The mean of the Tracy–Widom distribution

Find a closed form for the mean μ ≈ −1.7710868074 of the Tracy–Widom distribution of the largest GUE eigenvalue.

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	1	+# The mean of the Tracy–Widom distribution
	2	+
	3	+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.
	4	+
	5	+## Problem
	6	+
	7	+Let $q$ be the Hastings–McLeod solution of the Painlevé II equation,
	8	+
	9	+$$
	10	+q''(s) = s\,q(s) + 2q(s)^3, \qquad q(s) \sim \operatorname{Ai}(s) \text{ as } s \to +\infty,
	11	+$$
	12	+
	13	+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
	14	+
	15	+$$
	16	+\mu_2 = \int_{-\infty}^{\infty} s \, dF_2(s) = -1.771086807411\ldots
	17	+$$
	18	+
	19	+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.
	20	+
	21	+## Background
	22	+
	23	+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.
	24	+
	25	+## What is known
	26	+
	27	+- 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].
	28	+- 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.
	29	+- No closed form is known for the mean or the variance. According to the source, the mean has resisted identification despite extensive numerical searches.
	30	+
	31	+## What would count as progress
	32	+
	33	+- 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.
	34	+- A proof of such an evaluation, for example through the asymptotics of the Painlevé II solution or of the Fredholm determinant.
	35	+
	36	+## References
	37	+
	38	+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](https://doi.org/10.1007/BF02100489)
	39	+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](https://doi.org/10.1007/BF00283254)
	40	+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](https://doi.org/10.1090/S0894-0347-99-00307-0)
	41	+4. F. Bornemann, "On the numerical evaluation of distributions in random matrix theory: a review", 2010. [arXiv:0904.1581](https://arxiv.org/abs/0904.1581)
	42	+5. F. Bornemann, "On the numerical evaluation of Fredholm determinants", Math. Comp. 79 (2010), 871–915. [doi:10.1090/S0025-5718-09-02280-7](https://doi.org/10.1090/S0025-5718-09-02280-7)
	43	+
	44	+## Source
	45	+
	46	+Imported from [HorizonMath tracy_widom_f2_mean](https://github.com/ewang26/HorizonMath/blob/4ef0b61a8a6d4c5ded661fab7c2e896af84abe6e/data/problems_full.json#L660-L670), 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.