\omega^{\Varphi}_{x+\varepsilon}(z)=\sum_{n\geq 0}\beta^{\Varphi}_{2n+1}(x+\varepsilon,y+\varepsilon,\ldots,y+\varepsilon,x+\varepsilon)z^{n+1} \to \sum_{n\geq 0}\beta^{\Varphi}_{2n+1}(x,y,\ldots,y,x)z^{n+1}

as formal series, since $x+\varepsilon 1_\mathcal{A}$ converges in distribution to $x$ as $\varepsilon$ goes to $0$.

We move on to prove a subordination formula for the $\eta$-transform of the product of two conditionally free random variables. The formula we obtain can be easily guessed from the rule of thumb that such formula is inferred from the subordination formula for the additive convolution (see equation (<ref>)) upon replacing the $F$-transform by the $\eta$-transform (and the addition of series by the Cauchy product of series).

Let $x,y \in \mathcal{A}$ be conditionally free random variables in a conditional probability space $(\mathcal{A}, \Psi, \Varphi)$, then the following equality of formal series holds:

\begin{align*}
\eta_{xy}^{\Psi}(z)\eta_{xy}^{\Varphi}(z) = \eta_x^{\Varphi}(\omega^{\Psi}_x(z))\eta_y^{\Varphi}(\omega^{\Psi}_y(z)),
\end{align*}

or else:

\begin{equation}
\label{eqn:toshowcondmult}
\rho_{xy}^{\Varphi}(z) = \rho_x^{\Varphi}(\omega^{\Psi}_x(z))\rho_y^{\Varphi}(\omega^{\Psi}_y(z)).
\end{equation}

We refer to [30] for the definition of the various transforms we will use. As before, we will first treat the case where $\Psi(x)\neq 0$ and $\Psi(y)\neq 0$. In particular we will be using the conditional $R$-transform $C^{\Varphi|\Psi}_x\in\mathbb{C}[[z]]$, defined by the following functional equation:

\begin{align*}
\end{align*}

The conditional cumulants $\kappa^{\Varphi | \Psi}_n(x)$ of $x$ are defined as the Taylor coefficients at $0$ of the (shifted) conditional $R$-tranform:

C_x^{\Varphi | \Psi}(z)=\sum_{n\geq 1} \kappa^{\Varphi|\Psi}_n(x)z^n.

From the previous equation $\tilde{C}_x^{\Varphi|\Psi}(z):=\frac{1}zC^{\Varphi |  \Psi}(z)$ is well-defined as a series in $\mathbb{C}[[z]]$. The conditional $S^{\Varphi|\Psi}_x$ transform  of $x$ is defined by:

\begin{align*}
(S_x^{\Varphi|\Psi}(z))^{-1} := \tilde{C}_x^{\Varphi|\Psi}[(C^{\Psi}_x)^{\langle -1 \rangle}(z)].
\end{align*}

Note that $(C^{\Psi}_x)^{\langle -1 \rangle}(z))^{\langle -1 \rangle}$ is well-defined since $\Psi(x)\neq 0$.
Recall that simple arithmetic manipulations yield:

$$M^{\Psi}_x(\frac{z}{1+C^{\Psi}_x(z)})=C^{\Psi}_x(z) \Leftrightarrow M^{\Psi}_x(w)=C_x^{\Psi}(w(1+M_x^{\Psi}(w))),$$

and further:

Substituting $w = (C_x^{\Psi})^{\langle -1 \rangle}(z)=(1+z)(M_x^{\Psi})^{\langle -1 \rangle}(z)$ in the previous formula yields:

(M^{\Psi}_x)^{\langle -1 \rangle}(z)\tilde{C}_x^{\Varphi|\Psi}[(C^{\Psi}_x)^{\langle -1 \rangle}(z)]= \frac{M^{\Varphi}_x[(M_x^{\Psi})^{\langle -1 \rangle}(z)]}{1+M^{\Varphi}_x[(M_x^{\Psi})^{\langle -1 \rangle}(z)]}.

We thus obtain for the conditional $S$-transform of $x$:

S_x^{\Varphi|\Psi}(z) = (1+M^{\Varphi}_x [(M_x^{\Psi})^{\langle -1 \rangle}(z)])\frac{(M_x^{\Psi})^{\langle -1 \rangle}(z)}{M^{\Varphi}_x[(M_x^{\Psi})^{\langle -1 \rangle}(z)]} = (\frac{z}{\eta^{\Varphi}_{x}(z)})\circ (M_x^{\Psi})^{\langle -1 \rangle}(z).

Since the conditional $S$-transform is multiplicative over conditionally free multiplicative convolution [30]:

S_{xy}^{\Varphi |\Psi}(z)=S^{\Varphi |\Psi}_x(z) S_y^{\Varphi |\Psi}(z)

\frac{z}{\eta_{xy}^{\Varphi}(z)}=[\frac{z}{\eta^{\Varphi}_x(z)}][(M_x^{\Psi})^{\langle -1 \rangle}\circ (M_{xy}^{\Psi})(z)] [\frac{z}{\eta^{\Varphi}_y(z)}][(M_{y}^{\Psi})^{\langle -1 \rangle}\circ (M_{xy}^{\Psi})(z)].

\begin{equation*}
\eta_{xy}^{\Varphi}(z)=\frac{z}{\omega^{\Psi}_x(z)\omega^{\Psi}_y(z)} \eta^{\Varphi}_x(\omega^{\Psi}_x(z)) \eta^{\Varphi}_y(\omega^{\Psi}_y(z)).
\end{equation*}

This concludes the proof since $\omega^{\Psi}_x(z) \omega^{\Psi}_y (z) = z \eta_{xy}^{\Psi}(z)$.

