First-vertex degree in the LCD model ($\mathrm{Pr}(d_1(n)\ge y\sqrt{n})\to e^{-y^2/4}$)

In a uniform linearized chord diagram, $\mathrm{Pr}(\sqrt{n}{R}_1\ge x)=(1-x^2/n{)}^n\to e^{-x^2}$. Conditional on the right endpoints, $d_1(n)=2+{\sum }_{i=2}^n{1}_{\{L_i\le R_1\}}$, with independent success probabilities $R_1/R_i$. A Chernoff bound gives $R_i=(1+o(1))\sqrt{i/n}$ uniformly for $i\ge n^{1/10}$. The early indicators are $o(\sqrt{n})$, and the remaining conditional mean is $(2+o(1))n{R}_1$. Their conditional variance is $O_{\mathbb{P}}(\sqrt{n})$. Hence $d_1(n)/\sqrt{n}-2\sqrt{n}{R}_1\to 0$ in probability, proving the displayed tail limit. A self-loop's contribution is included in the constant two.