[$=$\,Theorem~\ref{thm:QK-Whitney_A}\,$+$\,Theorem~\ref{thm:QK-Whitney_B}] \label{ithm:QK-Whitney}
\begin{enu}
\item In $QK_{T}(\FlC{n})[y]$, for $1 \le k \le n$, we have
\begin{equation} \label{eq:QK-Whitney_1_intro}
\lambda_{y}(\CS_{k-1}) \star \lambda_{y}(\CS_{k}/\CS_{k-1}) = \lambda_{y}(\CS_{k}) - y \frac{Q_{k-1}}{1-Q_{k-1}} [\CS_{k}/\CS_{k-1}] \star (\lambda_{y}(\CS_{k-1}) - \lambda_{y}(\CS_{k-2})).
\end{equation}

\item In $QK_{T}(\FlC{n})[y]$, for $1 \le k \le n$, we have
\begin{equation} \label{eq:QK-Whitney_2_intro}
\lambda_{y}(\CS_{k-1}^{\vee}) \star \lambda_{y}((\CS_{k}/\CS_{k-1})^{\vee}) = \lambda_{y}(\CS_{k}^{\vee}) - y \frac{Q_{k-1}}{1-Q_{k-1}} [(\CS_{k}/\CS_{k-1})^{\vee}] \star (\lambda_{y}(\CS_{k-1}^{\vee}) - \lambda_{y}(\CS_{k-2}^{\vee})).
\end{equation}

\item In $QK_{T}(\FlC{n})[y]$, we have
\begin{equation} \label{eq:QK-Whitney_3_intro}
\begin{split}
& \lambda_{y}(\CS_{n}) \star \lambda_{y}(\CS_{n}^{\vee}) = \lambda_{y}(\underline{\BC}^{2n}) \\
& \qquad - y^{2} \sum_{p=1}^{n} \frac{Q_{p} \cdots Q_{n}}{1-Q_{p-1}} (\lambda_{y}(\CS_{p-1}) - Q_{p-1} \lambda_{y}(\CS_{p-2})) \star (\lambda_{y}(\CS_{p-1}^{\vee}) - Q_{p-1} \lambda_{y}(\CS_{p-2}^{\vee})).
\end{split}
\end{equation}

\end{enu}

\begin{thm} \label{thm:Whitney_presentation}
There exists an $R(T)$-algebra isomorphism $R_{Q}^{\SW}/I_{Q}^{\SW} \xrightarrow{\sim} QK_{T}(\FlC{n})$ such that
\begin{align}
F_{d}^{k} & \mapsto \left[ \extprod^{d} \CS_{k} \right] & & \text{for $1 \le k \le n$ and $1 \le d \le k$}, \\
G_{d}^{k} & \mapsto \left[ \extprod^{d} \CS_{k}^{\vee} \right] & & \text{for $1 \le k \le n$ and $1 \le d \le k$}, \\
x_{j} & \mapsto [\CS_{j}/\CS_{j-1}] & & \text{for $1 \le j \le n$}, \\
y_{j} & \mapsto [(\CS_{j}/\CS_{j-1})^{\vee}] & & \text{for $1 \le j \le n$}.
\end{align}
\end{thm}

\begin{thm}[{\cite[Theorem~4.1]{GMSXZZ2}}] \label{thm:Nakayama}

Assume that we are given an isomorphism of $R(T)$-algebras
\begin{equation}
\Phi: R(T)[m_{1}, \ldots, m_{s}]/\langle P_{1}, \ldots, P_{r} \rangle \xrightarrow{\sim} K_{T}(G/P)
\end{equation}
where $m_{1}, \ldots, m_{s}$ are indeterminates, and the $P_{i}$ are polynomials in the variables $m_{j}$ with coefficients in $R(T)$.
Consider any power series
\begin{equation}
\ti{P}_{i} = \ti{P}_{i}(m_{1}, \ldots, m_{s}; Q_{1}, \ldots, Q_{k}) \in (R(T)[m_{1}, \ldots, m_{s}])\bra{Q_{1}, \ldots, Q_{k}}
\end{equation}
such that
\begin{itemize}
\item $\ti{P}_{i}(m_{1}, \ldots, m_{s}; 0, \ldots, 0) = P_{i}(m_{1}, \ldots, m_{s})$ for all $m_{1}, \ldots, m_{s}$ and all $i$, and
\item $\ti{P}_{i}(m_{1}, \ldots, m_{s}; Q_{1}, \ldots, Q_{k}) = 0$ in $QK_{T}(G/P)$, where the $m_{i}$ are regarded as elements in $QK_{T}(G/P)$ via the isomorphism $\Phi$.
\end{itemize}
Then $\Phi$ lifts to an isomorphism of $R(T)\bra{Q_{1}, \ldots, Q_{k}}$-algebras
\begin{equation}
\ti{\Phi}: (R(T)\bra{Q_{1}, \ldots, Q_{k}})[m_{1}, \ldots, m_{s}]/\langle \ti{P}_{1}, \ldots, \ti{P}_{r} \rangle \xrightarrow{\sim} QK_{T}(G/P).
\end{equation}
\end{thm}

\begin{ithm}[$=$\,Theorem~\ref{thm:Whitney_presentation}] \label{ithm:Whitney_presentation}
Define a ring $R_{Q}^{\SW}$ by
\begin{equation}
R_{Q}^{\SW} := (R(T)\bra{Q_{1}, \ldots, Q_{n}}) [F_{d}^{k}, G_{d}^{k}, x_{j}, y_{j} \mid 1 \le k \le n, \ 1 \le d \le k, \ 1 \le j \le n].
\end{equation}
Take an ideal $I_{Q}^{\SW}$ of $R_{Q}^{\SW}$ generated by
\begin{align}
& F_{d}^{k} - \left( F_{d}^{k-1} + \frac{1}{1-Q_{k-1}} (F_{d-1}^{k-1} - Q_{k-1} F_{d-1}^{k-2})x_{k} \right) \quad \text{for $1 \le k \le n$ and $1 \le d \le k$}, \\
& G_{d}^{k} - \left( G_{d}^{k-1} + \frac{1}{1-Q_{k-1}} (G_{d-1}^{k-1} - Q_{k-1} G_{d-1}^{k-2})y_{k} \right) \quad \text{for $1 \le k \le n$ and $1 \le d \le k$}, \\
\begin{split}
&\sum_{k = 0}^{d} F_{k}^{n} G_{d-k}^{n} - \Biggl( e_{d}(T_{1}, T_{2}, \ldots, T_{n}, T_{n}^{-1}, \ldots, T_{2}^{-1}, T_{1}^{-1}) \\
& \quad - \sum_{p = 1}^{n} \frac{Q_{p} \cdots Q_{n}}{1-Q_{p-1}} \sum_{r=0}^{d-2} (F_{r}^{p-1} - Q_{p-1} F_{r}^{p-2})(G_{d-2-r}^{p-1} - Q_{p-1}G_{d-2-r}^{p-2}) \Biggr) \quad \text{for $1 \le d \le n$},
\end{split} \\
& F_{1}^{1} - x_{1}, \\
& G_{1}^{1} - y_{1}, \\
& x_{j}y_{j} = (1-Q_{j-1})(1-Q_{j}) \quad \text{for $1 \le j \le n$};
\end{align}
here, $e_{d}$ denotes the $d$-th elementary symmetric polynomial, $T_{k} \in R(T)$ for $1 \le k \le 2n$ are obtained from the decomposition of $\BC^{2n}$ into one-dimensional $T$-modules; note that $T_{n+1} = T_{n}^{-1}$, $T_{n+2} = T_{n-1}^{-1}, \ldots$, $T_{2n} = T_{1}^{-1}$.
In addition, we understand that
$F_{d}^{-1} = F_{d}^{0} = 0$ for $d > 0$, $F_{d}^{k} = 0$ for $1 \le k \le n$ and $d > k$, $F_{0}^{k} = 1$ for $-1 \le k \le n$.

