Convergent Discovery of Critical Phenomena Mathematics Across Disciplines

TL;DR

This paper surveys how different scientific disciplines independently developed measures for detecting critical phenomena, revealing limited cross-disciplinary awareness and highlighting convergent mathematical concepts across fields.

Contribution

It provides a taxonomy of discovery types and quantitatively documents the convergence of critical phenomena measures across multiple disciplines over time.

Findings

Cross-domain citations were below random-mixing expectations (1987-2010).

Multiple disciplines independently derived similar measures for critical phenomena.

A taxonomy of discovery types and convergence patterns is presented.

Abstract

Techniques for detecting critical phenomena -- phase transitions where correlation length diverges and small perturbations have large effects -- have been developed across multiple fields over nine decades. We survey between six and twelve disciplines (depending on classification criteria) where researchers derived functionally corresponding measures of correlation scaling, with little documented awareness of each other's work. The physicist's correlation length $\xi$, the cardiologist's DFA scaling exponent $\alpha$, the financial analyst's Hurst exponent $H$, and the machine learning engineer's spectral radius $\chi$ all detect critical signatures under different notation. We classify each surveyed domain as independent derivation, domain transfer, or empirical precursor, and present citation network evidence that cross-domain citations remained significantly below random-mixing expectations (under a simple null model of domain mixing) during the formative period (1987--2010). The framework that motivated this investigation -- derived from distributed systems engineering and presented in the Supplementary Material -- is excluded from the survey count given the authors' dual role. Building on prior syntheses -- notably Sornette's 2004 textbook -- this paper contributes a taxonomy of discovery types and quantitative documentation of the convergence pattern.