32 Painlevé Transcendents Applications 32.13 Reductions of Partial Differential Equations 32.15 Orthogonal Polynomials

§32.14 Combinatorics

ⓘ

Keywords:: Gaussian unitary ensemble, Hermitian matrices, Painlevé transcendents, applications, combinatorics, distribution function, limiting distribution of eigenvalues, random matrix theory
Notes:: See Baik et al. (1999).
Permalink:: http://dlmf.nist.gov/32.14
See also:: Annotations for Ch.32

Let $S_{N}$ be the group of permutations $\boldsymbol{\pi}$ of the numbers $1,2,\dots,N$ (§26.2). With $1\leq m_{1}<\cdots<m_{n}\leq N$ , $\boldsymbol{\pi}(m_{1}),\boldsymbol{\pi}(m_{2}),\dots,\boldsymbol{\pi}(m_{n})$ is said to be an increasing subsequence of $\boldsymbol{\pi}$ of length $n$ when $\boldsymbol{\pi}(m_{1})<\boldsymbol{\pi}(m_{2})<\cdots<\boldsymbol{\pi}(m_{n})$ . Let $\ell_{N}(\boldsymbol{\pi})$ be the length of the longest increasing subsequence of $\boldsymbol{\pi}$ . Then

32.14.1		$\lim_{N\to\infty}\mathrm{Prob}\left(\frac{\ell_{N}(\boldsymbol{\pi})-2\sqrt{N}% }{N^{1/6}}\leq s\right)=F(s),$
ⓘ Symbols: $\boldsymbol{\pi}(m)$ : permutations, $\ell_{N}(\boldsymbol{\pi})$ : length and $F(s)$ : distribution function Permalink: http://dlmf.nist.gov/32.14.E1 Encodings: TeX, pMML, png See also: Annotations for §32.14 and Ch.32

where the distribution function $F(s)$ is defined here by

32.14.2		$F(s)=\exp\left(-\int_{s}^{\infty}(x-s)w^{2}(x)\,\mathrm{d}x\right),$
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\exp\NVar{z}$ : exponential function, $\int$ : integral, $x$ : real and $F(s)$ : distribution function Referenced by: §32.14 Permalink: http://dlmf.nist.gov/32.14.E2 Encodings: TeX, pMML, png See also: Annotations for §32.14 and Ch.32

and $w(x)$ satisfies $\mbox{P}_{\mbox{\scriptsize II}}$ with $\alpha=0$ and boundary conditions

32.14.3	$\displaystyle w(x)$	$\displaystyle\sim\operatorname{Ai}\left(x\right),$
$x\to+\infty$ ,
ⓘ Symbols: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $\sim$ : asymptotic equality and $x$ : real Permalink: http://dlmf.nist.gov/32.14.E3 Encodings: TeX, pMML, png See also: Annotations for §32.14 and Ch.32
32.14.4	$\displaystyle w(x)$	$\displaystyle\sim\sqrt{-\tfrac{1}{2}x},$
$x\to-\infty$ ,
ⓘ Symbols: $\sim$ : asymptotic equality and $x$ : real Permalink: http://dlmf.nist.gov/32.14.E4 Encodings: TeX, pMML, png See also: Annotations for §32.14 and Ch.32

where $\operatorname{Ai}$ denotes the Airy function (§9.2).

The distribution function $F(s)$ given by (32.14.2) arises in random matrix theory where it gives the limiting distribution for the normalized largest eigenvalue in the Gaussian Unitary Ensemble of $n\times n$ Hermitian matrices; see Tracy and Widom (1994).

See Forrester and Witte (2001, 2002) for other instances of Painlevé equations in random matrix theory.

32.13 Reductions of Partial Differential Equations 32.15 Orthogonal Polynomials

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