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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 $$ \frac{1}{n!^2}\int_0^1 \int_0^1\cdots\int_0^1 \frac{\Delta(s_1,s_2,\ldots,s_n) \Delta(t_1,t_2,\ldots,t_n)}{\prod_{i,j}(1-t_i s_j)} d s_1\ldots d s_n d t_1 \ldots d t_n$$

Checking numerically for $n\leq 25$, the decay seems to be like $e^{-2 n^2}$.

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Very interesting question! –  plusepsilon.de Mar 17 '12 at 12:48
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