Deterministic Ising approximation under a spectral hypothesis alone

Establish deterministic approximation of the zero-field Ising partition function under the spectral hypothesis alone, without the entrywise O(1/n) diffuseness condition.

Background

The Ising FPTAS in the paper assumes both a one-sided spectral bound, λ_max(J)≤1−κ, and entrywise interaction bounds |J_ij|≤β/n. The entrywise condition is essential to the truncation estimates and to the bounded-remainder asymptotic proved in the paper.

The authors point out that a fixed interaction bond already prevents the normalized correction from being 1+O(1/n), demonstrating that spectral control alone does not imply the paper’s asymptotic conclusion. Whether deterministic approximation remains possible under only the spectral condition is unresolved.