We will use the notations introduced by the author in [23]. Given two distributions $\mu_x,\mu_y\colon \mathbb{C}\langle x \rangle \to \mathbb{C}$, their multiplicative monotone convolution, denoted by $\mu_x \circlearrowright \mu_y$ is defined through the following relation on the $\eta$-transforms:

\eta_{\mu_x \circlearrowright \mu_y}=\eta_{\mu_x} \circ \eta_{\mu_y}.

The subordination series $\omega_{\mu_x}$ coincides with the $\eta$-series of the s-free convolution $\mu_x\boxslash\mu_y$ of the two distributions $\mu_x$ and $\mu_y$.
The Boolean convolution $\mu_x \boxasterisk \mu_y$ of the two distributions $\mu_x$ and $\mu_y$ is the distribution whose $\rho$-transform satisfies:

\rho_{\mu_x \boxasterisk \mu_y} = \rho_{\mu_x}\rho_{\mu_y},

The statement of the previous proposition <ref> can be restated as follows:

\mu^{\Varphi}_x \prescript{}{\mu_x^{\Psi}}{\boxtimes}{}_{\mu_y^{\Psi}} \mu^{\Varphi}_y = (\mu^{\Varphi}_x \angle \mu^{\Psi}_y \boxslash \mu^{\Psi}_x) \boxasterisk (\mu^{\Varphi}_y\angle \mu^{\Psi}_x \boxslash \mu^{\Psi}_y),

where $\angle$ is the orthogonal convolution of distributions [23], by definition:

\begin{equation*}
\rho_{\mu_x \angle \mu_y} (z)= \rho_{\mu_x} \circ \eta_{\mu_y}(z).
\end{equation*}

and recall that the s-free convolution is the solution to the fixed point equation $\mu^{\Psi}_x \angle \mu^{\Psi}_y \boxslash \mu^{\Psi}_x = \mu^{\Psi}_y \boxslash \mu^{\Psi}_x$ and

\mu^{\Psi}_x \boxtimes \mu^{\Psi}_y = (\mu^{\Psi}_y \boxslash \mu^{\Psi}_x) \boxasterisk (\mu^{\Psi}_x \boxslash \mu^{\Psi}_y).

The previous proposition is a statement on formal series. To promote the equality (<ref>) to an equality between holomorphic functions on the upper-half plane, one has to require $x,y$ to be positive operators or unitaries. In that case, the subordination series $\omega_x$ and $\omega_y$ are the expansion around the origin of anayltic functions defiend over the complex minus the positive half-line.

The motivation of the authors in [13] was to give a combinatorial interpretation and a multivariate extension of the result contained in Proposition [4] elucidating a link between conditional additive convolution and infinitesimal additive convolution (in a nutshell : taking distributional derivative intertwines conditionally free additive convolution and infinitesimal convolution). In particular,

G^{W(\Varphi)}_x(z) = \frac{d}{dz} h_x^{\Varphi}(z),\quad  h_x^{\Varphi}(z) := \rho_x^{\Varphi}(\frac{1}{z}).

In view of the last two propositions, it is natural to ask whether there exists some linear functional ${\sf \tilde{W}}(\Psi,\Varphi)$ on $T(\mathcal{A})$ such that:

\frac{d}{dz} \ln(\rho_x^{\Varphi}(z)) = \partial^{{\sf \tilde{W}}(\Psi,\Varphi)}\rho_x(z) = \frac{M_x^{{\sf \tilde{W}}(\Psi,\Varphi)}}{(1+M_x^{\Psi})^2}.

For in that case, we would have, if $x$ and $y$ are infinitisimally free in $(\Psi, {\sf \tilde{W}}(\Psi,\Varphi))$ and conditionally free in $(\Psi,\Varphi)$ $\partial^{{\sf \tilde{W}}(\Psi,\Varphi)} \rho_{xy}(z)=\frac{d}{dz} {\rm ln}(\rho^{\Varphi}_{xy}(z)).$
Taking ${\sf \tilde{W}}(\Psi,\Varphi) = {\sf{W}}(\Psi,\Varphi)$ does not work.

For readability, we set $\delta = 1-\OmegA(1_\mathcal{A})$ in the next proposition.

Let $x,y \in \mathcal{A}$ be two cyclic c-free random variables in a cyclic-conditional probability space $(\mathcal{A},\Psi,\Varphi,\OmegA)$,

\begin{align*}
{M}^{\OmegA}_{xy}(z)=z\frac{d}{dz}{\rm log}(\omega^{\Psi}_x(z))&{M}^{\OmegA}_y(\omega^{\Psi}_x(z))+z\frac{d}{dz}{\rm log}(\omega^{\Psi}_y(z)){M}^{\OmegA}_x(\omega^{\Psi}_y(z))-z\frac{d}{dz}{\rm log}(\frac{1-\eta_{xy}^{\Varphi}(z)}{(1-\eta_{x}^{\Varphi}(\omega^{\Psi}_x(z))(1-\eta_{y}^{\Varphi}(\omega^{\Psi}_y(z))}) \\
&\hspace{8.5cm}-\delta z\frac{d}{dz}{\rm log}(1-\eta^{\Psi}_{xy}(z))
\end{align*}

or else:

\begin{align*}
\tilde{M}^{\OmegA}_{xy}(z)=\frac{d}{dz}\omega^{\Psi}_x(z)&\tilde{M}^{\OmegA}_y(\omega_x(z))+\frac{d}{dz}\omega^{\Psi}_y(z)\tilde{M}^{\OmegA}_x(\omega_y(z))-\frac{d}{dz}{\rm log}(\frac{1-z\rho_{xy}^{\Varphi}(z)}{(1-\omega^{\Psi}_x(z)\rho_{x}^{\Varphi}(\omega^{\Psi}_x(z))(1-\omega^{\Psi}_y(z)\rho_{y}^{\Varphi}(\omega^{\Psi}_y(z))}) \\
&\hspace{8.5cm}-\delta \frac{d}{dz}{\rm log}(1-\eta^{\Psi}_{xy}(z)).\end{align*}

