcal N}_{u'}$ with a set of vertices in $G_{j+1}$. In particular, $w\in {\mathcal N}_{u'}$.
  By symmetry, conditionally on $\V$, on the exploration process up to time $j$, and on the red/non-red particles at time $j+1$,   the probability that $w\in {\mathcal N}_{u'}$ is $\frac{\#{\mathcal N}_{u'}}{\#G_{j+1}}$.   Since $\sum_{u'} \#{\mathcal N}_{u'} \le N$, we obtain that the conditional probability that $v$ is colored purple, given it is not colored red, is less than
 \[
  \frac{N}{\#G_{j+1}}\le \frac{N}{\#G_0-N(j+1)}
  \]
as long as $N(j+1)<\#G_0$.  Since $\p(v \textrm{ is not blue} \mid \V) = \p(u \textrm{ is not blue} \mid \V)+\p(u \textrm{ is blue}, v \textrm{ is not blue} \mid \V)$, we find by induction that
 \begin{equation}\label{proba_red}
       \p( v \textrm{ is not blue} \mid \V) \le \sum_{k=0}^j \Big(\frac{n-\#G_0+N(k+1)}{n-1}  +\frac{N}{\#G_0-N(k+1)}\Big).
  \end{equation}

\subsection{Lower bound in Theorem \ref{t:main}}
The lower bound of Theorem \ref{t:main} is a consequence  of the following lemma.
\begin{lemma}
 Let $\alpha < \frac{\pi^2}{2}$ and $t<\frac{1}{\frac{\pi^2}{2}-\alpha}$. Choose $a\in (0,1)$ close enough to $1$ such that
\[
- a + \frac{\pi^2}{2 a^2} t < \alpha t .
\]
Apply the above exploration process with $G_0=K_n$ and $N_0=N=\lfloor n^a\rfloor$. With probability tending to $1$ as $n\to\infty$, there is an active vertex  at time $\lfloor t\ln^3 n\rfloor $ such that $V(u)\le \alpha t\ln n$.
\end{lemma}
 \begin{proof}
   For simplicity, suppose that $N=n^a$ and that $L:=t\ln^3 n= \frac{t}{a^3} \ln^3 N$ is an integer.  Note that $(-1+ \frac{\pi^2}2 \frac{t}{a^3}) \ln N= (-a + \frac{\pi^2}{2 a^2} t) \ln n $. Fix   $b\in (0,\alpha)$ such that $- a + \frac{\pi^2}{2 a^2} t<  b t$. By Proposition \ref{p:first_moment} (with $a=1$ there), there is  a particle $|u|=L$ in $\V^N$ with $\V^N(u)\le b t \ln n$ with probability going to $1$ as $n\to\infty$. By \eqref{proba_red}, the probability for $u$ to be blue also tends to $1$ as $n\to\infty$.  It remains to prove that $V(u)\le \alpha t \ln n$. By \eqref{eq:index-2},  $V(u)\le (bt \ln n)  \frac{n}{n-N} + \frac{N}{n-N} L \le \alpha t \ln n$ for $n$ large enough indeed.
\end{proof}

\begin{equation}    V(u^*_{j}) \le \frac{n}{n-N} \sum_{i=1}^{j} \eta_i + \frac{Nj}{n-N}, \qquad \forall\, j< \tau_1.\label{control-Vuj} \end{equation}

\noindent Hence $${\tt s}_1 \le \frac{n}{n-N} \sum_{i=1}^{\Delta-1} \max(\eta_i,0) + \frac{N (\Delta-1) }{n-N}.$$

 The term $\sum_{i=1}^{\Delta-1} \max(\eta_i,0)$ is stochastically smaller than the sum of $\Delta-1$ independent exponential r.v. with mean $2e$, as long as $G_{\Delta}$ has more than $\frac{n}{2}$ vertices. This condition is indeed satisfied thanks to \eqref{card_G}. Then for some positive constant $c$,  $$  \e_{G_{\tau_{i-1}}}[({\tt s}_1+\Delta)^2] \le c \Delta^2.$$

By Cauchy--Schwarz inequality and  \eqref{bound_eventE}, we obtain that $$  \e_{G_{\tau_{i-1}}}[ ({\tt s}_1+\Delta) {\bf 1}_{{\mathcal E^c_1}}]
\le
c^{1/2} \Delta \Big( (\frac{2L}{n-1} +\frac{1}{n-2NL})N\Delta\Big)^{1/2}
\le \Delta^{-4},
$$

