Large complex correlated Wishart matrices: Fluctuations and asymptotic independence at the edges

Abstract

We study the asymptotic behavior of eigenvalues of large complex correlated Wishart matrices at the edges of the limiting spectrum. In this setting, the support of the limiting eigenvalue distribution may have several connected components. Under mild conditions for the population matrices, we show that for every generic positive edge of that support, there exists an extremal eigenvalue which converges almost surely toward that edge and fluctuates according to the Tracy-Widom law at the scale N 2/3 . Moreover, given several generic positive edges, we establish that the associated extremal eigenvalue fluctuations are asymptotically independent. Finally, when the leftmost edge is the origin (hard edge), the fluctuations of the smallest eigenvalue are described by mean of the Bessel kernel at the scale N 2 .

Universal edge fluctuations of discrete interlaced particle systems

“…β n K n ((u n , r n ), (v n , s n )) =        J n − Φ n ; for cases (1-4), J n + Φ n ; for cases (5,6), −J n − Φ n ; for cases (7,8), −J n + Φ n ; for cases (9-12), where we define: ; for cases (9)(10)(11)(12).…”

“…(ii) {y ∈S 1,n ∪S 3,n : y ∈ Γ • n } equalsS 1,n for (1-6), and equalsS 3,n for (7-12). (iii) {y ∈ (S 1,n ∪S 3,n ) ∩ V n : y ∈ γ • n ∩ Γ • n } equals (V U (n) ) ∩ P n for (1-3) and (6) when v n ≥ u n , equals (V U (n) ) ∩ P n for (7) and (10)(11)(12) when v n + s n ≤ u n + r n , and is empty otherwise. (iv) {y ∈ V n \ U n : y ∈ γ • 2,n } equals V U (n) for (6) when v n ≥ u n , equals V U (n) for (7) when v n + s n ≤ u n + r n , is empty for (6) when u n > v n , and is empty for (7) when v n + s n > u n + r n .…”

“…(v) Either case (6) or (7) is satisfied when t ∈ (χ+η −1, χ) and L t ∈ {K (9) is satisfied when t < χ + η − 1 and L t = J 4 . (viii) Either case (10) or (11) is satisfied when t < χ + η − 1 and L t ∈ {K…”

“…(2) γ + 1,n ∩ B(t, n −θ |q n |) and Γ + 1,n ∩ B(t, n −θ |q n |) are straight lines from t to points d 1,n ∈ ∂B(t, n −θ |q n |) andã 1,n ∈ ∂B(t, n −θ |q n |) (respectively) for which, letting Arg(·) be the principal value of the argument, Arg(d 1,n − t) = π 3 + O(n − 1 3 +θ ) and Arg(ã 1,n − t) = 2π 3 + O(n − 1 3 +θ ) in cases (1,2,7,8,9,10), and Arg(d 1,n − t) = 2π 3 + O(n − 1 3 +θ ) and Arg(ã 1,n − t) = π 3 + O(n − 1 3 +θ ) in cases (3,4,5,6,11,12). (3) Re(f n (w)) ≤ Re(f n (d 1,n )) for all w ∈ γ + 1,n \ B(t, n −θ |q n |) and w ∈ γ + 2,n .…”

Deterministic Parallel Analysis: An Improved Method for Selecting Factors and Principal Components

“…See Hachem et al . () and Lee and Schnelli () for recent results. In Gaussian white noise, Johnstone () showed that the scale parameter isso fluctuations in the largest eigenvalue quickly become negligible.…”