We compute first:

\begin{align*}
\partial^{[\Varphi]} \eta_x(z)&:=\frac{\partial^{[\Varphi]} M_x(z)}{(M^{\Psi}_x(z)+1)^2}\\
&=(1-\eta_x^{\Psi}(z))^2(\partial^{[\Varphi]} M_x(z))\\
&=-(1-\eta_x^{\Psi}(z))^2z\frac{d}{dz}{\rm log}(1-\eta_x^{\Varphi}(z)).
\end{align*}

This implies, since $\ell \mapsto \partial^{\ell} \eta_x$ is linear and by setting $D\Psi : = \OmegA -[\Varphi] + \delta [\Psi]$ (as in Proposition <ref>):

\begin{align*}
\partial^{D\Psi}\eta_x(z)&=\partial^{\OmegA} \eta_x(z)+(1-\eta_x^{\Psi}(z))^2 z\frac{d}{dz}{\rm log}(1-\eta_x^{\Varphi}(z))-{\delta z}(1-\eta^{\Psi}_{x}(z))^2\frac{d}{dz}{\rm log}(1-\eta_x^{\Psi}(z)).
\end{align*}

Recall from Proposition <ref> that $x,y$ are cyclic c-free in $(\mathcal{A},\Psi,\Varphi,\OmegA)$ then then are cyclically-free in $(T(\mathcal{A}),\Psi, D\Psi)$). Remark <ref> and Proposition <ref> imply:

\begin{align}
\label{eqn:fondamental}
\partial^{D\Psi}\eta_{xy}(z)=\Big(z\frac{d}{dz}{\rm ln}(\omega^{\Psi}_x(z))\Big)(\partial^{D\Psi}\eta_x \circ \omega^{\Psi}_x(z))  + \Big(z\frac{d}{dz}{\rm ln}(\omega^{\Psi}_y(z))\Big)(\partial^{D\Psi} \eta_y \circ \omega^{\Psi}_y(z)),
\end{align}

\begin{align}
\label{eqn:relationun}
&z\frac{(\omega^{\Psi}_x)'(z)}{\omega^{\Psi}_x(z)}\partial \eta^{D\Psi}_x \circ \omega^{\Psi}_x(z) =z\frac{(\omega^{\Psi}_x)'(z)}{\omega^{\Psi}_x(z)}\partial \eta_{x}^{\OmegA}(\omega^{\Psi}_x(z))\\
&\hspace{2cm}+(1-\eta^{\Psi}_{xy}(z))^2z\frac{d}{dz}{\rm log}(1-\eta_{x}^{\Varphi}(\omega_x^{\Psi}(z))-{\delta z}(1-\eta^{\Psi}_{xy}(z))^2\frac{d}{dz}{\rm log}(1-\eta_{xy}^{\Psi}(z))
\end{align}

Of course, we have a similar relation when $x$ is replaced by $y$. By injecting (<ref>) in (<ref>), we deduce the first formula of the proposition.

As for additive convolution, we specialize the above proposition to two cases; cyclic Boolean independence between $x-1$ and $y-1$ and cyclic monotone independence between $x-1$ and $y$.

Assume that $x$ and $y$ are two random variables in a cyclic Boolean probability space
and assume that $x-1$ and $y-1$ are cyclic Boolean independent. One has the following relations between the Moments generating functions of $x$,$y$ and $xy$ with respect to $\OmegA$:

\begin{equation*}
M^{\OmegA}_{xy}(z)=M_x^{\OmegA}(z)+M_y^{\OmegA}(z)-z \frac{d}{dz} {\rm log}(\frac{1-z\rho^{\Varphi}_x(z)\rho^{\Varphi}_{y}(z)}{(1-z\rho^{\Varphi}_x(z))(1-z\rho^{\Varphi}_y(z))}) - \frac{z}{1-z}
\end{equation*}

From equation (<ref>), specializing the equation (<ref>) to the case where $x-1$ and $y-1$ are Boolean independent among to $\omega_x(z)=\omega_y(z)=z$ in (<ref>) and $\eta^{\Psi}_{x}(z)=\eta^{\Psi}_{y}(z)=\eta^{\Psi}_{xy}(z) = z$.

If setting, from a random variable $x$ in a cyclic probability space $(\mathcal{A},\Varphi,\OmegA)$:

\begin{align*}
\partial^{\OmegA | \Varphi}\Sigma_x(z):=M_x^{\OmegA}(z)+z\frac{d}{dz}\log(1-z\rho^{\Varphi}_{x}(z)) + \frac{z}{1-z},
\end{align*}

from Proposition <ref>, we infer the following:

