]$ all graphs of degree 2, etc and $\cZ[0]=\1$. Combining (\ref{eq:gen}) and (\ref{eq:Xd}) leads to
\begin{equation}
\label{eq:X}
\Z=\sum_{n=0}^\infty \beta^n \cZ[n],
\end{equation}
and
\begin{equation}\label{eq:logz0}
\ln \Z = -\sum_{n=1}^{\infty}\frac{1}{n} \left ( \halfacommute[-\frac{\Z-\1}{2}] \right )^n \1.
\end{equation}

Let us call $\Set[n]{m}$ the set of all sequences $s=\{s_i\}$ made of strictly positive integers $s_i$ such as $\text{Card}(s)=n$ and $\sum_{i \in s} s_i = m$. Notice that considering non-vanishing integer sequences, the minimal value of $\sum_{i \in s} s_i$ for $s \in \Set[n]{m}$ is $n$ and therefore $\Set[n]{m}$ is non-empty only for $m\geq n$. Moreover, $\Set[0]{m}$ is the empty set for all $m$. Using this notation,
\begin{equation}\label{eq:logz1}
\ln \Z = -\sum_{n=1}^{\infty} \sum_{m=n}^{\infty} \frac{(-1)^n}{n\,2^n} \beta^m \sum_{s \in \Set[n]{m}} \left ( \prod_{i=1}^{n} \halfacommute[{\cZ[s_i]}] \right ) \1,
\end{equation}
which, by inverting summations, can be written in turn as,
\begin{equation}\label{eq:logz2}
\ln \Z = \sum_{m=1}^\infty \beta^m \Zc[m],
\end{equation}
where the order-$m$ {\em non-commutative cumulant} $\Zc[m]$ is defined as,
\begin{equation}\label{eq:cn0}
\Zc[m]=-\sum_{n=1}^m \frac{(-1)^n}{n\,2^n}  \sum_{s \in \Set[n]{m}} \left ( \prod_{i=1}^{n} \halfacommute[{\cZ[s_i]}] \right ) \1,
\end{equation}
for $m>0$. By definition $\Zc[0]=\1$.



The explicit expressions of the first $\Zc[m]$ are:
\begin{align}\label{eq:cn_explicit0}
\Zc[1] &= \cZ[1] \\
\Zc[2] &= \cZ[2]-\frac{1}{4}\acommute[{\cZ[1]}]{\cZ[1]} \nonumber \\
\Zc[3] &= \cZ[3]-\frac{1}{2}\acommute[{\cZ[1]}]{\cZ[2]}+\frac{1}{12} \acommute[{\cZ[1]}]{\acommute[{\cZ[1]}]{\cZ[1]}} \nonumber \\
\Zc[4] &= \cZ[4]-\frac{1}{2}\acommute[{\cZ[1]}]{\cZ[3]}-\frac{1}{4}\acommute[{\cZ[2]}]{\cZ[2]} \nonumber \\
&+\frac{1}{6} \acommute[{\cZ[1]}]{\acommute[{\cZ[1]}]{\cZ[2]}} +\frac{1}{12} \acommute[{\cZ[2]}]{\acommute[{\cZ[1]}]{\cZ[1]}} \nonumber \\
&- \frac{1}{32} \acommute[{\cZ[1]}]{\acommute[{\cZ[1]}]{\acommute[{\cZ[1]}]{\cZ[1]}}} \nonumber
\end{align}


Interestingly, these expressions of $\Zc[m]$ in terms of $\cZ[n]$ can be inverted using the fact $\Z = \exp ( \ln (\Z))$. In the spirit of (\ref{eq:logz0}) and (\ref{eq:logz1}),
\begin{align}\label{eq:inv_logz0}
\Z &= \sum_{m=0}^{\infty}\frac{1}{m!} \left ( \halfacommute[\frac{\ln \Z}{2}] \right )^m \1 \\
  &= \sum_{m=0}^{\infty} \sum_{n=m}^{\infty} \frac{\beta^n}{m!\,2^m} \sum_{s \in \Set[m]{n}} \left ( \prod_{i=1}^{n} \halfacommute[{\cZ[s_i]}] \right ) \1 \\
  &= \sum_{n=0}^{\infty} \beta^n \sum_{m=0}^{n} \frac{1}{m!\,2^m} \sum_{s \in \Set[m]{n}} \left ( \prod_{i=1}^{n} \halfacommute[{\cZ[s_i]}] \right ) \1.
\end{align}

This last expression allows to identify,

\begin{equation}
\label{eq:cn_explicit}
\cZ[n] = \sum_{m=1}^{n} \frac{1}{m!\,2^m} \sum_{s \in \Set[m]{n}} \left ( \prod_{i=1}^{n} \halfacommute[{\Zc[s_i]}] \right ) \1.
\end{equation}

Here is the explicit form of the first terms,
\begin{align}\label{eq:cn_explicit1}
\cZ[1] &= \Zc[1] \\
\cZ[2] &= \Zc[2]+\frac{1}{4}\acommute[{\Zc[1]}]{\Zc[1]} \nonumber\\
\cZ[3] &= \Zc[3]+\frac{1}{2}\acommute[{\Zc[1]}]{\Zc[2]}+\frac{1}{24} \acommute[{\Zc[1]}]{\acommute[{\Zc[1]}]{\Zc[1]}} \nonumber\\
\cZ[4] &= \Zc[4]+\frac{1}{2}\acommute[{\Zc[1]}]{\Zc[3]}+\frac{1}{4}\acommute[{\Zc[2]}]{\Zc[2]}\nonumber \\
&+\frac{1}{12} \acommute[{\Zc[1]}]{\acommute[{\Zc[1]}]{\Zc[2]}} +\frac{1}{24} \acommute[{\Zc[2]}]{\acommute[{\Zc[1]}]{\Zc[1]}} \nonumber \\
&+ \frac{1}{192} \acommute[{\Zc[1]}]{\acommute[{\Zc[1]}]{\acommute[{\Zc[1]}]{\Zc[1]}}} \nonumber
\end{align}

\subsection{Diagrammatic notation}
\label{subsec:diagrammatic}

This Appendix is devoted to the introduction of a lattice-independent diagrammatic notation. Let us introduce the following diagrammatic notation :
\begin{equation}
\label{eq:diag_d}
\underbrace{\gd{NDisconnected}}_{n} = n! \, \cZ[n],
\end{equation}
and its cumulant counterpart,
\begin{equation}
\label{eq:diag_c}
\underbrace{\gdc{NDisconnected}}_{m} = m! \, \Zc[m],
\end{equation}

Defining, for any $s = \{ s_1,\ldots, s_m\} \in \Set[m]{n}$, the combinatorial factor
\begin{equation}
\label{eq:combinatorial}
\comb{s} = \frac{1}{m!} \frac{\left ( \sum_{p=1}^{m} s_p \right )!}{\prod_{p=1}^m s_p!},
\end{equation}
we rewrite equations (\ref{eq:cn0}) and (\ref{eq:cn_explicit}) as,
\begin{align}
\label{eq:zdiag1}
&\mean{\underbrace{\gdc{NDisconnected}}_{m}}=  \\
&