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 $\mathrm{\Gamma }={\sum }_{i=1}^N⟨G_i,\cdot ⟩e_i$ the standard gaussian ensemble. We show that, for any set $A\subset S^{d-1}$, with exponentially high probability, $$\underset{x\in A}{\sup }\frac{1}{N}\sum _{i=1}^N\mid (\mathrm{\Gamma }x{)}_i^{\mathrm{♯}}-q_i\mid \le c\frac{\mathbb{E}\underset{x\in A}{\sup }⟨G,x⟩+{\log }^{2}N}{\sqrt{N}}\mathrm{.}$$ Here each $q_i$ is the $\frac{i}{N+1}$-quantile of the standard normal distribution and $(\mathrm{\Gamma }x{)}^{\mathrm{♯}}$ denotes the monotone increasing rearrangement of the vector $\mathrm{\Gamma }x$. 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.