samples/texts/2952501/page_32.md

Convergence of Eulerian triangulations

If it does not leave this layer, then its length is bounded below by

$$\sum _{i=1}^{N_r(2m,lm)}\mathrm{D}\mathrm{i}\mathrm{a}\mathrm{m}({\mathcal{G}}_{2m}(i,lm))\mathrm{.}\backslash sum\_\{i=1\}^\{N\_r(2m,lm)\}\; \backslash mathrm\{Diam\}(\backslash mathcal\{G\}\_\{2m\}(i,lm)).$$

Then, from Lemma 7.3, and from the definition of f, we get that, for r/m large enough, for some c' > 0 independent of r, l, m,

so that

$$\mathbb{P}(\mathrm{D}\mathrm{i}\mathrm{a}\mathrm{m}({\mathcal{G}}_r)\le m\mathrm{/}3\wedge c\cdot c^{\mathrm{\prime }}(r\mathrm{/}m{)}^{2}f(2m))\le \frac{1}{4},\backslash mathbb\{P\}(\backslash mathrm\{Diam\}(\backslash mathcal\{G\}\_r)\; \backslash leq\; m/3\; \backslash wedge\; c\; \backslash cdot\; c\text{'}(r/m)^2\; f(2m))\; \backslash leq\; \backslash frac\{1\}\{4\},$$

which implies the desired bound, by the definition of f.

The details of the proof can be adapted from the proof of [14, Proposition 1]. $\square$

Theorem 7.2 is then a purely analytic consequence of Proposition 7.4, and its proof is a straightforward adaptation of that of [14, Theorem 5].

We can now use Theorem 7.2 to obtain the following lower bounds for the distances along the boundary of $\mathcal{L}$:

Proposition 7.5. For every $\varepsilon > 0$, there exists an integer $K > 0$ such that, for every $r \ge 1$,

$$\mathbb{P}(\underset{\mathrm{\mid }j\mathrm{\mid }\ge K{r}^2}{\min }{\vec{d}}_{\mathcal{L}}((0,0),(j,0))\ge r)\ge 1-\epsilon \mathrm{.}\backslash mathbb\{P\}\; \backslash left(\; \backslash min\_\{|j|\; \backslash ge\; K\; r^2\}\; \backslash vec\{d\}\_\{\backslash mathcal\{L\}\}((0,0),\; (j,0))\; \backslash ge\; r\; \backslash right)\; \backslash ge\; 1\; -\; \backslash varepsilon.$$

Consequently, for $K' = 9K$, we also have, for every $r \ge 1$,

$$\mathbb{P}(\underset{\mathrm{\mid }j\mathrm{\mid }\ge 2K^{\mathrm{\prime }}{r}^2}{\min }\underset{-K^{\mathrm{\prime }}{r}^2\le i\le K^{\mathrm{\prime }}{r}^2}{\min }{\vec{d}}_{\mathcal{L}}((i,0),(j,0))\ge r)\ge 1-2\epsilon \mathrm{.}\backslash mathbb\{P\}\; \backslash left(\; \backslash min\_\{|j|\; \backslash ge\; 2K\text{'}r^2\}\; \backslash min\_\{-K\text{'}r^2\; \backslash le\; i\; \backslash le\; K\text{'}r^2\}\; \backslash vec\{d\}\_\{\backslash mathcal\{L\}\}((i,0),\; (j,0))\; \backslash ge\; r\; \backslash right)\; \backslash ge\; 1\; -\; 2\backslash varepsilon.$$

Proof. Let us start with the first assertion. Let $\epsilon > 0$. Fix $r \ge 1$, and $K \ge 1$. Then, from (5.18), the number $N_{(K,r)}$ of trees that reach height $r$ between $(0,0)$ and $(j,0)$ is bounded below by a binomial variable of parameters $(Kr^2, 3/((r+2)^2-1))$, so that, using Chebyshev's inequality, for any $a > 0$,

$$\mathbb{P}(N_{(K,r)}\le \frac{3}{8}K-a)\le \frac{3K}{a^2}\mathrm{.}\backslash mathbb\{P\}\backslash left(N\_\{(K,r)\}\; \backslash le\; \backslash frac\{3\}\{8\}K\; -\; a\backslash right)\; \backslash le\; \backslash frac\{3K\}\{a^2\}.$$

(Note that the binomial variable in question has expectation greater than or equal to $3K/8$, with equality when $r=1$, and a variance smaller than $3K$.)

Taking $a = \sqrt{(6K/\epsilon)}$, for $K$ large enough that $a \le (1/8)K + 1$, we get

$$\begin{array}{cccc}& \mathbb{P}(N_{(K,r)}\le \frac{1}{4}K+1)\le \frac{ϵ}{2}\mathrm{.}& & \text{(7.1)}\end{array}\backslash mathbb\{P\}\backslash left(N\_\{(K,r)\}\; \backslash le\; \backslash frac\{1\}\{4\}K\; +\; 1\backslash right)\; \backslash le\; \backslash frac\{\backslash epsilon\}\{2\}.\; \backslash tag\{7.1\}$$

Now, on the event that $N_{(K,r)} > K/4$, for any $j \ge Kr^2$, we have

$${\vec{d}}_{\mathcal{L}}((0,0),(j,0))\ge \sum _{i=1}^{\lfloor \frac{K}{4}\rfloor +1}\mathrm{D}\mathrm{i}\mathrm{a}\mathrm{m}({\mathcal{G}}_r(i))\wedge r,\backslash vec\{d\}\_\{\backslash mathcal\{L\}\}((0,0),\; (j,0))\; \backslash geq\; \backslash sum\_\{i=1\}^\{\backslash lfloor\; \backslash frac\{K\}\{4\}\; \backslash rfloor\; +\; 1\}\; \backslash mathrm\{Diam\}(\backslash mathcal\{G\}\_r(i))\; \backslash wedge\; r,$$

so that, using Theorem 7.2,