  Melvin models with irrational twist parameter provide an interesting example
of conformal field theories with non-compact target space, and localized states
which are arbitrarily close to being delocalized. We study the torus partition
sum of these models, focusing on the properties of the regularized dimension of
the space of localized states. We show that its behavior is related to
interesting arithmetic properties of the twist parameter $\gamma$, such as the
Lyapunov exponent. Moreover, for $\gamma$ in a set of measure one the
regularized dimension is in fact not a well-defined number but must be
considered as a random variable in a probability distribution.
