\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

The stem defines the quantity and asks for the asymptotic regime, but it does not state the limit, scaling exponent, or mode of convergence. The condition alpha < \pi^2/2 gives contextual information rather than leaking the answer.

This is close to a theorem-recall item: it essentially asks for the exact limiting statement under the stated hypothesis. However, it is not a pure verbatim restatement because the student must distinguish among several nearby conclusions (sign, scaling, strength, and convergence mode).

Some reasoning is needed to select the strongest correct conclusion among close alternatives, especially to reject the wrong logarithmic scaling and the almost-sure version. Still, the item mainly tests recognition/recall of a known asymptotic result rather than deeper derivation.

The distractors are mathematically meaningful and target common failure modes: sign error in the constant, weaker-but-true asymptotic boundedness, incorrect logarithmic exponent, and overly strong convergence mode. They are distinct and plausible, though choice E has a minor typo.

A solid theorem-identification MCQ with good distractors and little answer leakage, but it leans more toward precise recall than genuine generative mathematical reasoning.


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 \(n\to\infty\)?

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 \(n\to\infty\),
\[
{\mathscr L}\!\left(n,\frac{\alpha}{\ln^2 n}\right)= C\ln^3 n+o(1).
\]

The stem defines the quantity and asks for its asymptotic behavior, but it does not reveal the limiting constant, scaling, or the correct option. There is no explicit answer leakage.

The item is essentially asking the test-taker to recognize the exact theorem conclusion under the stated hypotheses. The options do vary in meaningful ways, so it is not a pure verbatim restatement, but it is still close to theorem recall rather than a genuinely rederived conclusion.

There is some reasoning involved in comparing the logarithmic scale, the constant, and the mode of convergence. However, the problem mainly tests recall/recognition of a known asymptotic statement rather than substantial generative reasoning from the stem alone.

Most distractors are mathematically plausible and target common mistakes: wrong sign in the denominator, wrong log exponent, and an overly strong deterministic asymptotic. However, option C is a weaker statement that is also true if A is true, which creates ambiguity in a single-answer MCQ and lowers distractor quality.

No answer leakage and several plausible distractors, but the item is theorem-recognition-heavy and weakened by the presence of a weaker true alternative, making it somewhat ambiguous as a single-correct-answer MCQ.


\begin{thm}\label{thm:1.1} For $G=\operatorname{PGL}(3,\mathbb{F}_q(\!(t^{-1})\!))$ and $\Gamma=\operatorname{PGL}(3,\mathbb{F}_q[t])$, the type 1 chamber zeta function $Z_{\Gamma}(u)$ converges for sufficiently $u$, and it is given by
$$Z_{\Gamma}(u)=\frac{(1-q^4u^6)(1-q^2u^3)}{(1-q^3u^6)(1-q^3u^3)}
.$$
\end{thm}

\label{sec:1}
In this paper we define and compute a chamber zeta function for the standard non-uniform arithmetic quotient of the Bruhat--Tits building of $\mathrm{PGL}_3$. Our zeta function is defined via weighted tailless galleries of chambers and admits an Ihara--Bass type determinant formula in terms of a natural chamber transfer operator.

Ihara introduced a zeta function associated with prime elements in discrete subgroups of rank one $p$-adic groups \cite{I}. This zeta function has been generalized by Sunada \cite{Su}, Hashimoto \cite{Has}, and Bass \cite{B} to uncover the connections between number theory and representation theory.

 In recent years, the extension of zeta functions to infinite graphs has been studied \cite{CS,Cla,GZ,CJK,DK18}. In particular, Deitmar and Kang considered cycles obtained from geodesics in the universal covering tree of infinite weighted graphs. We defined a Selberg zeta function of graphs of groups and compared it with the zeta function introduced by Deitmar and Kang \cite{HK2}.

 Building on Ihara's insights, Kang and Li defined a zeta function of a finite quotient of a Bruhat-Tits building associated with $\PGL_3(F)$ over a non-archimedean field by a discrete cocompact torsion-free subgroup $\Gamma\subset \PGL_3(F)$, and found a necessary and sufficient condition for Ramanujan complexes \cite{KL}.
 This article aims to generalize zeta functions of infinite graphs and complexes of groups simultaneously.