\partial^{\OmegA | \Varphi}\Sigma_{xy}(z) = \partial^{\OmegA | \Varphi}\Sigma_{x}(z)+\partial^{\OmegA | \Varphi}\Sigma_{y}(z)

whenever $x-1$ and $y-1$ are cyclic Boolean independent. It is instructive to compute $\frac{1}{z}\partial^{\OmegA | \Varphi}\Sigma_x(\frac{1}{z})$; straightforward computations yield:

\begin{align*}
\frac{1}{z}\partial^{\OmegA | \Varphi}\Sigma_x(\frac{1}{z}) &= G^{\OmegA}_x(z)+\frac{d}{dz}{\rm log}(zG^{\Varphi}_x(z)) + \frac{1}{z(z-1)} \\
& = \mathfrak{h}^{\OmegA}_x +  \frac{1}{z(z-1)},
\end{align*}

where we have used the notations from [2] (we added a superscript). The transform $\mathfrak{h}^{\OmegA}$ linearizes additive cyclic Boolean convolution and related to the generating series $\mathfrak{c}$ of the cyclic Boolean cumulants via

\mathfrak{c}_x=\frac{1}{z}\mathfrak{h}_x (\frac{1}{z})+z\frac{d}{dz}\eta^{\Psi}_x(z)

see also Section <ref>.
As in immediate consequence of the above formula, we get the following corollary.
Let $x_1,\ldots,x_N$ be a sequence of identically distributed elements in a cyclic Boolean probability space, such that $x_1-1,\ldots,x_N-1$ are cyclic Boolean independent. Then:

\begin{equation}
M^{\OmegA}_{x_1\cdots x_N}(z)=NM^{\OmegA}_{x_1}(z)+z\frac{d}{dz}{\rm log}\Big(\frac{1-z\rho^{\Varphi}_{x_1}(z)^{N}}{(1-z\rho^{\Varphi}_{x_1}(z))^N}\Big) - N\frac{z}{z-1}.
\end{equation}

As a final remark about cyclic Boolean multiplicative convolution, we comment on infinite multiplicative infinite divisibility in the case of a unitary operator $u$ acting on some finite dimensional Hilbert space $H$. We say that $u$ is Boolean multiplicative infinitely divisible if for each integer $n \geq 1$, there exists a finite sequence $(u_1,\ldots,u_n)$ of unitaries of $B(H)$, all identically distributed under each of the functional $\Varphi$ and $\OmegA$ such that $u=u_1\cdots u_n$. In particular, $u$ is Boolean multiplicative infinitely divisible. There is not many of such of operators. In fact, if we assume that $u$ has two distinct eigenvalues $e^{{\sf i}b_1}$ and $e^{{\sf i}b_2}$, with multiplicities $m_1$ and $m_2$ then the $\eta$-transform $\eta^{\Varphi}$ of $u$ has $0$ in its range; $\eta_u^{\Varphi}(\frac{m_1}{{\rm dim}(H)}e^{{\sf i}b_1} + \frac{m_2}{{\rm dim}(H)}e^{{\sf i}b_2})=0$. From [15], this prevents $u$ from being infinitely divisible.

We move on to computing the moment series with respect to $\OmegA$ of two cyclically monotone random variables $x$ and $y$.
Let $x$ and $y$ be two random variables in a cyclic probability space $(\Varphi,\OmegA)$ such that $x-1$ is cyclic monotone independent from $y$, then:

\begin{align}
\tilde{M}_{xy}^{\OmegA}(z)=\tilde{M}_y^{\OmegA}(z) + (\tilde{M}_x^{\OmegA}\circ \eta_{y}^{\Varphi}(z))(\eta^{\Varphi}_{y})^{\prime}(z)
%+\frac{d}{dz}{\rm log}(1-\eta_y^{\Varphi}(z)).
\end{align}

where $\tilde{M}^{\Varphi}_x(z)=z^{-1}M_x^{\Varphi}(z)$.

It among to take $\omega^{\Psi}_x(z)=\eta^{\Psi}_y(z),\omega^{\Psi}_y(z)=z$, $\eta^{\Psi}_{xy}(z)=\eta_y^{\Varphi}(z),\eta^{\Psi}_x(z)=z,\eta_y^{\Psi} (z)=\eta_y^{\Varphi}(z)$ in the formula of Proposition <ref> and $\delta = 1$.

§.§ Limit theorems

In this section, we utilize the relation (<ref>) to prove limit theorems for sums of cyclically conditionally free random variables. For the Central Limit Theorem, we rely on the Central Limits theorems for conditionally free random variables and for infinitesimally free random variables, which we recall now.
Let $(X_i)_{i\leq 0}$ be a sequence of random variables in a conditional probability space $(\mathcal{A},\Psi,\Varphi)$ identically distributed with respect to both states $\Psi$ and $\Varphi$ and centred. We notate:

\begin{equation*}
\alpha^2 := \Psi(X_1^{2}),\quad \beta^2 = \Varphi(X_1^2).
\end{equation*}

As $N$ tends to infinity, $S_N:= \frac{1}{\sqrt{N}}\sum_{i=1}^N X_i$ converges toward an element $s_{\alpha,\beta}$ whose $\Varphi$ distribution has Cauchy transform:

$$G^{\Varphi}_{\alpha^2,\beta^2}(z)=\frac{z( \frac{\beta^2}{2}-\alpha^2)+\frac{\beta^2}{2}\sqrt{z^2-4\alpha^2})}{(z^2(\beta^2-\alpha^2)-\beta^4)}$$

and is distributed under $\Psi$ as a semi-circular element with variance $\alpha^2$.

We use the notations of the previous proposition.
Let $(X_i)_{i\geq 0}$ be a sequence of infinitesimally free, centred identically distributed element of an infinitesimal probability space $(\Psi,\partial \Psi)$ and set

\alpha^2 := \Psi(X_i^2),~\alpha^{\prime} := \partial \Psi(X_1^2).

