.png)
4 days ago
1
Each week Quanta Magazine explains one of the most important ideas driving modern research. This week, math staff writer Joseph Howlett surveys phase transitions in mathematical systems.
In the real world, physical systems can undergo rapid, dramatic changes: Cool a liquid and it will crystallize into a solid; heat a magnet and it will suddenly lose its magnetism.
But it turns out that these abrupt changes, known as phase transitions, happen in abstract math contexts as well. When mathematicians build a simple system with just a few rules, they often discover that, at a certain point, surprising patterns suddenly emerge. These mathematical phase transitions can give mathematicians a window into how real physical systems work, while also providing them with important insights into how complex behavior can arise from the most straightforward laws.
Take percolation, a stripped-down mathematical model of how water might move through a sponge or some other porous medium. Start with an infinite grid of dots. Between each pair of adjacent dots, you might decide to draw a line, or edge. Use a weighted coin to determine your choice: If it lands on heads (which might happen with a probability of 0.01% or 1% or 10%, depending on how your coin is weighted), draw the edge; if it lands on tails, do nothing. Repeat this process for every pair of adjacent dots in your grid. What kinds of structures are you likely to end up with? Specifically, how likely is it that an infinitely long path will form on the grid?
The answer depends on the weight of your coin. Below a certain critical weight, it’s basically impossible for the grid to have an infinite path. (In the real world, the water will get stuck in the sponge.) But increase the weight ever so slightly above that threshold, and it becomes impossible for the grid not to have one. (The water will make it all the way through.)
At that threshold, then, a phase transition occurs. The behavior of the system below or above the threshold looks radically different.
Though easier to study than their real-life counterparts, these phase transitions — which pop up in all sorts of mathematical systems — reveal how order and chaos can arise side by side in even the simplest contexts.
What’s New and Noteworthy
Many questions about percolation remain open even after decades of progress. In 2023, mathematicians figured out precisely what happens at the transition point where the state of the system flips — a calculation sought since the 1970s. That same year, two mathematicians proved that for a certain three-dimensional version of percolation, you only need to study part of the grid to understand the whole. The grid’s local structure contains enough information about its global properties.
What happens if the coin flips aren’t independent, and the result of one flip influences the next? This situation gives mathematicians an even broader class of percolation problems to probe. But these are much harder. For a while, the field was stuck — until the pioneering work of a mathematician named Hugo Duminil-Copin revived it. He was awarded a Fields Medal, math’s highest honor, for this work in 2022. And he went on to prove that many of these systems exhibit a powerful suite of symmetries, called conformal invariance, at their critical point.
More generally, phase transitions tend to show up wherever probabilities are involved. You can use a similar coin-flipping procedure to build a graph — a collection of points, or nodes, connected by edges — entirely at random. And it turns out that once you add a certain number of edges, all sorts of structures suddenly emerge. Pass a certain threshold, and you can guarantee that your graph will contain a triangle, or a chain of edges called a Hamiltonian path, or basically any other pattern (so long as that pattern satisfies one simple property). In 2022, two young mathematicians at Stanford University proved a sweeping statement about these thresholds, called the Kahn-Kalai conjecture. The statement was so broad that many mathematicians thought it couldn’t possibly be true.
Phase transitions don’t have to involve points and edges, as they do in graphs and percolation systems. They exist in geometry as well. In the 1950s, for instance, the mathematician John Nash found a sharp transition point between smoothness and roughness in shapes. In particular, he studied a process by which shapes can be crumpled without creasing. Mathematicians have continued to study the thresholds at which shapes buckle and transform.
In all these cases, phase transitions beckon mathematicians toward the messiness of the real world. By examining these critical points of change, researchers can study the outermost fringe of mathematical order, where simplicity and complexity touch.
In this feature column from the American Mathematical Society, David Austin goes deep on percolation, summarizing its main open questions and explaining conformal invariance.
Percolation is also important for understanding social networks. This 2021 article from Scientific American explains what lessons percolation holds for the mathematics of human interaction.
I’ve only ever heard physicists (mis)pronounce Ernst Ising’s last name as “EYE-sing,” whereas mathematicians all seem to know to correctly say “EE-sing.” I still haven’t figured out why this difference persists, but it led me to this fascinating biographical writeup by his son Thomas Ising. Ernst Ising was only a student in the 1920s when he worked on his eponymous model, a sort of cartoon picture of magnetic behavior. Shortly afterward, his life was upended — as a Jewish mathematician, he was fired when Hitler came to power and eventually fled to the United States.
