[2603.20092] How Out-of-Equilibrium Phase Transitions can Seed Pattern Formation in Trained Diffusion Models

Computer Science > Machine Learning arXiv:2603.20092 (cs) [Submitted on 20 Mar 2026] Title:How Out-of-Equilibrium Phase Transitions can Seed Pattern Formation in Trained Diffusion Models Authors:Luca Ambrogioni View a PDF of the paper titled How Out-of-Equilibrium Phase Transitions can Seed Pattern Formation in Trained Diffusion Models, by Luca Ambrogioni View PDF HTML (experimental) 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 t...