Skewness and excess kurtosis of the Tracy-Widom distributions

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Numbers

$\beta$

$n$

$\lambda_n(\beta)$

GOE

skewness:

0.29346452408

GOE

excess kurtosis:

0.1652429384

GUE

skewness:

0.224084203610

GUE

excess kurtosis:

0.0934480876

GSE

skewness:

0.16550949435

GSE

excess kurtosis:

0.0491951565

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 standardized cumulants $\lambda_n(\beta)=\kappa_n/\kappa_2^{n/2}$ for $n=3,4$, where $\kappa_j$ is the $j$-th cumulant of $X_\beta$.

Parameters

$\beta$

— Dyson index

$n$

— cumulant order

Formulas

(1)

$\lambda_n(\beta)=\kappa_n(F_\beta)/\kappa_2(F_\beta)^{n/2}$.

(2)

$\lambda_3$ is the skewness and $\lambda_4$ is the excess kurtosis.

Comments

(3)

The $\beta=4$ rows use the soft-edge scaling in [1], where $F_4(k;s)=F_1(2k;s)$. The alternative $2N\times2N$ complex-matrix GSE convention rescales $s$ by $\sqrt2$; the standardized cumulants in this table are unchanged by that positive rescaling.

(4)

The $n=4$ rows are excess kurtosis, not Pearson kurtosis. Add $3$ to get the fourth standardized central moment.

Programs

(P1)

Python

values = {
    (1, 3): "0.29346452408",
    (1, 4): "0.1652429384",
    (2, 3): "0.224084203610",
    (2, 4): "0.0934480876",
    (4, 3): "0.16550949435",
    (4, 4): "0.0491951565",
}
values[(2, 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

Cumulants $\kappa_n$ of the Tracy-Widom distributions — gives the cumulants standardised here

Values of the Tracy-Widom distribution functions $F_\beta(s)$ — tabulates the same soft-edge laws by their distribution functions

Values of the Hastings-McLeod solution $q(s)$ of Painleve II — gives the Painleve-II solution used in the representation of $F_\beta$

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

Random-matrix factors $g_G(k)$ in the moments of characteristic polynomials of unitary, orthogonal and symplectic matrices — both tables store constants from random matrix theory, but this one concerns soft-edge distributions rather than characteristic-polynomial moments

Data properties

Entries are of type: real number

How well the digits are known: measured

Table is complete: yes

How they were obtained:

The entries are transcribed from the correctly truncated soft-edge statistical-property tables in [1]. The $\beta=1$ rows use the $F_1(1;s)$ row of Bornemann's Table 9, the $\beta=2$ rows use the $F_2(1;s)$ row of Table 10, and the $\beta=4$ rows use the $F_1(2;s)$ row of Table 9 through the identity $F_4(1;s)=F_1(2;s)$.

Before the draft was filled, these six values were compared with the values printed in the current Wikipedia article for the Tracy-Widom distribution; the stored digits agree.