  Local and non-local properties of Hawking radiation in the presence of short
distance dispersion are computed using connection formulae. The robustness of
the spectrum and that of the two-point function are explained by showing that
the leading deviations from the relativistic expressions decrease with the
inverse of the spatial extension of the near horizon region. This region
corresponds to a portion of de Sitter space with a preferred frame. We show
that the phases of the Bogoliubov coefficients are relevant for the two-point
function in black and white holes, and also for the black hole laser effect. We
also present an unexpected relation between the spectra obtained with sub and
with superluminal dispersion and we apply our formalism to massive fields. Our
predictions are validated by numerical analysis.
