  Recently, based on a novel analysis of the Planck satellite data, a hint of a
uniform rotation of the polarization of cosmic microwave background photons,
called isotropic cosmic birefringence, has been reported. The suggested
rotation angle of polarization of about $0.2-0.4$ degrees strongly suggests
that it is determined by the fine structure constant, which can be naturally
explained over a very wide parameter range by the domain walls of axion-like
particles. Interestingly, the axion-like particle domain walls predict not only
isotropic cosmic birefringence but also anisotropic one that reflects the
spatial distribution of the axion-like particle field on the last scattering
surface. In this Letter, we perform lattice simulations of the formation and
evolution of domain walls in the expanding universe and obtain for the first
time the two-point correlation function and power spectrum of the scalar field
that constitutes the domain walls. We find that while the power spectrum is
generally consistent with analytical predictions based on random wall
distributions, there is a predominant excess on the scale corresponding to the
Hubble radius. Applying our results to the anisotropic cosmic birefringence, we
predict the power spectrum of the rotation angles induced by the axion-like
particle domain walls and show that it is within the reach of future
observations of the cosmic microwave background.
