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

§32.14 Combinatorics

Notes:: See Baik et al. (1999).
Keywords:: Gaussian unitary ensemble, Hermitian matrices, Painlevé transcendents, combinatorics, distribution function, random matrix theory
Permalink:: http://dlmf.nist.gov/32.14

Let $S_{N}$ be the group of permutations $\boldsymbol{\pi}$ of the numbers $1,2,\dots,N$ (§26.2). With $1\leq m_{1}<\dots<m_{n}\leq N$ , $\boldsymbol{\pi}(m_{1}),\boldsymbol{\pi}(m_{2}),\dots,\boldsymbol{\pi}(m_{n})$ is said to be an increasing subsequence of $\boldsymbol{\pi}$ of length when $\boldsymbol{\pi}(m_{1})<\boldsymbol{\pi}(m_{2})<\dots<\boldsymbol{\pi}(m_{n})$ . 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 : distribution function
Permalink:: http://dlmf.nist.gov/32.14.E1
Encodings:: TeX, pMML, png

where the distribution function is defined here by

32.14.2 $F(s)=\mathop{\exp\/}\nolimits\!\left(-\int_{s}^{\infty}(x-s)w^{2}(x)dx\right),$

Symbols:: : differential of , $\mathop{\exp\/}\nolimits z$ : exponential function, $\int$ : integral, : real and : distribution function
Referenced by:: §32.14
Permalink:: http://dlmf.nist.gov/32.14.E2
Encodings:: TeX, pMML, png

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

32.14.3 $w(x)\sim\mathop{\mathrm{Ai}\/}\nolimits\!\left(x\right),$ $x\to+\infty$ ,

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $\sim$ : asymptotic equality and : real
Permalink:: http://dlmf.nist.gov/32.14.E3
Encodings:: TeX, pMML, png

32.14.4 $w(x)\sim\sqrt{-\tfrac{1}{2}x},$ $x\to-\infty$ ,

Symbols:: $\sim$ : asymptotic equality and : real
Permalink:: http://dlmf.nist.gov/32.14.E4
Encodings:: TeX, pMML, png

where $\mathop{\mathrm{Ai}\/}\nolimits$ denotes the Airy function (§9.2).

The distribution function 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.

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 32.13 Reductions of Partial Differential Equations 32.15 Orthogonal Polynomials