Approximating spin-glass partition functions
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For a finite random spin system, can an algorithm approximate log Z, where Z sums the Boltzmann weights of all configurations, with a controlled error and failure probability in polynomial resources? Temperature, disorder and external field specify different tasks.
Why it matters and what counts as progress
Why it matters: The partition function determines thermodynamic quantities. Counting guarantees also clarify which parts of a disordered physical model can be computed efficiently.
What counts as progress: Improve accuracy, success probability, temperature or field range under the same interaction model. Separate exact-operation counts from numerical precision and bit complexity.
How the problem developed
- 1987 · Research result [1]
Michael Aizenman, Joel Lebowitz and David Ruelle Developed high-temperature cluster-expansion methods underlying the later direct-summation approach.
- 2026 · Research result [1]
Concurrent counting researchers Developed deterministic Gaussian zero-field counting algorithms. Wu’s comparison distinguishes their thresholds, disorder models and precision guarantees.
- 27 Oct 2026 · AI contribution [1]
Qiang Wu with GPT models Reported polynomial exact-operation counting with weak homogeneous fields and symmetric disorder; model-generated algorithmic ideas are allocated in Section 1.8.
AI contribution
Approximate Counting for Mixed p-Spin Glasses in Weak External Fields via Cluster Expansion
For SK and finite mixed Ising p-spin models with all interaction orders at least three, at a fixed margin below the second-moment threshold and homogeneous fields |h|≤ρ₀N^{-1/4}, algorithms approximate log Z_N to additive error ε using a number of exact operations polynomial jointly in N, ε^{-1} and δ^{-1}. Symmetric unit-variance disorder is allowed; failure is at most δ plus an explicit disorder-tail term, which vanishes with a finite fourth moment.
Sources
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Publication dates and editorial changes.
Approximate Counting for Mixed p-Spin Glasses in Weak External Fields via Cluster Expansion
Approximate Counting for Mixed p-Spin Glasses in Weak External Fields via Cluster Expansion
Approximate Counting for Mixed p-Spin Glasses in Weak External Fields via Cluster Expansion