\noindent where the last inequality holds for all small $\varepsilon$ and large $n$ (see \eqref{eq:NLDelta}). It follows from \eqref{cond-Gi} that $$ \e \Big[\sum_{i=1}^{a_1L/\Delta}   ({\tt s}_i+\Delta)   {\bf 1}_{{\mathcal E}_i^c}\Big]
\le a_1 L \Delta^{-3}.
$$

\noindent Since $L \Delta^{-3}= o(\beta_n L)$, the Markov inequality yields \eqref{sum-S-tau<Delta-2}
and completes the proof of the lemma.
\end{proof}

\section{Proofs of Lemmas \ref{main:upper}, \ref{main2:upper},  and of Proposition  \ref{p:first_moment}}\label{s:rw-brw}

Recall that $(Y_n)_{n\ge 0}$ is a centered random walk starting at $0$ with step distribution $\mbox{Exp}(1)-1$. In the first subsection, we present several estimates on $(Y_n)_{n\ge 0}$ and provide  the proofs of Lemmas \ref{main:upper}, \ref{main2:upper}. The second subsection is devoted to the study of the branching random walks $\V$ and $\V^N$, and to the proof of Proposition  \ref{p:first_moment}.

\subsection{Estimates of random walks: Proofs of Lemmas \ref{main:upper} and \ref{main2:upper}}\label{sub:rw}

\begin{lemma}\label{l:mogulski}
 Let $\upsilon>0$ and $0<d<\frac{\pi^2}{2}$. There exists $\Delta_0>0$ such that for all $\Delta\ge \Delta_0$,   $r_0\le 0\le r_0+\Delta$ and  $\ell\ge 1$,
\begin{equation}
    \p(r_0  \le Y_j\le r_0+\Delta,\, \forall \, j\in \lb 1,  \ell\rb) \le e^{\Delta^\upsilon-\frac{d}{\Delta^2}\ell}. \label{mogulskii_upper0}
\end{equation}

We consider the complete graph $K_n$ of the set of vertices $\llbracket 1, n\rrbracket=\{1,2,\ldots,n\}$. Following Aldous \cite{aldous}, we  assign each edge  an independent exponential random variable with mean $n$. In this setting, Aldous proved the existence of a phase transition for the size of  the largest tree with average weight below a given threshold \cite[Theorem 1]{aldous}. We are interested in the analogous result for paths, rather than trees, established in  \cite[Proposition 3.7]{aldous}. In order to state this result, we introduce the following notation.
For ease of notation, in this paper, we will suppose that the labels $X(u,v)_{u\neq v \in \llbracket 1, n\rrbracket}$  on the (non-oriented) edges of $K_n$ are i.i.d. copies of
\begin{equation}\label{def:X}
X=ne \mathrm{Exp}(1)-1
\end{equation}
where $\mathrm{Exp(1)}$ denotes an exponential random variable with mean $1$.
A path $\gamma$ is a sequence of vertices $(\gamma_0,\gamma_1,\ldots,\gamma_\ell) \in  \llbracket 1, n\rrbracket^{\ell+1}$ which is made of distinct adjacent edges $(\gamma_i,\gamma_{i+1})$. We call $|\gamma|=\ell$ the length of the path, and let ${\mathcal P}_\ell$ be the collection of paths of length $\ell$. A path of length $0$ is reduced to a single vertex.

For any $\gamma \in {\mathcal P}_\ell$, we let $S_0(\gamma)=0$ and
\begin{equation}
S_i(\gamma) := X(\gamma_0,\gamma_1)+\ldots+X(\gamma_{i-1},\gamma_i),\qquad  i\in \llbracket 1,  \ell\rrbracket. \label{Sigamma}
\end{equation}
We write $S(\gamma):=(S_1(\gamma),\ldots,S_\ell(\gamma))$ and call $\frac{S_{\ell}(\gamma)}{\ell}$ the average weight of $\gamma$. For $\lambda\in (-1, \infty)$, let ${\mathscr L}(n,\lambda)$ be the length of the longest path with average weight below $\lambda$:
\begin{equation}
{\mathscr L}(n,\lambda):=\sup\{ \ell \ge 1\colon \exists \, \gamma \in  {\mathcal P}_\ell \textrm{ such that } S_{\ell}(\gamma) \le \lambda \ell \}.
\label{def-Lnlambda} \end{equation}

