Dynamical Regimes of Discrete Diffusion Models

[2604.10961] Dynamical Regimes of Discrete Diffusion Models Skip to main content Search Submit Donate Log in Condensed Matter > Statistical Mechanics arXiv:2604.10961 (cond-mat) [Submitted on 13 Apr 2026 ( v1 ), last revised 17 Aug 2026 (this version, v3)] Title: Dynamical Regimes of Discrete Diffusion Models Authors: Tomoei Takahashi , Takashi Takahashi , Yoshiyuki Kabashima View PDF HTML (experimental) Abstract: Diffusion models generate high-dimensional data such as images by learning a process that gradually removes noise from corrupted data. Recent studies have shown that the backward dynamics of diffusion models exhibit two characteristic transitions: the speciation transition, at which generated samples begin to capture the global structure of the training data, and the collapse transition, at which the generation dynamics starts committing to individual training samples. While these transitions have been theoretically analyzed for continuous data, the same theoretical criteria have not been applied for discrete diffusion models, which are diffusion models for discrete data. In this work, we propose a simple effective model for discrete diffusion models trained on two-class Ising variable data with a general mixture ratio and analyze its backward dynamics using methods from statistical mechanics. We show that, as in the previous study on continuous data, the speciation transition can be determined through a second-order phase transition analysis using high-temperature expansion, while the collapse transition corresponds to a condensation transition described by the Random Energy Model. An analytical expression for the speciation time is obtained, and we show that its scaling becomes consistent with that of the continuous case when the noise increases with time as in practical diffusion models. These theoretical predictions are confirmed by numerical simulations and experiments with trained discrete diffusion models on real datasets. These results suggest that the original theoretical framework for continuous data remain valid for discrete data, and may provide a useful starting point for the statistical-mechanics analysis of discrete generative diffusion in more realistic settings. Comments: 36 pages, 7 figures Subjects: Statistical Mechanics (cond-mat.stat-mech) Cite as: arXiv:2604.10961 [cond-mat.stat-mech] (or arXiv:2604.10961v3 [cond-mat.stat-mech] for this version) https://doi.org/10.48550/arXiv.2604.10961 Focus to learn more arXiv-issued DOI via DataCite Journal reference: Tomoei Takahashi et al J. Stat. Mech. (2026) 083402 Related DOI : https://doi.org/10.1088/1742-5468/ae9930 Focus to learn more DOI(s) linking to related resources Submission history From: Tomoei Takahashi [ view email ] [v1] Mon, 13 Apr 2026 03:56:29 UTC (6,799 KB) [v2] Mon, 10 Aug 2026 01:30:01 UTC (6,801 KB) [v3] Mon, 17 Aug 2026 08:27:16 UTC (6,801 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: cond-mat.stat-mech < prev | next > new | recent | 2026-04 Change to browse by: cond-mat References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) IArxiv recommender toggle IArxiv Recommender ( What is IArxiv? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. 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