In \cite{KLW10} and \cite{KL}, the authors considered not only edge zeta function but also the chamber zeta function. Given a torsion-free cocompact lattice $\Gamma$ of $\operatorname{PGL}(3,F)$, the type 1 chamber zeta function for $\Gamma$ is defined as follows. Let $X=\Gamma\backslash \mathcal{B}$ be the quotient complex. Every vertex $v$ has a color $\tau(v)$ (see Section \ref{sec:2}). The action of $\Gamma$ does not preserve the color of vertices whereas it preserves the color difference between the endpoints of each directed edge. This enables us to define the type of oriented edges and pointed chambers. A pointed chamber $c=\{v_1,v_2,v_3\}$ is called \textit{type} $k$ if  for any $i\in \mathbb{Z}/3\mathbb{Z}$, $$\tau(v_{i+1})-\tau(v_i)=k.$$
A sequence of type $k$ pointed chambers $(c_1,\cdots,c_n)$ is called a \textit{type $k$ gallery} if the sequence consists of the edge-adjacent pointed chambers, i.e. for any $ i$, $$v_{2,i}=v_{1,i+1},v_{3,i}=v_{2,i+1},$$
where $c_i=\{v_{1,i},v_{2,i},v_{3,i}\}.$ A gallery $\mathcal{G}=(c_1,\cdots,c_n)$ is called \textit{tailless} if $v_{1,i}\neq v_{3,i+1}$ for any $i$.
A gallery $\mathcal{G}=(c_1,\cdots,c_n)$ is called \emph{closed} if $v_{1,1}=v_{2,n}$ and $v_{2,1}=v_{3,n}.$ The \emph{shift map} of a gallery is defined by $\sigma(\mathcal{G})=(c_2,\ldots,c_n,c_1)$.  Two closed galleries $\mathcal{G}_1$ and $\mathcal{G}_2$ are equivalent if $\mathcal{G}_1=\sigma^i(\mathcal{G}_2)$ for some $i$. An equivalence class $\mathcal{C}=[\mathcal{G}]$ of closed galleries is called a
\emph{closed gallery class}, or simply a \emph{class} when no confusion arises. The power $\mathcal{C}^n$ of a closed gallery class $\mathcal{C}$ is the class obtained by winding around the same gallery for $n$ times. A class $\mathcal{C}_0$ is called \emph{primitive} if it is not a power of a shorter one. For each class $\mathcal{C}$, there exists a unique primitive class $\mathcal{C}_0$ satisfying $\mathcal{C}=\mathcal{C}_0^m$. The number $m=m(\mathcal{C})$ is called the \emph{multiplicity} of $\mathcal{C}$. As an analogy of edge zeta function, the \emph{type 1 chamber zeta function} for $\Gamma$ is defined by
$$Z_{\Gamma}:=\prod_{\mathcal{C}}\left(1-u^{\ell(\mathcal{C})}\right)^{-1},$$
where the product runs over all tailless type 1 primitive classes and and $\ell(\mathcal{C})$ is the length (the number of chambers) of the class $\mathcal{C}$. Let $T$ be the edge adjacency operator on $L^2$-space on the set of type 1 pointed chambers in $X$ defined by
$$Tf(c):=\sum_{c'}f(c'),$$
where the sum is taken over the edge adjacency type 1 pointed chamber of $c$ in $X$.
The authors of \cite{KLW10} and \cite{KL} proved that the type 1 chamber zeta function and the operator $T$ also satisfy the determinant formula
$$Z_{\Gamma}(u)=\frac{1}{\det(I-uT)}.$$

By extending the idea in \cite{DK18} to higher dimension, we have defined the edge zeta function for non-compact weighted complexes.
Combining the ideas in \cite{DK18} and \cite{KL}, we regarded classes of galleries as image of the tailless galleries in the Bruhat-Tits building $\mathcal{B}_3$ associated to $\operatorname{PGL}_3(F).$ Similar to \cite{HK1} and \cite{HK3}, we define the weight $w$ of two type 1 pointed chambers.
For a pointed chamber $\widetilde{c}$ of $\mathcal{B}_3$ and a pointed chamber $c'$ of $X$, let
$$w(\widetilde{c},c'):=\#\{\widetilde{c}'\colon\widetilde{c}\textrm{ is a lift }\textrm {of }c',(\widetilde{c},\widetilde{c}') \textrm{ is a talless gallery in }\mathcal{B}\}.$$
For a pointed chamber $c$ of $X$, let $\widetilde{c}$ be a lift of $c$ in $\mathcal{B}$ and $w(c,c')=w(\widetilde{c},c')$. The weight $w(c,c')$ does not depend on the specific choice of the preimage $\widetilde{c}$. Thus it is well-defined.

The weight of a closed gallery $\mathcal{G}=(c_1,c_2,\ldots,c_n)$ is defined by
$$w(\mathcal{G}):=\prod_{j\textrm{ mod }n}w(c_j,c_{j+1}).$$
Then, the type 1 chamber zeta function for $\Gamma$ is defined by
$$Z_{\Gamma}(u):=\prod_C\left(1-w(\mathcal{C})u^{\ell(\mathcal{C})}\right)^{-1},$$
where the product is taken over all primitive type 1 closed gallery classes, and $\ell(\mathcal{C})$ is the length of the class $\mathcal{C}$.

\label{sec:1}
In this paper we define and compute a chamber zeta function for the standard non-uniform arithmetic quotient of the Bruhat--Tits building of $\mathrm{PGL}_3$. Our zeta function is defined via weighted tailless galleries of chambers and admits an Ihara--Bass type determinant formula in terms of a natural chamber transfer operator.

Ihara introduced a zeta function associated with prime elements in discrete subgroups of rank one $p$-adic groups \cite{I}. This zeta function has been generalized by Sunada \cite{Su}, Hashimoto \cite{Has}, and Bass \cite{B} to uncover the connections between number theory and representation theory.

Building on Ihara's insights, Kang and Li defined a zeta function of a finite quotient of a Bruhat-Tits building associated with $\PGL_3(F)$ over a non-archimedean field by a discrete cocompact torsion-free subgroup $\Gamma\subset \PGL_3(F)$, and found a necessary and sufficient condition for Ramanujan complexes \cite{KL}.
 This article aims to generalize zeta functions of infinite graphs and complexes of groups simultaneously.