  Translated in our setting, \cite[Proposition 3.7]{aldous} states that ${\mathscr L}(n, \lambda)=o(n)$ if $\lambda<0$ and  ${\mathscr L}(n,\lambda)=\Omega(n)$ if $\lambda>0$.
 Later, Ding \cite{ding} proved that there is a critical window of size $(\ln n)^{-2}$ around $\lambda=0$:  there exist some constants $c'> c >0$ and $C'>C$ such that

 (i) if $ -1\le \alpha \le c $, then  \begin{equation} \mathbb P\Big( C \ln^3 n \le {\mathscr L}\big(n,\frac{\alpha}{\ln^2 n}\big)  \le C' \ln^3 n\Big) \to 1, \qquad n \to \infty;  \label{Ding-1}\end{equation}

(ii) if $\alpha \ge c'$, then \begin{equation} \mathbb P\Big( n^{1/4} \le {\mathscr L}\big(n,\frac{\alpha}{\ln^2 n}\big)  \le C' \frac{n}{\ln^2 n}\Big) \to 1, \qquad n \to \infty.  \label{Ding-2}\end{equation}

The constant $-1$ in (i) is arbitrary, see \cite[Theorem 1.3]{ding}. To the best of our knowledge, it remains open to identify the critical value at which  the longest path undergoes a transition from  order $\ln^3 n$ to   polynomial order, as well as the precise asymptotic behavior of ${\mathscr L}\big(n,\frac{\alpha}{\ln^2 n}\big)$. Another natural problem concerns the near-critical case $\lambda = \varepsilon$ for small $\varepsilon > 0$, which was further investigated by Ding and Goswami \cite{ding_goswami}: there exists some $\varepsilon_0>0$ such that for all $\varepsilon < \varepsilon_0$, \begin{equation} \mathbb P\Big( n e^{- C' \varepsilon^{-1/2}} \le {\mathscr L}(n, \varepsilon) \le   n e^{- C \varepsilon^{-1/2}}\Big) \to 1.  \label{Ding-Goswami}\end{equation}

  The following theorems give sharp asymptotics of ${\mathscr L}(n,\lambda)$ in these regimes.

For any $\gamma \in {\mathcal P}_\ell$, we let $S_0(\gamma)=0$ and
\begin{equation}
S_i(\gamma) := X(\gamma_0,\gamma_1)+\ldots+X(\gamma_{i-1},\gamma_i),\qquad i\in \llbracket 1, \ell\rrbracket. \label{Sigamma}
\end{equation}
We write $S(\gamma):=(S_1(\gamma),\ldots,S_\ell(\gamma))$ and call $\frac{S_{\ell}(\gamma)}{\ell}$ the average weight of $\gamma$. For $\lambda\in (-1, \infty)$, let ${\mathscr L}(n,\lambda)$ be the length of the longest path with average weight below $\lambda$:
\begin{equation}
{\mathscr L}(n,\lambda):=\sup\{ \ell \ge 1\colon \exists \, \gamma \in {\mathcal P}_\ell \textrm{ such that } S_{\ell}(\gamma) \le \lambda \ell \}.
\label{def-Lnlambda} \end{equation}

Translated in our setting, \cite[Proposition 3.7]{aldous} states that ${\mathscr L}(n, \lambda)=o(n)$ if $\lambda<0$ and ${\mathscr L}(n,\lambda)=\Omega(n)$ if $\lambda>0$.
 Later, Ding \cite{ding} proved that there is a critical window of size $(\ln n)^{-2}$ around $\lambda=0$: there exist some constants $c'> c >0$ and $C'>C$ such that

(i) if $ -1\le \alpha \le c $, then \begin{equation} \mathbb P\Big( C \ln^3 n \le {\mathscr L}\big(n,\frac{\alpha}{\ln^2 n}\big) \le C' \ln^3 n\Big) \to 1, \qquad n \to \infty; \label{Ding-1}\end{equation}

(ii) if $\alpha \ge c'$, then \begin{equation} \mathbb P\Big( n^{1/4} \le {\mathscr L}\big(n,\frac{\alpha}{\ln^2 n}\big) \le C' \frac{n}{\ln^2 n}\Big) \to 1, \qquad n \to \infty. \label{Ding-2}\end{equation}

