Continuum versus periodic lattice Monte Carlo approach to classical field theory

We compare the momentum space with the standard periodic lattice approach to Monte Carlo calculations in classical $\phi^4$ field theory. We show that the mismatch in the initial value of $\phi^2_{\text{cl}}(t)$, results in a shift in the ``thermalized'' value, at large times. The two approaches converge to the same result in the continuum limit.