EP/D505534/1.

Multiple Laguerre polynomials and the correlation kernel {#se:MOP}
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The main tool in our analysis involves the use of multiple orthogonal polynomials and the Riemann-Hilbert problem associated with them. In this section we shall recall the results in [@BK1] and [@DF] and express the correlation kernel $K_{M,N}(x,y)$ in (\[eq:mpoint\]) in terms of multiple Laguerre polynomials. In Section \[se:RHP\], we will apply Riemann-Hilbert analysis to obtain the asymptotics of these multiple Laguerre polynomials and use them to prove Theorem \[thm:main2\].

We shall not define the multiple Laguerre polynomials in the most general setting, but only define the ones that are relevant to our set up. Readers who are interested in the theory of multiple orthogonal polynomials can consult the papers [@Ap1], [@Ap2], [@BK1], [@vanGerKuij]. Let $L_{n_1,n_2}(x)$ be the monic polynomial such that $$\label{eq:multiop}
\begin{split}
&\int_{0}^{\infty}L_{n_1,n_2}(x)x^{i+M-N}e^{-Mx}dx=0,\quad i=0,\ldots,n_1-1,\\
&\int_0^{\infty}L_{n_1,n_2}(x)x^{i+M-N}e^{-Ma^{-1}x}dx=0,\quad
i=0,\ldots, n_2-1.
\end{split}$$ and let $Q_{n_1,n_2}(x)$ be a function of the form $$\label{eq:qform}
Q_{n_1,n_2}(x)=A^1_{n_1,n_2}(x)e^{-Mx}+A^a_{n_1,n_2}(x)e^{-Ma^{-1}x},$$ where $A^1_{n_1,n_2}(x)$ and $A^a_{n_1,n_2}(x)$ are polynomials of degrees $n_1-1$ and $n_2-1$ respectively, and that $Q_{n_1,n_2}(x)$ satisfies the following $$\label{eq:qorth}
\int_0^{\infty}x^iQ_{n_1,n_2}(x)x^{M-N}dx=\left\{
                                            \begin{array}{ll}
                                              0, & \hbox{$i=0,\ldots,n_1+n_2-2$;} \\
                                              1, & \hbox{$i=n_1+n_2-1$.}
                                            \end{array}
                                          \right.$$ The polynomial $L_{n_1,n_2}(x)$ is called the multiple Laguerre polynomial of type II and the polynomials $A^1_{n_1,n_2}(x)$ and $A^a_{n_1,n_2}(x)$ are 