Maximal boundary excursion of eigenvalues

Determine whether, for non-Hermitian ensembles with density proportional to exp{-N Tr W(M,M*)} dM, connected limiting support with regular boundary, and limiting density positive and continuous on the boundary, the maximal distance D_N of an eigenvalue from the limiting spectral support satisfies D_N ~ sqrt(log N/(4πNρ_*)), where ρ_* is the minimum boundary density.

Background

The conjecture concerns the largest outward excursion among all eigenvalues. The paper argues heuristically that a single-eigenvalue displacement by distance d outside the support has probability of order exp{-2πNρ(z_0)d²}, where ρ(z_0) is the limiting density at the nearest boundary point.

Because only eigenvalues in a boundary layer of width of order N^{-1/2} are effective candidates, the number of relevant candidates is estimated to be of order √N rather than N. Balancing this candidate count against the one-particle tail yields the proposed sqrt(log N/N) scale, with the smallest boundary density determining the dominant excursion location.