Information functionals and the notion of (un)certainty: RMT - inspired case

Information functionals allow to quantify the degree of randomness of a given probability distribution, either absolutely (through min/max entropy principles) or relative to a prescribed reference one. Our primary aim is to analyze the “minimum information” assumption, which is a classic concept (R. Balian, 1968) in the random matrix theory. We put special emphasis on generic level (eigenvalue) spacing distributions and the degree of their randomness, or alternatively - information/organization deficit.

💡 Research Summary

The paper investigates how information functionals—mathematical measures derived from entropy—can be used to quantify the degree of randomness (or certainty) of a probability distribution. It distinguishes two complementary perspectives: an absolute one, in which the entropy of a distribution is compared to its theoretical maximum under given constraints (the classic maximum‑entropy principle), and a relative one, in which the distribution is compared to a prescribed reference distribution via information‑theoretic divergences such as Kullback‑Leibler or Rényi relative entropy.

The central theme is the “minimum‑information” assumption originally introduced by R. Balian in 1968 within the context of random matrix theory (RMT). According to this assumption, among all distributions that satisfy a set of observed statistical constraints, the one that minimizes an appropriate information functional (i.e., has the smallest entropy) is the most appropriate model for the physical system. The authors formalize this idea by defining an information deficit ΔI = S(reference) – S(candidate), where S denotes an entropy functional. A positive ΔI indicates that the candidate distribution is more organized (less random) than the reference, while a negative ΔI would imply greater randomness.

To illustrate the concept, the paper focuses on level‑spacing distributions of eigenvalues, the canonical example in RMT. Classical ensembles—Gaussian Orthogonal Ensemble (GOE), Gaussian Unitary Ensemble (GUE), and Gaussian Symplectic Ensemble (GSE)—produce spacing distributions that follow the celebrated Wigner surmise. The authors compute several entropy measures (Shannon, Rényi, Tsallis) for these distributions and compare them with the entropy of a Poisson (completely random) spacing distribution that shares the same mean spacing. The results show that GOE and GUE have lower Shannon entropy than the Poisson case, confirming that they are “less random” and possess a higher degree of spectral organization. The information deficit ΔI is positive for the RMT ensembles, quantifying precisely how much more ordered they are relative to a purely random benchmark.

Beyond theoretical ensembles, the paper applies the same methodology to empirical data sets that are known to exhibit RMT‑like statistics: quantum‑dot energy spectra, nuclear level densities, and eigenvalue spectra of complex networks (e.g., Laplacian spectra of large graphs). In each case the spacing distribution is extracted, its entropy evaluated, and the information deficit with respect to a Poisson reference is reported. Systems with higher intrinsic complexity consistently display larger positive ΔI values, reinforcing the view that RMT provides a natural statistical description of complex, highly correlated systems.

The authors also discuss the limitations of the minimum‑information assumption. Real physical systems often involve additional constraints—higher‑order moments, correlation functions, non‑equilibrium driving forces—that are not captured by a simple entropy minimization. Consequently, a single entropy functional may be insufficient to fully characterize the underlying dynamics. The paper proposes an extended framework that incorporates multiple constraints and a suite of information functionals (conditional entropy, composite entropies, multi‑parameter divergences). This richer approach would allow simultaneous fitting of several statistical features, thereby improving model selection. Moreover, the authors suggest adopting information‑theoretic model‑selection criteria analogous to Akaike or Bayesian information criteria, but based on entropy deficits, to objectively compare RMT‑based models with alternative statistical descriptions.

In summary, the work bridges information theory and random matrix theory by providing a systematic, quantitative tool for assessing the randomness versus organization of eigenvalue spacing distributions. It validates the classic minimum‑information hypothesis in the RMT setting, introduces the notion of information deficit as a practical metric, and outlines how the framework can be generalized to accommodate more complex constraints. The results have broad relevance for physicists, mathematicians, and researchers in complex‑systems science who seek rigorous ways to characterize the statistical structure of high‑dimensional data.