A broad introduction to my research area
Phase transitions
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.
Other models in my research area
Percolation is one model that researchers in lattice models are interested in, but there are many others. The properties of the phase transition can be very different for different models; the idea is that they cover different phenomena in the real world. Below, we simply give a few more models, and list some of their interesting features.