In \cite{KLW10} and \cite{KL}, the authors considered not only edge zeta function but also the chamber zeta function. Given a torsion-free cocompact lattice $\Gamma$ of $\operatorname{PGL}(3,F)$, the type 1 chamber zeta function for $\Gamma$ is defined as follows. Let $X=\Gamma\backslash \mathcal{B}$ be the quotient complex. Every vertex $v$ has a color $\tau(v)$ (see Section \ref{sec:2}). The action of $\Gamma$ does not preserve the color of vertices whereas it preserves the color difference between the endpoints of each directed edge. This enables us to define the type of oriented edges and pointed chambers. A pointed chamber $c=\{v_1,v_2,v_3\}$ is called \textit{type} $k$ if for any $i\in \mathbb{Z}/3\mathbb{Z}$, $$\tau(v_{i+1})-\tau(v_i)=k.$$
A sequence of type $k$ pointed chambers $(c_1,\cdots,c_n)$ is called a \textit{type $k$ gallery} if the sequence consists of the edge-adjacent pointed chambers, i.e. for any $ i$, $$v_{2,i}=v_{1,i+1},v_{3,i}=v_{2,i+1},$$
where $c_i=\{v_{1,i},v_{2,i},v_{3,i}\}.$ A gallery $\mathcal{G}=(c_1,\cdots,c_n)$ is called \textit{tailless} if $v_{1,i}\neq v_{3,i+1}$ for any $i$.
A gallery $\mathcal{G}=(c_1,\cdots,c_n)$ is called \emph{closed} if $v_{1,1}=v_{2,n}$ and $v_{2,1}=v_{3,n}.$ The \emph{shift map} of a gallery is defined by $\sigma(\mathcal{G})=(c_2,\ldots,c_n,c_1)$. Two closed galleries $\mathcal{G}_1$ and $\mathcal{G}_2$ are equivalent if $\mathcal{G}_1=\sigma^i(\mathcal{G}_2)$ for some $i$. An equivalence class $\mathcal{C}=[\mathcal{G}]$ of closed galleries is called a
\emph{closed gallery class}, or simply a \emph{class} when no confusion arises. The power $\mathcal{C}^n$ of a closed gallery class $\mathcal{C}$ is the class obtained by winding around the same gallery for $n$ times. A class $\mathcal{C}_0$ is called \emph{primitive} if it is not a power of a shorter one. For each class $\mathcal{C}$, there exists a unique primitive class $\mathcal{C}_0$ satisfying $\mathcal{C}=\mathcal{C}_0^m$. The number $m=m(\mathcal{C})$ is called the \emph{multiplicity} of $\mathcal{C}$. As an analogy of edge zeta function, the \emph{type 1 chamber zeta function} for $\Gamma$ is defined by
$$Z_{\Gamma}:=\prod_{\mathcal{C}}\left(1-u^{\ell(\mathcal{C})}\right)^{-1},$$
where the product runs over all tailless type 1 primitive classes and and $\ell(\mathcal{C})$ is the length (the number of chambers) of the class $\mathcal{C}$. Let $T$ be the edge adjacency operator on $L^2$-space on the set of type 1 pointed chambers in $X$ defined by
$$Tf(c):=\sum_{c'}f(c'),$$
where the sum is taken over the edge adjacency type 1 pointed chamber of $c$ in $X$.
The authors of \cite{KLW10} and \cite{KL} proved that the type 1 chamber zeta function and the operator $T$ also satisfy the determinant formula
$$Z_{\Gamma}(u)=\frac{1}{\det(I-uT)}.$$

By extending the idea in \cite{DK18} to higher dimension, we have defined the edge zeta function for non-compact weighted complexes.
Combining the ideas in \cite{DK18} and \cite{KL}, we regarded classes of galleries as image of the tailless galleries in the Bruhat-Tits building $\mathcal{B}_3$ associated to $\operatorname{PGL}_3(F).$ Similar to \cite{HK1} and \cite{HK3}, we define the weight $w$ of two type 1 pointed chambers.
For a pointed chamber $\widetilde{c}$ of $\mathcal{B}_3$ and a pointed chamber $c'$ of $X$, let
$$w(\widetilde{c},c'):=\#\{\widetilde{c}'\colon\widetilde{c}\textrm{ is a lift }\textrm {of }c',(\widetilde{c},\widetilde{c}') \textrm{ is a talless gallery in }\mathcal{B}\}.$$
For a pointed chamber $c$ of $X$, let $\widetilde{c}$ be a lift of $c$ in $\mathcal{B}$ and $w(c,c')=w(\widetilde{c},c')$. The weight $w(c,c')$ does not depend on the specific choice of the preimage $\widetilde{c}$. Thus it is well-defined.

The weight of a closed gallery $\mathcal{G}=(c_1,c_2,\ldots,c_n)$ is defined by
$$w(\mathcal{G}):=\prod_{j\textrm{ mod }n}w(c_j,c_{j+1}).$$
Then, the type 1 chamber zeta function for $\Gamma$ is defined by
$$Z_{\Gamma}(u):=\prod_C\left(1-w(\mathcal{C})u^{\ell(\mathcal{C})}\right)^{-1},$$
where the product is taken over all primitive type 1 closed gallery classes, and $\ell(\mathcal{C})$ is the length of the class $\mathcal{C}$.

Hence, we have
$$Z_{\Gamma}(u)=\frac{1}{\det(I-uT)}=\exp\left(\sum_{n=1}^{\infty}\frac{N_n(\Gamma\backslash\mathcal{B})}{n}u^{n}\right)$$ which implies the following corollary. (See Section~\ref{sec:4} for the detail.)
\begin{coro}[Counting closed galleries]\label{coro:cycle_intro} Let $\Gamma$ be the discrete subgroup $\operatorname{PGL}(3,\mathbb{F}_q[t])$ of $\operatorname{PGL}(3,\mathbb{F}_q(\!(t^{-1})\!))$ and $N_n(\Gamma\backslash\mathcal{B})$ be defined as above. Then, we have
$$N_{m}=\left\{\begin{array}{ll}3q^{3r}-3q^{2r} & \textrm{ if }n=3r\textrm{ and }n\not\equiv 0 \,(\operatorname{mod}\,{6}) \\
3q^{6r}-9q^{4r}+3q^{3r} & \textrm{ if }n=6r \\ 0
& \textrm{ otherwise}\end{array}\right..$$\end{coro}

