  Using the parameterisation of the deformation space of GHMC anti-de Sitter
structures on $S \times \mathbb{R}$ by the cotangent bundle of the
Teichm\"uller space of $S$, we study how some geometric quantities, such as the
Lorentzian Hausdorff dimension of the limit set, the width of the convex core
and the H\"older exponent, degenerate along rays of quadratic differentials.