The constant $-1$ in (i) is arbitrary, see \cite[Theorem 1.3]{ding}. To the best of our knowledge, it remains open to identify the critical value at which the longest path undergoes a transition from order $\ln^3 n$ to polynomial order, as well as the precise asymptotic behavior of ${\mathscr L}\big(n,\frac{\alpha}{\ln^2 n}\big)$. Another natural problem concerns the near-critical case $\lambda = \varepsilon$ for small $\varepsilon > 0$, which was further investigated by Ding and Goswami \cite{ding_goswami}: there exists some $\varepsilon_0>0$ such that for all $\varepsilon < \varepsilon_0$, \begin{equation} \mathbb P\Big( n e^{- C' \varepsilon^{-1/2}} \le {\mathscr L}(n, \varepsilon) \le n e^{- C \varepsilon^{-1/2}}\Big) \to 1. \label{Ding-Goswami}\end{equation}

The following theorems give sharp asymptotics of ${\mathscr L}(n,\lambda)$ in these regimes.

(i) if $ -1\le \alpha \le c $, then \begin{equation} \p\Big( C \ln^3 n \le \L\big(n,\frac{\alpha}{\ln^2 n}\big) \le C' \ln^3 n\Big) \to 1, \qquad n \to \infty; \label{Ding-1}\end{equation}

(ii) if $\alpha \ge c'$, then \begin{equation} \p\Big( n^{1/4} \le \L\big(n,\frac{\alpha}{\ln^2 n}\big) \le C' \frac{n}{\ln^2 n}\Big) \to 1, \qquad n \to \infty. \label{Ding-2}\end{equation}

The constant $-1$ in (i) is arbitrary, see \cite[Theorem 1.3]{ding}. To the best of our knowledge, it remains open to identify the critical value at which the longest path undergoes a transition from order $\ln^3 n$ to polynomial order, as well as the precise asymptotic behavior of $\L\big(n,\frac{\alpha}{\ln^2 n}\big)$. Another natural problem concerns the near-critical case $\lambda = \varepsilon$ for small $\varepsilon > 0$, which was further investigated by Ding and Goswami \cite{ding_goswami}: there exists some $\varepsilon_0>0$ such that for all $\varepsilon < \varepsilon_0$, \begin{equation} \p\Big( n e^{- C' \varepsilon^{-1/2}} \le \L(n, \varepsilon) \le n e^{- C \varepsilon^{-1/2}}\Big) \to 1. \label{Ding-Goswami}\end{equation}

Note that taking $\alpha=0$ in Theorem \ref{t:main} answers \cite[{Equation (2)}]{ding}.

\begin{theorem}\label{t:main2}
 For each $\varepsilon>0$, let $\beta_n$ be a sequence with $\beta_n \in (0,\varepsilon)$ for all $n$ and such that $\liminf_{n\to\infty} \frac{\sqrt{2\beta_n}}{\pi} \ln n > 1$. As $n\to \infty$ then $\varepsilon\to 0$, $$\sqrt{\beta_n}\Big(\ln \L(n,\beta_n) -\ln n\Big) \, \cvp\, -\frac{\pi}{\sqrt{2}}.$$
 In other words, for any $a>0$,
\[
\lim_{\varepsilon\downarrow0}\liminf_{n\to\infty}\p\Big(\frac{1}{n}\L(n,\beta_n) \in [e^{-\frac{\pi}{\sqrt{2\beta_n}}(1+a)}, e^{-\frac{\pi}{\sqrt{2\beta_n}}(1-a)}]\Big)=1.
\]
\end{theorem}

Taking $\beta_n= \frac{\alpha}{\ln^2 n}$ with $\alpha > \frac{\pi^2}{2}$, we see that \begin{equation} \L\big(n,\frac{\alpha}{\ln^2 n}\big)= n^{1- \frac{\pi}{\sqrt{2 \alpha}} +o_p(1)},\label{alpha:polynomial}\end{equation}

\noindent where $o_p(1)$ denotes some quantity which converges to $0$ in probability as $n\to\infty$. This, together with Theorem \ref{t:main}, shows that there is a phase transition at $\lambda_c=\frac{\pi^2}{2}\frac1{\ln^2 n}$. We also note that \eqref{alpha:polynomial} is consistent with \eqref{Ding-2} when $\alpha$ is taken sufficiently large.

