This is a completely unmotivated question, but what happens to the 1point 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?).
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 onedimensional 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 $N1$. Small $N$ values of $q_N$ are $$ q_1(x)=1,\qquad q_2(x)=x+d,\qquad q_3(x)=x^2+d(d+2). $$

$\begingroup$ Hi Didier, Could you give a reference to the unitary polynomials you mentioned? I have never encountered them before. Also do you think the point process defined in my question is actually determinantal? Thanks. $\endgroup$ – John Jiang Apr 7 '11 at 19:31

1$\begingroup$ Hello John. Unfortunately I cannot suggest a reference, I did it myself. Determinantal or not? I do not know. Sorry (twice). $\endgroup$ – Did Apr 7 '11 at 23:35
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 Coulombgaz 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 $$ \iint\log\frac{1}{\ xy\}d\mu(x)d\mu(y) +\frac{1}{2}\int \x\^2d\mu(x) $$ over probability measures $\mu$ on $\mathbb{R}^3$ (or higher). I guess that $\z_1\$ should converge towards $\max \big(Supp(\mu^*)\cap \mathbb{R}\big)$...

$\begingroup$ Hi Adrien, I guess if you have a closed form for the determinantal structure, then you can compute the marginal distribution directly right? $\endgroup$ – John Jiang Jan 25 '12 at 21:36

$\begingroup$ Hi John, I don't see what you mean exactly. Could you elaborate ? $\endgroup$ – Adrien Hardy Jan 26 '12 at 11:02