Theory
For X (p×n) with i.i.d. N(0,σ²/n) entries, the empirical spectral distribution of XX^T → MP law as p,n→∞ with p/n → λ.
ρ_MP(x) = √[(λ₊-x)(x-λ₋)] / (2πλσ²x)
λ± = σ²(1 ± √λ)²
BBP transition (Baik-Ben Arous-Péché): spikes with strength > σ/√λ escape the bulk — detectable signal!