\begin{proof}[Proof of the upper bound in Theorem \ref{t:main}] Recall by Claim 2.4 of \cite{ding} that if there is a path of length greater than $\ell$ with average weight below $\lambda$, then there is a path of length in $\lb \ell,2\ell\rb$ with average weight below $\lambda$. Then for $\lambda= \frac{\alpha}{\ln^2 n}$, $t> \frac{1}{\frac{\pi^2}{2}-\alpha}$, and $\ell= \lfloor t \ln^3 n\rfloor$, \begin{align} & \p\Big(\L(n, \lambda) \ge \ell\Big)
 \nonumber \\
 \le & \p\Big(\cup_{L=\ell}^{2 \ell}\{ \exists \gamma \in {\mathcal P}_L: S_L(\gamma) \le \lambda L \}\Big) \nonumber
\\
 \le & \p\Big(\cup_{L=\ell}^{2 \ell}\{ \exists \gamma \in {\mathcal P}_L: S_L(\gamma) \le \lambda L, \min_{0\le i \le j \le L} (S_j(\gamma)- S_i(\gamma)) > -b_n\}\Big) + o(1), \label{eq:L>ell-bn}
\end{align}

\begin{equation} \p\Big( n^{1/4} \le \L\big(n,\frac{\alpha}{\ln^2 n}\big)  \le C' \frac{n}{\ln^2 n}\Big) \to 1, \qquad n \to \infty.  \label{Ding-2}\end{equation}

\begin{equation}    \L\big(n,\frac{\alpha}{\ln^2 n}\big)= n^{1- \frac{\pi}{\sqrt{2 \alpha}} +o_p(1)},\label{alpha:polynomial}\end{equation}

\begin{theorem}\label{t:main}
If $\alpha<\frac{\pi^2}{2}$, then as $n\to\infty$, $$\frac{1}{\ln^3 n} \L\big(n,\frac{\alpha}{\ln^2 n}\big) \, \cvp\, \frac{1}{\frac{\pi^2}{2}-\alpha}.$$
\end{theorem}

Note that taking $\alpha=0$ in Theorem \ref{t:main} answers \cite[{Equation (2)}]{ding}.

\begin{theorem}\label{t:main2}
 For each $\varepsilon>0$, let $\beta_n$ be a sequence with $\beta_n \in (0,\varepsilon)$ for all $n$ and such that $\liminf_{n\to\infty}  \frac{\sqrt{2\beta_n}}{\pi} \ln n > 1$.  As $n\to \infty$ then $\varepsilon\to 0$, $$\sqrt{\beta_n}\Big(\ln {\mathscr L}(n,\beta_n) -\ln n\Big) \, \cvp\,  -\frac{\pi}{\sqrt{2}}.$$
 In other words, for any $a>0$,
\[
\lim_{\varepsilon\downarrow0}\liminf_{n\to\infty}\mathbb P\Big(\frac{1}{n}{\mathscr L}(n,\beta_n) \in [e^{-\frac{\pi}{\sqrt{2\beta_n}}(1+a)}, e^{-\frac{\pi}{\sqrt{2\beta_n}}(1-a)}]\Big)=1.
\]
\end{theorem}

Taking $\beta_n= \frac{\alpha}{\ln^2 n}$ with $\alpha > \frac{\pi^2}{2}$, we see that \begin{equation}    {\mathscr L}\big(n,\frac{\alpha}{\ln^2 n}\big)= n^{1- \frac{\pi}{\sqrt{2 \alpha}} +o_p(1)},\label{alpha:polynomial}\end{equation}

\noindent where $o_p(1)$ denotes some quantity which converges to $0$ in probability as $n\to\infty$. This, together with Theorem \ref{t:main}, shows that there is a phase transition at $\lambda_c=\frac{\pi^2}{2}\frac1{\ln^2 n}$. We also note that \eqref{alpha:polynomial} is consistent with \eqref{Ding-2} when $\alpha$ is taken sufficiently large.

This critical value $\lambda_c$ also appears in a related problem concerning the minimum average weight of a cycle in $K_n$.
Ding, Sun and Wilson \cite{ding_sun_wilson} proved that conditionally on the minimum being nonnegative, its value is $\frac{\pi^2}{2 \ln^2 n}  + O(\frac{1}{\ln^3 n})$ and that the length of the corresponding cycle is of order $\ln^3 n$. It would be interesting to investigate further the connection between these two models.

