Improved sampling thresholds across the entire AT region

Improve the field-strength and parameter thresholds imposed by the high-magnetization sampling approach so as to establish efficient sampling for the Sherrington–Kirkpatrick model throughout the entire Almeida–Thouless region.

Background

The paper’s unconditional high-field sampler operates above a threshold asymptotically larger than the AT line, while the oracle-centered sampler reaches the AT scale only under access to the signs of the mean magnetization vector and with additional parameter restrictions.

The authors identify the remaining discrepancy as arising partly from low-order terms in their definition of the required field strength and partly from the sparsity requirements of their sampler. They explicitly pose improving these thresholds to enable efficient sampling throughout the full AT region as an open problem.

References

Additionally, our definition of $h(\beta)$ includes potentially unnecessary low-order terms, arising due to a discrepancy between the weak AT region's definition (which gives $1 - q(\beta, h) \le \beta{-2}$) with the requirements of our sampler in Corollary~\ref{cor:fast_mixing_sparse_set_dobrushin_gen} (which requires a sparsity parameter $\rho \approx (\beta2 \log \beta){-1}$). It is also an interesting open problem to improve the thresholds imposed by our approach, with the goal of efficient sampling across the entire AT region.

— High-Magnetization Sampling at Low Temperatures: Ising Models and Bayesian Sparse Linear Regression  (2609.08873 - Kumar et al., 8 Sep 2026) in Appendix, Section “Sampling around the mean,” final paragraph after Theorem (oracle-centered sampling)