This is a completely unmotivated question, but what happens to the 1-point marginal distribution for the following $N$-point joint distribution: $$\displaystyle p(z_1,\ldots, z_N) = C_N \exp(-\sum_{j=1}^N exp\left(-\sum_{j=1}^N \|z_j\|^2) |z_j\|^2\right) \prod_{j < k} \|z_j - z_k\|^2$$ Here $z_j$'s are points in $\R^3$ \mathbb{R}^3$ or higher. Presumably one can no longer write the Vandermonde as a determinant so orthogonal polynomial theory breaks down. But I am interested in the distribution of $\|z_1\|$ as $N$ goes to infinity in this case, which shouldn't need the full machinery of orthogonal polynomials and determinantal point processes (but maybe it is still determinantal?).

This is a completely unmotivated question, but what happens to the 1-point marginal distribution for the following $N$-point joint distribution: $$\displaystyle p(z_1,\ldots, z_N) = C_N \exp(-\sum_{j=1}^N \|z_j\|^2) \prod_{j < k} \|z_j - z_k\|^2$$ Here $z_j$'s are points in $\R^3$ or higher. Presumably one can no longer write the Vandermonde as a determinant so orthogonal polynomial theory breaks down. But I am interested in the distribution of $\|z_1\|$ as $N$ goes to infinity in this case, which shouldn't need the full machinery of orthogonal polynomials and determinantal point processes (but maybe it is still determinantal?).