Talk:Tracy–Widom distribution

Images
Tracy-Widom distributions is skewed in the positive direction, since it's far more likely for the largest eigenvalue to have a spurious fluctuation up rather than down, since then it would have to move against the bulk of the eigenvalues. Can somebody provide an image of a TW distribution? --74.66.20.170 (talk) 15:49, 3 June 2013 (UTC)

Also, as Melcombe pointed out four years ago, there are multiple versions of TW distributions, corresponding to different universality classes of random matrices, such as the Gaussian Unitary Ensemble (GUE) or the Gaussian Orthogonal Ensemble (GOE). This really should be fixed. This article needs some expert attention. --74.66.20.170 (talk) 15:49, 3 June 2013 (UTC)

Clarifications
It also appears as solution of the KPZ equation. Phys. Rev. Lett. 104, 230602 (2010), https://arxiv.org/abs/1002.1883 — Preceding unsigned comment added by Luis araras (talk • contribs) 18:04, 19 July 2019 (UTC)

There is a basic issue in the definition section. Nowhere is $$\lambda_{max}$$ defined as the largest eigenvalue of a random Hermitian matrix; even if that is clear from context, nowhere is the ensemble actually specified. One could use the GUE in the definition, and then ideally mention universality results somewhere as well.

Can something be done regarding the following:
 * Is this a continuous distribution? The formulation given (without looking elsewhere) leaves open the possibility of a discrete component at zero.
 * Can something be said about the asymptotic behaviour of the distribution function at zero and infinity, using simple functions (not something as unusual as the Airy function)?
 * Versions of distributions and naming. What specifically is a "Tracy-Widom distribution". The article starts by specifying "the" Tracy-Widom distribution, but then goes on to mention others also apparently called "Tracy-Widom distributions". So is there more than one distribution function that can be called a ""Tracy-Widom distribution"? If there are, can they be specified here in relevant detail?

Melcombe (talk) 09:52, 12 March 2009 (UTC)

bounded above and below
In the definition section, the article says
 * The normalization factors (a shift by $$\sqrt{2N}$$ and a multiplication by $$(\sqrt{2})n^{1/6}$$) are used to keep the distributions in the limit centered at 0 with a standard deviation that is $$o(1)$$ (i.e. bounded above and below) as the limit is taken.

This makes no sense to me. Saying that something is o(1) means that it goes to zero in the specified limit, i.e. that it is not bounded below. Something that is bounded both above and below would be $$\Theta(1)$$. Can someone else clarify whether the o(1) is correct and the text is confused, or vice versa? —David Eppstein (talk) 07:29, 1 November 2014 (UTC)
 * I messed up on the little-o notation, which should probably just be removed. The source says "[R]oughly, ... the largest eigenvalue is within o(1) of 2σ$$\sqrt{N}$$, and so this is thought ofas the right edge of the spectrum. In fact, the largest eigenvalue is within $$O(N^{-1/6})$$ of 2σ$$\sqrt{N}$$". So the line about o(1) should probably just be deleted.Brirush (talk) 12:01, 1 November 2014 (UTC)


 * Here's an alternate version:


 * The shift by $$\sqrt{2N}$$ is used to keep the distributions centered at 0. The multiplication by $$(\sqrt{2})n^{1/6}$$) is used because the widths of the distributions scale as $$n^{-1/6}$$.


 * I will implement this change unless you have any different ideas. Thanks for catching it!

Brirush (talk) 12:08, 1 November 2014 (UTC)