Approximating the partition function of planar two-state spin systems

•We study approximation of the hard-core partition function.•Our graphs are planar with degree at most 4.•We show that the problem is intractable for sufficiently high activity.•However, the logarithm of the partition function can be approximated.

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 λ   is sufficiently large, there is no fully polynomial randomised approximation scheme for evaluating the partition function unless NP=RPNP=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.