Staircase phase transitions for the largest eigenvalue of heavy-tailed sample correlation matrices

Abstract

We establish staircase phase transitions for the largest eigenvalue of heavy-tailed sample correlation matrices formed from a $p_n \times n$ data matrix with i.i.d. real entries of mean zero and unit variance, allowing an infinite fourth moment. In the proportional regime $p_n / n \to \phi \in (0, \infty)$, the first-order asymptotics depend jointly on the aspect ratio and the entry tail. The transitions are driven by collisions of large entries in distinct rows of a common column. The first collision order capable of producing a separated upper outlier is $k_* (\phi) = \lfloor \sqrt{\phi} \rfloor + 2$, yielding the critical tail exponent $\alpha_* (\phi) = 2 + 2 / k_* (\phi)$. This exponent decreases in steps as $\phi$ increases, creating a staircase boundary between convergence to the upper Mar\v{c}enko--Pastur edge and successive outlier levels. At exact critical tail scales, the point process of eigenvalues above the upper edge or the preceding deterministic level converges to a Poisson point process. The resulting nondegenerate limiting laws for the largest eigenvalue connect adjacent phases and have a positive atom at this baseline. If every fixed collision order is supercritical, the largest eigenvalue diverges in probability despite finite entry variance.