  We present a perfect marginal sampler of the unique Gibbs measure of a spin
system on $\mathbb Z^2$. The algorithm is an adaptation of a previous `lazy
depth-first' approach by the authors, but relaxes the requirement of strong
spatial mixing to weak. It exploits a classical result in statistical physics
relating weak spatial mixing on $\mathbb Z^2$ to strong spatial mixing on
squares. When the spin system exhibits weak spatial mixing, the run-time of our
sampler is linear in the size of sample. Applications of note are the
ferromagnetic Potts model at supercritical temperatures, and the ferromagnetic
Ising model with consistent non-zero external field at any non-zero
temperature.
