Tracy-Widom Distribution

A Tracy-Widom Distribution is a Probability Distribution of the normalized largest eigenvalue of a random Hermitian matrix.

• • …
• Counter-Example(s):
  • a Gaussian Probability Distribution Family.
• See: Gaussian Unitary Ensemble, Multivariate Statistics, Probability Distribution, Eigenvalue, Longest Increasing Subsequence, Permutation, Asymmetric Simple Exclusion Process, Longest Common Subsequence.

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References

2016

• (Wikipedia, 2016) ⇒ https://en.wikipedia.org/wiki/Tracy–Widom_distribution Retrieved:2016-10-28.
  • The Tracy–Widom distribution, introduced by , is the probability distribution of the normalized largest eigenvalue of a random Hermitian matrix.

    In practical terms, Tracy–Widom is the crossover function between the two phases of weakly versus strongly coupled components in a system.

    It also appears in the distribution of the length of the longest increasing subsequence of random permutations , in current fluctuations of the asymmetric simple exclusion process (ASEP) with step initial condition, and in simplified mathematical models of the behavior of the longest common subsequence problem on random inputs . See () for experimental testing (and verifying) that the interface fluctuations of a growing droplet (or substrate) are described by the TW distribution [math]\displaystyle{ F_2 }[/math] (or [math]\displaystyle{ F_1 }[/math] ) as predicted by (...).

    The distribution F₁ is of particular interest in multivariate statistics . For a discussion of the universality of F_β, β = 1, 2, and 4, see . For an application of F₁ to inferring population structure from genetic data see .

2016

• https://www.wired.com/2014/10/tracy-widom-mysterious-statistical-law/
  • QUOTE: The central limit theorem, which was finally made rigorous about a century ago, certifies that test scores and other “uncorrelated” variables — meaning any of them can change without affecting the rest — will form a bell curve.

... the Tracy-Widom curve appears to arise from variables that are strongly correlated, such as interacting species, stock prices and matrix eigenvalues. The feedback loop of mutual effects between correlated variables makes their collective behavior more complicated than that of uncorrelated variables like test scores. While researchers have rigorously proved certain classes of random matrices in which the Tracy-Widom distribution universally holds, they have a looser handle on its manifestations in counting problems, random-walk problems, growth models and beyond.

So far, researchers have characterized three forms of the Tracy-Widom distribution: rescaled versions of one another that describe strongly correlated systems with different types of inherent randomness. But there could be many more than three, perhaps even an infinite number, of Tracy-Widom universality classes.

On the right, Majumdar and Dean were surprised to find that the distribution dropped off at a rate related to the number of eigenvalues, N; on the left, it tapered off more quickly, as a function of N2.

In May’s ecosystem model, for example, the critical point at √2N separates a stable phase of weakly coupled species, whose populations can fluctuate individually without affecting the rest, from an unstable phase of strongly coupled species, in which fluctuations cascade through the ecosystem and throw it off balance. In general, Majumdar and Schehr believe, systems in the Tracy-Widom universality class exhibit one phase in which all components act in concert and another phase in which the components act alone.

The asymmetry of the statistical curve reflects the nature of the two phases. Because of mutual interactions between the components, the energy of the system in the strong-coupling phase on the left is proportional to N2. Meanwhile, in the weak-coupling phase on the right, the energy depends only on the number of individual components, N.

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