# GUE, tau-function of Painlevé II, and an article of Forrester-Witte

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

P. J. Forrester, N. S. Witte, Application of the $$\tau$$-Function Theory of Painlevé Equations to Random Matrices: P-IV, P-II and the GUE (2001), Comm. Math. Phys. 219(2):357-398.

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.