Subquadratic counting for the hardcore model up to the critical threshold

Determine whether there exists an expected o(n^2)-time fully polynomial randomized approximation scheme (FPRAS) for the partition function of the hardcore model on n-vertex graphs of maximum degree Δ for all fugacities λ in (0, λ_c(Δ)), where λ_c(Δ) = ((Δ−1)^{Δ−1})/((Δ−2)^{Δ}).

Background

The paper presents subquadratic-time approximate counting algorithms for several spin systems and, for the hardcore model, achieves subquadratic time when λ < 1/(Δ−1), which is a constant-factor away from the uniqueness threshold. Prior to this work, subquadratic counting for the hardcore model was known only for λ = o(Δ^{-3/2}).

For anti-ferromagnetic two-spin systems, including the hardcore model, polynomial-time approximate counting is known in the uniqueness regime, up to the critical threshold λ_c(Δ) ≈ e/(Δ−1). The authors develop a black-box reduction from perfect marginal sampling to subquadratic counting and ask whether subquadratic counting can match the full uniqueness regime for the hardcore model.