The trace of $T^n$ is described by
$$\operatorname{Tr}(T^n)=\sum_{c }\sum_{\mathcal{C}\colon c\in \mathcal{C}}w(\mathcal{C}),$$
where $c$ runs over the type 1 pointed chambers and $\mathcal{C}$ runs over the closed gallery classes containing $c$.
For given a class $\mathcal{C}$ of length $n$, the weight of $\mathcal{C}$ appears $\ell(\mathcal{C}_0)$ times in the above sum. Thus we have the proof of Lemma \ref{lem:4.1}
\end{proof}
\begin{prop}\label{prop:4.2} The type 1 chamber zeta function is formally described by
$$Z_\Gamma(u)=\exp\biggr(\sum_{n=1}^\infty \frac{\operatorname{Tr}(T^n)}{n}u^n\biggr).$$
\end{prop}
\begin{proof}Given a closed gallery class $\mathcal{C}$, let $\mathcal{C}_0$ be the primitive class of $\mathcal{C}$. As in \cite{DK18} and \cite{HK3}, we have
\begin{equation*}
\begin{split}
Z_{\Gamma}(u)^{-1}=&\,\prod_{\mathcal{C}_0}\bigr(1-w(\mathcal{C}_0)u^{\ell(\mathcal{C}_0)}\bigr)=\exp\left(\sum_{\mathcal{C}_0}\log(1-w(\mathcal{C}_0)u^{\ell(\mathcal{C}_0)})\right)\\
=&\,\exp\left(-\sum_{\mathcal{C}_0}\sum_{n=1}^{\infty}\frac{w(\mathcal{C}_0)^nu^{n\ell(\mathcal{C}_0)}}{n}\right)=\exp\left(-\sum_{\mathcal{C}}\frac{w(\mathcal{C})u^{\ell(\mathcal{C})}}{\ell(\mathcal{C})}\ell(\mathcal{C}_0)\right)\\=&\,\exp\left(-\sum_{n=1}^{\infty}\frac{u^{n}}{n}\sum_{\mathcal{C}\colon \ell(\mathcal{C})=n}w(\mathcal{C})\ell(\mathcal{C}_0)\right)=\exp\biggr(-\sum_{n=1}^\infty \frac{\operatorname{Tr}(T^n)}{n}u^n\biggr).
\end{split}
\end{equation*}
This proves Proposition \ref{prop:4.2}.
\end{proof}
Following the proof of Lemma 4.2 in \cite{DK18} and Proposition 4.3 in \cite{HK3}, we have the following proposition.
\begin{prop}For sufficiently small $u$, the type 1 chamber zeta function satisfies
$$Z_\Gamma(u)=\frac{1}{\det(I-uT)}.$$
\end{prop}

\begin{coro}\label{coro:rationality}
For $\Gamma=\mathrm{PGL}_3(\mathbb{F}_q[t])$ acting on the Bruhat--Tits building $\mathcal{B}$ of $G=\mathrm{PGL}_3(\mathbb{F}_q(\!(t^{-1})\!))$, the type~1 chamber zeta function
\[
Z_{\Gamma}(u)
=
\prod_{\mathcal{C}}
\bigl(1-w(\mathcal{C})u^{\ell(\mathcal{C})}\bigr)^{-1}
\]
converges for $|u|$ sufficiently small and extends to a rational function in $u$. In particular, $Z_{\Gamma}(u)$ admits meromorphic continuation to the projective line $\mathbb{P}^1(\mathbb{C})$, with possible poles contained in the spectrum of the chamber transfer operator $T$.
\end{coro}

