The One‐Dimensional Log‐Gas Free Energy Has a Unique Minimizer

Source abstract

We prove that, at every positive temperature, the infinite‐volume free energy of the one‐dimensional log‐gas, or beta‐ensemble, has a unique minimizer, which is the Sine‐beta process arising from random matrix theory. We rely on a quantitative displacement convexity argument at the level of point processes, and on the screening procedure introduced by Sandier‐Serfaty. © 2021 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC.