Phase transition for the smallest eigenvalue of high-dimensional sample correlation matrices

Abstract

We study the smallest nonzero eigenvalue of the sample correlation matrix $\mathbf{R}_n$ formed from a $p_n \times n$ data matrix with i.i.d. real entries $\xi$ of mean zero and unit variance, in the high-dimensional regime $p_n / n \to \phi \in (0, \infty) \setminus \{1\}$. In the tall regime $\phi>1$, we prove almost-sure convergence of $\lambda_n (\mathbf{R}_n)$ to the lower Mar\v{c}enko--Pastur edge $\lambda_- = (1 - \sqrt{\phi})^2$ without additional moment assumptions. In the wide regime $\phi<1$, we establish a phase transition at the third-order tail scale. If $t^3 \mathbb{P}\{\lvert \xi \rvert>t\} \to 0$ as $t \to \infty$, the smallest eigenvalue $\lambda_{p_n} (\mathbf{R}_n)$ converges in probability to $\lambda_-$. While if $t^3 \mathbb{P}\{\lvert \xi \rvert>t\} \to \infty$, then $\lambda_{p_n} (\mathbf{R}_n)$ converges in probability to zero. At the critical scale $t^3 \mathbb{P}\{\lvert \xi \rvert>t\} \to \kappa \in (0, \infty)$, the point process of eigenvalues in the lower gap $(0, \lambda_-)$ converges in distribution to a Poisson random measure with explicit intensity. In this critical regime, we also identify the nondegenerate limiting distribution of $\lambda_{p_n} (\mathbf{R}_n)$, which has a continuous density on $(0, \lambda_-)$ and a positive atom at $\lambda_-$.