Scalable Generative Sampling and Multilevel Estimation for Lattice Field Theories Near Criticality

TL;DR

This paper introduces a multiscale generative sampler inspired by renormalization-group ideas to efficiently sample lattice field theories near criticality, significantly reducing autocorrelation times and enabling unbiased physical observable estimation.

Contribution

The authors develop a novel multilevel generative sampling method that models the Boltzmann distribution across scales, improving efficiency and accuracy in critical lattice field theory simulations.

Findings

Achieves autocorrelation times much smaller than HMC on large volumes.

Maintains high importance-sampling efficiency compared to other generative models.

Reproduces unbiased physical observables consistent with long HMC simulations.

Abstract

Sampling lattice field theories near criticality is severely hindered by critical slowing down, which makes standard Markov chain methods increasingly inefficient at large lattice volumes. We introduce a multiscale generative sampler, inspired by renormalization-group ideas, that models the Boltzmann distribution through a coarse-to-fine hierarchy across length scales. At each level, a conditional Gaussian mixture model captures the main local dependence of newly introduced variables on the already-sampled coarse field, while a masked continuous normalizing flow refines the remaining conditional structure. Coarse levels encode the dominant long-wavelength modes, and finer levels progressively add short-distance fluctuations. In addition, because the architecture preserves coarse fields exactly during refinement, it provides exact restriction maps at no additional computational cost and directly enables unbiased Multilevel Monte Carlo (MLMC) variance reduction. For the two-dimensional scalar $\phi^4$ theory at criticality, the method achieves integrated autocorrelation times orders of magnitude smaller than Hybrid Monte Carlo (HMC) on large volumes, maintains high importance-sampling efficiency relative to other generative baselines, and reproduces unbiased physical observables in statistical agreement with long HMC simulations.