Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 22 (2026), 047, 20 pages      arXiv:2506.18673      https://doi.org/10.3842/SIGMA.2026.047

Asymptotic Expansions of Gaussian and Laguerre Ensembles at the Soft Edge III: Generating Functions

Folkmar Bornemann
Department of Mathematics, Technical University of Munich, 80290 Munich, Germany

Received January 08, 2026, in final form April 27, 2026; Published online May 13, 2026

Abstract
We conclude our work on asymptotic expansions at the soft edge for the classical $n$-dimensional Gaussian and Laguerre ensembles, now studying the gap-probability generating functions. We show that the correction terms in the asymptotic expansion are multilinear forms of the higher-order derivatives of the leading-order term, with certain rational polynomial coefficients that are independent of the dummy generating function variable. In this way, the same multilinear structure, with the same polynomial coefficients, is inherited by the asymptotic expansion of any linearly induced quantity such as the distribution of the $k$-th largest level. Whereas the results for the unitary ensembles are presented with proof, the discussion of the orthogonal and symplectic ones is based on some hypotheses. To substantiate the hypotheses, we check the result for the $k$-th largest level in the orthogonal ensembles against simulation data for choices of $n$ and $k$ that require as many as four correction terms to achieve satisfactory accuracy.

Key words: generating functions; soft-edge scaling limit; Gaussian and Laguerre ensembles; Wishart distribution; asymptotic expansions; Painlevé II; Fredholm determinants.

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