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 $e^{-2 n^2}$.



Asymptotics of a Selberg-type integral
You can follow large deviation-type estimates from random matrix theory, starting with Ben Arous & Guionnet's paper: <https://link.springer.com/article/10.1007/s004400050119>


You will eventually obtain that your Selberg-type integral behaves like 
$$
\frac{1}{(n!)^2}e^{-n^2 E\_\*}
$$
where $E\_\*$ is the minimum of the functional 
\begin{multline}
E(\mu,\nu)=\iint \log\frac1{|x-y|}d\mu(x)d\mu(y)\\+\iint \log\frac1{|x-y|}d\nu(x)d\nu(y)\\-2\iint\log\frac1{|1-xy|}d\mu(x)d\nu(y)
\end{multline}
where $(\mu,\nu)$ ranges over pairs of Borel probability measures supported in $[0,1]$. I guess you can work out the existence and unicity of a minimizer for $E$ and obtain Euler-Lagrange equations to characterize this minimiser and, hopefully, compute explicitly the minimal value $E\_\*$.


There is quite some technical work to do to fill the gaps, and sorry for the self-advertisement, but if you follow this approach let me tell you that I had to struggle with a similar two type particle problem in the paper <http://www.worldscientific.com/doi/abs/10.1142/S2010326312500165> written with Arno Kuijlaars. I hope this can help. 


By the way, minimizing $E$ is referred as to a vector equilibrium problem in potential theory. 


ADDENDUM: After thinking further, it seems more natural to make the changes of variables $s\_i=1/x\_i$, which leads to the same asymptotics but with $E\_\*$ the minimum of 
\begin{multline}
\tilde E(\mu,\nu)=\iint \log\frac1{|x-y|}d(\mu-\nu)(x)d(\mu-\nu)(y)+\int\_1^\infty \log(x)d\mu(x)
\end{multline}
where $\mu,\nu$ are Borel probability measures living on $[1,\infty)$ and $[0,1]$ respectively. 


