Exact autoregressive sampling of planar Ising spin glasses via the Kac--Ward theory

What the math gives to ML

The paper provides a constructive exact autoregressive factorization for planar Ising spin glasses, despite the fact that conditioning on a prefix creates boundary magnetic fields that normally invalidate zero-field Pfaffian/Kac–Ward evaluation. Its transferable asset is an exact oracle for every conditional probability, obtained by encoding the induced fields in a planarity-preserving auxiliary-spin graph and evaluating the resulting zero-field partition function through a Kac–Ward determinant. The most useful neural-network application is exact teacher distillation and calibration of autoregressive samplers: train a model against oracle conditionals, use exact likelihoods to detect exposure-bias errors, and benchmark generated samples without MCMC uncertainty. The method predicts a sharp zero-KL and zero-autocorrelation reference signature, while its computational cost provides a concrete regime where learned amortization should eventually outperform repeated exact oracle calls.