Diffusion Models for Sampling Near Criticality in Lattice Field Theories

We investigate generative diffusion models as denoising samplers for two- and three-dimensional lattice $\phi^4$ theory across the symmetric, near-critical, and broken phases. Validated against ensembles generated by Fourier-accelerated HMC combined with Wolff cluster updates, the reverse-SDE sampler reproduces scalar observables and the momentum-space propagator $G(|k|)$, with residual bias concentrated in the zero-mode and, in three dimensions, the action density. We introduce two local diagnostics and an HMC-referenced effective sample size (ESS), which probe the learned drift directly, through a Metropolis-adjusted Langevin acceptance rate, and through observable-level bias and variance. Exploiting a fully convolutional architecture with weights shared across different volumes ($V=L^D$), we show that cross-volume training transfers to unseen sizes, matching or slightly improving in-distribution training in the two-dimensional symmetric and broken phases. A three-dimensional model trained on $L \in \{4, 8, 16, 32\}$ reproduces the propagator and most scalar observables at the unseen lattice size $L = 64$ across the phase diagram, with the residual susceptibility excess in the broken phase as the main exception, and improves several critical observables relative to in-distribution $L = 64$ training. This establishes cross-volume generalization as a viable mechanism for large-volume sampling, and the score learned from many cheap small-lattice configurations transfers to the target volume without retraining.