  We explain and exploit the random matrix formulation of the Loschmidt echo
for the XX spin chain, valid for multiple domain wall initial states and also
for a XX spin chain generalized with additional interactions to more
neighbours. For models with interactions decaying as $e^{-\alpha \left\vert
l-j\right\vert }/\left\vert l-j\right\vert ^{p+1}$, with $p$ integer or natural
number and $\alpha \geq 0$, we show that there are third order phase
transitions in a double scaling limit of the complex-time Loschmidt echo
amplitudes. For the long-range version of the chain, we use an exact result for
Toeplitz determinants with a pure Fisher-Hartwig singularity, to obtain exactly
the Loschmidt echo for complex times and discuss the associated Stokes
phenomena. We also study the case of a finite chain for one of the generalized
XX models.
