Then for any $x_0\in(\lambda_1,\lambda_2)$ and $m\in\mathbb{N}$, we have $$\label{eq:bulk}
\begin{split}
\lim_{N,M\rightarrow\infty}&\left(\frac{1}{M\rho(x_0)}\right)^m
\mathcal{R}_m^{(M,N)}\left(x_0+\frac{u_1}{M\rho(x_0)},\ldots,x_0+\frac{u_m}{M\rho(x_0)}\right)\\
&=\det\left(\frac{\sin
\pi(u_i-u_j)}{\pi(u_i-u_j)}\right)_{i,j=1}^{m}.
\end{split}$$ uniformly for any $(u_1,\ldots,u_m)$ in compact subsets of $\mathbb{R}^m$.

On the other hand, let $x_0=\lambda_k$, $k=1,2$, then for any $m\in\mathbb{N}$, we have $$\label{eq:edge}
\begin{split}
\lim_{N,M\rightarrow\infty}&\left(\frac{1}{\left(M\rho_k\right)^{\frac{2}{3}}}\right)^m
\mathcal{R}_m^{(M,N)}\left(\lambda_k+(-1)^k\frac{u_1}{\left(M\rho_k\right)^{\frac{2}{3}}},\ldots,
\lambda_k+(-1)^k\frac{u_m}{\left(M\rho_k\right)^{\frac{2}{3}}}\right)\\
&=\det\left(\frac{\mathrm{Ai}(u_i)\mathrm{Ai}^{\prime}(u_j)
-\mathrm{Ai}^{\prime}(u_i)\mathrm{Ai}(u_j)}{u_i-u_j}\right)_{i,j=1}^m,
\end{split}$$ uniformly for any $(u_1,\ldots,u_m)$ in compact subsets of $\mathbb{R}^m$, where $\mathrm{Ai}(z)$ is the Airy function and $\rho_k$, $k=1,2$ are the constants in (\[eq:rhok\]).

Recall that the Airy function is the unique solution to the differential equation $v^{\prime\prime}=zv$ that has the following asymptotic behavior as $z\rightarrow\infty$ in the sector $-\pi+\epsilon\leq \arg(z)\leq \pi-\epsilon$, for any $\epsilon>0$. $$\label{eq:asymairy}
\mathrm{Ai}(z)=\frac{1}{2\sqrt{\pi}z^{\frac{1}{4}}}e^{-\frac{2}{3}z^{\frac{3}{2}}}\left(1+O(z^{-\frac{3}{2}})\right)
,\quad -\pi+\epsilon\leq \arg(z)\leq \pi-\epsilon,\quad
z\rightarrow\infty.$$ where the branch cut of $z^\frac{3}{2}$ in the above is chosen to be the negative real axis.

Since the limiting kernel takes the form of the Airy kernel (\[eq:edge\]), by a well-known result of Tracy and Widom [@TW1], we have the following

\[thm:TW\] Let $y_1$ be the largest eigenvalue of $B_N$, then we have $$\lim_{M,N\rightarrow\infty}
\mathbb{P}\left(\left(y_1-\lambda_2\right)\left(M\rho_2\right)^{\frac{2}{3}}<s\right)
=TW(s),$$ where $TW(s)$ is the Tracy-Widom distribution $$\label{eq:TW}
TW(s)=\exp\left(-\int_{s}^{\infty}(t-s)q^2(t)dt\right),$$ and $q(s)$ is the solution of Painlevé II equation $$q^{\prime\prime}(s)=sq(s)+2q^3(s),$$ with the following asymptotic behavior as $s\rightarrow\infty$. $$q(s)\sim-\mathrm{Ai}(s),\quad s\rightarrow +\infty.$$

The results obtained in this paper are obtained through 