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

Abstract Independent samples from the Boltzmann distribution are valuable for estimating spin-glass observables and evaluating approximate sampling methods. Local Markov chain Monte Carlo methods typically generate temporally correlated samples and can encounter critical slowing down or metastable trapping, while neural autoregressive samplers such as variational autoregressive networks generate independent but approximate samples. Here we present an exact autoregressive sampling algorithm for planar Ising spin glasses based on the Kac--Ward theory. Under the chain-rule factorization, sequentially fixing spins induces boundary-localized external fields, which destroy the zero-field structure required for exact evaluation. By encoding these fields with a planarity-preserving auxiliary spin construction, the conditional partition functions are mapped to an extended zero-field Ising model and exactly evaluated using the Kac--Ward determinant formula. The method generates strictly independent and identically distributed samples with exact normalized likelihoods at a computational cost of $\mathcal{O}(N^{5/2})$ for $N$ spins, thereby providing an exact baseline for benchmarking neural autoregressive samplers.