  We compute the renormalized expectation value of the square of a massless,
conformally coupled, quantum scalar field on the brane of a higher-dimensional
black hole. Working in the AADD brane-world scenario, the extra dimensions are
flat and we assume that the compactification radius is large compared with the
size of the black hole. The four-dimensional on-brane metric corresponds to a
slice through a higher-dimensional Schwarzschild-Tangherlini black hole
geometry and depends on the number of bulk space-time dimensions. The quantum
scalar field is in a thermal state at the Hawking temperature. An exact,
closed-form expression is derived for the renormalized expectation value of the
square of the quantum scalar field on the event horizon of the black hole.
Outside the event horizon, this renormalized expectation value is computed
numerically. The answer depends on the number of bulk space-time dimensions,
with a magnitude which increases rapidly as the number of bulk space-time
dimensions increases.
