The random walks of a Schwarzschild black hole

We show that spinless and neutral black holes in thermal equilibrium with radiation undergo fluctuations of charge and angular momentum. The corresponding spreads for a black hole in contact with charged scalar particles is calculated. The angular momentum spread turns out to grow with the size of the cavity. Charge spread does not depend either on the size of the cavity nor on the elementary charge of the field. Then, either the scale of the elementary chage is fixed by black hole physics $\alpha \sim 1/4\pi$ or the underlying physics of these fluctuations is not the random absorption and emission of charged particles from the bath, it remains unknown and must be clarified.