On the structure of marginals in high dimensions

Let $G, G_1,\dots,G_N$ be independent copies of a standard gaussian random vector in $\mathbb{R}^d$ and denote by $\Gamma = \sum_{i=1}^N \langle G_i,\cdot\rangle e_i$ the standard gaussian ensemble. We show that, for any set $A\subset S^{d-1}$, with exponentially high probability, $$\sup_{x\in A} \frac{1}{N}\sum_{i=1}^N \big| (\Gamma x)^\sharp_i - q_i\big| \le c \frac{ \mathbb{E} \sup_{x\in A} \langle G,x\rangle + \log^2N }{\sqrt N }.$$ Here each $q_i$ is the $\frac{i}{N+1}$-quantile of the standard normal distribution and $(\Gamma x)^\sharp$ denotes the monotone increasing rearrangement of the vector $\Gamma x$. The estimate is sharp up to a possible logarithmic factor and significantly extends previously known bounds. Moreover, we show that similar estimates hold in much greater generality: after replacing the gaussian quantiles by the appropriate ones, the same phenomenon persists for a broad class of random vectors.