Wishart Ensemble & Marchenko–Pastur Law

Random matrix eigenvalue distributions · bulk universality · Tracy-Widom edge

Parameters

Spectral Bounds

λ₋ (lower edge):-
λ₊ (upper edge):-
Empirical max:-
Empirical min:-
Variance σ²:-
Marchenko-Pastur (1967):
ρ(λ) = √[(λ₊-λ)(λ-λ₋)] / (2πγλ)
where λ± = (1 ± √γ)²

For γ < 1: eigenvalue support is a single arc. For γ ≥ 1: a point mass at 0 with weight 1−1/γ appears.

Tracy-Widom: largest eigenvalue fluctuates as n^(−2/3) with TW₁/TW₂ distributions.