Explicit characterization of the Gaussian Wigner sparse-eigenvalue limit

Determine an explicit expression for the function E(c) governing the first-order limit of the largest eigenvalue among principal minors of a Gaussian Wigner matrix when the minor-size ratio satisfies k/ncin (0,1].

Background

For Gaussian Wigner matrices, the paper proves that when the principal-minor size k grows proportionally to the matrix dimension n, with k/n tending to c in (0,1], the normalized maximum M_{n,k}/sqrt(n) converges in probability to a deterministic function E(c). The function is shown to be increasing and concave, with E(1)=2 and specified asymptotic behavior as c approaches 0 and 1.

Despite these qualitative and asymptotic properties, the paper does not derive a closed-form or otherwise explicit representation of E(c). An explicit characterization would refine the established first-order limit and clarify the interpolation between sparse extreme-value behavior and the full-matrix spectral edge.