Then there exists an $R(T)$-algebra isomorphism $R_{Q}^{\SW}/I_{Q}^{\SW} \xrightarrow{\sim} QK_{T}(\FlC{n})$ such that
\begin{align}
F_{d}^{k} & \mapsto \left[ \extprod^{d} \CS_{k} \right] & & \text{for $1 \le k \le n$ and $1 \le d \le k$}, \\
G_{d}^{k} & \mapsto \left[ \extprod^{d} \CS_{k}^{\vee} \right] & & \text{for $1 \le k \le n$ and $1 \le d \le k$}, \\
x_{j} & \mapsto [\CS_{j}/\CS_{j-1}] & & \text{for $1 \le j \le n$}, \\
y_{j} & \mapsto [(\CS_{j}/\CS_{j-1})^{\vee}] & & \text{for $1 \le j \le n$}.
\end{align}
\end{ithm}

\begin{ithm}[$=$\,Theorem~\ref{thm:QK-Whitney_A}\,$+$\,Theorem~\ref{thm:QK-Whitney_B}] \label{ithm:QK-Whitney}
\begin{enu}
\item In $QK_{T}(\FlC{n})[y]$, for $1 \le k \le n$, we have
\begin{equation} \label{eq:QK-Whitney_1_intro}
\lambda_{y}(\CS_{k-1}) \star \lambda_{y}(\CS_{k}/\CS_{k-1}) = \lambda_{y}(\CS_{k}) - y \frac{Q_{k-1}}{1-Q_{k-1}} [\CS_{k}/\CS_{k-1}] \star (\lambda_{y}(\CS_{k-1}) - \lambda_{y}(\CS_{k-2})).
\end{equation}

\item In $QK_{T}(\FlC{n})[y]$, for $1 \le k \le n$, we have
\begin{equation} \label{eq:QK-Whitney_2_intro}
\lambda_{y}(\CS_{k-1}^{\vee}) \star \lambda_{y}((\CS_{k}/\CS_{k-1})^{\vee}) = \lambda_{y}(\CS_{k}^{\vee}) - y \frac{Q_{k-1}}{1-Q_{k-1}} [(\CS_{k}/\CS_{k-1})^{\vee}] \star (\lambda_{y}(\CS_{k-1}^{\vee}) - \lambda_{y}(\CS_{k-2}^{\vee})).
\end{equation}

\item In $QK_{T}(\FlC{n})[y]$, we have
\begin{equation} \label{eq:QK-Whitney_3_intro}
\begin{split}
& \lambda_{y}(\CS_{n}) \star \lambda_{y}(\CS_{n}^{\vee}) = \lambda_{y}(\udBC^{2n}) \\
& \qquad - y^{2} \sum_{p=1}^{n} \frac{Q_{p} \cdots Q_{n}}{1-Q_{p-1}} (\lambda_{y}(\CS_{p-1}) - Q_{p-1} \lambda_{y}(\CS_{p-2})) \star (\lambda_{y}(\CS_{p-1}^{\vee}) - Q_{p-1} \lambda_{y}(\CS_{p-2}^{\vee})).
\end{split}
\end{equation}

\end{enu}
\end{ithm}

Let $X$ be a partial flag manifold of arbitrary type.
Givental \cite{Givental} and Lee \cite{Lee} introduced the \emph{torus-equivariant quantum $K$-ring} $QK_{T}(X)$ of the flag manifold $X$,
which is a deformation of the ordinary torus-equivariant $K$-ring $K_{T}(X)$; here $T$ denotes the torus.
The product $\star$ of $QK_{T}(X)$, called the \emph{quantum product}, is defined by using \emph{$K$-theoretic Gromov--Witten invariants}.
To understand the quantum product, presentations of $QK_{T}(X)$ with generators and relations have been studied, including Borel-type, Coulomb branch, and Whitney presentations. In this paper, we focus on the Whitney presentation.

For a $T$-equivariant vector bundle $\mathcal{E}$ over $X$, we denote by $[\mathcal{E}]$ the class of $\mathcal{E}$ in $K_{T}(X)$.
The \emph{Hurzebruch $\lambda_{y}$-class} $\lambda_{y}(\mathcal{E}) \in K_{T}(X)[y]$ of a $T$-equivariant vector bundle $\mathcal{E}$, introduced in \cite{Hirzebruch}, is defined as
\begin{equation}
\lambda_{y}(\mathcal{E}) := 1 + y [\mathcal{E}] + y^{2} \left[ \bigwedge\nolimits^{2} \mathcal{E} \right] + \cdots + y^{\rk(\mathcal{E})} \left[ \bigwedge\nolimits^{\rk(\mathcal{E})} \mathcal{E} \right],
\end{equation}
where $\rk(\mathcal{E})$ denotes the rank of $\mathcal{E}$.
For a short exact sequence $0 \rightarrow \CE_{1} \rightarrow \CE_{2} \rightarrow \CE_{3} \rightarrow 0$ of torus-equivariant vector bundles over $X$, we have
\begin{equation} \label{eq:Whitney}
\lambda_{y}(\CE_{1}) \cdot \lambda_{y}(\CE_{3}) = \lambda_{y}(\CE_{2})
\end{equation}
in $K_{T}(X)[y]$, where $\cdot$ denotes the product of $K_{T}(X)$ (or $K_{T}(X)[y]$).
The identity \eqref{eq:Whitney} is called the \emph{$K$-theoretic Whitney relation}.

