samples/texts/2952501/page_37.md

a distinguished vertex $o_n$. The hull $B_r^\bullet(\bar{T}_n^{(1)})$ is well-defined when $\vec{d}(\rho_n, o_n) > r + 1$, otherwise we set it to be $\bar{T}_n^{(1)}$.

Lemma 8.5. There exists a constant $\bar{c} > 0$ such that, for every $n,r,p \ge 1$ and every $\Delta \in C_{1,r}$ with top boundary half-length $p$, such that $n > N(\Delta) + p$,

$$P(B_r^{\bullet }({\bar{T}}_n^{(1)})=\mathrm{\Delta })\le \bar{c}{\left(\frac{n}{n-N(\mathrm{\Delta })+1}\right)}^{3\mathrm{/}2}\cdot P(B_r^{\bullet }({\mathcal{T}}_{\mathrm{\infty }}^{(1)})=\mathrm{\Delta })\mathrm{.}(8.1)P(B\_r^\backslash bullet(\backslash bar\{T\}\_n^\{(1)\})\; =\; \backslash Delta)\; \backslash le\; \backslash bar\{c\}\; \backslash left(\; \backslash frac\{n\}\{n\; -\; N(\backslash Delta)\; +\; 1\}\; \backslash right)^\{3/2\}\; \backslash cdot\; P(B\_r^\backslash bullet(\backslash mathcal\{T\}\_\backslash infty^\{(1)\})\; =\; \backslash Delta).\; \backslash quad\; (8.1)$$

Proof. The proof of this lemma is very similar to that of [13, Lemma 22], with the additional subtlety that, like for Lemma 5.3, we do not start with explicit expressions for probabilities in finite triangulations, as shown in (8.3).

Fix $r \ge 1$ and $\Delta \in C_{1,r}$ with top boundary half-length $p$. We will write $V$ for $#V(\Delta)$ to simplify notation. Using (5.4) and the fact that $\mathcal{T}_\infty^{(1)}$ is the local limit of $\mathcal{T}_n^{(1)}$, we have

$$P(B_r^{\bullet }({\mathcal{T}}_{\mathrm{\infty }}^{(1)})=\mathrm{\Delta })=\frac{C(p)}{C(1)}{8}^{-N(\mathrm{\Delta })}\mathrm{.}(8.2)P(B\_r^\{\backslash bullet\}(\backslash mathcal\{T\}\_\{\backslash infty\}^\{(1)\})\; =\; \backslash Delta)\; =\; \backslash frac\{C(p)\}\{C(1)\}\; 8^\{-N(\backslash Delta)\}.\; \backslash qquad\; (8.2)$$

On the other hand, (5.3) gives the formula

$$P(B_r^{\bullet }({\bar{T}}_n^{(1)})=\mathrm{\Delta })=\frac{B_{n-N,p}}{B_{n,1}}\cdot \frac{\mathrm{\#}\text{inner vertices in }{\mathcal{T}}_n^{(1)}\setminus \mathrm{\Delta }}{\mathrm{\#}\text{inner vertices in }{\mathcal{T}}_n^{(1)}}(8.3)P\; \backslash left(\; B\_r^\backslash bullet\; (\backslash bar\{T\}\_n^\{(1)\})\; =\; \backslash Delta\; \backslash right)\; =\; \backslash frac\{B\_\{n-N,p\}\}\{B\_\{n,1\}\}\; \backslash cdot\; \backslash frac\{\backslash \#\backslash text\{inner\; vertices\; in\; \}\; \backslash mathcal\{T\}\_n^\{(1)\}\; \backslash setminus\; \backslash Delta\}\{\backslash \#\backslash text\{inner\; vertices\; in\; \}\; \backslash mathcal\{T\}\_n^\{(1)\}\}\; \backslash quad\; (8.3)$$

$$\le \frac{B_{n-N,p}}{B_{n,1}}\cdot \frac{n-V}{n},(8.4)\backslash leq\; \backslash frac\{B\_\{n-N,p\}\}\{B\_\{n,1\}\}\; \backslash cdot\; \backslash frac\{n-V\}\{n\},\; \backslash quad\; (8.4)$$

where the last inequality is given by Euler's formula and the fact that at most $p$ vertices of $\partial^*\Delta$ are identified together in $\mathcal{T}_n^{(1)}$. (We still need $n > N + p$ since $\mathcal{T}_n^{(1)} \setminus \Delta$ will have $n - N - p$ inner vertices if none of these identifications occur.)

Then, using the bounds of (4.5) and the asymptotics of (4.3), we get that

$$P(B_r^{\bullet }({\bar{\mathcal{T}}}_n^{(1)})=\mathrm{\Delta })\le c^{\ast }C(p){\left(\frac{n}{n-N}\right)}^{3\mathrm{/}2}{8}^{-N}P(B\_r^\{\backslash bullet\}(\backslash overline\{\backslash mathcal\{T\}\}\_n^\{(1)\})\; =\; \backslash Delta)\; \backslash le\; c^*\; C(p)\; \backslash left(\backslash frac\{n\}\{n-N\}\backslash right)^\{3/2\}\; 8^\{-N\}$$

for some constant $c^*$. Comparing the last bound with (8.2) gives the desired result. $\square$

Proof of Proposition 8.4. Fix $\varepsilon > 0$ and $\nu > 0$. Its suffices to prove that, for all $n$ sufficiently large, we have

Indeed, as detailed in Proposition 4.1, the sequence $n^{-1/4}\vec{d}(\rho_n, o_n)$ is bounded in probability, so that the statement of the proposition will follow from (8.5). Note that, in [13], the equivalent tightness is obtained as a consequence of the convergence of usual planar triangulations to the Brownian map: in our case, we had to use the weaker result of Theorem 2.12, as we obviously do not have a convergence at the level of maps yet.

To obtain (8.5), we want to transfer the results of Proposition 8.3 on the UIPET to large finite triangulations. This necessitates the bounds of Lemma 8.5, together with the statement of Proposition 4.2 on the profile of distances in large finite triangulations (which is once again a consequence of Theorem 2.12, as we cannot rely on a convergence to the Brownian map).

We omit the details of the proof of (8.5), as they can be straightforwardly adapted from the equivalent statement in the proof of Proposition 21 in [13], replacing once again $d_{gr}$ by $\vec{d}$, and $d_{fpp}$ by $d$. $\square$