\bigskip
Now, we outline the main ideas of the proofs.
 Heuristically, we  explore the graph in the following way. Let $N=o(n)$ be an integer. We start with an initial set of $N$ vertices, called generation $0$. We examine all adjacent edges of these vertices
  and select the $N$ edges with the  smallest labels. The endpoints of the selected edges form generation $1$, and the associated labels stand for their positions. We then delete the initial set of vertices and all their adjacent edges, and continue the exploration from the vertices of generation $1$. Each incident edge of a vertex at generation $1$ carries a weight. Add this weight to the position of the vertex,
  then select the $N$ edges associated to the $N$ smallest obtained numbers. The endpoints of the selected edges form the vertices of generation $2$ and the associated numbers stand for their positions.  This process can be repeated for roughly $\frac{n}{N}$ steps before the graph is exhausted. The weights of the paths obtained by this strategy resemble an (inhomogeneous)  branching random walk  in which only the $N$ leftmost particles are retained at each step. This model was introduced by Brunet and Derrida \cite{brunet_derrida2} who conjectured that the selection mechanism was shifting the speed of the population by a correction term of order $\frac{1}{\ln^2 N}$, a result later proved by B\'erard and Gou\'er\'e \cite{berard_gouere}.   We refer to Maillard \cite{maillard}
  for more  results on the model in continuous time. In our setting, the minimum of the exploration process at time $t$ is of order
\[
-\ln N + \frac{\pi^2}{2 \ln^2 N} t.
\]
Taking  $N=n^{1-o(1)}$  and $N=e^{\frac{\pi}{\sqrt{2\beta_n}}}$ yields Theorem \ref{t:main} and Theorem \ref{t:main2}, respectively. A slight variation of this strategy is used to establish the lower bound for ${\mathscr L}(n,\lambda)$. The upper bound follows a similar approach to that of \cite{ding,ding_goswami,ding_sun_wilson} by computing suitable truncated first moments.

\bigskip

Finally, we mention the recent preprint by Jorritsma, Maillard and M\"orters \cite{JMM} in which the authors make use of a killed branching random walk to explore growing sparse random graphs. Even if the models are different, our results  show similarity with several features of  their model, see \cite{JMM} and the references therein.

\bigskip

The paper is organized as follows. Sections \ref{s:upper} and \ref{s:lower} give the proofs of the upper bounds and lower bounds for ${\mathscr L}(n,\lambda)$ respectively.  Section \ref{s:rw-brw} collects results on real-valued random walks and  branching random walks that are used in Sections \ref{s:upper} and \ref{s:lower}.

Heuristically, the proof uses an exploration of the graph that “resembles an (inhomogeneous) branching random walk in which only the $N$ leftmost particles are retained at each step.” One starts with “generation $0$” consisting of $N=o(n)$ vertices, then repeatedly: examine all adjacent edges, “select the $N$ edges with the smallest labels,” take their endpoints as the next generation, delete previously explored vertices/edges, and iterate. At later generations, each incident edge weight is “add[ed] … to the position of the vertex,” and again one selects the $N$ smallest resulting values; the associated numbers are the new “positions.” This can be repeated for about $\frac{n}{N}$ steps.

In this model, “the minimum of the exploration process at time $t$ is of order
\[
-\ln N + \frac{\pi^2}{2 \ln^2 N} t.
\]
Taking $N=n^{1-o(1)}$ … yields Theorem~\ref{t:main} (and taking $N=e^{\frac{\pi}{\sqrt{2\beta_n}}}$ yields Theorem~\ref{t:main2}). A “slight variation of this strategy is used to establish the lower bound for ${\mathscr L}(n,\lambda)$.” For the upper bound, the paper “follows a similar approach to that of \cite{ding,ding_goswami,ding_sun_wilson} by computing suitable truncated first moments.”

Heuristically, the proof uses an exploration of the graph that “resembles an (inhomogeneous) branching random walk in which only the $N$ leftmost particles are retained at each step.” One starts with “generation $0$” consisting of $N=o(n)$ vertices, then repeatedly: examine all adjacent edges, “select the $N$ edges with the smallest labels,” take their endpoints as the next generation, delete previously explored vertices/edges, and iterate. At later generations, each incident edge weight is “add[ed] … to the position of the vertex,” and again one selects the $N$ smallest resulting values; the associated numbers are the new “positions.” This can be repeated for about $\frac{n}{N}$ steps.

