A generative sampler's effective sample size collapses with volume in lattice field theory, yet long-distance observables barely move
Synopsis
Lattice field theory usually judges generative samplers by effective sample size (ESS), a measure of overlap between generated and target samples, but a small global ESS need not mean inaccurate long-distance observables. A scale-resolved overlap diagnostic shows that small errors spread over many short-distance modes can dominate ESS; in two-dimensional φ⁴ theory near criticality, trained diffusion samplers show path-weight ESS fractions falling rapidly with volume while low moments of the magnetization stay close to benchmark values. Given the target marginal, a Metropolis test on a few collective variables keeps an acceptance rate of order one in the tests, and exact local updates then reduce the remaining mismatch.
Interpretation
Whether a generative sampler is usable should be judged by where its error sits across scales, not by a single global effective sample size. Such samplers have commonly been evaluated by effective sample size (ESS), with a small global ESS read as a sign that long-distance observables are inaccurate. In the exactly solvable free field, two samplers with comparable global ESS give a biased and an unbiased susceptibility respectively: a mass error biases the susceptibility, while a short-distance lattice artifact leaves it unbiased.
The approach compares model and target overlap layer by layer in cutoff scale and applies an acceptance test to only a few collective variables. The overlap flow gives the global ESS at the lattice cutoff, and lowering the cutoff isolates long-wavelength errors; the marginal test uses only the density ratio of a few variables and never needs the model density. In two-dimensional φ⁴ theory near criticality at volumes 16², 32² and 64², the path-weight ESS fraction falls to the low values associated with 1.38 down to 28.5, while the marginal-test acceptance rate on the magnetization falls only from 0.92 to 0.78.
Perspective
The discussion is limited to scalar fields: the free-field part is exact analytic work, and the interacting part is a numerical test in two-dimensional φ⁴ theory near criticality at volumes 16², 32² and 64² against public high-precision Monte Carlo ensembles. The marginal test takes the target marginal as input; in the tests that distribution came from the benchmark ensembles or from constrained simulations at fixed magnetization.
The composite chain is generally biased at finite relaxation length; untested low modes relax slowly near criticality, and errors in the estimated marginal densities leave a systematic bias. Interactions can transfer a proposal's ultraviolet error into the tested zero mode during relaxation, so the ultraviolet is best relaxed before the infrared is corrected. The samplers' own weights give a reliable magnetization marginal only at the smallest volume.