Hawking radiation with dispersion: The broadened horizon paradigm
  To identify what replaces the key notion of black hole horizon when working with theories which break Lorentz invariance at high energy, we study the modes responsible for the Hawking effect in the presence of high frequency dispersion. We show that they are regularized across the horizon over a short length which only depends on the scale of dispersion and the surface gravity. Moreover, outside this width, short and long wavelength modes no longer mix. These results can be used to show that the spectrum is hardly modified by dispersion as long as the background geometry does not vary significantly over this length. For relevant frequencies, the regularization differs from the usual WKB resolution of wave singularity near a turning point. 
Antonin Coutant and Renaud Parentani
gr-qc hep-th