Tracy–Widom distribution

The Tracy–Widom distribution, introduced by Craig Tracy and Harold Widom (1993, 1994), is the probability distribution of the normalized largest eigenvalue of a random Hermitian matrix. The Tracy–Widom distribution, introduced by Craig Tracy and Harold Widom (1993, 1994), 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 Takeuchi & Sano (2010) and Takeuchi et al. (2011) for experimental testing (and verifying) that the interface fluctuations of a growing droplet (or substrate) are described by the TW distribution F 2 {displaystyle F_{2}} (or F 1 {displaystyle F_{1}} ) as predicted by Prähofer & Spohn (2000). The distribution F1 is of particular interest in multivariate statistics. For a discussion of the universality of Fβ, β = 1, 2, and 4, see Deift (2007). For an application of F1 to inferring population structure from genetic data see Patterson, Price & Reich (2006).In 2017 it was proved that the distribution F is not infinitely divisible. The Tracy–Widom distribution is defined as the limit: where λ max {displaystyle lambda _{max }} denotes the largest eigenvalue of the random matrix. The shift by 2 n {displaystyle {sqrt {2n}}} is used to keep the distributions centered at 0. The multiplication by ( 2 ) n 1 / 6 {displaystyle ({sqrt {2}})n^{1/6}} is used because the standard deviation of the distributions scales as n − 1 / 6 {displaystyle n^{-1/6}} . The cumulative distribution function of the Tracy–Widom distribution can be given as the Fredholm determinant of the operator As on square integrable functions on the half line (s, ∞) with kernel given in terms of Airy functions Ai by