TITLE: Asymptotics of a Selberg-type integral
QUESTION [23 upvotes]: Let $\Delta(s_1,s_2,\ldots,s_n) := \prod_{i<j}(s_i-s_j)^2$. Is there a standard way to estimate the decay of the Selberg-type integral
$$ I_n:=      \frac{1}{n!^2}\int_0^1 \int_0^1\cdots\int_0^1 \frac{\Delta(s_1,s_2,\ldots,s_n) \Delta(t_1,t_2,\ldots,t_n)}{\prod_{i,j}(1-s_i t_j)^2} d s_1\ldots d s_n d t_1 \ldots d t_n$$ as $n\to\infty$ ?
Checking numerically for $n\leq 25$, the decay seems to be like $e^{-2 n^2}$.

REPLY [4 votes]: This is very late to answer this question, but let me mention a rather different approach.
You can use the Schur function expansion
$$\frac{1}{\prod_{ij}(1-s_it_j)^2}=\sum_{\lambda,\mu}S_\lambda(s)S_\lambda(t)S_\mu(s)S_\mu(t),$$
where the infinite sums are over partitions, to get
$$I_n=\frac{1}{n!^2} \sum_{\lambda,\mu}(g_{\lambda,\mu})^2,$$ with
$$g_{\lambda,\mu}=\int_0^1 \Delta(s) S_\lambda(s)S_\mu(s)ds.$$
The above integral is known (see L.K. Hua, Harmonic analysis of functions of several complex variables in the complex domains, Translations of mathematical monographs, Vol. 6) to be
$$g_{\lambda,\mu}=n!\frac{\prod_{1\le i<j\le n}(\lambda_i-\lambda_j+j-i)(\mu_i-\mu_j+j-i)}{\prod_{1\le k,r\le n}(2n+\lambda_k+\mu_r-k-r+1)}.$$
I am not sure how to extract the asymptotics from the above expression, though.