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\title{Metric Geometry}
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\section{Introduction to the three-point function}

\noindent \textbf{The mutli-pointed bijection}. Let $(Q,s_1,s_2,s_3)$ be a planar quadrangulation with $3$ distinct distinguished vertices, we call these vertices the \emph{sources} of the quadrangulation. We also impose that the graph distance in $Q$ between any of the sources is larger than $2$ (that is no sources are neighbors) and that the sources are not aligned (none of the sources lies on a geodesic path between the two others). The object $(Q,s_1,s_2,s_3)$ is then called \emph{a triply pointed quadrangulation}. Together with the sources we are also given integer \emph{delays}, $\tau_1, \tau _2$ and $\tau_3$. We impose that the delays satisfy 
$$  (\star) \left \{\begin{array}{l}\displaystyle |\tau_i-\tau_j| < \op{d}_{\op{gr}}(s_i,s_j), \ i,j \in \{1,2,3\}\\
\displaystyle \tau_i -\tau_j + \op{d}_{\op{gr}}(s_i,s_j) \mbox{ is even }, \ i,j\in \{1,2,3\}.
\end{array}\right.$$
\begin{enumerate}
\item Show that we can always find delays satisfying $(\star)$. 
\end{enumerate}
We associate a labeling $\ell$ to the vertices of the map $Q$ by putting 
$$ \ell(v) = \min_{i \in \{1,2,3\}}\big( \tau_i + \op{d_{gr}}(v,s_i)\big).$$
\begin{enumerate}
\setcounter{enumi}{1}
\item Show that the label of any source is equal to its delays and that two neighboring vertices in the map $Q$ have labels that differ by $1$ or $-1$.
\end{enumerate}
Hence the faces of the quadrangulation $Q$ can be decomposed into two subsets: The faces such that the labels  of the vertices are $(i,i+1,i+2,i+1)$ or those satisfying $(i,i+1,i,i+1)$. We add a ``red'' edge in each face following the rule given by the figure below.
\begin{figure}[h]
\begin{center} 
\includegraphics[height=3cm]{schaeffer} \end{center}
\end{figure}
\begin{enumerate}
\setcounter{enumi}{2}
\item Apply this construction to quadrangulation of Fig.\,1. (taken from Bouttier and Guitter).
\item Verify that you are left with a labeled map with $3$ faces such that for any $i \in\{1,2,3\}$
$$ \min_{v \mbox{ \small incident to face }i} \ell(v) = \tau_i +1.$$
Verify also that the labels can vary by $-1,0$ or $1$ along an edge, we say that the map is \emph{well-labeled}. Like in the classical Schaeffer construction, check that if a vertex $v$ is incident to a face $i$ for $i\in \{1,2,3\}$ then 
$$ \op{d_{gr}}(v,s_i) = \ell(v)-\tau_i.$$ Given the well-labeled map with three faces, how can you reconstruct the original quadrangulation? (No proof)
\end{enumerate}
\noindent \textbf{Three-point function}.
\begin{enumerate}
\setcounter{enumi}{4}
\item Can you give the pairwise distances between the sources in the quadrangulation associated to the well-labeled planar map of Fig.\,2 without reconstructing the whole quadrangulation? 
\end{enumerate} Let $(Q,s_1,s_2,s_3)$ be a triply-pointed quadrangulation. Now we choose particular values for the delays associated to the sources of a triply-pointed quadrangulation. We let $\tau_1, \tau_2$ and $\tau_3$ such that 
$$ (\clubsuit)\left\{ \begin{array}{l} \tau_1+ \tau_2 = -\op{d_{gr}}(s_1,s_2), \\ \tau_1+ \tau_3 = -\op{d_{gr}}(s_1,s_3), \\ \tau_2+ \tau_3 = -\op{d_{gr}}(s_2,s_3). \end{array}\right.$$
\begin{enumerate}
\setcounter{enumi}{5}
\item Show that these labels satisfy $(\star)$.
\item Show that the map with three faces obtained with Miermont's construction is such that any two faces have a non-empty boundary and satisfy
$$ (\heartsuit)\left\{\begin{array}{cc}
\min \big\{\ell(v): {v \mbox{ incident to face }1}\big\} = 1+\tau_1, & \min\big\{\ell(v) : v \mbox{ incident to faces } 1 \mbox{ and }2 \big\} = 0 \\
\min \big\{\ell(v): {v \mbox{ incident to face }2}\big\} = 1+\tau_2, & \min\big\{\ell(v) : v\mbox{ incident to faces } 1 \mbox{ and }3 \big\} = 0\\
\min \big\{\ell(v): {v \mbox{ incident to face }3}\big\} = 1+\tau_3, & \ \min\big\{\ell(v) : v\mbox{ incident to faces } 2 \mbox{ and }3 \big\} = 0.
\end{array} \right.$$
\end{enumerate}

\begin{theorem} The construction presented above is a bijection between on the one hand triply-pointed quadrangulations with $n$ faces such that the sources $s_1, s_2 $ and $s_3$ are not aligned and on the other hand well-labeled planar maps with $3$ faces with $n$ edges satisfying $(\heartsuit)$ for the delays related to the distances between the sources by $(\clubsuit)$. \end{theorem}

\begin{enumerate} 
\setcounter{enumi}{7}
\item Let $(Q_n,s_1,s_2,s_3)$ a triply-pointed quadrangulation with $n$ faces. How \emph{would you} show that the triplet 
$$ n^{-1/4} \big( \op{d_{gr}}(s_1,s_2), \op{d_{gr}}(s_1,s_3), \op{d_{gr}}(s_2,s_3)\big),$$
converge in distribution as $n\to \infty$?
\end{enumerate}

\begin{exo} Who are these charming gentlemen ? \begin{figure}[h] \begin{center} \includegraphics[height=4cm]{bouttier} \hspace{1cm}
\includegraphics[height=4cm]{guitter}\hspace{1cm}
\end{center} \end{figure}\end{exo}
\nocite{Mie09}
\nocite{BG08}
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\clearpage
\begin{figure}
\begin{center}
Fig.\,1. \includegraphics[height=12cm]{miermont}
\end{center}
\end{figure} \begin{figure}
\begin{center}
Fig.\,2. \includegraphics[height=11cm]{troisfaces}
\end{center}
\end{figure}


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