= First-vertex degree in the LCD model
{title2=$\Pr(d_1(n)\ge y\sqrt n)\to e^{-y^2/4}$}
In a <uniform linearized chord diagram>, $\Pr(\sqrt nR_1\ge x)=(1-x^2/n)^n\to e^{-x^2}$. Conditional on the right endpoints, $d_1(n)=2+\sum_{i=2}^n\mathbf1_{\{L_i\le R_1\}}$, 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 $(2+o(1))nR_1$. Their conditional <variance> is $O_{\mathbb P}(\sqrt n)$. Hence $d_1(n)/\sqrt n-2\sqrt nR_1\to0$ in <probability>, proving the displayed tail limit. A <self-loop>'s contribution is included in the constant two.
Back to article page
