Mathematicians prove the symmetry of phase transitions

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The presence of conformal immutability has a direct physical meaning: It indicates that the global behavior of the system will not change even if you adjust the microscopic details of the substance. It also hints at a certain mathematical elegance that is placed, in short, in between, just as the whole system breaks its all-encompassing form and becomes something else.

The first evidence

In 2001, Smirnov produced the first rigorous mathematics proof conformal invariances in the physical model. It was applied to the percolation model, which is the process of passing liquid through a labyrinth in a porous medium, like a stone.

Smirnov watched percolation on a triangular lattice, where water was allowed to flow only through peaks that were “open.” Initially, each peak has the same probability of being open to water flow. When the probability is small, the chances of water passing through the stone are small.

But as you slowly increase the probability, there comes a point when enough peaks are open to create the first path that extends through the rock. Smirnov proved that at the critical threshold, a triangular lattice is conformally invariant, which means that percolation occurs regardless of how you transform it with conformal symmetries.

Five years later, at the International Congress of Mathematicians, Smirnov announced that he again proved conformal immutability, this time in Ising’s model. Combined with the 2001 proof, this groundbreaking work earned him the Fields Medal, the greatest mathematical honor.

In the years that followed, other evidence merged from case to case, establishing conformal invariance for certain models. No one came close to proving the universality that Poljakov imagined.

“Previous evidence that worked has been tailored to specific models,” he said Federico Camia, a mathematical physicist from Abu Dhabi University in New York. “You have a very specific tool to prove it for a very specific model.”

Smirnov himself admitted that both of his proofs relied on some kind of “magic” that was present in the two models he worked with, but was not usually available.

“Because he used magic, it only works in situations where magic exists, and we haven’t been able to find magic in other situations,” he said.

The new work is the first to disrupt this pattern – proving that rotational invariance, a key characteristic of conformal invariance, exists widely.

One by one

Duminil-Copin first began thinking about proving universal conformal invariance in the late 2000s, when he was a Smirnov graduate student at the University of Geneva. He had a unique understanding of the brilliance of his mentor’s techniques – as well as their limitations. Smirnov bypassed the need to prove all three symmetries separately and instead found a direct path to establishing conformal invariance – like a shortcut to the top.

“He is an incredible problem solver. He proved the conformal immutability of two models of statistical physics by finding the entrance to this huge mountain, like this kind of essence he went through, “said Duminil-Copin.

Years after his graduate studies, Duminil-Copin worked to create a series of pieces of evidence that could eventually enable him to overcome Smirnov’s work. When he and his co-authors seriously began work on conformal invariance, they were willing to take a different approach from Smirnov. Instead of risking magic, they returned to the original hypotheses about conformal immutability put forward by Polyakov and later physicists.

Hugo Duminil-Copin of the Institute for Advanced Scientific Studies and the University of Geneva and his collaborators take a symmetrical approach in proving the universality of conformal invariance.Photo: IHES / MC Vergne

Physicists sought proof in three steps, one for each symmetry present in conformal invariance: translational, rotational, and scale. Prove each of them separately and as a result you will get a conformal invariance.

With this in mind, the authors first tried to prove the invariance of scale, believing that rotational invariance would be the most difficult symmetry and knowing that translational invariance is simple enough and will not require its own proof. By trying this, they realized instead that they could prove the existence of rotational invariance at a critical point in a large number of different percolation models on square and rectangular grids.

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