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The Tracy-Widom law describes, among other things, the fluctuations of maximal eigenvalues of many random large matrix models. Because of its universal character, it obtained his position on the podium of very famous laws in probability theory. I'd like to discuss what are the ingredients to be present in order expect his apparition.

More precisely, the Tracy-Widom law has for cumulative distribution the Fredholm determinant $$ F(s)=\det(I-A_s) $$ where the operator $A_s$ acts on $L^2(s,+\infty)$ by $$ A_sf(x)=\int A(x,y)dy,\qquad A(x,y)=\frac{Ai(x)Ai'(y)-Ai(y)Ai'(x)}{x-y}, $$ $Ai$ being the Airy function. It is moreover possible to rewrite $F$ in a more explicit (?) form, involving a solution of the Painlevé II equation. It is known that this distribution describes the fluctuations of the maximal value of the GUE, and actually of a large class of Wigner Matrices. It curiously also appears in many interacting particle processes, such as ASEP, TASEP, longest increasing subsequence of uniformly random permutations, polynuclear growth models ... (For an introduction, see http://arxiv.org/abs/math-ph/0603038 and references inside. You may jump at (30) if you are in a hurry, and read more about particles models in Section 3). A natural (but ambitious) question is

  • You have $N$ interacting random points $(x_1,\ldots,x_N)$ on $\mathbb{R}$, when can you predict that $x_{\max}^{(N)}=\max_{i=1}^N x_i$ will fluctuate (up to a rescaling) according to Tracy-Widom law around its large $N$ limiting value ?

Assume that the limiting distribution of the $x_i$'s $$ \mu(dx)=\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{i=1}^N\delta_{x_i}\qquad \mbox{(in the weak topology)} $$ admits a density $f$ on a compact support $S(\mu)$, and note $x_\max=\max S(\mu)$ (which can be assumed to be positive by translation). I have the impression that a necessary condition for the appearance of Tracy-Widom is to satisfy the three following points :

1) (strong repulsion) There exists a strong repulsion between the $x_i$'s (typically, the joint density of the $x_i$'s has a term like $\prod_{i\neq j}|x_i-x_j|$, or at least the $x_i$'s form a determinantal point process).

2) (no jump for $x_\max^{(N)} $) $x_\max^{(N)}\rightarrow x_\max$ a.s. when $N\rightarrow\infty$.

3) (soft edge) The density of $\mu$ vanishes like a square root around $x_\max$, i.e. $f(x)\sim (x_\max-x)^{1/2}$ when $x\rightarrow x_\max$.

For TASEP and longest increasing subsequence models, one can see that 1), 2) and 3) hold [since these models are somehow discretizations of random matrix models where everything is explicit (Wishart and GUE respectively)]. For the Wigner matrices, 2) and 3) clearly hold [Wigner's semicircular law], and I guess 1) is ok [because of the local semicircular law]. For ASEP, 1) clearly holds [because of the E of ASEP], 2) and 3) are not so clear to me, but sound reasonable.

  • Do you know any interacting particle model where Tracy-Widom holds but where one of the previous points is cruelly violated ?

Of course the condition 1) is pretty vague, and would deserve to be defined precisely. It is a part of the question !

NB : I have a pretty weak physical background, so if by any chance a physicist was lost on MO, I'd love to hear his/her criteria for Tracy-Widom...

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Nice question! I, for one, would appreciate the insertion of a few references. –  Stephen Griffeth Jul 27 '11 at 14:13
SPG : Thank you for your interest. As I will edit my post in order to include some references (it was hard to find something packed and semi-exhaustive !). –  Adrien Hardy Jul 29 '11 at 3:25
I am not a physicist, but since you wanted to hear from some, take a look at this paper. I saw the first author's talk in MSRI last year. The authors are are experimental physicists; they set up an experiment in which they fired a laser into a dish of a certain kind of liquid crystal, held very near a phase transition. The laser pulse nucleated a local phase change, and the fluctuations of the edge of the resulting ordered region agreed very well with the Tracy-Widom distribution. daisy.phys.s.u-tokyo.ac.jp/student/kazumasa/publications/… –  Benjamin Young Dec 31 '11 at 21:22

1 Answer 1

I allow myself to make the following comment as an answer.

There is this paper of Baik, Ben Arous and Péché, "phase transition of the largest eigenvalue for nonnull complex sample covariance matrices" [The Annals of Probability 2005, Vol. 33, No. 5, 1643–1697] that I had in mind for some time and which answers a parallel question of mine, and I thought I should share this observation with the people interested in the question.

This paper has a concrete example of particle system (which is the eigenvalues of a finite rank perturbation of random non-correlated covariance matrices) for which, if certain parameters linked to this system are somehow in a critical regime, then the maximal particle satisfies the three conditions of my question, but DOES NOT fluctuate according to the Tracy-Widom distribution (but according to a close relative. Nevertheless, it turns out it is possible to cook up a more complicated matrix model for which such fluctuations might be of really different nature).

Thus, when my question was about the necessity of this three conditions, I want to stress that they are not sufficient for the appearance of the Tracy-Widom law.

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