In this model, “the minimum of the exploration process at time $t$ is of order
\[
-\ln N + \frac{\pi^2}{2 \ln^2 N} t.
\]
Taking $N=n^{1-o(1)}$ … yields the main theorem (and taking $N=e^{\frac{\pi}{\sqrt{2\beta_n}}}$ yields the following theorem.
\begin{theorem}\label{t:main2}
 For each $\varepsilon>0$, let $\beta_n$ be a sequence with $\beta_n \in (0,\varepsilon)$ for all $n$ and such that $\liminf_{n\to\infty}  \frac{\sqrt{2\beta_n}}{\pi} \ln n > 1$.  As $n\to \infty$ then $\varepsilon\to 0$, $$\sqrt{\beta_n}\Big(\ln \L(n,\beta_n) -\ln n\Big) \, \cvp\,  -\frac{\pi}{\sqrt{2}}.$$
 In other words, for any $a>0$,
\[
\lim_{\varepsilon\downarrow0}\liminf_{n\to\infty}\p\Big(\frac{1}{n}\L(n,\beta_n) \in [e^{-\frac{\pi}{\sqrt{2\beta_n}}(1+a)}, e^{-\frac{\pi}{\sqrt{2\beta_n}}(1-a)}]\Big)=1.
\]
\end{theorem}
A “slight variation of this strategy is used to establish the lower bound for ${\mathscr L}(n,\lambda)$.” For the upper bound, the paper “follows a similar approach to that of \cite{ding,ding_goswami,ding_sun_wilson} by computing suitable truncated first moments.”

\label{t:main}
If $\alpha<\frac{\pi^2}{2}$, then as $n\to\infty$, $$\frac{1}{\ln^3 n} {\mathscr L}\big(n,\frac{\alpha}{\ln^2 n}\big) \, \cvp\, \frac{1}{\frac{\pi^2}{2}-\alpha}.$$,

Question:
Fix a real number \(\alpha<\pi^2/2\). For \(\lambda\in(-1,\infty)\), define
\[
{\mathscr L}(n,\lambda):=\sup\{\ell\ge 1:\ \exists\text{ a path }\gamma=(\gamma_0,\ldots,\gamma_\ell)\text{ with }S_i(\gamma)=\sum_{k=1}^i X(\gamma_{k-1},\gamma_k)\text{ and }S_\ell(\gamma)\le \lambda \ell\}.
\]
Thus \({\mathscr L}(n,\lambda)\) is the maximum path length for which there exists a path whose average weight \(S_\ell(\gamma)/\ell\) is at most \(\lambda\). Under these assumptions, which statement about \({\mathscr L}(n,\alpha/\ln^2 n)\) holds as \(n\to\infty\)?

As \(n\to\infty\),
\[
\frac{1}{\ln^3 n}\,{\mathscr L}\!\left(n,\frac{\alpha}{\ln^2 n}\right) \xrightarrow{\mathbb P} \frac{1}{\frac{\pi^2}{2}-\alpha}.
\]

Choices:
B.
As \(n\to\infty\),
\[
\frac{1}{\ln^3 n}\,{\mathscr L}\!\left(n,\frac{\alpha}{\ln^2 n}\right) \xrightarrow{\mathbb P} \frac{1}{\frac{\pi^2}{2}+\alpha}.
\]
C.
As \(n\to\infty\),
\[
\frac{{\mathscr L}\!\left(n,\frac{\alpha}{\ln^2 n}\right)}{\ln^3 n}=O_{\mathbb P}(1).
\]
D.
As \(n\to\infty\),
\[
\frac{1}{\ln^2 n}\,{\mathscr L}\!\left(n,\frac{\alpha}{\ln^2 n}\right) \xrightarrow{\mathbb P} \frac{1}{\frac{\pi^2}{2}-\alpha}.
\]
E.
For every fixed \(\alpha<\pi^2/2\), there exists a deterministic constant \(C=C(\alpha)>0\) such that, with probability tending to \(1\) as \(n\to\infty\),
\[
{\mathscr L}\!\left(n,\frac{\alpha}{\ln^2 n}\right)= C\ln^3 n+o(1).
\]

mode of approximation upgraded from relative/probabilistic scale to additive deterministic asymptotics