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3 Convergence to the Brownian map

We will now state and give the proof of the main result of this paper.

Before doing so, let us recall the construction of the Brownian map, and introduce some notation. As in Section 2.3, we write $\mathfrak{e}$ for a standard Brownian excursion, and $Z$ for the "head" of the Brownian snake driven by $\mathfrak{e}$, i.e., conditionally on $\mathfrak{e}$, $Z$ a continuous, centered Gaussian process on $[0, 1]$ with covariance

$$\mathrm{C}\mathrm{o}\mathrm{v}(Z_s,Z_t)={\mathfrak{e}}_{s,t},s,t\in [0,1]\mathrm{.}\backslash mathrm\{Cov\}(Z\_s,\; Z\_t)\; =\; \backslash mathfrak\{e\}\_\{s,t\},\; \backslash quad\; s,\; t\; \backslash in\; [0,\; 1].$$

The Brownian excursion $\mathfrak{e}$ encodes the Continuum Random Tree $(\mathcal{T}{\mathfrak{e}}, d{\mathfrak{e}})$, defined by:

$$\begin{array}{c}{\displaystyle d_{\mathfrak{e}}(s,t)=\mathfrak{e}(s)+\mathfrak{e}(t)-2{\mathfrak{e}}_{s,t}}\\ {\displaystyle {\mathcal{T}}_{\mathfrak{e}}=[0,1]\mathrm{/}\{d_{\mathfrak{e}}=0\}\mathrm{.}}\end{array}\backslash begin\{gathered\}\; d\_\{\backslash mathfrak\{e\}\}(s,t)\; =\; \backslash mathfrak\{e\}(s)\; +\; \backslash mathfrak\{e\}(t)\; -\; 2\backslash mathfrak\{e\}\_\{s,t\}\; \backslash \backslash \; \backslash mathcal\{T\}\_\{\backslash mathfrak\{e\}\}\; =\; [0,\; 1]/\backslash \{d\_\{\backslash mathfrak\{e\}\}\; =\; 0\backslash \}.\; \backslash end\{gathered\}$$

The function $d_{\mathfrak{e}}$, which is a pseudo-distance on $[0, 1]$, induces a true distance on $\mathcal{T}{\mathfrak{e}}$ via the canonical projection $p{\mathfrak{e}}: [0, 1] \to \mathcal{T}{\mathfrak{e}}$, to $\mathcal{T}{\mathfrak{e}}$.

Almost surely, there is a unique $s \in [0, 1]$ such that $Z_s = \inf Z$ [22]. We then denote this point by $s_*$, and $x_* = p_{\mathfrak{e}}(s_*)$ its projection on $\mathcal{T}_{\mathfrak{e}}$.

We define, for $s \le t \in [0, 1]$,

$$D^{\circ }(s,t)=D^{\circ }(t,s):=Z_s+Z_t-2\max (\underset{r\in [s,t]}{\min }{Z}_r,\underset{r\in [t,1]\cap [0,s]}{\min }{Z}_r)\mathrm{.}D^\{\backslash circ\}(s,t)\; =\; D^\{\backslash circ\}(t,s)\; :=\; Z\_s\; +\; Z\_t\; -\; 2\; \backslash max(\; \backslash min\_\{r\; \backslash in\; [s,t]\}\; Z\_r,\; \backslash min\_\{r\; \backslash in\; [t,1]\; \backslash cap\; [0,s]\}\; Z\_r\; ).$$

This function does not satisfy the triangle inequality, which leads us to introduce

$$D^{\ast }(s,t):=\inf \{\sum _{i=1}^k{D}^{\circ }(s_i,t_i)\mid k\ge 1,s_1=s,t_k=t,d_{\mathfrak{e}}(t_i,s_{i+1})=0\mathrm{\forall }i\in \{1,2,\dots ,k\}\}\mathrm{.}D^*(s,t)\; :=\; \backslash inf\; \backslash left\backslash \{\; \backslash sum\_\{i=1\}^k\; D^\{\backslash circ\}(s\_i,\; t\_i)\; \backslash mid\; k\; \backslash ge\; 1,\; s\_1\; =\; s,\; t\_k\; =\; t,\; d\_\{\backslash mathfrak\{e\}\}(t\_i,\; s\_\{i+1\})\; =\; 0\; \backslash quad\; \backslash forall\; i\; \backslash in\; \backslash \{1,\; 2,\; \backslash dots,\; k\backslash \}\; \backslash right\backslash \}.$$

We can now define the Brownian map, by setting $m_{\infty} = [0, 1]/{D^* = 0}$, and equipping this space with the distance induced by $D^*$, which we still denote by $D^*$.

Let $\mathcal{T}_n$ be a uniform random rooted Eulerian planar triangulation with $n$ black faces, equipped with its usual graph distance $d_n$, and its oriented pseudo-distance $\vec{d}_n$. Let $\overline{\mathcal{T}}_n$ be the triangulation $\mathcal{T}_n$ together with a distinguished vertex $o_n$ picked uniformly at random. Recall from Section 2.2 that $\overline{\mathcal{T}}_n$ is the image, by the BDG bijection, of a random labeled tree $\mathcal{T}_n$, uniformly distributed over the set of well-labeled rooted plane trees with $n$ edges. We denote by $l_n$ the labels of the vertices of $\mathcal{T}_n$, and enumerate as in Section 2.2 the vertices (or rather, the corners) of $\mathcal{T}_n$, by setting $u_i^{(n)}$ to be the $i$-th vertex visited by the contour process of $\mathcal{T}n$, for $0 \le i \le 2n$. As before, we denote by $L{(n)}$ the rescaled labels of the vertices of $\overline{\mathcal{T}}_n$.

We define the symmetrization $\overleftrightarrow{d_n}$ of $\vec{d}_n$, by

$$\overleftrightarrow{d_n}(u,v)=\frac{{\vec{d}}_n(u,v)+{\vec{d}}_n(v,u)}{2}\mathrm{.}\backslash overleftrightarrow\{d\_n\}(u,v)\; =\; \backslash frac\{\backslash vec\{d\}\_n(u,v)\; +\; \backslash vec\{d\}\_n(v,u)\}\{2\}.$$

We also define a rescaled oriented distance $\tilde{D}_{(n)}$ on $[0, 1]^2$, by first setting, for $i, j \in {0, 1, \dots, 2n}$:

$${\tilde{D}}_{(n)}(\frac{i}{n},\frac{j}{n})=\frac{{\vec{d}}_n(u_i^{(n)},u_j^{(n)})}{n^{1\mathrm{/}4}},\backslash tilde\{D\}\_\{(n)\}\; \backslash left(\; \backslash frac\{i\}\{n\},\; \backslash frac\{j\}\{n\}\; \backslash right)\; =\; \backslash frac\{\backslash vec\{d\}\_n(u\_i^\{(n)\},\; u\_j^\{(n)\})\}\{n^\{1/4\}\},$$

then linearly interpolating to extend $\tilde{D}_{(n)}$ to $[0, 1]^2$.

We define similarly $D_{(n)}$ from $d_n$, as well as $\overleftrightarrow{D}_{(n)}$ from $\overleftrightarrow{d}_n$.