  We consider the problem of approximating the partition function of the
hard-core model on planar graphs of degree at most 4. We show that when the
activity lambda is sufficiently large, there is no fully polynomial randomised
approximation scheme for evaluating the partition function unless NP=RP. The
result extends to a nearby region of the parameter space in a more general
two-state spin system with three parameters. We also give a polynomial-time
randomised approximation scheme for the logarithm of the partition function.