As $N$ tends to infinity, $S_N$ tends in infinitesimal distribution to an infinitesimal semi-circular element $s^{\prime}_{\alpha^2,\alpha^{\prime}}$ whose infinitesimal Cauchy transform is given by

$$\partial^{\Psi}G_{s_{\alpha^2,\alpha^{\prime}}}(z):= \partial \Psi((z-s_{\alpha^2,\alpha^{\prime}})^{-1})=\frac{\alpha^{\prime}}{\alpha^2}(\frac{1}{\sqrt{z^2-4\alpha^2}}-\frac{(z-\sqrt{z^2-4\alpha^2})}{2\alpha^2}).$$

Let $(X_i)_{i\geq 0}$ be a sequence of cyclic c-free random variables in a cyclic-conditional probability space $(\Psi, \Varphi,\OmegA)$ centred and identically distributed. Then, as $N$ tends to infinity, $S_N$ converges in distribution to a random variable $s_{\alpha^2,\beta^2,\gamma}$ whose distribution has for Cauchy transform with respect to $\OmegA$:

\begin{align*}G^{\OmegA}_{s_{\alpha,\beta^2,\gamma}}(z)= \frac{\alpha^{\prime}}{\alpha^2}(\frac{1}{\sqrt{z^2-4\alpha^2}}-\frac{z-\sqrt{z^2-4\alpha^2}}{2\alpha^2})-(1-\OmegA(1_\mathcal{A}))\frac{1}{\sqrt{z^2-4\alpha^2}}\\
\frac{\frac{\beta^2}{2}-\alpha^2 + \frac{\beta^2}{2}\frac{z}{\sqrt{z^2-4\alpha^2}}}{ z(\frac{\beta^2}{2}-\alpha^2)+\frac{\beta^2}{2}\sqrt{z^2-4\alpha^2}} - \frac{2z(\beta^2-\alpha^2)}{z^2(\beta^2-\alpha^2)-\beta^4} \right),
\end{align*}

where $\alpha^2=\Psi(X^2)$, $\beta^2=\Varphi(X^2)$, $\gamma^2=\OmegA(X_1^2)$ and $\alpha^{\prime}=[(1-\OmegA(1_\mathcal{A}))\alpha^2-\beta^2+\gamma^2)].$

Before proving the last proposition, let us derive the cyclic Boolean Central Limit Theorem, as in [2], Theorem 4.7. Formally, we want to set $\OmegA(1_{\mathcal{A}})=1$ and $\alpha=0,~\beta=1,\gamma^2=1$ which implies $\alpha^{\prime}=\gamma^2-1=0$
Straightforward computations show that:

\begin{align*}
&\frac{\frac{\beta^2}{2}-\alpha^2 + \frac{\beta^2}{2}\frac{z}{\sqrt{z^2-4\alpha^2}}}{ z(\frac{\beta^2}{2}-\alpha^2)+\frac{\beta^2}{2}\sqrt{z^2-4\alpha^2}} - \frac{2z(\beta^2-\alpha^2)}{z^2(\beta^2-\alpha^2)-\beta^4}=\frac{1}{z}-\frac{2z}{z^2-1} + o_{\alpha,0}(1)
\end{align*}

\begin{align*}
\frac{\alpha^{\prime}}{\alpha^2}\left(\frac{1}{\sqrt{z^2-4\alpha^2}}-\frac{z-\sqrt{z^2-4\alpha^2}}{2\alpha^2}\right)= \alpha^{\prime}(\frac{1}{z^3}) + o_{\alpha,0}(1).
\end{align*}

We obtain, with the notations of [2]:

\begin{align*}
\tilde{G}^{\OmegA}_{s_{0,1,\gamma^2}}=-\frac{2}{z} + \frac{2z}{z^2-1}+ \frac{\gamma^2-1}{z^3}=-\frac{2}{z} + \frac{2z}{z^2-1},
\end{align*}

which is the cyclic Boolean Central Limit Theorem.

We let $S = s_{\alpha^2,\beta^2,\gamma}$. Let $S_n=\frac{1}{\sqrt{n}}(a_1+\cdots+a_n)$,  we will denote by $S$ its possible limit.
Firstly, with respect to $\Psi$, $S_n$ satisfies the free CLT with variance $\alpha^2=\Psi(a^2)$. So

G^{\Psi}_{S_n}(z)\to \frac{z-\sqrt{z^2-4\alpha^2}}{2\alpha^2}=:G^{\Psi}_S(z),

and from an easy calculation we get:

Secondly, with respect to the pair $(\Psi,\Varphi)$, $S_n$ satisfies the conditionally free CLT. So

$$G^{\Varphi}_{S_n}(z)\to \frac{z(\frac{\beta^2}{2}-\alpha^2)+1/2 \beta^2\sqrt{z^2-4\alpha^2})}{(z^2(\beta^2-\alpha^2)-\beta^4)}=:G^{\Varphi}_{S}(z),$$

that satisfies

\frac{G^{\Varphi}_S(z)'}{G^{\Varphi}_S(z)}=\frac{\frac{\beta^2}{2}-\alpha^2 + \frac{\beta^2}{2}\frac{z}{\sqrt{z^2-4\alpha^2}}}{ z(\frac{\beta^2}{2}-\alpha^2)+\frac{\beta^2}{2}\sqrt{z^2-4\alpha^2}} - \frac{2z(\beta^2-\alpha^2)}{z^2(\beta^2-\alpha^2)-\beta^4}

