A stationary observer at infinity sees a particle of mass m falling in a supermassive Schwarzschild black hole. He observes an increasing redshift and sees the particle ceasing to progress when it approaches the black hole's horizon. What happens to the positional uncertainty of this particle in the reference frame of the distant observer?
A straightforward scaling argument (inserting the Hawking temperature into the equation for the thermal de Broglie wavelength for a particle of mass m) yields a thermal areal uncertainty scaling as the black hole circumference times the particle's Compton wavelength.
Is this the correct limiting behavior?