# Mathematicians prove the symmetry of phase transitions

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The presence of conformal immutability has a direct physical meaning: This 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 being set, for a brief intervention, just as the whole system is breaking its all-encompassing form and something else is being done.

The first evidence

In 2001 Smirnov produced the first rigorous mathematics evidence of conformal immutability in a physical model. Applied is applied to a penetration model, which is the process of fluid passing through a maze in a porous environment, like a rock.

Smirnov observed penetration into a triangular mesh, where water is allowed to flow only through roofs that are “open”. Initially, each canopy has the same probability of being open to water flow. When the probability is low, the chances of water having a path across the stone are low.

But as you slowly increase the probability, there comes a point where the verticals are open to create the first path that involves the stone. Smirnov proved that at the critical threshold, the triangular lattice is conformally immutable, meaning that penetration occurs regardless of how you transform it with conformal symmetry.

Five years later, at the 2006 International Congress of Mathematicians, Smirnov announced that he had again proved conformal invalidity, this time in the Ising model. Combined with his 2001 test, this groundbreaking work earned him the Fields Medal, the highest honor in mathematics.

Over the years, other evidence has flowed case by case, establishing conformal invariance for specific models. No one approached him to prove the universality that Polyakov envisioned.

“Previous trials that worked were tailored to specific models,” he said Federico Camia, a mathematical physicist at New York University Abu Dhabi. “You have a very specific tool to test it for a very specific model.”

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

“Since he used magic, it only works in situations where there is magic and we were not able to find magic in other situations,” he said.

New work is the first to break this pattern – proving that rotational invariance, an essential feature of conformal invalidity, is widespread.

One at a time

Duminil-Copin first started thinking about testing conformal universal invariance in the late 2000s, when he was a Smirnov graduate student at the University of Geneva. He had a unique sense of the brilliance of his mentor techniques – and also of their limitations. Smirnov bypassed the need to test the three symmetries separately and instead found a direct way to establish conformal immutability – as a shortcut to a summit.

“He is a great problem solver. He proved the conformal immutability of the two models of statistical physics by finding the entrance to this great mountain, as this kind of essence he passed, ”said Duminil-Copin.

For years after graduating from high school, Duminil-Copin worked on creating a series of proofs that could eventually allow him to go beyond Smirnov’s work. By the time he and his co-authors decided to work seriously on conform immutability, they were ready to take a different approach than Smirnov had. Instead of taking their chances with magic, they turned to the original hypotheses about the conformal immutability made by Polyakov and later physicists.

Physicists had required a test in three steps, one for each symmetry present in conformal invariance: translational, rotational, and scale invalidity. Try each of them separately, and you get conformal immutability as a result.

With this in mind, the authors decided to test scale invariance first, believing that rotational invariance would be the most difficult symmetry and knowing that translational invariance was quite simple and would not require its own proof. In trying to do this, they realized that they could prove the existence of rotational invariance at the critical point in a large variety of penetration patterns in square and rectangular grids.

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