A method of integration over matrix variables: IV

The $m$-point correlation function $$\int \left [\prod_{i=1}^n \mu (x_i)\right]\left[\prod_{1\le j<k\le n}|x_j-x_k|^{\beta}\right] {\rm d}x_{m+1}...{\rm d}x_n,$$ is calculated for the three values $\beta=1,2$ and 4, and integers $m$ and $n$ with $0\le m\le n$. For some applications one needs this integral when $\mu(x)=\exp [-V(x)]$, $V(x)$ an even polynomial, specially in the limit $n\to \infty$ keeping $m$ finite. A conjecture for this limit in the case $\beta=2$ is given when $V(x)=x^2+\gamma x^4$