A magnet is slowly heated. At low temperatures, all its microscopic arrow-like spins point in the same direction: that's order. At high temperatures, they dart about chaotically. But there is a special point, the critical point, where order is just about to vanish and the system freezes in indecision. It is precisely at this point that physicists have peered inside not one but two identical copies of the system.
They took two 'replicas'—two independent snapshots of the same magnet in its critical state, and asked: how much will their patterns coincide? The measure of coincidence is called the overlap. It turns out that in large systems, the overlap decreases according to a clear law: it is proportional to the cube root of the number of particles. And if you scale the overlap up by the same factor, its random distribution converges to a universal measure. This measure is described by the reflected Airy process — a mathematical object born from problems of crystal growth and airport queues.
This result is not just a pretty abstraction. It answers a concrete question posed by mathematician Talagrand: what is the expected squared fluctuation of the overlap? The answer is that it grows as the 2/3 power of the system size. For one of the models (the spherical one), the proof relies on a connection to the edge of the spectrum of random matrices — the very ones that describe the energy levels of complex nuclei. For the other (the Ising model), a clever trick is used: it turns out that at the critical point, the behavior of a system with a hard constraint (spins only up/down) and a system with a soft constraint (spins on a sphere) is practically indistinguishable. This is like comparing a crowd of people locked in a room with corners to a crowd on a circular plaza — at the moment of greatest uncertainty, their movements follow the same rhythm.
Thus, a freezing magnet reveals the secret not only of its own internal structure but also of the language nature uses to speak about disorder on the threshold of order.
The distribution governing the overlap first appeared in extreme value statistics (for example, to describe record floods or the tallest trees in a forest) but now it has been found in the world of spin glasses.