as $v\to\infty$. The limiting density is given by the function $$\label{eq:Q:def}
  Q(\eta)=\frac{1}{2\pi^{3/2}\eta}{{\mathfrak{Im}}}\int_{-\infty}^{\infty}\exp\left(-\frac{\xi^2}{4}-\frac{{\bar{L}}\eta m(\xi)^2}
{8}\right){\mathop{\rm erfc}}\left( \frac{\sqrt{{\bar{L}}\eta}m(\xi)}{2{{\mathrm i}}\sqrt{2}}\right){{\mathrm d}}\xi$$ which is continuous on $(0,\infty)$. Here $m(\xi)$ is as in theorem \[thm:4\]. $Q(\eta)$ has asymptotic expansion $$\label{eq:Q:asympt}
   Q(\eta)=\frac{\sqrt{2}}{\sqrt{{\bar{L}}}\pi^2\eta^{3/2}}\int_{-\infty}^{\infty}
\frac{{{\mathrm e}}^{-\xi^2/4}}{m(\xi)}{{\mathrm d}}\xi
+{{\mathrm O}}(\eta^{-5/2})\qquad$$ as $\eta\to\infty$.

The proofs of theorems \[thm:3\]–\[thm:6\] rely on an equidistribution result of Barra and Gaspard [@bar:lsd]. We review this work in section \[sec:3\].

The limit $v\to\infty$ is analogous to the semiclassical limit $\hbar\to 0$ [@kot:pot]. We note that theorem \[thm:6\] describes the wave functions on a vanishingly small fraction of the graph. It thus goes beyond the information provided by the Schnirelman theorem. It instead corresponds to the Gaussian value distribution for the wave functions of classically chaotic systems implied by the random wave model [@ber:rwm].

The value distribution of the eigenvector components of asymptotically large random matrices are particular cases of the $\chi^2_{\beta}$ density $$P_{\chi_{\beta}^2}(\eta)=\left(\frac{\beta}{2}\right)^{\beta/2}
\eta^{\beta/2-1}\Gamma^{-1}\left(\frac{\beta}{2}\right){{\mathrm e}}^{-\beta\eta/2},$$ where the parameter $\beta$ takes the values $1$, $2$ and $4$ in, respectively, the orthogonal, unitary and symplectic ensembles (see for example [@haa:rmt]). When $\beta=1$ the density is called the Porter-Thomas density. It is characterised by ${{\mathrm O}}(\eta^{-1/2})$ behaviour as $\eta\to0$ and ${{\mathrm O}}(\eta^{-1/2}{{\mathrm e}}^{-\eta/2})$ as $\eta\to\infty$. The limiting distribution 