Painlevé V and the distribution function of a discontinuous linear statistic in the Laguerre unitary ensembles

Estelle Basor, Yang Chen
2008 Journal of Physics A: Mathematical and Theoretical  
In this paper we study the characteristic or generating function of a certain discontinuous linear statistic of the Laguerre unitary ensembles and show that this is a particular fifth Painlevé transcendent in the variable t, the position of the discontinuity. The proof of the ladder operators adapted to orthogonal polynomial with discontinuous weight announced sometime ago [13] is presented here, followed by the nonlinear difference equations satisfied by two auxiliary quantities and the
more » ... ties and the derivation of the Painlevé equation. PACS numbers: 02.20.Gp, 02.30.Hq, 02.30.Ik, 02.50.Cw where w 0 (x) with x ∈ [a, b] say, is strictly positive and satisfies a Lipshitz condition and has finite moments, that is, the existence of the integrals, b a x j w 0 (x) dx, j ∈ {0, 1, 2, . . .}.
doi:10.1088/1751-8113/42/3/035203 fatcat:ahra5he6prelncglfu2ukcvylq