Our interest is to quantize the above $K$-theoretic Whitney relations.
More precisely, we study the quantum product $\lambda_{y}(\CE_{1}) \star \lambda_{y}(\CE_{3})$ in $QK_{T}(X)[y]$.
Let us consider the case $X = \Fl(r_{1}, \ldots, r_{k}; n)$ is a $k$-step flag manifold of type $A$ parametrizing all sequences $(0 \subset V_{1} \subset \cdots \subset V_{k} \subset \mathbb{C}^{n})$ of vector subspaces of $\mathbb{C}^{n}$ with $\dim V_{i} = r_{i}$ for $1 \le i \le k$.
Then we can take a tautological filtration $0 = \CS_{0} \subset \CS_{1} \subset \cdots \subset \CS_{k} \subset \underline{\BC}^{n}$ of the trivial bundle $\underline{\BC}^{n} = X \times \mathbb{C}^{n}$.
Gu, Mihalcea, Sharpe, Xu, Zhang, and Zou (\cite{GMSXZZ}) conjectured the result of the quantum product of $\lambda_{y}$-classes associated to tautological vector bundles as follows:
\begin{equation} \label{eq:QK-Whitney_typeA}
\lambda_{y}(\CS_{j}) \star \lambda_{y}(\CS_{j+1}/\CS_{j}) = \lambda_{y}(\CS_{j+1}) - y^{r_{j+1}-r_{j}} \dfrac{Q_{j}}{1-Q_{j}} \det(\CS_{j+1}/\CS_{j}) \star (\lambda_{y}(\CS_{j}) - \lambda_{y}(\CS_{j-1}))
\end{equation}
for $1 \le j \le k$, here $Q_{1}, \ldots, Q_{k}$ are Novikov variables of $QK_{T}(X)$, and $\lambda_{y}(\CS_{n+1}) = \lambda_{y}(\underline{\BC}^{n}) = \prod_{1 \le j \le n} (1 + yT_{j})$, where $T_{j} \in R(T)$ for $1 \le j \le n$ are given by the decomposition of $\mathbb{C}^{n}$ into one-dimensional $T$-modules (see \cite[Conjecture~1.1]{GMSXZZ}).
This identity is called the \emph{quantum $K$-theoretic Whitney relation} (the \emph{$QK$-Whitney relation} for short).
This $QK$-Whitney relation was proved for Grassmannians $X = \mathrm{Gr}(k, n) := \Fl(k; n)$ by Gu, Mihalcea, Sharpe, and Zou (\cite{GMSZ2}), for $1 \le k \le n$, for the incidence variety $X = \Fl(1, n-1; n)$ by Gu, Mihalcea, Sharpe, Xu, Zhang, and Zou (\cite{GMSXZZ}), and for arbitrary partial flag manifolds in type $A$ by Huq-Kuruvilla (\cite{HK}).
In addition, Gu, Mihalcea, Sharpe, Xu, Zhang, and Zou showed in \cite[Theorem~1.4]{GMSXZZ} that this conjecture holds under the quantum $K$-theoretic divisor axiom stated by Buch and Mihalcea, the identity for three-pointed $K$-theoretic Gromov--Witten invariants.
Recently, Lenart, Naito, Sagaki, and Xu proved the quantum $K$-theoretic divisor axiom in \cite{LNSX} (not only for $X = \Fl(r_{1}, \ldots, r_{k}; n)$ but also for arbitrary $X$).
As a benefit of describing $QK$-Whitney relations, we can obtain a presentation of $QK_{T}(\Fl(r_{1}, \ldots, r_{k}; n))$ as a quotient ring of a polynomial ring. In fact, $QK$-Whitney relations give a complete set of defining relations of $QK_{T}(\Fl(r_{1}, \ldots, r_{k}, n))$ (\cite[Corollary~5.4]{GMSXZZ2}).

Since the progress of the study of $QK$-Whitney relations is recent, explicit descriptions of $QK$-Whitney relations are known only for type $A$ partial flag manifolds at present; note that conjectures exist beyond type $A$ (for example, \cite{GMSZ}). The purpose of this paper is to describe $QK$-Whitney relations for the flag manifold in type $C$.
Let $X = \FlC{n}$ be the flag manifold of type $C_{n}$, the manifold parametrizing all sequences
\begin{equation}
(0 \subset V_{1} \subset \cdots \subset V_{n} \subset \mathbb{C}^{2n})
\end{equation}
of isotropic subspaces $V_{1}, \ldots, V_{n}$ with respect to a fixed symplectic form with $\dim V_{k} = k$ for $1 \le k \le n$.
Then we can take a tautological filtration
\begin{equation}
0 = \CS_{0} \subset \CS_{1} \subset \cdots \subset \CS_{n} \subset \underline{\BC}^{2n}
\end{equation}
of subbundles of the trivial bundle $\underline{\BC}^{2n} = \FlC{n} \times \mathbb{C}^{2n}$ over $\FlC{n}$ such that $\rk(\CS_{k}) = k$ for $1 \le k \le n$.
For a $T$-equivariant vector bundle $\mathcal{E}$ over $\FlC{n}$, we denote by $\mathcal{E}^{\vee}$ its dual bundle.
We denote by $Q_{1}, \ldots, Q_{n}$ the Novikov variables of $QK_{T}(\FlC{n})$. We understand that $Q_{0} = 0$ and $\CS_{-1} = 0$.
Our main results are as follows.

Since the progress of the study of $QK$-Whitney relations is recent, explicit descriptions of $QK$-Whitney relations are known only for type $A$ partial flag manifolds at present; note that conjectures exist beyond type $A$ (for example, \cite{GMSZ}). The purpose of this paper is to describe $QK$-Whitney relations for the flag manifold in type $C$.
Let $X = \FlC{n}$ be the flag manifold of type $C_{n}$, the manifold parametrizing all sequences
\begin{equation}
(0 \subset V_{1} \subset \cdots \subset V_{n} \subset \BC^{2n})
\end{equation}
of isotropic subspaces $V_{1}, \ldots, V_{n}$ with respect to a fixed symplectic form with $\dim V_{k} = k$ for $1 \le k \le n$.
Then we can take a tautological filtration
\begin{equation}
0 = \CS_{0} \subset \CS_{1} \subset \cdots \subset \CS_{n} \subset \udBC^{2n}
\end{equation}
of subbundles of the trivial bundle $\udBC^{2n} = \FlC{n} \times \BC^{2n}$ over $\FlC{n}$ such that $\rk(\CS_{k}) = k$ for $1 \le k \le n$.
For a $T$-equivariant vector bundle $\CE$ over $\FlC{n}$, we denote by $\CE^{\vee}$ its dual bundle.
We denote by $Q_{1}, \ldots, Q_{n}$ the Novikov variables of $QK_{T}(\FlC{n})$. We understand that $Q_{0} = 0$ and $\CS_{-1} = 0$.
Our main results are as follows.

\item In $QK_{T}(\FlC{n})[y]$, for $1 \le k \le n$, we have
\begin{equation} \label{eq:QK-Whitney_2_intro}
\lambda_{y}(\CS_{k-1}^{\vee}) \star \lambda_{y}((\CS_{k}/\CS_{k-1})^{\vee}) = \lambda_{y}(\CS_{k}^{\vee}) - y \frac{Q_{k-1}}{1-Q_{k-1}} [(\CS_{k}/\CS_{k-1})^{\vee}] \star (\lambda_{y}(\CS_{k-1}^{\vee}) - \lambda_{y}(\CS_{k-2}^{\vee})).
\end{equation}

\item In $QK_{T}(\FlC{n})[y]$, we have
\begin{equation} \label{eq:QK-Whitney_3_intro}
\begin{split}
& \lambda_{y}(\CS_{n}) \star \lambda_{y}(\CS_{n}^{\vee}) = \lambda_{y}(\udBC^{2n}) \\
& \qquad - y^{2} \sum_{p=1}^{n} \frac{Q_{p} \cdots Q_{n}}{1-Q_{p-1}} (\lambda_{y}(\CS_{p-1}) - Q_{p-1} \lambda_{y}(\CS_{p-2})) \star (\lambda_{y}(\CS_{p-1}^{\vee}) - Q_{p-1} \lambda_{y}(\CS_{p-2}^{\vee})).
\end{split}
\end{equation}

Then we observe that relations \eqref{eq:QK-Whitney_1_intro}, \eqref{eq:QK-Whitney_2_intro}, and \eqref{eq:QK-Whitney_3_intro} give a complete set of the defining relations of $QK_{T}(\FlC{n})$ and hence we obtain a presentation of $QK_{T}(\FlC{n})$, which we call the \emph{Whitney-type presentation}.

