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**Rights statement:**Copyright 2018 American Institute of Physics. The following article appeared in Journal of Mathematical Physics, 59 (10), 2018 and may be found at http://dx.doi.org/10.1063/1.5023128 This article may be downloaded for personal use only. Any other use requires prior permission of the author and the American Institute of Physics.Accepted author manuscript, 18.7 MB, PDF document

Available under license: CC BY-NC: Creative Commons Attribution-NonCommercial 4.0 International License

- https://aip.scitation.org/doi/10.1063/1.5023128
Final published version

Research output: Contribution to journal › Journal article

Published

Article number | 103301 |
---|---|

<mark>Journal publication date</mark> | 10/2018 |

<mark>Journal</mark> | Journal of Mathematical Physics |

Issue number | 10 |

Volume | 59 |

Number of pages | 28 |

Publication status | Published |

Early online date | 27/09/18 |

Original language | English |

In this paper, we study the probability density function, $\mathbb{P}(c,\alpha,\beta, n)\,dc$, of the center of mass of the finite $n$ Jacobi unitary ensembles with parameters $\alpha\,>-1$ and $\beta >-1$;
that is the probability that ${\rm tr}M_n\in(c, c+dc),$ where $M_n$ are $n\times n$ matrices drawn from the unitary Jacobi ensembles.
We first compute the exponential moment generating function of the linear statistics $\sum_{j=1}^{n}\,f(x_j):=\sum_{j=1}^{n}x_j,$ denoted by $\mathcal{M}_f(\lambda,\alpha,\beta,n)$.
The weight function associated with the Jacobi unitary ensembles reads $x^{\alpha}(1-x)^{\beta},\; x\in [0,1]$. The moment generating function is the $n\times n$ Hankel determinant $D_n(\lambda,\alpha,\beta)$
generated by the time-evolved Jacobi weight, namely,
$w(x;\lambda ,\alpha,\beta )=x^{\alpha}(1-x)^{\beta}\,{\rm e}^{-\lambda\:x},\,x\in[0,1],\,\alpha>-1,\,\beta>-1$. We think of $\lambda$ as the time
variable in the resulting Toda equations. The non-classical polynomials defined by the monomial expansion,
$P_n(x,\lambda)= x^n+ p(n,\lambda)\:x^{n-1}+\dots+P_n(0,\lambda)$, orthogonal with respect to $w(x,\lambda,\alpha,\beta )$ over $[0,1]$ play an important role.
Taking the time evolution problem studied in Basor, Chen and Ehrhardt (\cite{BasorChenEhrhardt2010}), with some change of variables, we obtain a
certain auxiliary variable $r_n(\lambda),$ defined by integral over $[0,1]$ of the product of the unconventional orthogonal polynomials of degree $n$ and $n-1$ and
$w(x,\lambda,\alpha,\beta )/x$. It is shown that $r_n(2\imath\/{\rm e}^{z})$ satisfies a Chazy $II$ equation. There is another auxiliary variable, denote as $R_n(\lambda),$ defined by an integral over $[0,1]$ of the product of two polynomials of degree $n$ multiplied by $w(x,\lambda)/x.$ Then $Y_n(-\lambda)=1-\lambda/R_n(\lambda)$ satisfies a particular Painlev\'{e} \uppercase\expandafter{\romannumeral 5}:
$P_{\rm V}(\alpha^2/2$, $ -\beta^2/2, 2n+\alpha+\beta+1,1/2)$.\\
The $\sigma_n$ function defined in terms of the $\lambda\:p(n,-\lambda)$ plus a translation in $\lambda$ is the Jimbo--Miwa--Okamoto
$\sigma$-form of Painlev\'{e} \uppercase\expandafter{\romannumeral 5}. In the continuum approximation, treating the collection of eigenvalues as a charged fluid as in the Dyson Coulomb Fluid,
gives an approximation for the moment generation function $\mathcal{M}_f(\lambda,\alpha,\beta,n)$ when $n$ is sufficiently large. Furthermore, we deduce a new expression of $\mathcal{M}_f(\lambda,\alpha,\beta,n)$ when $n$ is finite in terms the $\sigma$ function of this the Painlev\'{e} \uppercase\expandafter{\romannumeral 5}
An estimate shows that the moment generating function is a function of exponential type and of order $n$. From the Paley-Wiener theorem, one deduces that $\mathbb{P}(c,\alpha,\beta,n)$ has compact support $[0,n]$. This result is easily extended to the $\beta$ ensembles, as long as
$w$ the weight is positive and continuous over $[0,1].$

Copyright 2018 American Institute of Physics. The following article appeared in Journal of Mathematical Physics, 59 (10), 2018 and may be found at http://dx.doi.org/10.1063/1.5023128 This article may be downloaded for personal use only. Any other use requires prior permission of the author and the American Institute of Physics.