Local Universality in Biorthogonal Laguerre Ensembles

Abstract

We consider n particles $0\le x_1<x_2< \cdots < x_n < +\infty$, distributed according to a probability measure of the form

$$\begin{aligned} \frac{1}{Z_n}\prod _{1\le i <j \le n}(x_j-x_i)\prod _{1\le i <j \le n}(x_j^{\theta }-x_i^{\theta })\prod _{j=1}^nx_j^\alpha e^{-x_j}\,\mathrm {d}x_j, \quad \alpha >-1,\quad \theta >0, \end{aligned}$$

where $Z_n$ is the normalization constant. This distribution arises in the context of modeling disordered conductors in the metallic regime, and can also be realized as the distribution for squared singular values of certain triangular random matrices. We give a double contour integral formula for the correlation kernel, which allows us to establish universality for the local statistics of the particles, namely, the bulk universality and the soft edge universality via the sine kernel and the Airy kernel, respectively. In particular, our analysis also leads to new double contour integral representations of scaling limits at the origin (hard edge), which are equivalent to those found in the classical work of Borodin. We conclude this paper by relating the correlation kernels to those appearing in recent studies of products of M Ginibre matrices for the special cases $\theta =M\in \mathbb {N}$.

Appendix: The Meijer G-Function

For convenience of the readers, we give a brief introduction to the Meijer G-function in this appendix, which includes its definition and some properties used in this paper.

By definition, the Meijer G-function is given by the following contour integral in the complex plane:

$$\begin{aligned} \left. G^{m,n}_{p,q}\left( {\begin{array}{c} a_1,\ldots ,a_p \\ b_1,\ldots ,b_q \end{array}}\right| z\right) =\frac{1}{2\pi i}\int _\gamma \frac{\prod _{j=1}^m\Gamma (b_j+u)\prod _{j=1}^n\Gamma (1-a_j-u)}{\prod _{j=m+1}^q\Gamma (1-b_j-u)\prod _{j=n+1}^p\Gamma (a_j+u)}z^{-u} \,\mathrm {d}u,\quad \end{aligned}$$

(4.1)

where $\Gamma$ denotes the usual gamma function and the branch cut of $z^{-u}$ is taken along the negative real axis. It is also assumed that

• $0\le m\le q$ and $0\le n \le p$, where m, n, p and q are integer numbers;

• The real or complex parameters $a_1,\ldots ,a_p$ and $b_1,\ldots ,b_q$ satisfy the conditions

  $$\begin{aligned} a_k-b_j \ne 1,2,3, \ldots , \quad \text {for }k=1,2,\ldots ,n\text { and }j=1,2,\ldots ,m, \end{aligned}$$

  i.e., none of the poles of $\Gamma (b_j+u)$, $j=1,2,\ldots ,m$ coincides with any poles of $\Gamma (1-a_k-u)$, $k=1,2,\ldots ,n$.

The contour $\gamma$ is chosen in such a way that all the poles of $\Gamma (b_j+u)$, $j=1,\ldots ,m$ are on the left of the path, while all the poles of $\Gamma (1-a_k-u)$, $k=1,\ldots ,n$ are on the right, which is usually taken to go from $-i\infty$ to $i\infty$. For more details, we refer to the references [37, 41].

Most of the known special functions can be viewed as special cases of the Meijer G-functions. For instance, with the generalized hypergeometric function ${\; }_p F_q$ given in (3.9), one has [41, formula 16.18.1]

$$\begin{aligned} \left. \mathop {{{}_{p}F_{q}}}\nolimits \!\left( {\begin{array}{c} a_{1},\dots ,a_{p}\\ b_{1},\dots ,b _{q} \end{array}}\right| z\right) =\frac{\prod \limits _{k=1}^{q}\Gamma (b_k)}{\prod \limits _{k=1}^{p}\Gamma (a_k)} G^{1,p}_{p,q+1}\left. \left( {\begin{array}{c} 1-a_1,\ldots ,1-a_p \\ 0, 1-b_1,\ldots ,1-b_q \end{array}}\right| -z \right) . \end{aligned}$$

(4.2)

This, together with the fact that

$$\begin{aligned} \left. \left. z^{\alpha }G^{m,n}_{p,q}\left( {\begin{array}{c} a_1,\ldots ,a_p \\ b_1,\ldots ,b_q \end{array}}\right| z\right) =G^{m,n}_{p,q}\left( {\begin{array}{c} a_1+\alpha ,\ldots ,a_p+\alpha \\ b_1+\alpha ,\ldots ,b_q+\alpha \end{array}}\right| z\right) , \end{aligned}$$

(4.3)

gives us

$$\begin{aligned} x^\alpha e^{-x}=G^{1,0}_{0,1}\left. \left( {\begin{array}{c} - \\ \alpha \end{array}}\right| x\right) =\frac{1}{2\pi i}\int _\gamma \Gamma (\alpha +s)x^{-s}\,\mathrm {d}s. \end{aligned}$$

(4.4)