\subsection{The determinant of $I-uT$} For any $t\geq 2$, we obtain
$$a_{k,(1,t)}(u)=q^3u^4a_{k,(1,t-1)}(u)=q^6u^8a_{k,(1,t-2)}(u)=\cdots=q^{3t-3}u^{4t-4}a_{k,(1,1)}(u).$$
Using this, we have
\begin{equation*}
\begin{split}
a_{k,(1,1)}(u)&=-(q-1)u-q^2(q-1)u^4+a_{k,(1,1)}\sum_{i=1}^k q^{3i}(q-1)u^{6i}
\end{split}
\end{equation*}
and
\begin{equation*}
a_{k,(1,1)}(u)=\frac{-(q-1)u-q^2(q-1)u^4}{1-\displaystyle\sum_{i=1}^kq^{3i}(q-1)u^{6i}}.
\end{equation*}
The sequence $a_{k,(1,1)}$ converges to
$$a_{(1,1)}=\frac{-(1-q^3u^6)((q-1)u+q^2(q-1)u^4)}{1-q^4u^6}.$$
For any $s,t,$ with $s\geq 2$ and $t< s,$
$$a_{k,(s,t)}(u)=q^3u^4a_{k,(s,t-1)}(u)=q^6u^8a_{k,(1,t-2)}(u)=\cdots=q^{3t-3}u^{4t-4}a_{k,(s,1)}(u).$$
For any $s,t$ with $s\geq 2$ and $t\geq s$,
\begin{equation*}
\begin{split}
a_{k,(s,t)}(u)&=q^3u^4a_{k,(s,t-1)}(u)=\cdots=q^{3(t-s)}u^{4(t-s)}a_{k,(s,s)}(u)\\
&=-q^{3(t-s)}(q-1)u^{4(t-s)+1}+q^{3(t-s+1)}u^{4(t-s+1)}a_{k,(s,s-1)}(u)\\
&=-q^{3(t-s)}(q-1)u^{4(t-s)+1}+q^{3(t-1)}u^{4(t-1)}a_{k,(s,1)}(u).
\end{split}
\end{equation*}
Similarly,
\begin{equation*}
\begin{split}
a_{k,(s,1)}=&-q^{2}(q-1)u^{2s+2}+\displaystyle\sum_{i=1}^k a_{k,(s,i)}q^{3}(q-1)u^{2i+4}\\
=&-q^{2}(q-1)u^{2s+2}+\displaystyle\sum_{i=1}^k a_{k,(s,1)}q^{3i}(q-1)u^{6i}\\
&-\sum_{s}^kq^{3(i-s+1)}(q-1)^2u^{6i-4s+5}
\end{split}
\end{equation*}
and
\begin{equation*}
a_{k,(s,1)}=\frac{-q^{2}(q-1)u^{2s+2}-\displaystyle\sum_{s}^kq^{3(i-s+1)}(q-1)u^{6i-4s+5}}{1-\displaystyle\sum_{i=1}^k q^{3i}(q-1)^2u^{6i}}.
\end{equation*}
The sequence $a_{k,(s,1)}$ converges to
\begin{equation*}
a_{(s,1)}=\frac{-q^2(q-1)u^{2s+2}(1-q^3u^6)-q^3(q-1)^2u^{2s+5}}{1-q^4u^6}.
\end{equation*}
Using the limit of $\det(A_{k,0})$, the determinant of $I-uT$ is
\begin{equation*}
\lim_{k\rightarrow \infty}\det(A_{k,0})=\frac{(1-q^3u^6)(1-q^3u^3)}{(1-q^4u^6)(1-q^2u^3)}.
\end{equation*}
By determinant formula, we obtain the following theorem:
\begin{thm}\label{thm:5.1} Let $\Gamma=\PGL_3(\mathbb{F}_q[t])$. The type 1 chamber zeta function $Z_\Gamma(u)$ converges for sufficiently small $u$ and it is given by
$$Z_{\Gamma}(u)=\frac{(1-q^4u^6)(1-q^2u^3)}{(1-q^3u^6)(1-q^3u^3)}.$$
\end{thm}
Recall that we defined $$N_n(\Gamma\backslash\mathcal{B})=\sum_{c\colon\ell(\mathcal{C})=n}w(\mathcal{C})\ell(\mathcal{C}_0)$$ by the weighted number of type $1$ closed galleries in $\Gamma\backslash \mathcal{B}$ of length $n$.
The trace of the operator $T^m$ coincides with $N_m(\Gamma\backslash\mathcal{B})$. By Proposition \ref{prop:4.2}, we have
\begin{equation}\label{eq:5.1}
u\frac{d}{du}\log Z_{\Gamma}(u)=u\frac{Z'_{\Gamma}(u)}{Z_{\Gamma}(u)}=\sum_{m=1}^{\infty} N_m(\Gamma\backslash \mathcal{B})u^m.
\end{equation}
\begin{coro} Let $\Gamma=\operatorname{PGL}(3,\mathbb{F}_q[t])$ and $N_m(\Gamma\backslash\mathcal{B})$ as above. Then, we have
$$N_m(\Gamma\backslash\mathcal{B})=\left\{\begin{array}{ll}3q^{3r}-3q^{2r} & \textrm{ if }m=3r\textrm{ and }m\not\equiv 0 \,(\operatorname{mod}\,{6}) \\
3q^{6r}-9q^{4r}+6q^{3r} & \textrm{ if }m=6r \\ 0
& \textrm{ otherwise}\end{array}\right..$$
\end{coro}
\begin{proof}
By Theorem~5.1, we have
\[
Z_\Gamma(u)
=
\frac{(1-q^{4}u^{6})(1-q^{2}u^{3})}{(1-q^{3}u^{6})(1-q^{3}u^{3})}.
\]
Recall from \eqref{eq:5.1} that
\[
u\frac{d}{du}\log Z_\Gamma(u)
=
\sum_{m\ge1} N_m(\Gamma\backslash B)\,u^m.
\]
Taking the logarithmic derivative of $Z_\Gamma(u)$, we obtain
\begin{align*}
u\frac{d}{du}\log Z_\Gamma(u)
&=
u\frac{d}{du}\Bigl(
\log(1-q^{4}u^{6})
+
\log(1-q^{2}u^{3})
-
\log(1-q^{3}u^{6})
-
\log(1-q^{3}u^{3})
\Bigr) \\
&=
-\frac{6q^{4}u^{6}}{1-q^{4}u^{6}}
-\frac{3q^{2}u^{3}}{1-q^{2}u^{3}}
+\frac{6q^{3}u^{6}}{1-q^{3}u^{6}}
+\frac{3q^{3}u^{3}}{1-q^{3}u^{3}}.
\end{align*}
Equivalently,
\begin{equation}\label{eq:Nm-generating}
\sum_{m\ge1} N_m(\Gamma\backslash B)\,u^m
=
\frac{3q^{3}u^{3}}{1-q^{3}u^{3}}
-
\frac{3q^{2}u^{3}}{1-q^{2}u^{3}}
+
\frac{6q^{3}u^{6}}{1-q^{3}u^{6}}
-
\frac{6q^{4}u^{6}}{1-q^{4}u^{6}}.
\end{equation}
Each term on the right-hand side admits a geometric series expansion:
\[
\frac{3q^{3}u^{3}}{1-q^{3}u^{3}}
=
3\sum_{r\ge1} q^{3r}u^{3r},
\qquad
\frac{3q^{2}u^{3}}{1-q^{2}u^{3}}
=
3\sum_{r\ge1} q^{2r}u^{3r},
\]
\[
\frac{6q^{3}u^{6}}{1-q^{3}u^{6}}
=
6\sum_{r\ge1} q^{3r}u^{6r},
\qquad
\frac{6q^{4}u^{6}}{1-q^{4}u^{6}}
=
6\sum_{r\ge1} q^{4r}u^{6r}.
\]
It follows immediately that $N_m(\Gamma\backslash B)=0$ unless $3\,|\, m$.
If $m=3r$ and $m\not\equiv 0 \pmod{6}$, then only the first two series contribute,
and we obtain
\[
N_{3r}(\Gamma\backslash B)
=
3q^{3r}-3q^{2r}.
\]
If $m=6r$, then all four series in \eqref{eq:Nm-generating} contribute, yielding
\[
N_{6r}(\Gamma\backslash B)
=
\bigl(3q^{6r}-3q^{4r}\bigr)
+
\bigl(6q^{3r}-6q^{4r}\bigr)
=
3q^{6r}+6q^{3r}-9q^{4r}.
\]
This completes the proof.
\end{proof}

