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The presence of conformal variability has a direct physical meaning: it indicates that the overall behavior of the system will not change despite the microscopic details of the substance. It also conveys a kind of mathematical elegance that is established for a short interval, just as the whole system is breaking its main form and becoming something else.
First Evidence
In 2001 Smirnov created the first rigorous mathematics proof of the conformal immutability of a physical model. It was applied to a percolation model, which is the process of a liquid passing through a labyrinth on a porous support, like a stone.
Smirnov focused on the percolation of the triangular network, where water is allowed to flow only from the vertices that are “open”. Initially, each vertex has the same probability of being open to water flow. When the probability is low, the chances of the water having a path through the stone are small.
But as the probability slowly increases, there comes a time when there are enough vertices to create the first path that the stone crosses. Smirnov proved at a critical threshold that the triangular lattice is immutable, i.e., percolation occurs regardless of how it is transformed with conformal symmetries.
Five years later, at the 2006 International Congress of Mathematicians, Smirnov he announced that he proved the conformal immutability again, this time in the Ising model. Along with the 2001 proof, this innovative work earned him the Fields Medal, the highest honor in mathematics.
Since then, other evidence has emerged on a case-by-case basis, establishing a compliant variant for specific models. No one has come close to proving the universality that Poliakov envisioned.
“The previous evidence they worked on was tailored to specific models,” he said Federico Camia, A mathematical physicist at Abu Dhabi University in New York. “You have a very accurate tool to prove for a specific model.”
Smirnov himself admitted that his two proofs were based on a kind of “magic” that worked on both models but was not normally available.
“Because he used magic, it only works in situations where magic exists and in other situations we couldn’t find magic,” he said.
The new work is the first to break this pattern. Rotational variance, a key feature of conformal variability, proves that there are many.
One by one
Duminil-Copin began thinking about demonstrating universal conformal variability in the late 2000s when Smirnov was a graduate student at the University of Geneva. He also understood the brilliance of his tutor’s techniques and his limitations. Smirnov overcame the need to prove the three symmetries separately and instead found the right way to establish a compliant invariance – like a shortcut to the summit.
“It’s an amazing problem-solver. He proved the conformability of the two models of statistical physics to the conformity when he found the entrance to this giant mountain, like he experienced this kind of cruising,” Duminil-Copin said.
Duminil-Copin was doing graduate work and building a set of evidence that could go beyond Smirnov’s work in a few years. At a time when he and his colleagues were taking the unresolved issue seriously, Smirnov was ready to take a different approach. Instead of taking the risk with magic, they returned to the original hypotheses about the conformal variant made by Polyakov and later physicists.
Physicists needed a proof in three steps, one for each symmetry in composite variability: translation, rotation, and scale variance. Prove each of them separately, and you will get a consistent immutability as a result.
With this in mind, the authors first proceeded to prove the variance of the scale, believing that the rotation variance would be the most difficult symmetry, and knowing that the translational variance was simple enough and would not need its proof. When they tried this, they realized that at a critical point they could prove that a rotation variant existed in a variety of square and rectangular grid percolation models.
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