Simplified Version of the Work of Hugo Duminil-Copin, Fields Medal Winner 2022
Demystifying Phase Transitions
The Work of Hugo Duminil-Copin
Phase transitions are something we observe in everyday life - water freezing, magnets losing their magnetism at high temperatures, and so on. But mathematically explaining these transitions between different states of matter has stumped scientists for over a century.Â
Now, French mathematician Hugo Duminil-Copin, who has solved several long-standing mysteries in the field. His work acting as a detective to create order out of complexity.
The problems studied by Duminil-Copin relate to "statistical physics" models. These are mathematical representations of physical systems like magnets or porous materials. By studying simplified models, physicists aim to understand real phase transitions.
Two key models are the Ising model for magnetism and percolation model for porous media. The Ising model represents a material as many tiny magnets that influence each other. Percolation models materials as a lattice with some randomly blocked paths. Both undergo phase transitions - magnetized to unmagnetized, blocked to flowing.Â
Duminil-Copin advanced the rigorous mathematical theories of these models using new techniques. For magnetism, he introduced "random currents" to represent interactions between the tiny magnets. For percolation, he generalized methods from studying electrical networks.
Armed with these tools, he tackled long-unsolved problems. For example, he proved the transition in a 3D Ising model is smooth and continuous, resolving a 30 year old problem. He also showed transitions can be precisely pinpointed in the percolation model, rather than being gradual.
Beyond specific models, Duminil-Copin introduced broad techniques like "parafermions" - observables that blend properties of different particles. With parafermions, he could move between the physics models and theory to derive exact values for properties at transitions.
While the math is intense, the impact is profound. Duminil-Copin's work provides deep insights into how matter changes states. His solutions and creativity have advanced a field pivotal to physics and materials science. In revealing math underlying physical reality, he exemplifies science at its best.