Note that the above identities are similar to those for the type $A$ flag manifold; see \eqref{eq:QK-Whitney_typeA}.
On the other hand, the following identity, analogous to \eqref{eq:Whitney_classical_3}, is specific to the type $C$ case.
\begin{thm} \label{thm:QK-Whitney_B}
In $QK_{T}(\FlC{n})[y]$, we have
\begin{equation} \label{eq:QK-Whitney_3}
\begin{split}
& \lambda_{y}(\CS_{n}) \star \lambda_{y}(\CS_{n}^{\vee}) = \lambda_{y}(\udBC^{2n}) \\
& \qquad - y^{2} \sum_{p=1}^{n} \frac{Q_{p} \cdots Q_{n}}{1-Q_{p-1}} (\lambda_{y}(\CS_{p-1}) - Q_{p-1} \lambda_{y}(\CS_{p-2})) \star (\lambda_{y}(\CS_{p-1}^{\vee}) - Q_{p-1} \lambda_{y}(\CS_{p-2}^{\vee})).
\end{split}
\end{equation}
\end{thm}
We call identities \eqref{eq:QK-Whitney_1}, \eqref{eq:QK-Whitney_2}, and \eqref{eq:QK-Whitney_3} \emph{quantum $K$-theoretic Whitney relations} (\emph{$QK$-Whitney relations} for short) for $\FlC{n}$.

\subsection{Proof of Theorem~\ref{thm:QK-Whitney_A}}
First, we give a proof \eqref{eq:QK-Whitney_1}.
For this purpose, we show the following identity.
\begin{lem}
In $QK_{T}(\FlC{n})$, for $1 \le k \le n$ and $1 \le d \le k$, we have
\begin{equation} \label{eq:QK-Whitney_lem_1}
\left[ \extprod^{d} \CS_{k} \right] = \left[ \extprod^{d} \CS_{k-1} \right] + \frac{1}{1-Q_{k-1}} [\CS_{k}/\CS_{k-1}] \star \left( \left[ \extprod^{d-1} \CS_{k-1} \right] - Q_{k-1} \left[ \extprod^{d-1} \CS_{k-2} \right] \right).
\end{equation}
\end{lem}
\begin{proof}
We compute as follows:
\begin{align}
& \left[ \extprod^{d} \CS_{k} \right] \\
& = \sum_{\substack{J \subset [1, k] \\ |J| = d}} [\CO_{G/B}(-\ve_{J})] \\
& = \sum_{\substack{J \subset [1, k-1] \\ |J| = d}} [\CO_{G/B}(-\ve_{J})] + \sum_{\substack{J \subset [1, k-1] \\ |J| = d-1}} [\CO_{G/B}(-\ve_{J \sqcup \{k\}})] \\
\begin{split}
& = \left[ \extprod^{d} \CS_{k-1} \right] + \sum_{\substack{J \subset [1, k-2] \\ |J| = d-1}} [\CO_{G/B}(-\ve_{k})] \star [\CO_{G/B}(-\ve_{J})] \\
& \qquad + \frac{1}{1-Q_{k-1}} \sum_{\substack{J \subset [1, k-1] \\ k-1 \in J \\ |J| = d-1}} [\CO_{G/B}(-\ve_{k})] \star [\CO_{G/B}(-\ve_{J})]
\end{split} \qquad \text{by \eqref{eq:multiple_1}} \\
\begin{split}
& = \left[ \extprod^{d} \CS_{k-1} \right] + \frac{1}{1-Q_{k-1}} [\CO_{G/B}(-\ve_{k})] \star \sum_{\substack{J \subset [1, k-1] \\ |J| = d-1}} [\CO_{G/B}(-\ve_{J})] \\
& \qquad - \frac{Q_{k-1}}{1-Q_{k-1}} [\CO_{G/B}(-\ve_{k})] \star \sum_{\substack{J \subset [1, k-2] \\ |J| = d-1}} [\CO_{G/B}(-\ve_{J})]
\end{split} \\
\begin{split}
& = \left[ \extprod^{d} \CS_{k-1} \right] \\
& \qquad + \frac{1}{1-Q_{k-1}} [\CO_{G/B}(-\ve_{k})] \star \left( \sum_{\substack{J \subset [1, k-1] \\ |J| = d-1}} [\CO_{G/B}(-\ve_{J})] - Q_{k-1} \sum_{\substack{J \subset [1, k-2] \\ |J| = d-1}} [\CO_{G/B}(-\ve_{J})] \right)
\end{split} \\
&= \left[ \extprod^{d} \CS_{k-1} \right] + \frac{1}{1-Q_{k-1}} [\CS_{k}/\CS_{k-1}] \star \left( \left[ \extprod^{d-1} \CS_{k-1} \right] - Q_{k-1} \left[ \extprod^{d-1} \CS_{k-2} \right] \right),
\end{align}
as desired.
\end{proof}

\subsection{Proof of Theorem~\ref{thm:QK-Whitney_B}}
We prove \eqref{eq:QK-Whitney_3} by comparing coefficients of $y^{d}$ of both sides for each $1 \le d \le 2n$.
\begin{lem}
In $QK_{T}(\FlC{n})$, for $1 \le d \le 2n$, we have
\begin{equation} \label{eq:QK-Whitney_computation_lem1}
\begin{split}
& \sum_{\substack{k, l \ge 0 \\ k+l = d}} \left[ \extprod^{k} \CS_{n} \right] \star \left[ \extprod^{l} \CS_{n}^{\vee} \right] = \left[ \extprod^{d} \udBC^{2n} \right] \\
& \hspace{20mm} - \sum_{p = 1}^{n} \frac{Q_{p}Q_{p+1} \cdots Q_{n}}{1 - Q_{p}} \sum_{\substack{J, K \subset [1, n] \\ |J| + |K| = d \\ \max J = \max K = p}} [\CO_{G/B}(-\ve_{J})] \star [\CO_{G/B}(\ve_{K})].
\end{split}
\end{equation}
\end{lem}
\begin{proof}
In what follows, we set $\overline{K} := \{\overline{j} \mid j \in K \}$ for $K \subset [1, n]$.
Let $J, K \subset [1, n]$.
By \eqref{eq:multiple_1}, we have
\begin{align}
[\CO_{G/B}(-\ve_{J})] &= \left( \prod_{\substack{2 \le j \le n \\ j-1 \in J, j \in J}} \frac{1}{1-Q_{j-1}} \right) \left( \qprod_{j \in J} [\CO_{G/B}(-\ve_{j})] \right) \\
&= \left( \prod_{1 \le j \le n} \varphi_{J}(j) \right) \left( \qprod_{j \in J} [\CO_{G/B}(-\ve_{j})] \right).
\end{align}
Also, by \eqref{eq:multiple_2}, we have
\begin{align}
[\CO_{G/B}(\ve_{K})] &= \left( \prod_{\substack{2 \le j \le n \\ j-1\in K, j \in K}} \frac{1}{1-Q_{j-1}} \right) \left( \qprod_{j \in K} [\CO_{G/B}(\ve_{j})] \right) \\
&= \left( \prod_{\overline{n} \le j \le \overline{1}} \varphi_{\overline{K}}(j) \right) \left( \qprod_{j \in \overline{K}} [\CO_{G/B}(-\ve_{j})] \right).
\end{align}
Hence we have
\begin{equation}
[\CO_{G/B}(-\ve_{J})] \star [\CO_{G/B}(\ve_{K})] = \left( \prod_{1 \le j \le n} \varphi_{J}(j) \right) \left( \prod_{\overline{n} \le j \le \overline{1}} \varphi_{\overline{K}}(j) \right) \left( \ \qprod_{j \in J \sqcup \overline{K}} [\CO_{G/B}(-\ve_{j})] \right).
\end{equation}

