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The Density of Eigenvalues Seen from the Soft Edge of Random Matrices in the Gaussian $\beta $-ensembles
We characterize the phenomenon of "crowding" near the largest eigenvalue λ max of random N ×N matrices belonging to the Gaussian β-ensemble of random matrix theory, including in particular the Gaussian orthogonal (β = 1), unitary (β = 2) and symplectic (β = 4) ensembles.We focus on two distinct quantities: …