Random matrix ensembles have edges — boundaries of the eigenvalue distribution where the density drops to zero. At the soft edge, eigenvalues thin out gradually, and the local statistics are governed by the Tracy–Widom distribution. But the approach to this limit — the correction terms — contains information that the limit itself discards.
The paper on edge density expansions for Gaussian and Laguerre ensembles (arXiv: 2603.22974) uncovers integrable structure in these correction terms using scalar differential equations rather than integral representations.
The differential equation approach has a technical advantage: in the soft-edge scaling variable, the equation isolates the function of N (the matrix size) that serves as the expansion parameter. This makes the corrections computable term by term. At the hard edge of Laguerre ensembles — where eigenvalues pile up against zero — the same technique applies, yielding explicit second-order corrections for unitary symmetry and first-order corrections for orthogonal and symplectic cases.
The surprise is that integrable structure persists beyond the three classical symmetry classes. For Dyson index β = 6 (neither orthogonal, unitary, nor symplectic), the same asymptotic expansion features appear, suggesting that integrability extends to general β ensembles.
The through-claim: the corrections to universality are themselves universal. The Tracy–Widom limit is the zeroth-order answer. The correction terms — how the finite-N density departs from the limit — have their own integrable structure that transcends the specific ensemble. Integrability governs not just the limit but the approach to it.
2603.22974. Random matrix theory / soft edge / Tracy–Widom distribution / integrable structure / asymptotic expansions.