Asymptotics of Tracy-Widom distributions and the total integral of a Painleve II function

The Tracy-Widom distribution functions involve integrals of a Painlev'e II function starting from positive infinity. In this paper, we express the Tracy-Widom distribution functions in terms of integrals starting from minus infinity. There are two consequences of these new representations. The first is the evaluation of the total integral of the Hastings-McLeod solution of the Painlev'e II equation. The second is the evaluation of the constant term of the asymptotic expansions of the Tracy-Widom distribution functions as the distribution parameter approaches minus infinity. For the GUE Tracy-Widom distribution function, this gives an alternative proof of the recent work of Deift, Its, and Krasovsky. The constant terms for the GOE and GSE Tracy-Widom distribution functions are new.

💡 Research Summary

The paper revisits the celebrated Tracy‑Widom (TW) distribution functions, which describe the fluctuations of the largest eigenvalue in the edge scaling limit of large random matrices belonging to the Gaussian Unitary (GUE), Orthogonal (GOE) and Symplectic (GSE) ensembles. It is well known that each TW distribution can be expressed through the Hastings‑McLeod solution (q(s)) of the Painlevé II equation \