If you drop a ball into a rugged landscape and let it roll, it will eventually reach a minimum. What height will it arrive to? With experiments one can say for each specific landscape and initial condition, but with less work one can predict what happens on average. Such predictions are typical of dynamical mean field theory.
But what if you change the kind of ball, or replace the ball with a mouse or robot? One can treat each set of behavior in turn and produce a zoo of predictions. However, there are certain features of high dimensional random landscapes that constrain the height reached by a large variety of agents.
Very intelligent rollers will find themselves stuck at a height given by the overlap gap property, which bounds the performance of reasonable algorithms given reasonable time using geometric properties of configuration space. However, most approaches to these problems are not very intelligent: they behave more like the stupid ball. We try to predict or bound where such approaches will land.
The above is quite abstract, but such questions are important in the physics of glasses and spin glasses, in the statistics of large inference problems, and in machine learning. We work at the intersection of these fields.
The renormalization group of Kadanoff and Wilson famously explained Widom’s scaling, and has found application in nearly every field of physics. Mostly it is used to justify the blind application of Widom scaling in diverse settings, where critical exponents are fit by eyeballing an appropriate scaling collapse.
However, the renormalization group does not predict exponents: rather, exponents are a trait of the most commonly encountered type of critical point. But the scaling functions that describe critical singularities can take more diverse and strange forms than simple power laws.
In addition, there is little besides experience to justify the renormalization group’s application in general nonequilibrium systems. What kinds of nasty surprises await in their singular structure and corrections to scaling?
Other topics in statistical mechanics and physics more broadly interest us.
What trait do magnetic fluids and maximally correlated local hidden variable theories share? As it happens, a zoo of interesting morphological phases and an effective description in terms of a novel integral transformation.
Landau–Ginsburg theory sheds light on the possible electronic order of a mysterious phase transition in a heavy fermion material, and a deterministic cost function designed to find Hadamard matrices produces glassy behavior.
Surprises lurk in the correlation functions of Monet’s paintings…
(from my collaborator)