This paper presents a theoretical analysis of "critical windows" in diffusion models, which are narrow time intervals during which specific features of the final image emerge during the reverse process. The authors develop a formal framework to understand these windows, showing that for data from a mixture of strongly log-concave densities, critical windows can be bounded in terms of inter- and intra-group separation measures. They instantiate these bounds for concrete examples like well-conditioned Gaussian mixtures and use them to provide a rigorous interpretation of diffusion models as hierarchical samplers that progressively decide output features over discrete times. The authors validate their bounds with synthetic experiments and preliminary experiments on Stable Diffusion suggest that critical windows may serve as a useful tool for diagnosing fairness and privacy violations in real-world diffusion models. The paper also compares their work to related studies and discusses the implications of their findings for understanding the behavior of diffusion models in real-world applications.This paper presents a theoretical analysis of "critical windows" in diffusion models, which are narrow time intervals during which specific features of the final image emerge during the reverse process. The authors develop a formal framework to understand these windows, showing that for data from a mixture of strongly log-concave densities, critical windows can be bounded in terms of inter- and intra-group separation measures. They instantiate these bounds for concrete examples like well-conditioned Gaussian mixtures and use them to provide a rigorous interpretation of diffusion models as hierarchical samplers that progressively decide output features over discrete times. The authors validate their bounds with synthetic experiments and preliminary experiments on Stable Diffusion suggest that critical windows may serve as a useful tool for diagnosing fairness and privacy violations in real-world diffusion models. The paper also compares their work to related studies and discusses the implications of their findings for understanding the behavior of diffusion models in real-world applications.