How Out-of-Equilibrium Phase Transitions can Seed Pattern Formation in Trained Diffusion Models

Abstract

In this work, we propose a theoretical framework that interprets the generation process in trained diffusion models as an instance of out-of-equilibrium phase transitions. We argue that, rather than evolving smoothly from noise to data, reverse diffusion passes through a critical regime in which small spatial fluctuations are amplified and seed the emergence of large-scale structure. Our central insight is that architectural constraints, such as locality, sparsity, and translation equivariance, transform memorization-driven instabilities into collective spatial modes, enabling the formation of coherent patterns beyond the training data. Using analytically tractable patch score models, we show how classical symmetry-breaking bifurcations generalize into spatially extended critical phenomena described by softening Fourier modes and growing correlation lengths. We further connect these dynamics to effective field theories of the Ginzburg-Landau type and to mechanisms of pattern formation in non-equilibrium physics. Empirical results on trained convolutional diffusion models corroborate the theory, revealing signatures of criticality including mode softening and rapid growth of spatial correlations. Finally, we demonstrate that this critical regime has practical relevance: targeted perturbations, such as classifier-free guidance pulses applied at the estimated critical time, significantly improve generation control. Together, these findings position non-equilibrium critical phenomena as a unifying principle for understanding, and potentially improving, the behavior of modern diffusion models.