Hidden Patterns: Kevin Hu Navigates the Mathematics of Disordered Systems

The atoms in crystals, the neural networks that power AI and the people in social networks all have many ways in which they can change. These ‘degrees of freedom’ — such as the orientation of a crystal’s atom or the friendships on a social media site — make the dynamics of these systems incredibly difficult to understand. Simons Society of Fellows Junior Fellow Kevin Hu combines insights from math and physics to elucidate these dynamics.

Hu, an expert in statistical physics, studies structure and disorder in high-dimensional systems. He earned his doctorate in applied mathematics from Brown University in 2025 and is now a postdoctoral researcher mentored by Dan Lacker at Columbia University in New York City.

We recently sat down with Hu to discuss the inherent joy in science, the importance of following one’s own interests and the benefits of the Simons Foundation community.

What’s your research about?

I am interested in problems in probability theory, especially those with connections to systems in mathematical physics. These are sometimes referred to as ‘disordered’ systems, since they have some underlying randomness. A good example is the Ising model, which is a famous toy model of magnetism. Recently, I have been interested in some problems in quantum field theory and random matrix theory.  Many beautiful patterns emerge (seemingly out of nowhere) in disordered systems, which is quite exciting. Understanding these patterns could be useful in explaining complicated phenomena.

Why are you interested in this line of research?

The aggregation of many independent random effects tends to produce rather predictable behavior. This is the most important fact in probability theory. Think, for instance, of a sequence of coin flips. Each coin flip is totally random, but one knows that about half of them should come up heads, and the rest should be tails.

These ideas translate to much more complicated settings as well. Instead of thinking about a large sequence of coin flips, one can think of a large collection of water particles. Once again, each individual water molecule appears to move totally chaotically, but the behavior of the overall fluid appears deterministic and is in principle predictable (though charting fluid dynamics is by no means an easy task). In the last 40 years or so, these ideas have been applied to a whole bunch of systems, including neural networks, social networks, and so on. Of course, these systems are much harder to analyze than a sequence of coin flips, but miraculously, they are still tractable.

What did you learn about disordered systems during your Ph.D.?

During my Ph.D., I considered disordered systems that were spiked with some underlying structure, which we called “sparse networks.” This is part of a larger research program of my Ph.D. advisor, Kavita Ramanan, at Brown.

Some of the examples mentioned before, like neural and social networks, contain many random components but interact in a much more structured way than, say, particles in a fluid. We were interested in what kind of dynamics this combination of structure and disorder would produce in sparse networks.

A common paradigm in physics is energy minimization. A useful caricature is that of a ball rolling down a hill. As it moves down the hill, it converts its potential energy into kinetic energy. However, the kinetic energy will be dissipated by friction, turning into heat or sound and becoming unusable. In other words, the free energy of the ball decreases until it comes to a stop at the bottom of the hill, when it does not have any more usable energy.

In the language of statistical mechanics, the usable energy is known as free energy. Many systems behave in ways that minimize free energy, and by studying free energy, one can obtain a simple description of the systems’ dynamics. The last example was pretty easy, and it has a unique equilibrium state (the ball at the bottom of the hill). This is common, even in some more complicated settings: There is often a nice free energy function with a unique equilibrium.

We found, however, that the free energy function in sparse networks has to be modified in a suitable way to account for the influence of additional structure. This allows for richer behavior than in systems without structure. For instance, you may now have several equilibria. These complications are often the source of the patterns I mentioned before.

What’s a problem you are working on now?

I am still working on some of the models I studied during my Ph.D., but during my postdoc, I’m branching out to other areas of probability theory and mathematical physics.

I recently completed a project with Manuel Arnese, who is a Ph.D. student of Dan’s, on another kind of disordered system known as a spin glass. A spin glass is a material where small magnetic pieces interact in random, conflicting ways, so instead of forming a simple magnetic order, they get stuck in more complicated states.

The study of spin glasses has become a large area of research in math and physics with lots of big results, which led, for example, to a Nobel Prize for Giorgio Parisi and an Abel Prize for Michel Talagrand. We worked on a dynamic version of a spin glass model known as the Langevin spin glass, which has also been proposed as a model for neural networks.

These dynamic models have been studied much less than the usual spin glass model, and we were interested in studying the correlation properties of the Langevin spin glass. Building on some work from the ’90s, we found that while individual particles have complex behavior, they evolve in a way that is nearly independent from each other. This phenomenon is known as quenched propagation of chaos, and a big part of our work is quantifying how decorrelated the particles are. It was a really fun project. It was satisfying to learn a lot of new math and physics and apply some of my existing toolset to an interesting problem.

What’s your approach to science?

Recently, I came across the proceedings of the 70th birthday celebration for Paul Dirac, a British theoretical physicist who was a titan of the 20th century. One of the organizers, Jagdish Mehra, said in the opening address that “the characteristic of science, if not infancy, is perpetual adolescence.” I think Mehra was referring to how science itself is constantly evolving, but to me, he is also describing the experience of doing research. I am free to be curious and explore wonderful ideas, and in this sense, I feel like I am enjoying an extended childhood.

How does that sense of playfulness relate to your Simons experience?

I really enjoy being a fellow with the Simons Society of Fellows, because it’s great to be around people who are passionate about science. Our cohort is very close; we go out for drinks and see each other socially a lot.

When the fellowship began, I thought I would only understand the math lectures. In fact, the neuroscience and biology talks are also fascinating, and if I don’t understand something, I can ask about it at dinner afterward. It’s a great community.