Uniform logarithmic Sobolev inequalities for the 2D Coulomb gas at the diffusive temperature scale

What the math gives to ML

The paper proves a dimension-uniform logarithmic Sobolev inequality for a two-dimensional Coulomb gas whose quadratic confinement and logarithmic repulsion are scaled so that the interaction remains mean-field sized as the particle count grows. This gives a principled way to build particle banks, prototype sets, or mixture-component locations that resist collapse while retaining Langevin mixing behavior that should not deteriorate sharply with bank size. The most direct neural experiment is to use these particles for prototype-based routing or latent-variable initialization and compare their diversity, sampler mixing, and downstream accuracy against Gaussian initialization and ordinary repulsive regularization.