Let $ \mu $ be the Gaussian measure $ d\mu(x) = e^{-x^2/2} \frac{dx}{\sqrt{2\pi} } $. I am interested in the following random matrix integral defined for all $ s \in \mathbb{R} $, $ N \geq 1 $ and $ a \in \{0, 1, 2, \dots \} $ \begin{align*}%$ \tilde{E}_N(s ; a) := \frac{1}{N! Z_N} \int_{\mathbb{R}^N } \Delta(x_1, \dots, x_N)^2 \prod_{k = 1}^N (s - x_k)^a \boldsymbol{1}_{\{x_k \leq s\}} d\mu(x_k) \end{align*} where $ \Delta(t_1, \dots, t_N) := \prod_{1 \leq i < j \leq N} (t_i - t_j) $ is the Vandermonde determinant and $ Z_N $ is a rescaling constant whose value is not important. This is an expectation for the GUE measure, and a $ \tau $-function of an integrable system. For instance, $ \tilde{E}_N(s ; 0) = \mathbb{P}(\boldsymbol{\lambda}_{\max, N} \leq s ) $ where $ \boldsymbol{\lambda}_{\max, N} $ is the largest eigenvalue of a GUE-distributed random matrix.

This quantity has been studied amongst others by Forrester and Witte in the following article

Forrester and Witte use nevertheless the measure $ d\widetilde{\mu}(x) = e^{-x^2} dx $ but I don't think there is much difference.

**Question :** let $ b_N = 2\sqrt{N} $ and $ c_N = N^{-1/6} $. Can we find $ u(N) \in \mathbb{N} $ and $ d_{k, N} $ such that
\begin{align*}%$
d_{k, N} \frac{\tilde{E}_{k + u(N) }( b_N s + c_N ; a + 1) }{ \tilde{E}_{k + u(N) }(b_N s + c_N ; a ) }
\end{align*}
converges to a certain quantity, $ F_k(s ; a) $, say.

Is there a heuristic way to find $ u(N) $ ?

Remark : For $ a = 0 $, we have $\tilde{E}_{N}( b_N s + c_N ; 0 ) \to F_{TW_2}(s) $ which is the cumulative distribution function of the GUE Tracy-Widom distribution. A similar result exists for $ \tilde{E}_{N}( b_N s + c_N ; a ) $ with $ a \geq 1 $. It is proven in Forrester-Witte (eq 4.14) that \begin{align*}%$ \tilde{E}_N( s ; a ) = \tilde{E}_N( s_0 ; a ) \exp\left( \int_{s_0}^s H_{N, a}(t) dt \right) \end{align*} where $ H_{N, a} $ is a particular function satisfying an ODE related with Painlevé IV, that converges to a function related with Painlevé II at the limit.