\begin{prop}
In $QK_{T}(\FlC{n})$, for $1 \le d \le 2n$, we have
\begin{equation} \label{eq:QK-Whitney_lem_3}
\begin{split}
& \sum_{\substack{k, l \ge 0 \\ k+l = d}} \left[ \extprod^{k} \CS_{n} \right] \star \left[ \extprod^{l} \CS_{n}^{\vee} \right] = \left[ \extprod^{d} \udBC^{2n} \right] \\
& \qquad - \sum_{p = 1}^{n} \frac{Q_{p} Q_{p+1} \cdots Q_{n}}{1 - Q_{p-1}} \sum_{\substack{k, l \ge 0 \\ k+l = d-2}} \left( \left[ \extprod^{k} \CS_{p-1} \right] - Q_{p-1} \left[ \extprod^{k} \CS_{p-2} \right] \right) \\
& \hspace{80mm} \star \left( \left[ \extprod^{l} \CS_{p-1}^{\vee} \right] - Q_{p-1} \left[ \extprod^{l} \CS_{p-2}^{\vee} \right] \right).
\end{split}
\end{equation}
\end{prop}

While \eqref{eq:QK-Whitney_1_intro} and \eqref{eq:QK-Whitney_2_intro} are analogous to \eqref{eq:QK-Whitney_typeA}, \eqref{eq:QK-Whitney_3_intro} is specific to the type $C$ case.

Then we observe that relations \eqref{eq:QK-Whitney_1_intro}, \eqref{eq:QK-Whitney_2_intro}, and \eqref{eq:QK-Whitney_3_intro} give a complete set of the defining relations of $QK_{T}(\FlC{n})$ and hence we obtain a presentation of $QK_{T}(\FlC{n})$, which we call the \emph{Whitney-type presentation}.

\begin{ithm}[$=$\,Theorem~\ref{thm:Whitney_presentation}] \label{ithm:Whitney_presentation}
Define a ring $R_{Q}^{\mathsf{W}}$ by
\begin{equation}
R_{Q}^{\mathsf{W}} := (R(T)\bra{Q_{1}, \ldots, Q_{n}}) [F_{d}^{k}, G_{d}^{k}, x_{j}, y_{j} \mid 1 \le k \le n, \ 1 \le d \le k, \ 1 \le j \le n].
\end{equation}
Take an ideal $I_{Q}^{\mathsf{W}}$ of $R_{Q}^{\mathsf{W}}$ generated by
\begin{align}
& F_{d}^{k} - \left( F_{d}^{k-1} + \frac{1}{1-Q_{k-1}} (F_{d-1}^{k-1} - Q_{k-1} F_{d-1}^{k-2})x_{k} \right) \quad \text{for $1 \le k \le n$ and $1 \le d \le k$}, \\
& G_{d}^{k} - \left( G_{d}^{k-1} + \frac{1}{1-Q_{k-1}} (G_{d-1}^{k-1} - Q_{k-1} G_{d-1}^{k-2})y_{k} \right) \quad \text{for $1 \le k \le n$ and $1 \le d \le k$}, \\
\begin{split}
&\sum_{k = 0}^{d} F_{k}^{n} G_{d-k}^{n} - \Biggl( e_{d}(T_{1}, T_{2}, \ldots, T_{n}, T_{n}^{-1}, \ldots, T_{2}^{-1}, T_{1}^{-1}) \\
& \quad - \sum_{p = 1}^{n} \frac{Q_{p} \cdots Q_{n}}{1-Q_{p-1}} \sum_{r=0}^{d-2} (F_{r}^{p-1} - Q_{p-1} F_{r}^{p-2})(G_{d-2-r}^{p-1} - Q_{p-1}G_{d-2-r}^{p-2}) \Biggr) \quad \text{for $1 \le d \le n$},
\end{split} \\
& F_{1}^{1} - x_{1}, \\
& G_{1}^{1} - y_{1}, \\
& x_{j}y_{j} = (1-Q_{j-1})(1-Q_{j}) \quad \text{for $1 \le j \le n$};
\end{align}
here, $e_{d}$ denotes the $d$-th elementary symmetric polynomial, $T_{k} \in R(T)$ for $1 \le k \le 2n$ are obtained from the decomposition of $\mathbb{C}^{2n}$ into one-dimensional $T$-modules; note that $T_{n+1} = T_{n}^{-1}$, $T_{n+2} = T_{n-1}^{-1}, \ldots$, $T_{2n} = T_{1}^{-1}$.
In addition, we understand that
$F_{d}^{-1} = F_{d}^{0} = 0$ for $d > 0$, $F_{d}^{k} = 0$ for $1 \le k \le n$ and $d > k$, $F_{0}^{k} = 1$ for $-1 \le k \le n$.

