TracyWidom function

The Tracy-Widom Distributions

Density, distribution function, quantile function, and random generation for the Tracy-Widom distribution with order parameter beta.

dtw(x, beta=1, log = FALSE)
ptw(q, beta=1, lower.tail = TRUE, log.p = FALSE)
qtw(p, beta=1, lower.tail = TRUE, log.p = FALSE)
rtw(n, beta=1)

Arguments

• x,q: vector of quantiles.
• p: vector of probabilities.
• n: number of observations. If length(n) > 1, the length is taken to be the number required.
• beta: the order parameter (1, 2, or 4).
• log, log.p: logical; if TRUE, probabilities p are given as log(p).
• lower.tail: logical; if TRUE (default), probabilities are $P[X <= x]$, otherwise, $P[X > x]$.

Returns

dtw gives the density, ptw gives the distribution function, qtw gives the quantile function, and rtw generates random deviates.

Details

If beta is not specified, it assumes the default value of 1.

The Tracy-Widom law is the edge-scaled limiting distribution of the largest eigenvalue of a random matrix from the $beta$-ensemble. Supported values for beta are 1 (Gaussian Orthogonal Ensemble), 2 (Gaussian Unitary Ensemble), and 4 (Gaussian Symplectic Ensemble).

Source

The distribution and density functions are computed using a lookup table. They have been pre-computed at 769 values uniformly spaced between -10 and 6 using MATLAB's bvp4c solver to a minimum accuracy of about 3.4e-08. For all other points, the values are gotten from a cubic Hermite polynomial interpolation. The MATLAB software for computing the grid of values is part of RMLab, a package written by Momar Dieng.

The quantiles are computed via bisection using uniroot .

Random variates are generated using the inverse CDF.

References

Dieng, M. (2006). Distribution functions for edge eigenvalues in orthogonal and symplectic ensembles: representations. arXiv:math/0506586v2 [math.PR].

Tracy, C.A. and Widom, H. (1994). Level-spacing distributions and the Airy kernel. Communications in Mathematical Physics

159 , 151--174.

Tracy, C.A. and Widom, H. (1996). On orthogonal and symplectic matrix ensembles. Communications in Mathematical Phsyics

177 , 727--754.

Author(s)

Iain M. Johnstone, Zongming Ma, Patrick O. Perry and Morteza Shahram