Mathematicians Prove Garban–Vargas Conjecture, Illuminating Fractal Chaos

Randomness shapes phenomena across vastly different scales — from galaxies to weather systems to subatomic particles. A major open problem about how tiny irregularities can produce large-scale fractal patterns has now been resolved: a 2023 conjecture by Christophe Garban and Vincent Vargas has been proved, offering new insight into the structure of Gaussian multiplicative chaos (GMC).

Background: What Is Gaussian Multiplicative Chaos?

Gaussian multiplicative chaos (GMC) is a mathematical framework for describing multiscale, fractal-like randomness. First introduced by Jean-Pierre Kahane in 1985 and later revitalized by researchers including Vincent Vargas, GMC models how fluctuations at many nested scales combine to produce complex patterns. It appears across disciplines — in quantum chaos, Brownian motion, turbulence and even some approaches to number theory.

The Garban–Vargas Conjecture

In 2023 Garban and Vargas proposed an elegant relation connecting two different ways of measuring dimension in GMC systems: a correlation (or geometric) dimension that quantifies clustering, and a harmonic dimension derived from frequency analysis (harmonic analysis). They conjectured a precise equation linking these dimensions for GMC on a circle, but a rigorous proof remained open.

The 2024 Proof

In 2024 a team — Zhaofeng Lin and Yanqi Qiu (Hangzhou Institute for Advanced Study, University of Chinese Academy of Sciences) and Mingjie Tan (Wuhan University) — posted a proof on arXiv that confirms the Garban–Vargas formula for the circle setting. Their paper, not yet peer-reviewed at the time of posting, not only establishes the conjectured equality of dimensions but also clarifies why it holds.

The key idea is an interpretation of GMC as a scale-by-scale "fair betting game." In such a game, expected winnings remain balanced across scales; mathematically, this balance is expressed by a martingale property. Rather than using only one-dimensional martingales, the authors constructed higher-dimensional martingale structures suited to the fractal, multiscale context of GMC. Tracking how randomness accumulates at every level, they showed that the conserved, energy-like quantity produces precisely the decay and dimension relationship predicted by Garban and Vargas.

"I heard about this conjecture during an online workshop," Yanqi Qiu recounts. "My earlier work on martingales suggested they might be the right tool here."

Why This Matters

The proof provides a powerful new method to relate geometric and harmonic descriptions of fractal randomness. By demonstrating how higher-dimensional martingales enforce scale-by-scale balance, the work opens the door to proving analogous results in more complicated geometries and models. It strengthens the theoretical foundations of GMC, an object of central importance in modern probability theory.

Limitations and Open Questions

Important limits remain. The martingale-based method breaks down at the critical phase-transition point where the GMC measure collapses — the regime analogous to a material losing its structure at a phase change. That critical case is notoriously delicate and remains a major open problem; resolving it will likely require new ideas and techniques.

In short, the 2024 proof settles a prominent conjecture about fractal chaos on the circle and provides tools that may extend to broader settings, while leaving rich, unresolved problems at the critical threshold.