A broad introduction to my research area
This page gives an informal introduction to the mathematics of my research area for a wide audience. It is not a fully precise or historically accurate account, but it should give an idea of some intriguing concepts.
Phase transitions in physics
A phase transition occurs when a small change in one quantity causes a system to behave in a fundamentally different way. A familiar example is water freezing: as the temperature passes through zero degrees Celsius, liquid water turns into solid ice. Other examples include water boiling into steam and magnets losing their magnetism when heated. Although these systems look very different, mathematics reveals common ideas behind their sudden changes.
Left. Ice crystals, in which water molecules form an ordered solid structure. Right. A bottle of supercooled water freezes almost instantly when disturbed: a striking example of a sudden change from liquid to solid.
Mathematical models for phase transitions
To understand phase transitions, mathematicians often begin with a simplified model. One important example is percolation. Imagine a grid whose connections are chosen independently at random: each connection is either open or closed. The grid can represent water moving through porous material, links in a communications network, or information spreading through a social network.
We ask a simple question: If water is poured in at the top, can it reach the bottom? This happens precisely when the open connections contain a path from the top row to the bottom row. Click the button below to generate new random grids.
Simulation of the percolation model. Each connection is independently open with probability 50%. Blue shows every open connection that can be reached from the top.
Where is the phase transition?
Let p be the probability that any given connection is open. When p is small, long open paths are rare. When p is large, they are common. Between these two regimes lies a special value, called the critical probability, where the behavior changes sharply. Use the sliders below to vary both p and the size of the grid. As the grid grows, the contrast between probabilities below and above 50% becomes increasingly clear.
Percolation at different probabilities. Below 50%, water is increasingly unlikely to cross a large grid; above 50%, it is increasingly likely to do so. At exactly 50%, neither outcome dominates. These experiments suggest that the transition occurs at 50%, but a simulation is not a proof.
Why is 50% special?
For the infinite square grid, mathematicians have proved that the critical probability is exactly 50%. The complete proof requires ideas beyond this introduction, but a beautiful symmetry argument already explains why 50% is the natural candidate. It proves the following exact statement about the finite grids used on this page.
Theorem. For each grid shape used above, when p = 50%, the probability that water reaches the bottom is exactly 50%.
To see the symmetry, return to the original 9x10 grid and draw a second, interlaced grid. Whenever a water connection is closed, we open the perpendicular connection on this second, or dual, grid. The result can be viewed as a game between two players. In the simulation below, you can see how this happens; you can switch between a simple view showing only the water (equivalent to the first simulation above) and the enhanced view as a game.
The transportation game at 50%. The water player (blue) tries to cross from top to bottom along open water connections. The olive-oil player (green) tries to cross from left to right along dual connections. Every closed water connection creates an open olive-oil connection. Consequently, exactly one player wins each round.
At p = 50%, an olive-oil connection is open with probability 100% - 50% = 50%. After rotating the dual grid by a quarter turn, the olive-oil player faces the same random problem as the water player. The game therefore treats the two players symmetrically:
- exactly one player wins each round, and
- both players have the same probability of winning.
The two winning probabilities must add up to 100%, so each must equal 50%. This proves the theorem above and illustrates a powerful theme in mathematics: symmetry can turn a complicated random system into a short, exact argument.
The Ising model
Percolation is only one mathematical model for phase transitions, but there are many others. The properties of the phase transition can be very different from one model to another, which motivates the study of each model individually. Below, we present a few more interesting models, without giving a full mathematical definition.
The Ising model describes a material as a grid of tiny magnets. Each magnet can point in one of two directions, shown below by + and -. There are two competing effects: magnetic interactions make nearby magnets prefer the same direction, while thermal fluctuations push them in random directions. The parameter β controls the relative strength of these two effects. The simulation below shows how the picture changes as they compete.
Simulation of the Ising model. Each particle is a tiny magnet. The β slider controls how strongly nearby magnets influence each other: a small β gives more disorder, while a large β makes nearby magnets more likely to agree. The most interesting changes happen around β ≈ 0.44. Press “Run” to watch the magnets change direction over time.
When β is small, the magnets look mixed and disorderly. When β is large, most magnets point in the same direction. This explains why magnetic materials lose their magnetism at a certain temperature: when β is small, the “micro-magnets” point in random directions, so their effects cancel each other out. At large β, they “work together” by pointing in the same direction, producing a much larger magnetic effect. The phase transition occurs around β ≈ 0.44.
The XY model
The XY model is defined in much the same way as the Ising model. The difference is that its tiny magnets can point in any direction, rather than only up or down. This continuous freedom leads to a very different kind of phase transition. In particular, the arrows can form swirling patterns called vortices, which play a central role in the model's phase transition.
Simulation of the XY model. Each site carries an arrow that may point in any direction. Neighboring arrows prefer to point in similar directions. Unlike in the Ising model, the arrows never settle into one common direction across the whole grid. Nevertheless, their behavior changes sharply near β ≈ 1.12.
The square ice model
The square ice model was originally introduced to describe the arrangement of water (H2O) molecules in ice. An oxygen atom sits at each point of a square grid. Each oxygen atom has two attached hydrogen atoms, chosen from the four grid directions. At the same time, neighboring oxygen atoms cannot both claim the same hydrogen position. The square ice model studies random arrangements of molecules that satisfy these local rules. The simulation below gives a visual representation of the model.
Simulation of the square ice model. Each molecule arrangement can also be viewed as a “height function”: an assignment of numbers to the squares of the grid. The height function can be calculated as follows: when you walk around an oxygen atom in the clockwise direction, the numbers decrease by 1 when you cross a hydrogen atom, and increase by 1 when there is no hydrogen atom.
This model does not have a parameter, but it still exhibits behavior similar to a critical model (like percolation at 50%). In the heights view, the numbers fluctuate, and it is expected that on a large grid, the colored regions gain more symmetry: the model is discrete, but on the large scale these color clusters should become invariant under rotations! This means that if you rotated the picture, it would look qualitatively the same. This “symmetry gain” is something that mathematicians are hoping to prove, as it is a very physically relevant property. This question is still unresolved for the square ice model.