Let $X\_1,...,X\_n$ be iid normal random variables. 


I am looking for a strategy to establish the following limit for fraction of expectation values


$$\lim\_{N \rightarrow \infty} \frac{E(\prod\_{1\le i < j\le n} \vert X\_i-X\_j \vert^{1/n})}{E(\prod\_{1\le i < j\le n-1} \vert X\_i-X\_j \vert^{1/n})}=1.$$


Does anybody have any ideas what to use for this limit?



Convergence of fraction of expectation values
The [Mehta integral](https://en.wikipedia.org/wiki/Selberg_integral#Mehta's_integral) is 
$$M\_n(\gamma):=E\prod\_{1\le i<j\le n}|X\_i-X\_j|^{2\gamma}
=\prod\_{j=1}^n\frac{\Gamma(1+j\gamma)}{\Gamma(1+\gamma)}.$$
So, your fraction under the limit sign is 
$$\frac{M\_n(1/(2n))}{M\_{n-1}(1/(2n)}=\frac{\Gamma(3/2)}{\Gamma(1+1/(2n))}\to\Gamma(3/2)\approx0.886227.$$


