  We develop a discrete version of paracontrolled distributions as a tool for
deriving scaling limits of lattice systems, and we provide a formulation of
paracontrolled distribution in weighted Besov spaces. Moreover, we develop a
systematic martingale approach to control the moments of polynomials of i.i.d.
random variables and to derive their scaling limits. As an application, we
prove a weak universality result for the parabolic Anderson model: We study a
nonlinear population model in a small random potential and show that under weak
assumptions it scales to the linear parabolic Anderson model.
