Lattice Configuration Generation with a Self-Learning Diffusion Model

Abstract

We show that a diffusion sampler for lattice-field configurations can be self-trained without preparing target-ensemble training configurations using an external Monte Carlo calculation. Starting from exactly sampled configurations at $\beta =0$, we use action-difference weights to train the score at the next coupling. Proposals from a fixed model are Metropolis-Hastings corrected at every noise level, and the resulting chain supplies training configurations for the next stage. This procedure defines the self-learning diffusion sampler SLDiffusion. In the two-dimensional compact XY model, self-training proceeds from $\beta =0.30$ to $0.50$ at $L=4$ and extends to $L=6,8,12$ at $\beta =0.5$. The energy and vortex densities agree with independent Hybrid Monte Carlo calculations within $1.6$ combined standard errors. Their integrated autocorrelation times, measured in stored updates, remain below two at all volumes studied. These results demonstrate a diffusion sampler whose training can be initialized and continued without external target-coupling ensembles.

Comments: 34 pages, 11 figures