Multiscale Gibbs Measures

• Multiscale Gibbs measures are probability distributions defined by integrating hierarchical energy and multilevel entropy constraints.
• They are derived through a variational principle that maximizes full Shannon entropy while enforcing scale-dependent energy and conditional entropy conditions.
• These measures are applied to problems in statistical mechanics, multifractal analysis, and machine learning, providing rigorous foundations for hierarchical inference and regularization.

Multiscale Gibbs measures are probability distributions that incorporate multiple scales of structure, hierarchy, or resolution within their definition or construction. Originating from statistical mechanics and information theory, these measures generalize the classical Gibbs (Boltzmann) distributions by introducing explicit multilevel entropic or energetic constraints, probabilistic reinforcement mechanisms, or hierarchical dependence. They play a central role in the rigorous analysis of high-dimensional systems with scale-dependent features, ranging from degenerate energy landscapes to hierarchical random fields, multifractal dynamical systems, hierarchical point processes, and multiscale inference in machine learning.

1. Variational Principles and Entropic Constraints

The foundational construction of multiscale Gibbs measures is via variational principles that constrain not only the average energy but also conditional entropies at each level of a system's hierarchy. Given a product state space $X = X_r \times \cdots \times X_1$ and a Hamiltonian $H: X \to \mathbb{R}$, one seeks the probability law $p$ maximizing the full Shannon entropy while enforcing

1. An energy constraint: $⟨H⟩_p = E$;
2. Constraints on conditional entropies at each level $\ell$:

$$H_{cond}^\ell[p] = H\big(X_\ell \mid X_{\ell-1}, ..., X_1\big)$$

The resulting variational problem,

$$\Phi[p] = S[p] + μ⟨H⟩_p + \sum_{\ell=2}^r γ_\ell H_{cond}^\ell[p]$$

admits a unique maximizer, the multiscale Gibbs measure, which is recursively characterized by a backward chain of Gibbs conditionals with scale parameters $\zeta_r,\ldots,\zeta_1$. The solution displays strict asymmetry among the conditional entropies—unless all $\gamma_\ell$ are equal, the measure cannot be reduced to a single-temperature form and is genuinely multiscale (Camilli et al., 20 Dec 2025, Asadi et al., 2020).

2. Probabilistic Reinforcement and Large Deviations

A distinct probabilistic origin for multiscale structure is provided by reinforced multinomial processes (generalizing Pólya urns). In such models, empirical histograms of samples from a hierarchy of reinforced multinomial draws satisfy a large-deviation principle where the rate function is a weighted sum of Kullback-Leibler divergences at each scale:

$$I(p) = \sum_{\ell=1}^r (1 + γ_\ell) D_{KL}(p^{<\ell} \| q^{<\ell})$$

This entropy-imbalance formula matches exactly the unconstrained part of the variational functional described above. The most likely empirical measure under the reinforced process, conditioned on an energy constraint, converges to the multiscale Gibbs measure with corresponding scale parameters. This probabilistic mechanism demonstrates that the structural asymmetry of conditional entropies is not merely a mathematical artifact but arises from fundamentally reinforced sampling dynamics (Camilli et al., 20 Dec 2025).

3. Multiscale Laplace Asymptotics and Degenerate Minima

In continuous systems, multiscale Gibbs phenomena manifest in low-temperature limits when the underlying energy function (potential) $f: \mathbb{R}^d \to \mathbb{R}$ has degenerate minima. For $μ_t(dx) \propto \exp(-f(x)/t)dx$ with $t \to 0$, if the Hessian at the minimizer is not positive definite, one must analyze higher-order Taylor expansions, leading to a hierarchy of polynomial scales:

$$\mathbb{R}^d = E_1 \oplus E_2 \oplus \cdots \oplus E_p$$

with scaling exponents $\alpha_j = 1/(2j)$. The Laplace method in these nested coordinates yields a limiting measure proportional to $\exp(-g(y))\,dy$, where $g$ is a nontrivial polynomial. The approach extends cleanly to the multiple-well case, localizing the measure on the flattest minima and computing asymptotic weights by integrals over multiscale densities (Bras, 2021).

