Mathematicians Crack a Fractal Conjecture on Chaos

The world may seem orderly, but chance and chaos shape everything in the universe, from enormous galaxies to subatomic particles. Cover a cold window covering with ice: Even an oddly shaped snowflake can influence the final frosty pattern.

Understanding how random fluctuations can reverberate to produce global effects is what French mathematician Vincent Vargas of the University of Geneva in Switzerland set out to do more than 10 years ago. His first ideas about simple geometries appeared in a decade-old paper, but it wasn’t until 2023, while working with Christophe Garban of the University of Lyon in France, that the concept finally crystallized into what is now known as the Garban-Vargas conjecture. Now mathematicians have proven this conjecture using an insightful technique that should open the door to understanding much more complex systems.

The conjecture involves the behavior of a form of chance found in a wide range of domains, from quantum chaos to Brownian motion to air turbulence. Mathematicians use a mathematical “tape measure” called Gaussian Multiplicative Chaos, or GMC, to detect subtle patterns hidden in an otherwise impenetrable sea of ​​chance. GMC has even been used to find patterns in prime numbers. This topic constitutes one of the most important and fundamental ideas in probability theory today.

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French mathematician Jean-Pierre Kahane is credited with first developing GMC in 1985, although his pioneering work was quickly forgotten. “I was one of those who revived his work,” Vargas says. “I met him several times and he said he was amazed at the importance of the subject. [had] become. All over the planet, people are working on something related to Gaussian chaos.

Vargas first discovered this metric while studying turbulence and finance. He then found it again in a project on conformal field theory, used to study patterns that remain constant when you zoom in or out. Lately he has focused on studying its fundamental mathematical nature.

To understand GMC, imagine a turbulent fluid filled with vortices at many different scales. Huge vortices randomly divide into smaller vortices, which in turn divide into even smaller vortices, in a vast nested random hierarchy. GMC serves as a mathematical model that measures this type of multi-scale randomness: it captures the random fluctuations that persist at each scale of the observation. For this reason, we often speak of a fractal measurement.

Mathematicians have discovered surprising behavior in the types of chance governed by GMC. For example, events at the smallest scales can govern the entire system; the powerful tendrils of fractal structure shape chaos at every level. Therefore, these systems cannot be understood by looking at averages. Instead, the GMC rules produce a universal picture that applies at all scales.

But this fascinating picture only holds up to a critical threshold. If the underlying randomness becomes Also strong, the GMC measure collapses. Or, in the language of whirlpools, once enough randomness permeates the whirlpools, they become unstable, losing all their hidden order. Much like ice turning into liquid, this breakdown marks an important phase transition into chaos.

In 2023, Garban and Vargas introduced a new perspective to study GMC chaos. This comes from a mathematical field called harmonic analysis. Instead of looking directly at the swirls, they looked at the frequencies of the patterns hidden in the swirls, much like analyzing a complex sound by breaking it down into pure tones.

That’s when an idea came to them. If they could match two completely different physical descriptions – complexity and harmonics – they might learn something new. Mathematicians call this idea of ​​matching independent physical descriptions “dimensions.”

As an example, consider snowflakes falling to the ground. As the snow gently lands, two possible dimensions could be the number of patterns appearing in the distribution of snowflakes and the number of clumps formed at different scales. But is there a formula that can link the two dimensions of patterns (harmonics) and agglutination (correlations)?

“The key word is dimension,” Vargas explains. “That’s the name of the game. You have a lot of natural dimensions, but when do they coincide?”

After studying GMC-governed systems on a circle, the duo conjectured an extraordinarily elegant equation that matched the correlation dimension of a GMC system to its harmonic dimension.

Unfortunately, they were unable to prove their formula even for simple geometry. In 2023, they published their conjecture on the preprint server arXiv.org, and it subsequently became a major open problem.

In 2024, mathematicians Zhaofeng Lin and Yanqi Qiu of the Hangzhou Institute for Advanced Study, University of the Chinese Academy of Sciences, and Mingjie Tan of Wuhan University solved the conjecture. Their research, which was published as a preprint on arXiv.org and has not yet been peer-reviewed, not only confirmed the formula, but also revealed Why That works.

Mathematically, they compared GMC to a “fair betting game,” in which expected payoffs remain constant regardless of the size of the game. When applied to fractal fluctuations, this means that the system remains balanced as you zoom in and out, and each smaller scale contributes to randomness in a way that conserves energy.

Mathematicians call a process that exhibits this type of fair behavior, scale by scale, a martingale. However, unlike normal betting games, chaos “games” are much more complex and require larger martingales.

“I learned about this conjecture during an online math workshop,” says Qiu. “I had focused on martingales for my doctoral thesis a few years ago, and I had a hunch they would be the right tool here.”

The group used its higher-dimensional martingale structure to carefully track the accumulation of chance at each scale. And of course, by conserving energy, many little “fair games” combined to yield the same decay formula that Garban and Vargas had conjectured.

Qiu and colleagues’ evidence not only confirmed the conjecture, but also paved the way for further evidence on more complex fractal patterns. However, the path to a complete theory is not entirely free of obstacles. Even the new method fails when chance forces the system to reach its critical phase transition point. According to mathematicians, this phase transition is itself a rich and intriguing topic, with its own set of profound questions. But “to go further,” says Qiu, “we need new ideas.”