Let $N$ be an integer and let $\mathbb{P}$ be the probability measure on $\mathbb{R}^n$ that has density (proportional to) $(x_1,\cdots,x_n) \mapsto e^{-N\sum_i x^2_i} \Delta_N(x)^2$ where $\Delta$ denotes the Vandermonde determinant. Let $t$ be a real number.
I would like to obtain asymptotics on $\int e^{it(x_2-x_1)} d\mathbb{P}(x)$, when $N$ is large and $t$ can be large.
I have read lecture notes that prove that the marginals of $\mathbb{P}$ converge to the semi-circle law, however, it seems that this is far from enough to compute the integral.