Separator Geometry Sets the Serial Depth of Product-Round Sampling on Lattice Fields

Masked and parallel generators for images, and parallel samplers in spatial statistics, produce a field in rounds: each round chooses positions from the values revealed so far and samples them independently. We ask how many rounds such a decoder needs to reach a target accuracy, and show that the answer is set by the geometry of separators. On an m × m Gaussian Markov lattice field with n = m^2 sites and attractive nearest-neighbour coupling, every such decoder, with any learned conditionals and any value-adaptive choice of positions, needs Ω(ℓ) rounds in expectation to reach forward Kullback–Leibler divergence ε, where ℓ = min{m, 1 + log(n/ε)}, and a wall-and-dissect order, which first closes a grid of separating walls and then dissects the enclosed cells, reaches it in O(ℓ) rounds. The optimal depth is therefore Θ(ℓ), saturating at Θ(m) once log(n/ε) exceeds the side length; it is Θ(log ℓ) on a chain and Θ(ℓ^2) on a cube. Spacing positions apart is not enough: a synchronized-cell order, whose concurrent sites are as far apart as in wall-and-dissect but mostly stay joined by short unrevealed paths, needs Ω(min{n, log^2(n/ε)}) rounds, and mixing fixed schedules with exact conditionals at random does not change the two-dimensional order. A slab of thickness h needs Θ(ℓ min{h, ℓ}) rounds, and the lower bounds extend to ferromagnetic Ising fields and strictly attractive fields on ordered alphabets at every positive temperature. In exact computations at n = 16,384 and ε = 10^−2, wall-and-dissect needs 42 rounds, against 100 for the synchronized-cell order and 124 for a coarse-to-fine nearest-neighbour order of the kind used for parallel Gaussian-process sampling.