The mean of the Tracy–Widom distribution

The largest eigenvalue of a large random Hermitian matrix fluctuates according to the Tracy–Widom law $F_2$. So does the longest increasing subsequence of a random permutation. The mean of this law, $\mu_2 \approx -1.7710868$, is a basic constant of random matrix theory. It can be computed to high precision, but no closed form is known.

Problem

Let $q$ be the Hastings–McLeod solution of the Painlevé II equation,

$$q''(s) = s\,q(s) + 2q(s)^3, \qquad q(s) \sim \operatorname{Ai}(s) \text{ as } s \to +\infty,$$

and let $F_2(s) = \exp\Bigl(-\int_s^\infty (x - s)\, q(x)^2 \, dx\Bigr)$ [1, 2]. Equivalently, $F_2(s)$ is the Fredholm determinant of the Airy kernel on $(s, \infty)$ [1, 4]. Find a closed-form expression for the mean

$$\mu_2 = \int_{-\infty}^{\infty} s \, dF_2(s) = -1.771086807411\ldots$$

as a finite combination of algebraic numbers, standard constants such as $\pi$, $e$, $\gamma$, $G$ and $\zeta(n)$, and special functions such as $\Gamma$, polylogarithms and elliptic integrals at algebraic arguments. Painlevé connection constants are allowed if they are themselves explicit. The function $q$ itself and unevaluated integrals of $q$ do not count.

Background

Tracy and Widom identified $F_2$ in 1994 as the limiting distribution of the largest eigenvalue of the Gaussian Unitary Ensemble, suitably centred and scaled [1]. Baik, Deift and Johansson proved in 1999 that the length $\ell_N$ of the longest increasing subsequence of a uniformly random permutation of $N$ numbers satisfies $(\ell_N - 2\sqrt N)/N^{1/6} \to F_2$ in distribution and with all moments. In particular $(\mathbb{E}\,\ell_N - 2\sqrt N)/N^{1/6} \to \mu_2$ [3], so $\mu_2$ is the second-order term in Ulam's problem on increasing subsequences.

What is known

• Bornemann (2010) evaluated $F_2$ by quadrature of the Fredholm determinant [5]. He gave the first four moments with certified digits: mean $-1.771086807411$, variance $0.8131947928329$, skewness $0.224084203610$ and excess kurtosis $0.0934480876$ [4].
• A high-precision value. Evaluating the Airy-kernel determinant by Bornemann's method in 45-digit arithmetic, with Gauss–Legendre quadrature also for the moment integral, gives $\mu_2 = -1.77108\,68074\,11601\,62598\,87144\,2185\ldots$; two refinements of the discretisation agree to 34 digits. The 63-digit reference value stored by the source, $-1.77108\,68074\,11601\,62612\,69382\ldots$, agrees with this only to 18 significant digits.
• No closed form is known for the mean or the variance. According to the source, the mean has resisted identification despite extensive numerical searches.

What would count as progress

• A closed-form expression that agrees with $\mu_2$ to 30 or more significant digits, checked against an independent high-precision evaluation such as the one above. HorizonMath's evaluator asks for 20 matching digits against its stored reference, which is wrong from the 19th digit, so a correct answer would fail that check until the reference is fixed. The evaluator also rejects numerical shortcuts such as quadrature, root-finding, infinite series and fitted constants.
• A proof of such an evaluation, for example through the asymptotics of the Painlevé II solution or of the Fredholm determinant.

References

1. C. A. Tracy and H. Widom, "Level-spacing distributions and the Airy kernel", Comm. Math. Phys. 159 (1994), 151–174. doi:10.1007/BF02100489
2. S. P. Hastings and J. B. McLeod, "A boundary value problem associated with the second Painlevé transcendent and the Korteweg–de Vries equation", Arch. Ration. Mech. Anal. 73 (1980), 31–51. doi:10.1007/BF00283254
3. J. Baik, P. Deift and K. Johansson, "On the distribution of the length of the longest increasing subsequence of random permutations", J. Amer. Math. Soc. 12 (1999), 1119–1178. doi:10.1090/S0894-0347-99-00307-0
4. F. Bornemann, "On the numerical evaluation of distributions in random matrix theory: a review", 2010. arXiv:0904.1581
5. F. Bornemann, "On the numerical evaluation of Fredholm determinants", Math. Comp. 79 (2010), 871–915. doi:10.1090/S0025-5718-09-02280-7

Source

Imported from HorizonMath tracy_widom_f2_mean, curated by Erik Y. Wang, Sumeet R. Motwani et al. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.