When should we expect Tracy-Widom?

Question

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 $$ A_sf(x)=\int A(x,y)f(y)dy,\qquad A(x,y)=\frac{Ai(x)Ai'(y)-Ai(y)Ai'(x)}{x-y}, $$ $Ai$ 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...

Answer 1

Disclaimer - I have only recently come to know about this law after taking a class on Random Matrix Theory. And I am no Physicist; I study Computer Science. Thus I do not claim to know or understand the details in your question. However, I will answer your original question, "When should we expect the Tracy-Widom law?" and, "What are the ingredients to be present in order expect his apparition?" So, please pardon me if my answer is not up to the mark you might have been expecting.

The way I understand it best, Tracy-Widom (TW) law is convenient to understand in conjunction with the Marcenko-Pastur (MP) law. The MP law gives a bound for the histogram of the eigenvalues of a random matrix. It is also one of the universal distributions. It looks like follows:

As you can see, the maximum eigenvalue is bounded. However, if the matrix is not truly random, like for instance in stock price returns, i.e., when there is some correlation among the different stocks, then there are some eigenvalues that go beyond the bound given by the MP law, as shown in my MATLAB simulation below:

The first plot shows all the eigenvalues, including the maximum eigenvalue which is way over to the right. The second plot removes that maximum eigenvalue and still we can see a few eienvalues beyond the MP law bound.

The meaning of this is understood as follows: whichever stock the one isolated eigenvalue, the maximum eigenvalue, is coming from, it is very, very strongly correlated with all the other stocks.

Anyway, so here comes the Tracy-Widom law. The TW law can be thought of as giving the probability of that maximum eigenvalue lying within the MP law bound. If we look at its graph, it is heavy tailed to the right, meaning that the probability of the maximum eigenvalue being more to the right of the MP law bound becomes lesser and lesser as we keep going to the right, almost nil beyond a point.

So, what is the significance of all this? Well, we know that the MP law and the TW law hold for only random matrices. So, if there is any significant deviation for a matrix, it means that the matrix in question is not in fact a random matrix, like the one of stock returns shown above. This is helpful as the null hypothesis is to assume no correlations among the random vectors, as it was believed for prices of different stocks, until people started doing these spectral analysis discussed here for the covariance (and correlation) matrix of stock returns, and the null hypothesis was rejected based on these evidence. So, deviation from these laws suggests some structure, some pattern in the data. I am aware of this being used, apart from financial engineering, in finding patterns in genetic mutations, among others.

In summary, we should expect Tracy-Widom for truly random matrices, i.e., constructed from measurements of random vectors, without any correlations. So, "the ingredients to be present in order to expect his apparition" is true randomness. If there is some correlation among the random vectors, then there will be some "proportionate" deviation from the law.