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 $$q_1(x)=1,\qquad q_2(x)=x+d,\qquad q_3(x)=x^2+d(d+2).$$I might not be bold enough to conjecture that $q_N(x)=x^{N-1}+a_N$ for every $N$, for some explicit constant $a_N$ depending on $d$ (polynomial with degree $N-1$).
Using the normalization $\mathrm{e}^{-\|z\|^2/2}$ instead of your $\mathrm{e}^{-\|z\|^2}$ for points $z$ in $\mathbb{R}^d$, the density probability distribution of the first point one-dimensional marginal $z_1$ z$you are interested in is $$\kappa_N\mathrm{e}^{-\|z_1\|^2/2}q_{N}(\|z_1\|^2) kappa_N\mathrm{e}^{-\|z\|^2/2}q_{N}(\|z\|^2)\mathrm{d}z,$$ 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 $$q_1(x)=1,\qquad q_2(x)=x+d,\qquad q_3(x)=x^2+d(d+2).$$ I might not be bold enough to conjecture that ,$q_N(x)=x^{N-1}+a_N$for each every$N$,$q_N(x)=x^{N-1}+a_N(d)$for some explicit constant$a_N(d)$.a_N$ depending on $d$ (polynomial with degree $N-1$).
Using the normalization $\mathrm{e}^{-\|z\|^2/2}$ instead of your $\mathrm{e}^{-\|z\|^2}$ for points $z$ in $\mathbb{R}^d$, the density of the first point $z_1$ you are interested in is $$\kappa_N\mathrm{e}^{-\|z_1\|^2/2}q_{N}(\|z_1\|^2)$$ 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 $$q_2(x)=x+d,\qquad q_3(x)=x^2+d(d+2).$$ I might not be bold enough to conjecture that, for each $N$, $q_N(x)=x^{N-1}+a_N(d)$ for some explicit constant $a_N(d)$.