Finally, $S_n$ satisfies an Infinitesimal CLT for $D\Psi$ with variance

D\Psi(a^2)=\OmegA(a^2)-\Varphi(a^2)+(1-\OmegA(1_A))\Psi(a^2) = \gamma^2-\beta^2+(1-\OmegA(1_{\mathcal{A}}))\alpha^2,

which implies:

$$G^{D\Psi}_{S_n}(z)\to \frac{\alpha^{\prime}}{\alpha^2}\frac{1}{\sqrt{z^2-4\alpha^2}}-\frac{z-\sqrt{z^2-4\alpha^2}}{2\alpha^2}.$$

We say that $a\in(\mathcal{A},\Psi,\Varphi,\OmegA)$ is Bernoulli with parameters $((\lambda_{\Psi},\lambda_{\Varphi},\lambda_{\OmegA}),(\alpha_{\Psi},\alpha_{\Varphi},\alpha_{\OmegA}))$, $\lambda_{\Psi},\lambda_{\Varphi}\in [0,1]$ and $\lambda_{\OmegA} \in \mathbb{R}$, $\alpha_{\Psi},\alpha_{\Varphi},\alpha_{\OmegA} \in \mathbb{R}$ in the cyclic-conditional probability space $(\Psi, \Varphi, \OmegA)$
if it is a Bernoulli element in the conditional probability space $(\mathcal{A},\Psi,\Varphi)$ and an infinitesimal Bernoulli element in the cyclic probability space $(T(\mathcal{A}),\Psi,D\Psi)$. In particular, for $n\geq 1$

\Psi(a^n)=\lambda_{\Psi}\alpha_{\Psi}^n,~\Varphi(a^n)=\lambda_{\Varphi}\alpha_{\Varphi}^n,

D\Psi(a^n)=\lambda_{\OmegA}\alpha_{\Psi}^{n}+n\lambda_{\Psi}\alpha_{\Psi}^{n-1}\alpha_{\OmegA},~\Psi(a^n)= \lambda_{\Psi}\alpha_{\Psi}^n.

From the formulae:

\begin{equation*}
G_a^{\Psi}(z)=\frac{1-\lambda_{\Psi}}{z}+\frac{\lambda_{\Psi}}{z-\alpha_{\Psi},},~ -\frac{(G_a^{\Psi})^{\prime}(z)}{G_{a}^{\Psi}(z)}=\frac{1}{z}+\frac{1}{z-\alpha_{\Psi}},
\end{equation*}

we easily get that $a$ is Bernoulli if and only if, for any integer $n\geq 1$ ($\delta = 1-\OmegA(1_{\mathcal{A}}$ ):

\begin{equation*}
\OmegA(a^n)=\alpha_{\Psi}^n(\delta+\lambda_{\OmegA})-\alpha^n_{\Varphi}\lambda_{\Varphi}+n\lambda_{\Psi}\alpha_{\Psi}^{n-1}\alpha_{\OmegA}.
\end{equation*}

Assume that $\lambda_{\Psi}(N)N\to\lambda_{\Psi}$ and $\lambda_{\Varphi}(N)N\to\lambda_{\Varphi}$ and take for each $N\geq 1$ a sequence $(X^{(N)}_1,\ldots,X_N^{(N)})$ of i.i.d conditionally free Bernoulli elements with rates $(\lambda_{\Psi}(N),\lambda_{\Varphi}(N))$ and $\alpha_{\Psi}=\alpha_{\Varphi}=1$. Then,

B^{N}=\sum_{i=1}^N X_i^{(N)}

converges in non-commutative distribution to a conditional free Poisson random variable in $(\mathcal{A},\Varphi,\Psi)$ with parameters $((\lambda_{\Psi},\lambda_{\Varphi}), (1,1))$:

G_{B^{N}}^{\Psi}(z) \to \mathcal{P}_{\lambda_{\Psi}}(z):=\frac{1}{2z}\big({z}+(1-\lambda^{\Psi})-\sqrt{(z-(1+\lambda_{\Psi}))^2-4\lambda_{\Psi}})\big)

G^{\Varphi}_{B^{N}}(z)\to \mathcal{P}^c_{\lambda_{\Psi},\lambda_{\Varphi}}:=\frac{z(2\lambda_{\Psi}-\lambda_{\Varphi})+\lambda_{\Varphi}(1-\lambda_{\Psi})-\lambda_{\Varphi}\sqrt{(z-(1+\lambda_{\Psi}))^2-4\lambda_{\Psi}}}{2z(z(\lambda_{\Psi}-\lambda_{\Varphi})+\lambda_{\Varphi}(1-\lambda_{\Psi}+\lambda_{\Varphi}))}.

Recall that the free Poisson distribution with parameters $\alpha,\lambda$ has Cauchy transform:

\begin{equation*}
\mathcal{P}_{\alpha,\lambda}(z)=\frac{z-\alpha(1+\lambda)-\sqrt{(z-\alpha(1+\lambda))^2-4\lambda\alpha^2}}{2\alpha z}
\end{equation*}

In comparison with our previous notations, $\mathcal{P}_{1,\lambda}(z)=\mathcal{P}_{\lambda}(z)$.
In an infinitesimal probability space $(\theta,\theta^{\prime})$, for each $N\geq 1$ we pick i.i.d Bernoulli elements $(X_1^{(N)},\ldots,X_{N}^{(N)})$ with rates $\lambda_{\theta}(N)N\to \lambda_{\theta}$ and $\lambda_{\theta^{\prime}}(N)N\to \lambda_{\theta^{\prime}}$.
Then $B^{(N)}$ converges in distribution toward an infinitesimal Poisson distribution with parameter $(\lambda_{\theta},\lambda_{\theta^{\prime}}$):

G_{B_N}^{\theta}(z)\to \mathcal{P}_{\lambda_{\theta}}(z),~G_{B_N}^{\theta^{\prime}}(z)\to \frac{d}{du}_{|u=0}\mathcal{P}_{1+u,\lambda_{\theta}+\lambda^{\prime}_{\theta}u}(z).

