Discriminating chaotic, regular, and stochastic processes using joint extrema statistics

Determine whether joint statistics of extrema, particularly distributions of consecutive spacings, can distinguish chaotic, regular, and stochastic erratic processes even when their marginal nearest-neighbor spacing distributions are approximately identical.

Background

The paper distinguishes marginal spacing distributions from joint distributions of consecutive spacings. Marginal distributions can conceal correlations and statistical organization, whereas joint extrema statistics retain information about dependencies among neighboring spacings.

The authors propose comparing two processes with nearly identical nearest-neighbor PDFs but different joint PDFs as a direct test of whether the proposed hierarchy of extrema statistics can identify distinctions that are invisible at the one-spacing level.

References

An importatnt question that has to be further explored is: Can joint extrema statistics distinguish chaotic, regular, and stochastic erratic processes?

— Statistical Characterization of Erratic and Chaotic Processes through Their Extrema I. Thermodynamics  (2610.21620 - Sonnenschein, 15 Oct 2026) in Section Summary and future direction, final open-questions list

In quantum system the most important tool is the map between the energy spectrum of a given system and the eigenvalues of random matrix theory. In the approach of analyzing erratic functions this is valid only for one dimensional functions. Thus, a natural question is what replaces this map for the higher-dimensional functions. In the latter cases one can project the functions down to one dimension in various different ways. This may be related to projecting tensors down to matrices. This seems to hint that the spectra of spacings of the higher dimensional functions may be related to the eigenvalues of random tensors. At present we do not have evidence to support this conjecture.

— Statistical Characterization of Erratic and Chaotic Processes through Their Extrema I. Thermodynamics  (2610.21620 - Sonnenschein, 15 Oct 2026) in Section Summary and future direction, final open-questions list; related discussion in Section 1