Gaussian-width gradient complexity, reverse log-Sobolev inequalities and nonlinear large deviations

Understand

We prove structure theorems for measures on the discrete cube and on Gaussian space, which provide sufficient conditions for mean-field behavior.

• These conditions rely on a new notion of complexity for such measures, namely the Gaussian-width of the gradient of the log-density.
• On the cube $\{-1,1\}^n$, we show that a measure $\nu$ which exhibits low complexity can be written as a mixture of measures $\{\nu_\theta\}_{\theta \in \mathcal{I}}$ such that: i.
• for each $\theta$, the measure $\nu_\theta$ is a small perturbation of $\nu$ such that $\log \tfrac{d \nu_\theta}{d \nu}$ is a linear function whose gradient is small and, ii.