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

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 ϕ\in (0, \inft