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

Luca Ambrogioni

cs.LGMar 20, 2026

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

Authors: Luca Ambrogioni

Organizations: Radboud University, Donders Institute for Brain, Cognition, and Behaviour

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 softening of low-frequency modes in the theoretically predicted critical regime. Similar signatures are found in trained convolutional diffusion models on Fashion-MNIST and in large-scale ImageNet models, where pattern formation coincides with a pronounced weakening of spatial modes. Finally, intervention experiments show that applying guidance precisely at this critical stage improves class alignment compared to applying it at random times, demonstrating that this regime is not only descriptive but functionally important.

Figures & tables

Figure 1: Visualization of mode softening. Low frequency modes lose stability at the critical points, leading to long range pattern formation.

Figure 2: Quantitative analysis of pattern formation dynamics in a trained EDM2 model [ 12 , 13 ] . Top: Example generative denoised trajectory. Left: spatial correlation length $\xi_{x}(\sigma)$ along several reverse trajectories (transparent blue) and their median (black). The dashed red curves show the equilibrium correlation length estimate $\xi_{\rm eq}(\sigma)$ obtained from the normalized low-frequency dispersion of the reverse-drift Jacobian. Middle: derivative $d\xi_{x}/d\log\sigma$ highlighting the rapid growth of spatial coherence near the estimated critical scale. Right: magnitude of median directional derivatives (computed over $8$ trajectories) along a range of Fourier modes (colored) together with their median (black) and zero-th mode (dashed) showing the softening of spatial modes. Since EDM2 is variance exploding, all modes become arbitrarily low as $\sigma$ increases. However, the low frequency modes are strongly suppressed compared to high frequencies at the onset of pattern formation.

Figure 3: Five snapshots of the evolving configuration are shown at logarithmically spaced times along the reverse diffusion path.

Figure 4: Quantitative analysis of pattern formation dynamics in patch score model (top), two patterns empirical score (middle), trained Fashion MNIST ConvNet (bottom) Left: spatial correlation length $\xi_{x}(t)$ along several reverse trajectories (transparent blue) and their median (black). The dashed red curves show the equilibrium correlation length estimate $\xi_{\rm eq}(t)$ obtained from the normalized low-frequency dispersion of the reverse-drift Jacobian. Middle: derivative $d\xi_{x}/d\log t$ highlighting the rapid growth of spatial coherence near the critical time. Right: magnitude of median directional derivatives (computed over $8$ trajectories) along a range of Fourier modes (colored) together with their mean (black) and the zero-th frequency mode (dashed), showing the softening of spatial modes near the transition.

Figure 5: Quantitative tests of domain seeding near the symmetry-breaking point. (a) Predicted KZ crossing length (dashed), exact intrinsic lattice linearization (crosses), and nonlinear spectral estimates (points; $95\%$ trajectory-bootstrap intervals). (b) Nonlinear crossing spectra at three rates and the predicted scaling function, without fitted parameters; bars show $1.96$ ensemble standard errors. Rate colors apply to (a–d). (c) Spectral deviations from the prediction through time, with bootstrap intervals. (d) Relative RMS difference between nonlinear and exact linear trajectories driven by the same noise; purple denotes $100$ -fold stronger noise. (e) Crossing spectra at weak and stronger noise, with the same standard-error bars. (f) Nonlinear/paired-linear spectral ratios at later times, showing saturation and redistribution of power. (g–i) One weak-noise trajectory, smoothed by a one-site Gaussian filter before binarization for display; all spectra and lengths use unsmoothed fields. The fixed bars denote $\xi_{\times}^{\rm KZ}=9.02$ , rather than a final domain diameter. Weak-noise ensembles contain $64$ trajectories per rate; the stronger-noise test contains $32$ .

Figure 6: Results of pulse guidance experiment. A) Example image generated by the EDM2 model without critical guidance pulse (top) and with critical guidance pulse (bottom). B) Difference in class alignments (DINOV2 score) between critical pulse guidance and random pulse guidance for different classes. Critical pulses show higher alignment for almost all tested classes.

Appendix figures & tables9 assets

Supplementary material from the paper’s appendix.

Appendix

$\gamma$	$\hat{\xi}$	$\xi_{\times}^{\rm KZ}$	$\xi_{\times}^{\rm exact}$	$S(0,0)$
1e-04	8.492	9.020	9.009	0.004926
5e-05	10.098	10.727	10.717	0.006961
2e-05	12.698	13.489	13.481	0.010998
1e-05	15.101	16.041	16.034	0.015547

Appendix

Table 1: Exact-lattice covariance tests at

K=2

D=2\times 10^{-5}

. The exact length is the intrinsic spectral curvature, without a finite-

L

estimator.

