Minimal assumptions and oracles for infinite spin systems

Identify the minimal assumptions under which one can construct an exponential-time oracle that approximately computes the marginals of an infinite spin system with o(1/ε) dependence on the accuracy parameter ε, and develop a method to obtain an exact oracle for finite-volume projections of infinite-volume Gibbs measures so that the proposed perfect sampling reductions can be extended to this setting.

Background

The authors discuss desirable properties of perfect samplers for infinite-volume spin systems—such as sampling from finite-volume projections of infinite-volume Gibbs measures under strong spatial mixing—achieved by other specialized techniques.

To adapt their black-box reductions to this setting, suitable oracles are required: an approximate oracle for marginals with sublinear (in 1/ε) dependence on ε, and an exact oracle for finite projections. Clarifying the least assumptions enabling such oracles is a key obstacle to extending the framework to infinite systems.