Cumulants $\kappa_n$ of the Tracy–Widom distributions

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Numbers

$\beta$

$n$

$\kappa_n(\beta)$

Gaussian orthogonal ensemble

-1.2065335745820

Gaussian orthogonal ensemble

1.607781034581

Gaussian orthogonal ensemble

0.5982682569

Gaussian orthogonal ensemble

0.427146362

Gaussian unitary ensemble

-1.771086807411

Gaussian unitary ensemble

0.8131947928329

Gaussian unitary ensemble

0.1643248053

Gaussian unitary ensemble

0.0617958907

Gaussian symplectic ensemble

-2.306884893241

Gaussian symplectic ensemble

0.5177237207726

Gaussian symplectic ensemble

0.06165524210

Gaussian symplectic ensemble

0.0131861640

Definition

For $\beta\in\{1,2,4\}$, let $X_\beta$ have the Tracy–Widom distribution [2] in the soft-edge scaling used by [1]. This table gives the cumulants $\kappa_n(\beta)$ defined by $\log\mathbb E(\exp(tX_\beta))=\sum_{n\geq1}\kappa_n(\beta)t^n/n!$.

Parameters

$\beta$

— Dyson index ($\beta\in\{1,2,4\}$)

$n$

— cumulant order ($n\geq1$)

Formulas

(1)

If $\gamma_1$ is the skewness and $\gamma_2$ is the excess kurtosis, then $\kappa_3=\gamma_1\kappa_2^{3/2}$ and $\kappa_4=\gamma_2\kappa_2^2$.

Comments

(2)

The entries with $n=1$ and $n=2$ are the mean and variance respectively.

(3)

Bornemann's table lists mean, variance, skewness and excess kurtosis. The entries with $n=3$ and $n=4$ are the corresponding raw cumulants, not the standardised skewness and excess kurtosis.

(4)

The $\beta=4$ row uses Bornemann's Tracy–Widom scaling $F_4(s)=F_4(1;\sqrt2\,s)$. In the convention without the $\sqrt2$ in the argument, the $\beta=4$ cumulant $\kappa_n$ is multiplied by $2^{n/2}$.

Programs

(P1)

Python

from decimal import Decimal, getcontext

getcontext().prec = 50
variance = Decimal("0.8131947928329")
skewness = Decimal("0.224084203610")
kappa_3 = skewness * variance.sqrt() ** 3

References

[1]

Folkmar Bornemann, On the numerical evaluation of distributions in random matrix theory: a review, Markov Processes and Related Fields 16 (2010), 803-866. (arXiv)

Links

[2]

Wikipedia: Tracy-Widom distribution

Similar tables

Values of the Airy function of the first kind $\operatorname{Ai}(x)$ — the Hastings-McLeod solution in the Painlevé-II representation of $F_\beta$ is asymptotic to $\operatorname{Ai}$

Random-matrix factors in the moments of characteristic polynomials of unitary, orthogonal and symplectic matrices — random-matrix constants from compact groups rather than soft-edge distribution cumulants

Data properties

Entries are of type: real number

How well the digits are known: measured

Table is complete: no (it holds the first four cumulants, $1\leq n\leq4$, for $\beta\in\{1,2,4\}$)

How they were obtained:

The mean, variance, skewness and excess kurtosis were transcribed from [1], Example 9.4.1. The generator converts the published skewness and excess kurtosis to the raw cumulants using (1); the stored digits are no longer than the published input digits justify.

The $\beta=2$ values were compared with a direct Painlevé-II integration from the screening run, and the $\beta=4$ scaling was checked by recomputing the GSE mean and variance from the same Painlevé-II representation.