[@clt]. Considering only the asymptotic solution we need to investigate only the first two cumulants ($n=1,2$) in detail. We will first discuss the structure of $\kappa_1$ and $\kappa_2$, and then deduce a few properties of Eq.(\[SimpleOmega\]).

The first order cumulant $\kappa_1$ of $\mathcal{Z}^{N_{\Omega},Q,E,\vec
  P}$ gives GCE expectation values for particle density $\kappa_1^{N_{\Omega}}$, charge density $\kappa_1^{Q}$, energy density $\kappa_1^{E}$, and expectation values of momentum $\kappa_1^{p_x}$, etc. Since we are only interested in a static source we find due to the antisymmetric momentum integral (see Appendix \[Calc\]) $\kappa_1^{p_x} = \kappa_1^{p_y} = \kappa_1^{p_z}=0$. The general form of the first cumulant $\kappa_1$ is then: $$\begin{aligned}
\kappa_1 = 
\begin{pmatrix}
\kappa_1^{N_{\Omega}}, & \kappa_1^{Q}, & \kappa_1^{E}, & 0, & 0, &  0 
\end{pmatrix}~. \label{vector}\end{aligned}$$

The second cumulant $\kappa_2$ of $\mathcal{Z}^{N_{\Omega},Q,E,\vec
  P}$ contains information about correlations due to different conserved quantities. A detailed discussion of correlation terms only involving Abelian charges and/or energy, e.g. $\kappa_2^{Q,Q}$, $\kappa_2^{Q,E}$, and $\kappa_2^{E,E}$, can be found in [@clt]. Again, due to the antisymmetric nature of the momentum integral, all cumulant entries involving an odd order in one of the momenta, e.g. $\kappa_2^{E,p_x}$, $\kappa_2^{p_x,p_y}$, or $\kappa_2^{Q,p_x}$ are equal to zero. The general second order cumulant $\kappa_2$ thus reads: $$\begin{aligned}
\kappa_2 = 
\begin{pmatrix}
\kappa_2^{N_{\Omega},N_{\Omega}} & 