$Z(k,{{\mathbf L}})$. Such a correspondence was noted in [@bar:lsd] for the spacing distribution of the eigenvalues of quantum graphs.

In [@snaith:1] it was shown that the value distributions of the real and imaginary parts of the logarithm of the characteristic polynomial of a random unitary matrix drawn from the circular ensembles of random matrix theory tend independently to a Gaussian distribution in the limit as the matrix size tends to infinity, subject to appropriate normalisation. The distribution that appears in theorems \[thm:1\] and \[thm:2\] is known as the Cauchy distribution. It is related to the Gaussian distribution by the fact that both are examples of a larger class of distributions known as stable distributions. Such distributions share the property that the sum of two random variables from a stable distribution is distributed like a random variable from the same distribution. Theorem \[thm:1\] is a consequence of this fact. We also note that the density in theorems \[thm:1\] and \[thm:2\] is independent of $v$ when $Z(k,{{\mathbf L}})$ is normalised as indicated.

We next consider the distribution of values taken by $Z'(k)$ when $k=k_n$, $n=1,2,\ldots$.

\[thm:3\] Let the components of ${{\mathbf L}}$ be linearly independent over ${\mathbb Q}$. Then there exists a 