For each $N\geq 1$, let $X^{(N)}_1,\ldots,X_N^{(N)}$ be a finite sequence of size $N$ of i.i.d Bernoulli elements in $(\mathcal{A},\Psi,\Varphi,\OmegA)$. Under the assumptions:

\lambda_{\Psi}(N)N \to \lambda_{\Psi},\quad\lambda_{\Varphi}(N)N \to \lambda_{\Varphi},\quad\lambda_{\OmegA}(N)N \to \lambda_{\OmegA},

the Cauchy transform under $\OmegA$ of the sum $B_N$ of the $X_i^{(N)}, ~ 1 \leq i \leq N$ converges toward  $G^{\OmegA}_{p_{\lambda_{\Psi},\lambda_{\Varphi},\lambda_{\OmegA}}}$ that satisfies:

\frac{d}{du}_{|u=0}\mathcal{P}_{1+u,\lambda_{\Psi}+\lambda_{\OmegA}u}(z) =G^{\OmegA}_{p_{\lambda_{\Psi},\lambda_{\Varphi},\lambda_{\OmegA}}}(z)-(1-\OmegA(1_{\mathcal{A}}))\frac{(\mathcal{P}_{\lambda_\Psi})^{\prime}(z)}{\mathcal{P}_{\lambda_\Psi}(z)}+\frac{(\mathcal{P}^c_{\lambda_{\Psi},\lambda_{\Varphi}})^{\prime}(z)}{\mathcal{P}^c_{\lambda_{\Psi},\lambda_{\Varphi}}(z)}.

In the context of cyclic Boolean independence, that is when $ \lambda_{\Psi}=0,~\OmegA(1_{\mathcal{A}})=0$ then:

\begin{align*}
&\mathcal{P}^c_{0,\lambda_{\Varphi}}(z)= \frac{1}{z-(1-\lambda_{\Varphi})}, \quad \frac{d}{dz}{\rm log}\mathcal{P}^c_{0,\lambda_{\Varphi}}(z) = -\frac{1}{z-(1-\lambda_{\Varphi})},\\
&\mathcal{P}_{0}(z) = \frac{1}{z},\\
\end{align*}

and we thus conclude that

We conclude these computations with the following corollary of Proposition <ref>.

With the notations introduced so far, for each $N\geq 1$ we pick cyclic Boolean independent and identically distributed Bernoulli elements $(X_1^{(N)},\ldots,X_{N}^{(N)})$ in a cyclic probability space $(\mathcal{B}(\mathcal{H}^{(N)}),\Varphi,\OmegA={\rm Tr}_{\mathcal{H}^{(N)}})$ with jump rate parameters $(\lambda_{\Varphi}(N),\lambda_{\OmegA}(N))$:

\Varphi((X_{i}^{(N)})^{n}) = \lambda_{\Varphi}(N),\quad\Psi((X_{i}^{(N)})^{n}) = 1+\lambda_{\OmegA}(N)-\lambda_{\Varphi}(N).

Then, under the assumptions that the sequences $\lambda_{\Varphi}(N)N$ and $\lambda_{\OmegA}(N)N$ converge respectively toward $\lambda_{\Varphi}$ and $\lambda_{\OmegA}$ with $\lambda_{\Varphi} \in [0,1[$ and $\lambda_{\OmegA}\in\mathbb{N}^{\star}$, as $N$ tends to infinity,

* the $\lambda_{\OmegA}$ largest eigenvalues of $B^{(N)}$ converge to $1$,

* the following eingenvalue in the decreasing order has, asymptotically, multiplicity one and converges to $1-\lambda_{\Varphi}$,

* the remaining part of the spectrum of $B^{(N)}$ converges to $0$.

§ PRODUCTS OF GRAPHS AND CYCLIC-CONDITIONAL FREENESS

In this last section we want to show how some known graph products maybe generalized to include models for the cyclic freeness and for cyclic-conditional freeness. In order to do this, we will work with rooted graphs.

§.§ Rooted graphs and Adjacency matrices

By a rooted graph we understand a pair $(\mathcal{G},e)$, where $\mathcal{G}=(V,E)$ is a undirected graph with set of vertices
$V=V(\mathcal{G})$, and set of edges $E=E(\mathcal{G})\subseteq \{(x,x'):x,\ x'\in V,\ x\neq x'\}$ and $e\in V$ is
a distinguished vertex called the root. For rooted graphs we will use the notation $V^0=V\backslash \{e\}$.
Two vertices $v,v'\in V$ are called adjacent if $(v,v')\in E$,  i.e.
vertices $v,v'$ are connected with an edge. Then we write $v \sim v'$.
The degree of $v\in V$ is defined by $d(v)=|\{v'\in V:v'\sim x\}|$. We will assume that $G$ is locally finite, that is, $deg(v)<\infty$.

The adjacency matrix $A=A({\mathcal G})$ of ${\mathcal G}$ is the matrix defined by

\begin{equation*}
\begin{array}{ll}
1 & {\rm if} \;\; v\sim v'\\
0 & {\rm otherwise.}
\end{array}
\right.
\end{equation*}

We identify $A$ with the densely defined symmetric operator on $l^{2}(V)$ defined by

\begin{equation}
A\delta(x)=\sum_{v\sim v'}\delta(v'),
\end{equation}

for $v\in V$, where for $\delta(v)$ denotes the indicator function on $\{v\}$.

