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

TL;DR

This paper explains how pattern formation in trained diffusion models results from out-of-equilibrium phase transitions, linking data symmetries and architectural constraints to the emergence of spatial patterns.

Contribution

It introduces a theoretical framework connecting phase transitions to pattern formation in diffusion models and validates it through experiments on various datasets.

Findings

Sharp increase in correlation length at critical time

Softening of low-frequency modes during pattern formation

Guidance at critical stage improves class alignment

Abstract

Diffusion models generate structure by progressively transforming noise into data, yet the mechanisms underlying this transition remain poorly understood. In this work, we show that pattern formation in trained diffusion models can be explained as an out-of-equilibrium phase transition driven by instabilities in the denoising dynamics. We develop a theoretical framework linking data symmetries and architectural constraints, such as locality and translation equivariance, to the emergence of collective spatial modes. In this view, structure arises when low-frequency modes become unstable, triggering a rapid growth of spatial correlations that organizes noise into coherent patterns. We validate this theory through a combination of analytical models and experiments. In a controlled patch-based model, we observe a sharp increase in correlation length and a simultaneous softening of low-frequency modes at a well-defined critical time, accurately predicted by theory. Similar signatures are found in trained convolutional diffusion models on Fashion-MNIST and in large-scale ImageNet models, where pattern formation coincides with a peak in estimated correlation length and a pronounced weakening of spatial modes. Finally, intervention experiments show that applying guidance precisely at this critical stage significantly improves class alignment compared to applying it at random times, demonstrating that this regime is not only descriptive but functionally important.