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From entropic constraints to reinforced processes: a probabilistic origin of multiscale measures

Francesco Camilli, Pierluigi Contucci, Emanuele Mingione

TL;DR

The paper investigates multiscale Gibbs measures by connecting variational principles with entropic constraints to a novel probabilistic mechanism. It introduces a reinforced multinomial process and proves a large-deviation principle for its empirical histogram, showing that the resulting rate function reproduces the characteristic entropy imbalance that defines multiscale measures. By analyzing the two-scale case and generalizing to multiple scales, the work unifies variational and probabilistic descriptions and reveals deep links to Poisson-Dirichlet processes and Derrida-Ruelle cascades. These results provide a rigorous probabilistic foundation for multiscale Gibbs structures with potential applications across physics, inference, and complex systems.

Abstract

We investigate multiscale Gibbs measures from a variational and probabilistic viewpoint, focusing on the structural asymmetry among conditional entropies that characterizes their construction. We show how this asymmetry emerges both from variational principles with entropic constraints and from stochastic processes with reinforcement. We thus introduce the reinforced multinomial process and prove a large-deviation principle for its empirical histogram. The associated rate function reproduces precisely the entropy imbalance defining multiscale measures, thereby providing a genuine probabilistic mechanism for their emergence. The reinforced multinomial process thus offers a simple and rigorous stochastic foundation for multiscale Gibbs structures.

From entropic constraints to reinforced processes: a probabilistic origin of multiscale measures

TL;DR

The paper investigates multiscale Gibbs measures by connecting variational principles with entropic constraints to a novel probabilistic mechanism. It introduces a reinforced multinomial process and proves a large-deviation principle for its empirical histogram, showing that the resulting rate function reproduces the characteristic entropy imbalance that defines multiscale measures. By analyzing the two-scale case and generalizing to multiple scales, the work unifies variational and probabilistic descriptions and reveals deep links to Poisson-Dirichlet processes and Derrida-Ruelle cascades. These results provide a rigorous probabilistic foundation for multiscale Gibbs structures with potential applications across physics, inference, and complex systems.

Abstract

We investigate multiscale Gibbs measures from a variational and probabilistic viewpoint, focusing on the structural asymmetry among conditional entropies that characterizes their construction. We show how this asymmetry emerges both from variational principles with entropic constraints and from stochastic processes with reinforcement. We thus introduce the reinforced multinomial process and prove a large-deviation principle for its empirical histogram. The associated rate function reproduces precisely the entropy imbalance defining multiscale measures, thereby providing a genuine probabilistic mechanism for their emergence. The reinforced multinomial process thus offers a simple and rigorous stochastic foundation for multiscale Gibbs structures.

Paper Structure

This paper contains 10 sections, 2 theorems, 82 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Definitions
  3. Generating functions
  4. Multiscale measures from variational principles
  5. Two temperature thermodynamics
  6. A large deviation perspective
  7. Reinforced multinomial process
  8. Generalization to an arbitrary number of scales
  9. Multiscale measure and Poisson Dirichlet processes
  10. Conclusions

Key Result

Proposition 1

Let $X$ be as above. The solution of the following variational principle with $\mu,(\gamma_\ell)_{2\leq\ell\leq r}$ a collection of Lagrange multipliers, is an $r$-level multiscale measure in the sense of Definition def:multiscale measure. Furthermore, the functional $\phi$ evaluated at the optimal distribution equals $P_0=\log Z_0$.

Figures (2)

  • Figure 1: Hierarchical tree of depth $r=3$. An element (leaf of the tree) ${\mathbf x}=(x_3,x_2,x_1)\in X$ uniquely determines a path reaching the root cluster; intermediate nodes represent clusters.
  • Figure 2: Schematic representation of a two--scale reinforced multinomial process. The space $X_{1}$ is partitioned into three equal parent cells; each parent cell contains four child cells. Hence, every child cell is labeled by a couple of indices in $X_{2}\times X_{1}$. Blue and yellow arrows represent balls cast into the parent (or child) cells. As we see, from the first to the second level, the duplication process takes place.

Theorems & Definitions (8)

  • Definition 1: Multiscale measure
  • Definition 2
  • Proposition 1
  • Remark 1
  • proof
  • Proposition 2: Ellis99
  • Remark 2
  • Remark 3