Then there exists an $R(T)$-algebra isomorphism $R_{Q}^{\mathsf{W}}/I_{Q}^{\mathsf{W}} \xrightarrow{\sim} QK_{T}(\FlC{n})$ such that
\begin{align}
F_{d}^{k} & \mapsto \left[ \bigwedge\nolimits^{d} \CS_{k} \right] & & \text{for $1 \le k \le n$ and $1 \le d \le k$}, \\
G_{d}^{k} & \mapsto \left[ \bigwedge\nolimits^{d} \CS_{k}^{\vee} \right] & & \text{for $1 \le k \le n$ and $1 \le d \le k$}, \\
x_{j} & \mapsto [\CS_{j}/\CS_{j-1}] & & \text{for $1 \le j \le n$}, \\
y_{j} & \mapsto [(\CS_{j}/\CS_{j-1})^{\vee}] & & \text{for $1 \le j \le n$}.
\end{align}
\end{ithm}

To prove Theorem~\ref{ithm:QK-Whitney}, we use the theory of semi-infinite flag manifolds.
Kato (\cite{Kato}) proved that the $K$-group of the semi-infinite flag manifold is isomorphic to the quantum $K$-ring of the flag manifold as $R(T)$-modules.
Through this isomorphism, we can translate certain quantum products in $QK_{T}(\FlC{n})$ that appear in the computation of the left-hand side of $QK$-Whitney relations into tensor products in the $K$-group of the semi-infinite flag manifold.
Since tensor products in the $K$-group of the semi-infinite flag manifold satisfy a multiplicative property, we can obtain identities of line bundle classes in $QK_{T}(\FlC{n})$ (Corollary~\ref{cor:multiple_line_bundles}).
These identities enable us to compute $\lambda_{y}(\CS_{k-1}) \star \lambda_{y}(\CS_{k}/\CS_{k-1})$ and $\lambda_{y}(\CS_{k-1}^{\vee}) \star \lambda_{y}((\CS_{k}/\CS_{k-1})^{\vee})$.
To consider the product $\lambda_{y}(\CS_{n}) \star \lambda_{y}(\CS_{n}^{\vee})$, we use an additional method, the Borel-type presentation given in \cite{KN2}. This gives a relation in $QK_{T}(\FlC{n})$ (Proposition~\ref{prop:QK-Borel_rel}) which enables us to simplify such a product.

To prove Theorem~\ref{ithm:Whitney_presentation}, we use Nakayama-type lemma established by Gu, Mihalcea, Sharpe, Xu, Zhang, and Zou (\cite{GMSXZZ2}). This lemma shows that the isomorphism for the classical $K$-ring $K_{T}(X)$ lifts to that for the quantum $K$-ring $QK_{T}(X)$ under conditions (Theorem~\ref{thm:Nakayama}). By applying this lemma, our Whitney-type presentation reduces to the well-known presentation of $K_{T}(\FlC{n})$, and hence we obtain the theorem.

This paper is organized as follows. In Section~\ref{sec:preliminaries}, we fix notation for root systems; in particular, for the type $C$ root system.
Also, we briefly review the flag manifold and its $K$-ring.
In Section~\ref{sec:QK-Whitney}, we give explicit descriptions of $QK$-Whitney relations in $QK_{T}(\FlC{n})$ (Theorem~\ref{ithm:QK-Whitney}) and the Whitney-type presentation of $QK_{T}(\FlC{n})$ (Theorem~\ref{ithm:Whitney_presentation}), which are the main results of this paper.
In Section~\ref{sec:method}, we review the semi-infinite flag manifold and its $K$-group. Then we compute some quantum products necessary to study $QK$-Whitney relations, where some proofs are deferred to Appendix~\ref{sec:appendix}.
In Section~\ref{sec:QK-Whitney_computation}, we give a proof of Theorem~\ref{ithm:QK-Whitney}. In the computation of $\lambda_{y}(\CS_{n}) \star \lambda_{y}(\CS_{n}^{\vee})$, we review Borel-type relations of $QK_{T}(\FlC{n})$.
In Section~\ref{sec:Nakayama}, we review the Nakayama-type lemma and give a proof of Theorem~\ref{ithm:Whitney_presentation}.

\subsection*{Acknowledgments}
The author is grateful to Takeshi Ikeda and Satoshi Naito for helpful discussions.
This work was supported by JSPS Grant-in-Aid for Research Activity Start-up 24K22842.

To prove the main theorem, we use the theory of semi-infinite flag manifolds. Kato (\cite{Kato}) proved that the $K$-group of the semi-infinite flag manifold is isomorphic to the quantum $K$-ring of the flag manifold as $R(T)$-modules. Through this isomorphism, we can translate certain quantum products in $QK_{T}(\FlC{n})$ that appear in the computation of the left-hand side of the $QK$-Whitney relations into tensor products in the $K$-group of the semi-infinite flag manifold. Since tensor products in the $K$-group of the semi-infinite flag manifold satisfy a multiplicative property, we can obtain identities of line bundle classes in $QK_{T}(\FlC{n})$. We first record the following corollary.

\begin{cor} \label{cor:multiple_line_bundles}
Assume that $G = \Sp_{2n}(\BC)$.
Let $2 \le k \le n$ and take $J \subset [1, k-1]$. Then we have
\begin{align}
[\CO_{G/B}(-\ve_{J \sqcup \{k\}})] &= \begin{cases} \dfrac{1}{1-Q_{k-1}} [\CO_{G/B}(-\ve_{J})] \star [\CO_{G/B}(-\ve_{k})] & \text{if $k-1 \in J$}, \\[5mm]
[\CO_{G/B}(-\ve_{J})] \star [\CO_{G/B}(-\ve_{k})] & \text{if $k-1 \notin J$}, \end{cases} \label{eq:multiple_1} \\
[\CO_{G/B}(\ve_{J \sqcup \{k\}})] &= \begin{cases} \dfrac{1}{1-Q_{k-1}} [\CO_{G/B}(\ve_{J})] \star [\CO_{G/B}(\ve_{k})] & \text{if $k-1 \in J$}, \\[5mm]
[\CO_{G/B}(\ve_{J})] \star [\CO_{G/B}(\ve_{k})] & \text{if $k-1 \notin J$}. \end{cases} \label{eq:multiple_2}
\end{align}
\end{cor}

These identities enable us to compute $\lambda_{y}(\CS_{k-1}) \star \lambda_{y}(\CS_{k}/\CS_{k-1})$ and $\lambda_{y}(\CS_{k-1}^{\vee}) \star \lambda_{y}((\CS_{k}/\CS_{k-1})^{\vee})$. To consider the product $\lambda_{y}(\CS_{n}) \star \lambda_{y}(\CS_{n}^{\vee})$, we use an additional method, the Borel-type presentation. This gives a relation in $QK_{T}(\FlC{n}) enabling us to simplify such a product; more precisely, we use the following proposition.

\begin{prop}[{\cite[Corollary~6.4]{KN2}}] \label{prop:QK-Borel_rel}
In $QK_{T}(\FlC{n})$, for $0 \le d \le 2n$, we have
\begin{equation} \label{eq:QK-Borel_rel}
\sum_{\substack{J \subset [1, \overline{1}] \\ |J|=d}} \left( \prod_{1 \le j \le \overline{1}} \varphi_{J}(j) \right) \left( \qprod_{j \in J} [\CO_{G/B}(-\ve_{j})] \right) = \left[ \extprod^{d} \udBC^{2n} \right].
\end{equation}
\end{prop}

[$=$\,\begin{thm} \label{thm:QK-Whitney_A}
\begin{enu}
\item In $QK_{T}(\FlC{n})[y]$, for $1 \le k \le n$, we have
\begin{equation} \label{eq:QK-Whitney_1}
\lambda_{y}(\CS_{k-1}) \star \lambda_{y}(\CS_{k}/\CS_{k-1}) = \lambda_{y}(\CS_{k}) - y \frac{Q_{k-1}}{1-Q_{k-1}} [\CS_{k}/\CS_{k-1}] \star (\lambda_{y}(\CS_{k-1}) - \lambda_{y}(\CS_{k-2})).
\end{equation}

