A Tracy-Widom Central Limit Theorem

Let $(a_1,a_2,\ldots)$ be an i.i.d. sequence, and let $X$ denote an operation that produces a sequence $(X_1,X_2,\ldots)$ from it. The question is to characterize such an operation probabilistically so that, as $n\to\infty$, $X_n$ converges in distribution to the Gaudin distribution or a Tracy–Widom distribution. The formulation gives no definition of $X$, no further assumptions on the input law, and no choice between the named limiting distributions; identifying the operation and the relevant conditions is the research direction. The motivating comparison is the central limit theorem, in which centered partial sums are scaled as

$$S_n=\frac{\sum_{i=1}^n a_i-n\mu}{\sqrt n}$$

and converge to a Gaussian law under the usual hypotheses. Deift, Some Open Problems in Random Matrix Theory and the Theory of Integrable Systems (2007)

Background

The question asks for a probabilistic counterpart to the central limit theorem: which transformations of independent input variables produce the characteristic edge laws of random matrix theory? In the 2007 formulation, Deift names independent work of Baik–Suidan and Bodineau–Martin as important progress, while describing the broad characterization problem as challenging. That historical statement does not specify a single operation or settle the status of the question today. Deift, Some Open Problems in Random Matrix Theory and the Theory of Integrable Systems (2007)

A concrete connection is the longest increasing subsequence of a uniformly random permutation. Its length has Tracy–Widom fluctuations under the edge scaling: if $\ell_N$ denotes the length for a permutation of $N$ elements, the cited result gives the limiting distribution of $(\ell_N-2\sqrt N)/N^{1/6}$. This is a specific model, not a definition of the general operation sought in the question. On the Distribution of the Length of the Longest Increasing Subsequence of Random Permutations

Known results

Baik and Suidan prove a GUE Tracy–Widom limit for a last-passage statistic built from an array $\{X_i^j\}_{i,j\ge1}$ of i.i.d. random variables with mean zero, variance one, and finite fourth moment. For integers $N,k$, define

$$L(N,k)=\sup_{0=i_0\le i_1\le\cdots\le i_k=N}\sum_{j=1}^k\sum_{i=i_{j-1}+1}^{i_j}X_i^j, \qquad R(N,k)=\inf_{0=i_0\le i_1\le\cdots\le i_k=N}\sum_{j=1}^k\sum_{i=i_{j-1}+1}^{i_j}X_i^j.$$

When $k,N\to\infty$ with $k=o(N^\alpha)$ for $\alpha<3/14$, both centered and scaled statistics converge to the GUE Tracy–Widom law $F_{GUE}$:

$$\left(\frac{L(N,k)}{\sqrt N}-2\sqrt{k}\right)k^{1/6}\Longrightarrow F_{GUE}, \qquad \left(-\frac{R(N,k)}{\sqrt N}-2\sqrt{k}\right)k^{1/6}\Longrightarrow F_{GUE}.$$

For Gaussian inputs satisfying the same moment conditions, the stated range extends to $k=o(N^\alpha)$ with $\alpha<3/7$. These are substantial examples of transformations of independent variables yielding Tracy–Widom fluctuations, but their hypotheses and two-parameter regime are specific; they do not identify a universal operation for the broad question. A GUE Central Limit Theorem and Universality of Directed First and Last Passage Site Percolation

The 2007 discussion also credits Bodineau–Martin with independent progress, but does not state its result in detail. The last-passage theorem supplies a concrete result relevant to the proposed probabilistic characterization; it should not be read as resolving the unspecified general operation or as establishing a present-day status for that broader question. Deift, Some Open Problems in Random Matrix Theory and the Theory of Integrable Systems (2007) A GUE Central Limit Theorem and Universality of Directed First and Last Passage Site Percolation