Question

This is a completely unmotivated question, but what happens to the 1-point marginal distribution for the following $N$-point joint distribution: $$\displaystyle p(z_1,\ldots, z_N) = C_N \exp\left(-\sum_{j=1}^N \|z_j\|^2\right) \prod_{j < k} \|z_j - z_k\|^2$$ Here $z_j$'s are points in $\mathbb{R}^3$ 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?).

Answer 1

Using the normalization $\mathrm{e}^{-\|z\|^2/2}$ instead of your $\mathrm{e}^{-\|z\|^2}$ for points $z$ in $\mathbb{R}^d$, the probability distribution of the one-dimensional marginal $z$ you are interested in is $$ \kappa_N\mathrm{e}^{-\|z\|^2/2}q_{N}(\|z\|^2)\mathrm{d}z, $$ where $\kappa_N$ is a positive constant and $q_N$ is a unitary polynomial of degree $N-1$. Small $N$ values of $q_N$ are $$ q_1(x)=1,\qquad q_2(x)=x+d,\qquad q_3(x)=x^2+d(d+2). $$

Answer 2

By $\|z_1\|$, I understand you mean the maximal modulus of the $z_i$'s.

If you are interested in the process of the $\|z_i\|$'s, you have no chance for a determinantal structure since you may have two different $z_i$'s with same modulus with non zero probability.

Concerning the process of the $z_i$'s, note that even if the Ginibre Ensemble (that is the case $\mathbb{R}^2$) is indeed determinantal, its kernel is related to the polynomials orthogonal with $\exp(-\|x\|^2/2)$, that is the $(z^k)_{k\geq 0}$ ... which have trivial zeros ! My point is that except on $\mathbb{R}$ you won't get so much information concerning $\|z_1\|$ from a determinantal structure.

I don't know how prove the convergence of $\|z_1\|$, but note that from your density expression, once renormalized $z_i\rightarrow z_i/\sqrt{N}$, you still can use the Coulomb-gaz approach to characterize the global distribution of the $z_i$'s (for example by proving a large deviation principle for the empirical measure) : It is given by the unique minimizer $\mu^*$ of the functional $$ \iint\log\frac{1}{\| x-y\|}d\mu(x)d\mu(y) +\frac{1}{2}\int \|x\|^2d\mu(x) $$ over probability measures $\mu$ on $\mathbb{R}^3$ (or higher). I guess that $\|z_1\|$ should converge towards $\max \big(Supp(\mu^*)\cap \mathbb{R}\big)$...