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 $N-1$).
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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 $N-1$). |
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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 $a_N(d)$. |
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