samples/texts/2952501/page_25.md

hence

Therefore, for some constant $C_3 > 0$,

$$\mathbb{P}(L_r=p)\le \frac{C_1}{h(1)}\sqrt{p}\frac{9(r+2)}{((r+2{)}^2-1)((r+2{)}^2-4)}{(1-\frac{3}{(r+2{)}^2-1})}^{p-1}\le \frac{C_3}{r^2}\sqrt{\frac{p}{r^2}}{e}^{-3p\mathrm{/}{r}^2}\mathrm{.}\backslash mathbb\{P\}(L\_r\; =\; p)\; \backslash leq\; \backslash frac\{C\_1\}\{h(1)\}\; \backslash sqrt\{p\}\; \backslash frac\{9(r+2)\}\{((r+2)^2-1)((r+2)^2-4)\}\; \backslash left(1\; -\; \backslash frac\{3\}\{(r+2)^2-1\}\backslash right)^\{p-1\}\; \backslash leq\; \backslash frac\{C\_3\}\{r^2\}\; \backslash sqrt\{\backslash frac\{p\}\{r^2\}\}\; e^\{-3p/r^2\}.$$

The bound (5.16) immediately follows. As for (5.17), since the function $x \mapsto \sqrt{x}e^{-3x}$ is decreasing for $x \ge 1/6$, we have, for $\alpha \ge 1/6$, for some constant $C_4 > 0$,

$$\mathbb{P}(L_r>\alpha r^2)\le \sum _{p=\alpha r^2+1}^{\mathrm{\infty }}\frac{C_3}{r^2}\sqrt{\frac{p}{r^2}}{e}^{-3p\mathrm{/}{r}^2}\le \frac{C_3}{r^2}{\int }_{\alpha r^2}^{\mathrm{\infty }}\sqrt{\frac{x}{r^2}}{e}^{-3x\mathrm{/}{r}^2}dx\le C_4{e}^{-\alpha \mathrm{/}4}\mathrm{.}\mathrm{\square }\backslash mathbb\{P\}(L\_r\; >\; \backslash alpha\; r^2)\; \backslash leq\; \backslash sum\_\{p=\backslash alpha\; r^2+1\}^\{\backslash infty\}\; \backslash frac\{C\_3\}\{r^2\}\; \backslash sqrt\{\backslash frac\{p\}\{r^2\}\}\; e^\{-3p/r^2\}\; \backslash leq\; \backslash frac\{C\_3\}\{r^2\}\; \backslash int\_\{\backslash alpha\; r^2\}^\{\backslash infty\}\; \backslash sqrt\{\backslash frac\{x\}\{r^2\}\}\; e^\{-3x/r^2\}\; dx\; \backslash leq\; C\_4\; e^\{-\backslash alpha/4\}.\; \backslash quad\; \backslash square$$

We now fix a positive constant $a \in (0, 1)$. For every integer $r \ge 1$, let $N_r^{(a)}$ be uniformly random in ${|ar^2|+1, \dots, [a^{-1}r^2]}$. We also consider a sequence $\tau_1, \tau_2, \dots$ of independent Galton-Watson trees with offspring distribution $\theta$, independent of $N_r^{(a)}$. For every integer $j \ge 0$, we write $[\tau_i]_j$ for the tree $\tau_i$ truncated at generation $j$.

Using the same arguments that yield Proposition 5 from Lemma 4 in [13], the above lemma implies the following bound:

Proposition 5.7. There exists a constant $C_1$, which only depends on $a$, such that, for every sufficiently large integer $r$, for every choice of $s \in {r+1, r+2, \dots}$, for every choice of integers $p$ and $q$ with $ar^2 < p, q \le a^{-1}r^2$, for every forest $\mathcal{F} \in F''_{p,q,s-r}$,

$$\mathbb{P}({\tilde{\mathcal{F}}}_{r,s}^{(1)}=\mathcal{F})\le C_1\mathbb{P}(([{\tau }_1{]}_{s-r},\dots ,[{\tau }_{N_r^{(a)}}{]}_{s-r})=\mathcal{F})\mathrm{.}(5.19)\backslash mathbb\{P\}(\backslash tilde\{\backslash mathcal\{F\}\}\_\{r,s\}^\{(1)\}\; =\; \backslash mathcal\{F\})\; \backslash le\; C\_1\; \backslash mathbb\{P\}(([\backslash tau\_1]\_\{s-r\},\; \backslash dots,\; [\backslash tau\_\{N\_r^\{(a)\}\}]\_\{s-r\})\; =\; \backslash mathcal\{F\}).\; \backslash quad\; (5.19)$$

5.3 Leftmost mirror geodesics

We now define a type of paths in Eulerian cylinder triangulations that will be useful in the sequel.

Let $\Delta$ be an Eulerian cylinder triangulation of height $r \ge 1$. Let $x$ be a type-$j$ vertex of $\partial_j \Delta$, with $1 \le j \le r$. We define the leftmost mirror geodesic from $x$ to the bottom cycle in the following way. Enumerate in clockwise order around $x$ all the half-edges incident to it, starting from the half-edge of $\partial_j \Delta$ that is to the right of $x$. The first edge on the leftmost mirror geodesic starting from $x$ is the last edge connecting $x$ to $\partial_{j-1} \Delta$ arising in this order. The path is then continued by induction (see Figure 14). Note that, taken in the reverse order, this path is an oriented geodesic, hence the name mirror geodesic. (Such a precision is not necessary in [13], that deals with proper, symmetric distances.)

The coalescence of leftmost geodesics from distinct vertices can be characterized by the skeleton of $\Delta$. Indeed, let $u, v$ be two distinct type-$r$ vertices of $\partial^\Delta$. Let $\mathcal{F}$ be the skeleton of $\Delta$, $\mathcal{F}'$ the subforest of $\mathcal{F}$ consisting of the trees rooted between $u$ and $v$ left-to-right in $\partial^\Delta$, and $\mathcal{F}''$ be the rest of the trees in $\mathcal{F}$. Then, for any $k \in {1, 2, \dots, r}$, the leftmost mirror geodesics from $u$ and $v$ merge before step $k$ (possibly exactly at step $k$) if and only if at least one of the two forests $\mathcal{F}'$ and $\mathcal{F}''$ have height strictly smaller than $k$.