  We consider an operator $P_V=(1+V)P$ on $\ell^2(Z^d)$, where $P$ is the
transition operator of a symmetric irreducible random walk, and $V$ is a
``sparse'' potential. We first characterize the essential spectra of this
operator. Secondly, we prove that all the eigenfunctions which correspond to
discrete spectra decay exponentially fast. Thirdly, we give a sufficient
condition for this operator to have an absolute spectral gap at the right edge
of the spectra. Finally, as an application of the absolute spectral gap and the
exponential decay of the eigenfunctions, we prove a limit theorem for the
random walk under the Gibbs measure associated to the potential $V$.