\begin{proof}[Idea of Proof] Given a class $\mathcal{C}$, we obtain that $\mathcal{C}=\mathcal{C}_0^n$ for some primitive class $\mathcal{C}_0$. The type 1 chamber zeta function formally satisfies
\begin{align*}
Z_{\Gamma}(u)^{-1}=\exp\left(-\sum_{n=1}^{\infty}\frac{u^{n}}{n}\sum_{\mathcal{C}\colon \ell(\mathcal{C})=n}w(\mathcal{C})\ell(\mathcal{C}_0)\right).
\end{align*}
Let $\operatorname{C}(\Gamma\backslash \mathcal{B})$ be the set of all type 1 pointed chambers in $\Gamma\backslash \mathcal{B}$ and let $S(\operatorname{C}(\Gamma\backslash \mathcal{B}))$ be the formal vector space defined by
$$S(\operatorname{C}(\Gamma\backslash \mathcal{B})):=\displaystyle\bigoplus_{c\in\operatorname{C}(\Gamma\backslash \mathcal{B})}\mathbb{C}c.$$
The operator $T\colon S(\operatorname{C}(\Gamma\backslash \mathcal{B}))\to S(\operatorname{C}(\Gamma\backslash \mathcal{B}))$ is given by
$$Tc=\sum_{c'}w(c,c')c'$$ where the sum runs over all type 1 pointed chambers $c'$ adjacent to $c$. We will prove the operator $T^n$ is traceable, and the trace is represented by
$$\operatorname{Tr}(T_k^n)=\sum_{c\colon\ell(\mathcal{C})=n}w(\mathcal{C})\ell(\mathcal{C}_0).$$
Thus, we obtain that $Z_{\Gamma,k}(q^{-s})^{-1}=\exp\bigr(\!\operatorname{Tr}(\log(1-uT))\bigr)$. Fubini trick and truncation by certain horizontal direction allows us to compute this determinant, which enables us to complete the proof.
\end{proof}

\noindent
\begin{remark}[Idea of the traceable argument]
The traceable property relies on a monotonicity phenomenon along the cuspidal directions of the quotient complex. Certain directed steps force the height to strictly increase (or decrease), and once such a step occurs, returning to the starting point requires traversing a path whose length is bounded below by a uniform constant. Consequently, for each fixed length $n$, only finitely many closed admissible gallery classes can occur.
\end{remark}

The trace of the operator $T^n$ coincides with the weighted number
$$N_n(\Gamma\backslash\mathcal{B})=\sum_{c\colon\ell(\mathcal{C})=n}w(\mathcal{C})\ell(\mathcal{C}_0)$$ of type~$1$ closed gallery classes in $\Gamma\backslash \mathcal{B}$ of length $n$.

Hence, we have
$$Z_{\Gamma}(u)=\frac{1}{\det(I-uT)}=\exp\left(\sum_{n=1}^{\infty}\frac{N_n(\Gamma\backslash\mathcal{B})}{n}u^{n}\right)$$ which implies the following corollary. (See Section~\ref{sec:4} for the detail.)
\begin{coro}[Counting closed galleries]\label{coro:cycle_intro} Let $\Gamma$ be the discrete subgroup $\operatorname{PGL}(3,\mathbb{F}_q[t])$ of $\operatorname{PGL}(3,\mathbb{F}_q(\!(t^{-1})\!))$ and $N_n(\Gamma\backslash\mathcal{B})$ be defined as above. Then, we have
$$N_{m}=\left\{\begin{array}{ll}3q^{3r}-3q^{2r} & \textrm{ if }n=3r\textrm{ and }n\not\equiv 0 \,(\operatorname{mod}\,{6}) \\
3q^{6r}-9q^{4r}+3q^{3r} & \textrm{ if }n=6r \\ 0
& \textrm{ otherwise}\end{array}\right..$$\end{coro}

\begin{remark}[Position within the higher-rank zeta program]
The present work fits into a broader ``higher-rank zeta'' program that originates from the Ihara--Bass theory for regular graphs and its extensions to graphs of groups and Bruhat--Tits buildings. Very roughly, one can distinguish two axes:
\begin{itemize}
\item[(i)] the nature of the quotient $\Gamma\backslash X$ (finite/cocompact versus non-uniform of finite volume);
\item[(ii)] the combinatorial level on which the zeta function is defined (vertex/edge versus higher-dimensional cells such as chambers).
\end{itemize}
On the cocompact side, Ihara's original zeta function and its refinements may be viewed as edge zeta functions for finite regular graphs, while the works of Kang--Li and Kang--Li--Wang treat chamber zeta functions for finite quotients of the $\widetilde{A}_2$-building by cocompact lattices in $\mathrm{PGL}_3(F)$ \cite{KL,KLW10,KLW18}. On the non-uniform side, the edge zeta function for the standard non-uniform complex attached to
\[
G=\mathrm{PGL}_3\bigl(\mathbb{F}_q(\!(t^{-1})\!)\bigr),\qquad
\Gamma=\mathrm{PGL}_3(\mathbb{F}_q[t]),
\]
was recently analyzed in detail, providing explicit determinant formulas and rational expressions \cite{HK3}.

