Minimal assumptions for oracle access in infinite spin systems and finite-volume projections

Identify the minimal assumptions under which one can construct (i) an exponential-time oracle that approximately computes marginals of an infinite spin system with running time dependence o(1/ε) in the accuracy parameter ε, and (ii) an exact oracle for finite projections of infinite-volume Gibbs measures; and develop a method to obtain such oracles.

Background

The authors highlight desirable properties of perfect samplers in settings involving infinite-volume Gibbs measures and finite-volume projections, noting that such capabilities do not automatically follow from black-box reductions based solely on approximate samplers. Extending their framework to these settings would require suitable oracle access for marginals and projections.

They emphasize that the least assumptions needed to realize such oracles, and explicit constructions achieving the required o(1/ε) dependence for marginals or exactness for finite projections, are currently unclear.