Approximating the partition function of planar two-state spin systems

TL;DR

This paper proves the computational hardness of approximating the partition function for certain planar two-state spin systems with large activity, while providing an efficient approximation for its logarithm.

Contribution

It establishes NP-hardness results for approximating the partition function in specific parameter regimes and offers a polynomial-time scheme for approximating its logarithm.

Findings

No efficient randomized approximation scheme exists for large activity values unless NP=RP.

A polynomial-time randomized scheme approximates the logarithm of the partition function.

Results extend to a broader class of two-state spin systems with three parameters.

Abstract

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.