\item In $QK_{T}(\FlC{n})[y]$, for $1 \le k \le n$, we have
\begin{equation} \label{eq:QK-Whitney_2}
\lambda_{y}(\CS_{k-1}^{\vee}) \star \lambda_{y}((\CS_{k}/\CS_{k-1})^{\vee}) = \lambda_{y}(\CS_{k}^{\vee}) - y \frac{Q_{k-1}}{1-Q_{k-1}} [(\CS_{k}/\CS_{k-1})^{\vee}] \star (\lambda_{y}(\CS_{k-1}^{\vee}) - \lambda_{y}(\CS_{k-2}^{\vee})).
\end{equation}
\end{enu}
\end{thm}\,+$\,\begin{thm} \label{thm:QK-Whitney_B}
In $QK_{T}(\FlC{n})[y]$, we have
\begin{equation} \label{eq:QK-Whitney_3}
\begin{split}
& \lambda_{y}(\CS_{n}) \star \lambda_{y}(\CS_{n}^{\vee}) = \lambda_{y}(\udBC^{2n}) \\
& \qquad - y^{2} \sum_{p=1}^{n} \frac{Q_{p} \cdots Q_{n}}{1-Q_{p-1}} (\lambda_{y}(\CS_{p-1}) - Q_{p-1} \lambda_{y}(\CS_{p-2})) \star (\lambda_{y}(\CS_{p-1}^{\vee}) - Q_{p-1} \lambda_{y}(\CS_{p-2}^{\vee})).
\end{split}
\end{equation}
\end{thm}] \label{ithm:QK-Whitney}
\begin{enu}
\item In $QK_{T}(\FlC{n})[y]$, for $1 \le k \le n$, we have
\begin{equation} \label{eq:QK-Whitney_1_intro}
\lambda_{y}(\CS_{k-1}) \star \lambda_{y}(\CS_{k}/\CS_{k-1}) = \lambda_{y}(\CS_{k}) - y \frac{Q_{k-1}}{1-Q_{k-1}} [\CS_{k}/\CS_{k-1}] \star (\lambda_{y}(\CS_{k-1}) - \lambda_{y}(\CS_{k-2})).
\end{equation}

\item In $QK_{T}(\FlC{n})[y]$, for $1 \le k \le n$, we have
\begin{equation} \label{eq:QK-Whitney_2_intro}
\lambda_{y}(\CS_{k-1}^{\vee}) \star \lambda_{y}((\CS_{k}/\CS_{k-1})^{\vee}) = \lambda_{y}(\CS_{k}^{\vee}) - y \frac{Q_{k-1}}{1-Q_{k-1}} [(\CS_{k}/\CS_{k-1})^{\vee}] \star (\lambda_{y}(\CS_{k-1}^{\vee}) - \lambda_{y}(\CS_{k-2}^{\vee})).
\end{equation}

\item In $QK_{T}(\FlC{n})[y]$, we have
\begin{equation} \label{eq:QK-Whitney_3_intro}
\begin{split}
& \lambda_{y}(\CS_{n}) \star \lambda_{y}(\CS_{n}^{\vee}) = \lambda_{y}(\underline{\BC}^{2n}) \\
& \qquad - y^{2} \sum_{p=1}^{n} \frac{Q_{p} \cdots Q_{n}}{1-Q_{p-1}} (\lambda_{y}(\CS_{p-1}) - Q_{p-1} \lambda_{y}(\CS_{p-2})) \star (\lambda_{y}(\CS_{p-1}^{\vee}) - Q_{p-1} \lambda_{y}(\CS_{p-2}^{\vee})).
\end{split}
\end{equation}

\end{enu}

Question: Let \(X=\mathrm{Fl}^C_n\) be the type \(C_n\) flag manifold parametrizing complete isotropic flags \(0\subset V_1\subset\cdots\subset V_n\subset \mathbb C^{2n}\), and let \(0=\mathcal S_0\subset \mathcal S_1\subset\cdots\subset \mathcal S_n\) be its tautological filtration. In the torus-equivariant quantum \(K\)-ring \(QK_T(X)\) with quantum product \(\star\) and Novikov variables \(Q_i\), define the Hirzebruch \(\lambda_y\)-class of a \(T\)-equivariant vector bundle \(\mathcal E\) by \(\lambda_y(\mathcal E)=\sum_{i=0}^{\operatorname{rk}(\mathcal E)} y^i[\wedge^i\mathcal E]\). Which explicit identities hold in \(QK_T(X)[y]\) for the tautological bundles, their quotients, and their duals?

The following three identities hold in \(QK_T(\mathrm{Fl}^C_n)[y]\):
\[
\text{for }1\le k\le n,\qquad
\lambda_y(\mathcal S_{k-1})\star \lambda_y(\mathcal S_k/\mathcal S_{k-1})
= \lambda_y(\mathcal S_k)
- y\,\frac{Q_{k-1}}{1-Q_{k-1}}\,[\mathcal S_k/\mathcal S_{k-1}]\star\bigl(\lambda_y(\mathcal S_{k-1})-\lambda_y(\mathcal S_{k-2})\bigr),
\]
\[
\text{for }1\le k\le n,\qquad
\lambda_y(\mathcal S_{k-1}^{\vee})\star \lambda_y\bigl((\mathcal S_k/\mathcal S_{k-1})^{\vee}\bigr)
= \lambda_y(\mathcal S_k^{\vee})
- y\,\frac{Q_{k-1}}{1-Q_{k-1}}\,[(\mathcal S_k/\mathcal S_{k-1})^{\vee}]\star\bigl(\lambda_y(\mathcal S_{k-1}^{\vee})-\lambda_y(\mathcal S_{k-2}^{\vee})\bigr),
\]
and
\[
\lambda_y(\mathcal S_n)\star \lambda_y(\mathcal S_n^{\vee})
= \lambda_y(\underline{\mathbb C}^{2n})
- y^2\sum_{p=1}^n \frac{Q_p\cdots Q_n}{1-Q_{p-1}}
\bigl(\lambda_y(\mathcal S_{p-1})-Q_{p-1}\lambda_y(\mathcal S_{p-2})\bigr)
\star
\bigl(\lambda_y(\mathcal S_{p-1}^{\vee})-Q_{p-1}\lambda_y(\mathcal S_{p-2}^{\vee})\bigr).
\]

