Tracy–Widom distribution
It is the 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.
[1] It also appears in the distribution of the length of the longest increasing subsequence of random permutations,[2] as large-scale statistics in the Kardar-Parisi-Zhang equation,[3] in current fluctuations of the asymmetric simple exclusion process (ASEP) with step initial condition,[4] and in simplified mathematical models of the behavior of the longest common subsequence problem on random inputs.
to inferring population structure from genetic data see Patterson, Price & Reich (2006).
It can be defined as a law of large numbers, similar to the central limit theorem.
For example:[10] where the matrix is sampled from the gaussian unitary ensemble with off-diagonal variance
("Airy kernel") on square integrable functions on the half line
can also be given as an integral in terms of a solution[note 1] of a Painlevé equation of type II with boundary condition
Other than in random matrix theory, the Tracy–Widom distributions occur in many other probability problems.
be the length of the longest increasing subsequence in a random permutation sampled uniformly from
[13] The Tracy–Widom distribution exhibits a third-order phase transition in the large deviation behavior of the largest eigenvalue of a random matrix.
This transition occurs at the edge of the Wigner semicircle distribution, where the probability density of the largest eigenvalue follows distinct scaling laws depending on whether it deviates to the left or right of the edge.
denote the rate function governing the large deviations of the largest eigenvalue
The discontinuity in the third derivative of the free energy marks a fundamental change in the behavior of the system, where fluctuations transition between different scaling regimes.
This third-order transition has also been observed in problems related to the maximal height of non-intersecting Brownian excursions, conductance fluctuations in mesoscopic systems, and entanglement entropy in random pure states.
[12] To interpret this as a third-order transition in statistical mechanics, define the (generalized) free energy density of the system as
lower end can be interpreted as the strongly interacting regime, where
objects are interacting strongly pairwise, so the total energy is proportional to
The Tracy–Widom distribution phase transition then occurs at the point as the system switches from strongly to weakly interacting.
To find the distribution of the largest eigenvalue, we take a wall and push against the Coulomb gas.
, then most of the gas remains unaffected, and we are in the weak interaction regime.
, then the entire bulk of the Coulomb gas is affected, and we are in the strong interaction regime.
These approximation techniques were further analytically justified in Bejan (2005) and used to provide numerical evaluation of Painlevé II and Tracy–Widom distributions (for
These distributions have been tabulated in Bejan (2005) to four significant digits for values of the argument in increments of 0.01; a statistical table for p-values was also given in this work.
Bornemann (2010) gave accurate and fast algorithms for the numerical evaluation of
These algorithms can be used to compute numerically the mean, variance, skewness and excess kurtosis of the distributions
For a simple approximation based on a shifted gamma distribution see Chiani (2014).
Shen & Serkh (2022) developed a spectral algorithm for the eigendecomposition of the integral operator
th largest level at the soft edge scaling limit of Gaussian ensembles, to machine accuracy.
scaling of the one-dimensional KPZ equation with fixed time.
