  Hawking radiation sets stringent constraints on Primordial Black Holes (PBHs)
as a dark matter candidate in the $M \sim 10^{16} \ \mathrm{g}$ regime based on
the evaporation products such as photons, electrons, and positrons. This
motivates the need for rigorous modeling of the Hawking emission spectrum.
Using semi-classical arguments, Page [Phys. Rev. D 16, 2402 (1977)] showed that
the emission of electrons and positrons is altered due to the black hole
acquiring an equal and opposite charge to the emitted particle. The Poisson
fluctuations of emitted particles cause the charge $Z|e|$ to random walk, but
since acquisition of charge increases the probability of the black hole
emitting another charged particle of the same sign, the walk is biased toward
$Z=0$, and $P(Z)$ approaches an equilibrium probability distribution with
finite variance $\langle Z^2\rangle$. This paper explores how this ``stochastic
charge'' phenomenon arises from quantum electrodynamics (QED) on a
Schwarzschild spacetime. We prove that (except for a small Fermi blocking term)
the semi-classical variance $\langle Z^2 \rangle$ agrees with the variance of a
quantum operator $\langle \hat{\cal Z}^2 \rangle$, where $\hat{\cal Z}$ may be
thought of as an ``atomic number'' that includes the black hole as well as
charge near it (weighted by a factor of $2M/r$). In QED, the fluctuations in
$\hat{\cal Z}$ do not arise from the black hole itself (whose charge remains
fixed), but rather as a collective effect in the Hawking-emitted particles
mediated by the long-range electromagnetic interaction. We find the rms charge
$\langle Z^2\rangle^{1/2}$ asymptotes to 3.44 at small PBH masses $M \lesssim
2\times 10^{16}\,$g, declining to 2.42 at $M=5.2\times 10^{17}\,$g.