The present paper occupies the remaining natural corner of this picture: we define and study the type~1 chamber zeta function for the same standard non-uniform complex $\Gamma\backslash\mathcal{B}$ and show that it admits an Ihara--Bass type determinant formula together with an explicit rational expression. From this point of view, one may regard our result as completing the first explicit higher-rank case in which both edge and chamber zeta functions are understood for a non-uniform arithmetic quotient of a Bruhat--Tits building. This suggests several possible generalizations, for instance to higher rank groups such as $\mathrm{PGL}_d$ with $d\ge4$ or to other non-uniform complexes of groups, where systematic comparisons between edge- and chamber-level zeta functions remain largely unexplored.
\end{remark}

\begin{figure}[ht]
\centering
\begin{tikzpicture}[>=latex, node distance=3.4cm, font=\small]

\node at (-4.8,3.6) {quotient type};
\node[rotate=90] at (-4.95,1.7) {non-uniform};
\node[rotate=90] at (-4.9,-1.8) {finite / cocompact};

\node at (-2.0,-3.6) {edge-level};
\node at (3.7,-3.6) {chamber-level};

\draw[dashed] (0.77,4) -- (0.77,-4);

\draw[dashed] (-5.2,0) -- (6.2,0);

\node[anchor=south] at (-2,3.3) {edge zeta};
\node[anchor=south] at (3.4,3.3) {chamber zeta};

\node[draw, rounded corners, align=center, minimum width=3.2cm, minimum height=1.6cm] (Q1) at (-2,-1.8)
{finite graphs,\\
(regular or irregular)\\
(Ihara, Bass, Hashimoto\\
Sunada, etc.)};

\node[draw, rounded corners, align=center, minimum width=3.2cm, minimum height=1.6cm] (Q2) at (3.5,-1.8)
{$\widetilde{A}_2$-buildings,\\
cocompact lattices in\\
$\mathrm{PGL}_3(F)$, chamber zeta\\
(\cite{KL},\cite{KLW10},\cite{KLW18})};

\node[draw, rounded corners, align=center, minimum width=3.2cm, minimum height=1.6cm] (Q3) at (-2,1.8)
{weighted graphs (\cite{DK18}),\\standard non-uniform\\
complex for $\mathrm{PGL}_3$\\
 (\cite{HK3})
};

\node[draw, rounded corners, thick, align=center, minimum width=3.2cm, minimum height=1.6cm, fill=gray!10] (Q4) at (3.5,1.8)
{standard non-uniform\\
complex for $\mathrm{PGL}_3$,\\
type~1 chamber zeta\\
\textbf{(this paper)}};

\draw[->, thick] (Q3.east) -- (Q4.west);
\draw[->, thick] (Q2.north) -- (Q4.south);

\end{tikzpicture}
\caption{A schematic position of this work among edge and chamber zeta functions for cocompact and non-uniform quotients.}
\end{figure}

This paper is organized as follows. We provide the preliminary notions on Bruhat-Tits building for $\operatorname{PGL}_3$ in Section~\ref{sec:2}.
In Section~\ref{sec:3}, we deal with the fundamental domain of $\Gamma$ and the image of tailless galleries in the quotient space by the standard non-uniform lattice $\Gamma$ of $G$. In Section~\ref{sec:4}, we define the type 1 chamber zeta function for $\Gamma\backslash \mathcal{B}$ and prove that the zeta function satisfies the determinant formula. In Section \ref{sec:5}, we present the proof of Theorem \ref{thm:1.1} and Corollary \ref{coro:cycle_intro}.

Given a class $\mathcal{C}$, one has $\mathcal{C}=\mathcal{C}_0^n$ for some primitive class $\mathcal{C}_0$, and the type 1 chamber zeta function "formally satisfies"
\[
Z_{\Gamma}(u)^{-1}=\exp\left(-\sum_{n=1}^{\infty}\frac{u^{n}}{n}\sum_{\mathcal{C}:\, \ell(\mathcal{C})=n}w(\mathcal{C})\,\ell(\mathcal{C}_0)\right).
\]
Introduce the formal vector space
\[
S(\operatorname{C}(\Gamma\backslash \mathcal{B})):=\bigoplus_{c\in\operatorname{C}(\Gamma\backslash \mathcal{B})}\mathbb{C}c
\]
of type 1 pointed chambers, and define an operator $T$ by
\[
Tc=\sum_{c'} w(c,c')\,c'
\]
summing over adjacent type 1 pointed chambers $c'$. One "will prove the operator $T^n$ is traceable" and that its trace is
\[
\operatorname{Tr}(T_k^n)=\sum_{\mathcal{C}:\,\ell(\mathcal{C})=n} w(\mathcal{C})\,\ell(\mathcal{C}_0).
\]
Hence $Z_{\Gamma,k}(q^{-s})^{-1}=\exp\bigl(\operatorname{Tr}(\log(1-uT))\bigr)$, yielding the determinant expression
\[
Z_{\Gamma}(u)=\frac{1}{\det(I-uT)}=\exp\left(\sum_{n=1}^{\infty}\frac{N_n(\Gamma\backslash\mathcal{B})}{n}u^{n}\right),\qquad
N_n(\Gamma\backslash\mathcal{B})=\sum_{\mathcal{C}:\,\ell(\mathcal{C})=n}w(\mathcal{C})\,\ell(\mathcal{C}_0).
\]
To justify traceability, the "traceable property relies on a monotonicity phenomenon along the cuspidal directions": certain directed steps force a height to strictly increase/decrease, and returning to the start then requires a path of uniformly bounded-below length, so for each fixed $n$ "only finitely many closed admissible gallery classes can occur." Finally, a "Fubini trick and truncation by certain horizontal direction allows us to compute this determinant," which completes the proof of Theorem~\ref{thm:1.1} (producing the explicit rational expression).

