Probability and Mathematical Physics Seminar

---

Area and Perimeter Laws in Lattice Yang–Mills Theories

Speaker: Ron Nissim, MIT

Location: Warren Weaver Hall 1302

Date: Friday, October 2, 2026, noon

Synopsis:

Lattice Yang–Mills theory was introduced by Ken Wilson in 1974, in part to give a mathematical framework for understanding quark confinement—the physical phenomenon that quarks are not observed in isolation. Wilson proposed the area law for Wilson loop expectations as a criterion for confinement: for large rectangular loops, the expectation should decay exponentially in the area enclosed by the loop, rather than merely in its perimeter.

I will start by briefly reviewing joint work with Scott Sheffield and Sky Cao in which we substantially expanded the regime of parameters where an area law is proved. A key ingredient in this work is the center symmetry of the model. The main focus of the talk will be recent and upcoming solo work in the opposite direction: when the relevant Wilson loop is insensitive to the center of the gauge group, we prove a perimeter-law lower bound, ruling out Wilson's confinement criterion. The result holds in dimensions three and higher and, importantly, extends to the weak-coupling regime relevant to the continuum theory. I will also briefly discuss a Lean formalization of the proof and the role that AI tools played in its development.