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

TL;DR

This paper analyzes the asymptotic expansions of Gaussian and Laguerre ensembles at the soft edge, focusing on generating functions and correction terms, with results supported by simulations.

Contribution

It introduces a multilinear form structure for correction terms in asymptotic expansions of gap-probability generating functions for classical ensembles.

Findings

Correction terms are multilinear forms of higher derivatives with rational polynomial coefficients.

The same multilinear structure applies to linearly induced quantities like the distribution of the k-th largest level.

Simulation data supports the hypotheses for orthogonal ensembles, validating the theoretical results.

Abstract

We conclude our work [arXiv:2403.07628, arXiv:2503.12644] 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.