Given a class $\mathcal{C}$, one has $\mathcal{C}=\mathcal{C}_0^n$ for some primitive class $\mathcal{C}_0$, and the type 1 chamber zeta function "formally satisfies"
\[
Z_{\Gamma}(u)^{-1}=\exp\left(-\sum_{n=1}^{\infty}\frac{u^{n}}{n}\sum_{\mathcal{C}:\, \ell(\mathcal{C})=n}w(\mathcal{C})\,\ell(\mathcal{C}_0)\right).
\]
Introduce the formal vector space
\[
S(\operatorname{C}(\Gamma\backslash \mathcal{B})):=\bigoplus_{c\in\operatorname{C}(\Gamma\backslash \mathcal{B})}\mathbb{C}c
\]
of type 1 pointed chambers, and define an operator $T$ by
\[
Tc=\sum_{c'} w(c,c')\,c'
\]
summing over adjacent type 1 pointed chambers $c'$. One "will prove the operator $T^n$ is traceable" and that its trace is
\[
\operatorname{Tr}(T_k^n)=\sum_{\mathcal{C}:\,\ell(\mathcal{C})=n} w(\mathcal{C})\,\ell(\mathcal{C}_0).
\]
Hence $Z_{\Gamma,k}(q^{-s})^{-1}=\exp\bigl(\operatorname{Tr}(\log(1-uT))\bigr)$, yielding the determinant expression
\[
Z_{\Gamma}(u)=\frac{1}{\det(I-uT)}=\exp\left(\sum_{n=1}^{\infty}\frac{N_n(\Gamma\backslash\mathcal{B})}{n}u^{n}\right),\qquad
N_n(\Gamma\backslash\mathcal{B})=\sum_{\mathcal{C}:\,\ell(\mathcal{C})=n}w(\mathcal{C})\,\ell(\mathcal{C}_0).
\]
To justify traceability, the "traceable property relies on a monotonicity phenomenon along the cuspidal directions": certain directed steps force a height to strictly increase/decrease, and returning to the start then requires a path of uniformly bounded-below length, so for each fixed $n$ "only finitely many closed admissible gallery classes can occur." Finally, a "Fubini trick and truncation by certain horizontal direction allows us to compute this determinant," which completes the proof of the main theorem (producing the explicit rational expression).

Question:
Let \(\mathcal B\) be the Bruhat--Tits building of \(G=\operatorname{PGL}(3,\mathbb F_q((t^{-1})))\), and let \(\Gamma=\operatorname{PGL}(3,\mathbb F_q[t])\). For the quotient \(\Gamma\backslash \mathcal B\), define the type 1 chamber zeta function by
\[
Z_\Gamma(u)=\prod_{\mathcal C}\left(1-w(\mathcal C)u^{\ell(\mathcal C)}\right)^{-1},
\]
where the product runs over tailless type 1 primitive closed gallery classes \(\mathcal C\), \(\ell(\mathcal C)\) is the gallery length, and for a representative closed gallery \((c_1,\dots,c_n)\), its weight is \(w(\mathcal C)=\prod_j w(c_j,c_{j+1})\), with \(w(c,c')\) the number of lifts of \(c'\) that form a tailless type 1 gallery with a lift of \(c\). Which statement is valid for \(Z_\Gamma(u)\)?

\(Z_\Gamma(u)\) converges for sufficiently small \(u\), and
\[
Z_\Gamma(u)=\frac{(1-q^4u^6)(1-q^2u^3)}{(1-q^3u^6)(1-q^3u^3)}.
\]

Choices:
B.
\(Z_\Gamma(u)\) converges for sufficiently small \(u\), and
\[
Z_\Gamma(u)=\frac{(1-q^3u^6)(1-q^2u^3)}{(1-q^4u^6)(1-q^3u^3)}.
\]
C. \(Z_\Gamma(u)\) converges for sufficiently small \(u\), and \(Z_\Gamma(u)\) is a rational function of \(u\).
D.
\(Z_\Gamma(u)\) converges for sufficiently small \(u\), and
\[
Z_\Gamma(u)=\frac{1}{\det(I-uT)},
\]
where \(T\) is the weighted type 1 chamber transfer operator on type 1 pointed chambers of \(\Gamma\backslash\mathcal B\).
E.
\(Z_\Gamma(u)\) converges for sufficiently small \(u\), and
\[
Z_\Gamma(u)=\frac{(1-q^4u^3)(1-q^2u^6)}{(1-q^3u^3)(1-q^3u^6)}.
\]

explicit determinant evaluation factors placed on wrong sides

dropped the explicit rational formula while keeping convergence and rationality

formal determinant identity for transfer operator treated as the final theorem in place of the computed closed formula

length divisibility structure in exponents 3 and 6 interchanged

The stem does not explicitly reveal the formula or uniquely signal choice A. It names the objects and asks for the explicit statement, but does not leak the answer beyond identifying the theorem's topic.

This is essentially a direct recall of a specific explicit theorem/formula for the zeta function in the stated setting. The task is mainly to recognize the exact closed form rather than derive or compare genuinely different mathematical conclusions.

There is some reasoning pressure because several choices are close variants of the correct formula, and one option is a weaker true statement. However, solving it mainly depends on remembering the exact identity rather than generating a conclusion from underlying principles.

The distractors are plausible and mathematically targeted: they alter exponents, swap numerator/denominator factors, omit numerator terms, or offer a weaker true statement. These align with realistic failure modes in recalling or manipulating the formula.

A technically well-constructed recall-style MCQ with strong distractors, but it is largely theorem-restatement rather than a genuine test of generative mathematical reasoning.