4. Hierarchical Models and Renormalization Structures

Several classes of models exhibit explicit multiscale Gibbs measure structures:

• Hierarchical Cube Models: Configurations of non-overlapping cubes at all dyadic scales are equipped with scale-dependent activities. The corresponding Gibbs measure is supported on hierarchically non-overlapping configurations and evolves by a top-down recursion. Existence and uniqueness are governed by sharp scale-summability conditions. Correlation decay exhibits ultrametric features, often exponential or stretched-exponential in the ultrametric distance (Jansen et al., 10 Jun 2024).
• Multiscale Regularization in Machine Learning: In variational inference and regularization (e.g., neural network posteriors), multiscale relative entropy terms are introduced, maximizing a sum of scale-weighted entropies or Kullback-Leibler divergences between marginals at multiple coarse-grainings. Closed-form solutions reveal a recursive (marginalize–tilt–renormalize) structure, exactly analogous to Wilsonian renormalization group steps, with provable generalization advantages over single-scale regularization (Asadi et al., 2020).

Model type	Multiscale mechanism	Reference
Degenerate Laplace	Nested polynomial scaling/subspaces	(Bras, 2021)
Hierarchical cubes	Tree recursive hard-core constraints	(Jansen et al., 10 Jun 2024)
Micro/Macrocanonical	Multiscale energies via wavelets or scattering	(Bruna et al., 2018)
Neural net regularization	Multiscale entropy/KL objectives	(Asadi et al., 2020)

5. Multiscale Sparse and Fractal Gibbs Measures

In texture synthesis, non-Gaussian process approximation, and multifractal analysis, multiscale energy vectors (e.g., wavelet $\ell^1$, $\ell^2$, and scattering coefficients) define the sufficient statistics of the system. Macrocanonical (Gibbs) and microcanonical ensembles can be defined on these vectors, and gradient flows enforcing these constraints yield convergence to multiscale Gibbs measures. These models outperform single-scale approaches when capturing long-range dependencies, sparsity, or multifractal properties, as shown by

• superior synthesis of textures and critical Ising spin configurations (Bruna et al., 2018),
• rigorous multifractal spectra for random weak-Gibbs measures via thermodynamic formalism, where the $L^q$-spectrum and Hausdorff dimensions of level sets are determined by multiscale Legendre transforms (Yuan, 2016).

6. Hierarchical Random Fields and Ruelle Probability Cascades

In the study of log-correlated and hierarchical fields (notably the 2D GFF with scale-inhomogeneous variance), the Gibbs measure admits a representation that converges—under ultrametric scaling—to a measure indistinguishable from a Ruelle probability cascade (RPC). The ultrametric overlap structure, extended Ghirlanda–Guerra identities, and atomic pure-overlap distributions emerge naturally from the multiscale variance profile, echoing the structure of the Generalized Random Energy Model (GREM). This formalism elucidates the geometry of extremes in disordered systems and enables explicit hierarchical decomposition of the resulting Gibbs measure (Ouimet, 2017).

7. Significance, Applications, and Theoretical Context

Multiscale Gibbs measures unify a broad class of constructions across probability, physics, and statistical learning:

• Statistical mechanics: Provide rigorous foundations for renormalization procedures, hierarchical models, and systems with multiscale energy landscapes.
• Information theory and statistics: Yield new classes of Maximum Entropy and Bayesian posteriors with fine-grained control over multiscale features.
• Machine learning: Enable improved generalization by decomposing regularization across network depth or feature hierarchy.
• Stochastic processes and fractal geometry: Describe systems exhibiting multifractal spectra and nontrivial clustering at multiple scales.

These measures are essential to the modern mathematical theory of multiscale structure, bringing together variational, probabilistic, and algorithmic perspectives in a unified framework (Camilli et al., 20 Dec 2025, Bras, 2021, Bruna et al., 2018, Asadi et al., 2020, Yuan, 2016, Jansen et al., 10 Jun 2024, Ouimet, 2017).