  By solving a master equation in the Sierpinski lattice and in a planar
random-resistor network, we determine the scaling with size L of the shot noise
power P due to elastic scattering in a fractal conductor. We find a power-law
scaling P ~ L^(d_f-2-alpha), with an exponent depending on the fractal
dimension d_f and the anomalous diffusion exponent alpha. This is the same
scaling as the time-averaged current I, which implies that the Fano factor
F=P/2eI is scale independent. We obtain a value F=1/3 for anomalous diffusion
that is the same as for normal diffusion, even if there is no smallest length
scale below which the normal diffusion equation holds. The fact that F remains
fixed at 1/3 as one crosses the percolation threshold in a random-resistor
network may explain recent measurements of a doping-independent Fano factor in
a graphene flake.