$\gamma$	$\xi_{\times}^{\rm KZ}$	$\xi_{\rm fit}$ [interval]	$\xi_{\rm fit}^{\rm linear}$	$E_{0}$ (%)	$\delta_{0}$ (%)
1e-04	9.02	9.02 [8.25, 9.91]	9.07	5.12	0.170
4e-05	11.34	10.09 [9.25, 11.26]	10.21	4.68	0.290
1e-05	16.04	15.64 [13.64, 19.01]	16.12	9.49	0.673

Appendix

Table 2: Crossing seed-scale tests in the full nonlinear score,

64

trajectories per weak-noise rate. The measured length fits amplitude and shape independently; bracketed values are

95\%

trajectory-bootstrap intervals. The matched linear estimate uses the same realizations and noise.

E_{0}

is the no-fit mean absolute shell deviation, and

\delta_{0}

is the mean paired RMS difference divided by the mean nonlinear field RMS, at

y=0

$y$	$\gamma=10^{-4}$	$\gamma=4\times 10^{-5}$	$\gamma=10^{-5}$	Stronger noise
-1	4.7	5.1	9.0	14.8
0	5.1	4.7	9.5	25.5
1	4.9	6.4	7.8	44.8
1.5	4.8	7.6	6.7	53.0
2	7.6	10.8	14.3	62.3
3	35.3	39.2	49.5	80.3

Appendix

Table 3: Mean absolute no-fit shell deviations

100E(y)

in percent. The first three columns use

D=2\times 10^{-5}

and

64

trajectories each; the final column uses

D=2\times 10^{-3}

and

32

trajectories at

\gamma=10^{-4}

. These positive deviations include sampling noise.

Figure 7: Receptive-field dependence at fixed

\gamma=10^{-4}

. Left: predicted crossing scale and exact intrinsic lattice curvature. Right: the relative separation from the patch radius approaches a constant, despite the increasing absolute length.

Figure 8: Controlled wall tests and stochastic domain formation. (a) Planar width estimates against the cubic prediction. (b) Circular shrinkage relative to the leading curvature law, at three radii. (c) Sinusoidal-wall decay relative to the leading prediction. Dotted curves in (b,c) show the independent full-tanh continuum mobility. (d,e) Inverse sign-boundary bond density during slow weak-noise and native generation, with Gaussian smoothing of width $0,1,2$ sites; bands are $1.96$ trajectory standard errors. (f,i) Representative terminal fields. (g,h) Later measured squared lengths compared with a coarsening law calibrated on earlier observations. These morphology checks have fitted coefficients and must be distinguished from the controlled no-fit predictions.

Figure 9: Driven circles and limits of network extrapolation. Top: changes in zero-contour squared radius for two initial radii and three ramp rates; solid curves are full-score dynamics, dashed curves integrate the leading stiffness, and dotted curves integrate the independently calculated full-tanh wall mobility. Bottom left and center: linear stiffness and nonlinear wall mobility along the slow and native schedules. The latter is a formal continuum extrapolation in the native regime, where thin walls invalidate its spatial-gradient assumptions; it is not a validated native transport coefficient. Bottom right: independent

L=256

validation of the late-window coarsening closure, either transferring both coefficients from

L=128

or fitting on the designated

L=256

training interval. Bars are

95\%

trajectory-bootstrap intervals. This calibrated network closure is distinct from the no-fit controlled geometry tests.

Figure 10: Patch dictionary (top) and a representative reverse denoising trajectory (bottom). Five snapshots of the evolving configuration are shown at logarithmically spaced times along the reverse diffusion path. Initially the field is dominated by noise, while at later times large coherent domains form as the dynamics selects one of the dominant global patterns.

Figure 11: Pattern-correlation and Fourier-response diagnostics for the binarized Fashion-MNIST model. The methods settings are described in Appendix D .

Figure 12: Quantitative analysis of pattern formation dynamics on non-binarized ConvNets trained on FashionMNIST. The critical line uses the binary-model reference and is only approximate for continuous-valued data.

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Diffusion models undergo a phase transition in a critical time window during generation dynamics, with two complementary diagnoses of criticality. The symmetry breaking picture views the critical window as when trajectories bifurcate into different semantic minima of the energy landscape, whereas the nonlocality picture views the critical window as when local denoising fails. We study whether two notions of such phase transitions are concurrent in modern diffusion transformers. By evaluating the dynamics and outcomes of the generation trajectory, we observe a near-simultaneous occurrence of the non-locality and symmetry breaking critical times. Our work is the first to unify the two notions of phase transitions in practice: it provides a concrete diagnostic for when and why diffusion models rely on conditioning and global denoising, enabling principled evaluation of model efficiency and guiding the design of architectures and sampling schemes that avoid unnecessary computation.

Abstract

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