Limiting spectral distributions of large consistent rank correlation matrices

We study random matrices whose entries are obtained by applying consistent rank correlations, such as Hoeffding’s $D$, pairwise to a high-dimensional random vector with mutually independent components. Prior work has shown that, in the proportional high-dimensional regime, the empirical spectral distributions of large Kendall’s tau and Spearman’s rho matrices converge weakly almost surely to the Marchenko–Pastur law. By contrast, we prove that for consistent rank correlations such as Hoeffding’s $D$, the limiting spectral distribution is given by the semicircle law. Our result thus generalizes a recent work of Dong, Han, and Yao (2025), who considered the special case of Chatterjee’s rank correlation and established the first semicircle law for a large correlation matrix in the proportional regime.