Choices:
B.
The following three identities hold in \(QK_T(\mathrm{Fl}^C_n)[y]\):
\[
\text{for }1\le k\le n,\qquad
\lambda_y(\mathcal S_{k-1})\star \lambda_y(\mathcal S_k/\mathcal S_{k-1})
= \lambda_y(\mathcal S_k)
- y\,\frac{Q_k}{1-Q_k}\,[\mathcal S_k/\mathcal S_{k-1}]\star\bigl(\lambda_y(\mathcal S_{k-1})-\lambda_y(\mathcal S_{k-2})\bigr),
\]
\[
\text{for }1\le k\le n,\qquad
\lambda_y(\mathcal S_{k-1}^{\vee})\star \lambda_y\bigl((\mathcal S_k/\mathcal S_{k-1})^{\vee}\bigr)
= \lambda_y(\mathcal S_k^{\vee})
- y\,\frac{Q_k}{1-Q_k}\,[(\mathcal S_k/\mathcal S_{k-1})^{\vee}]\star\bigl(\lambda_y(\mathcal S_{k-1}^{\vee})-\lambda_y(\mathcal S_{k-2}^{\vee})\bigr),
\]
and
\[
\lambda_y(\mathcal S_n)\star \lambda_y(\mathcal S_n^{\vee})
= \lambda_y(\underline{\mathbb C}^{2n})
- y^2\sum_{p=1}^n \frac{Q_p\cdots Q_n}{1-Q_p}
\bigl(\lambda_y(\mathcal S_{p-1})-Q_p\lambda_y(\mathcal S_{p-2})\bigr)
\star
\bigl(\lambda_y(\mathcal S_{p-1}^{\vee})-Q_p\lambda_y(\mathcal S_{p-2}^{\vee})\bigr).
\]
C.
The following two identities hold in \(QK_T(\mathrm{Fl}^C_n)[y]\):
\[
\text{for }1\le k\le n,\qquad
\lambda_y(\mathcal S_{k-1})\star \lambda_y(\mathcal S_k/\mathcal S_{k-1})
= \lambda_y(\mathcal S_k)
- y\,\frac{Q_{k-1}}{1-Q_{k-1}}\,[\mathcal S_k/\mathcal S_{k-1}]\star\bigl(\lambda_y(\mathcal S_{k-1})-\lambda_y(\mathcal S_{k-2})\bigr),
\]
\[
\text{for }1\le k\le n,\qquad
\lambda_y(\mathcal S_{k-1}^{\vee})\star \lambda_y\bigl((\mathcal S_k/\mathcal S_{k-1})^{\vee}\bigr)
= \lambda_y(\mathcal S_k^{\vee})
- y\,\frac{Q_{k-1}}{1-Q_{k-1}}\,[(\mathcal S_k/\mathcal S_{k-1})^{\vee}]\star\bigl(\lambda_y(\mathcal S_{k-1}^{\vee})-\lambda_y(\mathcal S_{k-2}^{\vee})\bigr).
\]
D.
The following three identities hold in \(QK_T(\mathrm{Fl}^C_n)[y]\):
\[
\text{for }1\le k\le n,\qquad
\lambda_y(\mathcal S_{k-1})\star \lambda_y(\mathcal S_k/\mathcal S_{k-1})
= \lambda_y(\mathcal S_k)
- y\,\frac{Q_{k-1}}{1-Q_{k-1}}\,[\mathcal S_k/\mathcal S_{k-1}]\star\bigl(\lambda_y(\mathcal S_{k-1})-\lambda_y(\mathcal S_{k-2})\bigr),
\]
\[
\text{for }1\le k\le n,\qquad
\lambda_y(\mathcal S_{k-1}^{\vee})\star \lambda_y\bigl((\mathcal S_k/\mathcal S_{k-1})^{\vee}\bigr)
= \lambda_y(\mathcal S_k^{\vee})
- y\,\frac{Q_{k-1}}{1-Q_{k-1}}\,[(\mathcal S_k/\mathcal S_{k-1})^{\vee}]\star\bigl(\lambda_y(\mathcal S_{k-1}^{\vee})-\lambda_y(\mathcal S_{k-2}^{\vee})\bigr),
\]
and
\[
\lambda_y(\mathcal S_n)\star \lambda_y(\mathcal S_n^{\vee})
= \lambda_y(\underline{\mathbb C}^{2n})
- y\sum_{p=1}^n \frac{Q_p\cdots Q_n}{1-Q_{p-1}}
\bigl(\lambda_y(\mathcal S_{p-1})-Q_{p-1}\lambda_y(\mathcal S_{p-2})\bigr)
\star
\bigl(\lambda_y(\mathcal S_{p-1}^{\vee})-Q_{p-1}\lambda_y(\mathcal S_{p-2}^{\vee})\bigr).
\]
E.
The following three identities hold in \(QK_T(\mathrm{Fl}^C_n)[y]\):
\[
\text{for }1\le k\le n,\qquad
\lambda_y(\mathcal S_{k-1})\star \lambda_y(\mathcal S_k/\mathcal S_{k-1})
= \lambda_y(\mathcal S_k)
- y\,\frac{Q_{k-1}}{1-Q_{k-1}}\,[\mathcal S_k/\mathcal S_{k-1}]\star\bigl(\lambda_y(\mathcal S_{k-1})-\lambda_y(\mathcal S_{k-2})\bigr),
\]
\[
\text{for }1\le k\le n,\qquad
\lambda_y(\mathcal S_{k-1}^{\vee})\star \lambda_y\bigl((\mathcal S_k/\mathcal S_{k-1})^{\vee}\bigr)
= \lambda_y(\mathcal S_k^{\vee})
- y\,\frac{Q_{k-1}}{1-Q_{k-1}}\,[(\mathcal S_k/\mathcal S_{k-1})^{\vee}]\star\bigl(\lambda_y(\mathcal S_{k-1}^{\vee})-\lambda_y(\mathcal S_{k-2}^{\vee})\bigr),
\]
and
\[
\lambda_y(\mathcal S_n)\star \lambda_y(\mathcal S_n^{\vee})
= \lambda_y(\underline{\mathbb C}^{2n})
- y^2\sum_{p=1}^n \frac{Q_p\cdots Q_n}{1-Q_{p-1}}
\bigl(\lambda_y(\mathcal S_{p-1})-Q_{p-1}\lambda_y(\mathcal S_{p-2})\bigr)
\star
\bigl(\lambda_y(\mathcal S_{p-1})-Q_{p-1}\lambda_y(\mathcal S_{p-2})\bigr).
\]

shifted-index denominator and Novikov dependence

dropped the third identity for \(\lambda_y(\mathcal S_n)\star\lambda_y(\mathcal S_n^{\vee})\)

power of \(y\) in the global correction term

dual factor in the second parenthesis of the third identity


\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


Question:
Let ${\mathscr L}(n,\lambda)$ denote the length of the longest path whose average weight is at most $\lambda$, namely
\[
{\mathscr L}(n,\lambda):=\sup\Big\{\ell\ge 1:\exists\,\gamma=(\gamma_0,\dots,\gamma_\ell)\in {\mathcal P}_\ell\text{ such that }S_\ell(\gamma)\le \lambda \ell\Big\},
\]
where ${\mathcal P}_\ell$ is the family of paths of length $\ell$ and
\[
S_\ell(\gamma)=\sum_{i=1}^\ell X(\gamma_{i-1},\gamma_i)
\]
is the total weight of the path. Fix a real number $\alpha<\pi^2/2$. As $n\to\infty$, which limiting statement holds for ${\mathscr L}\bigl(n,\alpha/\ln^2 n\bigr)$?

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

precise convergence in probability to the explicit constant

logarithmic scaling exponent coming from the exploration-time balance