If $(G,e)$ is a rooted graph with adjacency matrix $A$ acting on $\oplus_{v\in\mathbb{V}(G)} \mathbb{C}v$, the distribution of $G$ in the root state is $\mu_G\colon \mathbb{C}[X]\to \mathbb{C}$, $\mu(X^m)=\langle e, A^n(e) \rangle$.

§.§ Star, comb, orthogonal and free products of graphs

For details about the material exposed in the first part of this section, we refer the reader to [1].
Let $G_1 = (V_1, E_1)$ and $G_2 = (V_2, E_2)$ be two
rooted graphs
with distinguished vertices $e_1\in V_1$ and $e_2\in V_2$.

The star product of $G_1$ with $G_2$  is the graph $G_1\star G_2 =(V_1 \times V_2, E)$ such that for $(v_1, w_1), (v_2, w_2) \in V_1\times V_2$
the edge $e = (v_1, w_1) \sim (v_2, w_2)\in E$ if and only if one of the following holds:
i) $v_1 = v_2 = e_1$ and $w_1\sim w_2$
ii) $v_1 \sim v_2$ and $w_1 = w_2 = e_2$.

The comb product of $G_1$ with $G_2$ is the graph $G_1\rhd G_2 =(V_1 \times V_2, E)$ such that for $(v_1, w_1), (v_2, w_2) \in V_1\times V_2$
the edge $e = (v_1, w_1) \sim (v_2, w_2)\in E$ if and only if one of the following holds:
i) $v_1 = v_2 $ and $w_1\sim w_2$
ii) $v_1 \sim v_2$ and $w_1 = w_2 = e_2$.

A rooted set $(V,e)$ is simply a set with distinguished element $e$. For a countable family of rooted vertex sets $(V_i,e_i)$, $i\in I$, the free product of  $\{(V_i,e_i)\}_{i\in I}$,
is the rooted set $(*_{i\in I}V_i, e)$, where

$$*_{i\in I}V_i=\{e\}\cup \{v_1 v_2 \cdots v_m : v_k\in V_{i_k}^0,\ \text{and}\ i_1\neq i_2\neq \cdots \neq i_m,\ m\in \mathbb{N}\},$$

and $e$ is the empty word.
The (non-associative) orthogonal product of the rooted graphs $(G_1, e_1)$ and $(G_2,e_2)$ denoted $G_1 \vdash G_2$ resembles the comb product $G_1 \rhd G_2$ of the two graphs, to the exception that $G_2$ is not attached to the root of $G_1$;
$V(G_1 \vdash G_2)={V}(G_1)\times V(G_2)$ and the set of edges of $G_1 \vdash G_2$ consist of the pair of vertices $(v_1, w_1), (v_2, w_2) \in V_1\times V_2$
such that one of the following holds:
i) $v_1 = v_2\neq e_1$ and $w_1\sim w_2$
ii) $v_1 \sim v_2$ and $w_1 = w_2 = e_2$.

The free product of rooted graphs $(\mathcal{G}_i,e_i),\ i\in I$, is the  rooted graph $(*_{i\in I}\mathcal{G}_i,e)$
with vertex set $*_{i\in I}V_i$ and edge set $*_{i\in I}E_i$, defined by

$$*_{i\in I}E_i:= \{(vu,v'u):(v,v')\in \bigcup_{i\in I}E_i\ \text{and}\ u,\ vu,\ v'u\in *_{i\in I}V_i\}.$$

We denote this product by $*_{i\in I}(\mathcal{G}_i,e_i)$ or $*_{i\in I}\mathcal{G}$ if no confusion arises.

The free product $G_1 * G_2$ decomposes as
${G_1 * G_2} = \Gamma_1 \star \Gamma_{2}$, where the two rooted graphs $\Gamma_1$ and $\Gamma_2$ are the subordination branches associated to $G_1$ and $G_2$ respectively. Each of the subordination branch is the limit, in the appropriate sense, of a sequence of graphs $(\Gamma_1^{(n)})_{n\geq 1}$ and $(\Gamma_2^n)_{n\geq 1}$ defined by

\Gamma^{(n)}_1 := (\Gamma^{(n-1)}_1 \vdash G_2) \vdash G_1,~\Gamma^{(1)} := G_1

\Gamma^{(n)}_2 := (\Gamma^{(n-1)}_2 \vdash G_1) \vdash G_2,~\Gamma^{(2)} := G_2.

§.§ Free product of graphs and Voiculescu's Freeness

Let $(G_1,e_1)$ and $(G_2,e_2)$ be two rooted graphs. We consider the free product $G_1*G_2.$ of the two graphs $G_1$ and $G_2$. We denote by $\mu_1$ and $\mu_2$ the spectral distributions in their root state of $G_1$ and $G_2$ respectively. The spectral distribution in the root state of $G_1*G_2$ is given the free additive convolution $\mu_1 \boxplus \mu_2$ of the distributions of $\mu_1$ and $\mu_2$.
Moreover, the adjacency matrix $A$ of $G_1*G_2$ can be be written as a sum:

with $A^{(1)}$ and $A^{(2)}$ having distribution $\mu_1$ and $\mu_2$, respectively, in the root state of $G_1\star G_2$. Denoting by
$$\Psi:\mathbb{C}\langle X_1,X_2\rangle\to \mathbb{C}$$
the joint distribution of $(A^{(1)},A^{(2)})$ (in the root state), the variables $X_1$ and $X_2$ are free with respect to $\Psi$ [1]

§.§ Conditional free product of graphs

Let $(G_1,e_1)$ and $(G_2,e_2)$ be two rooted graphs.
Let $(H_1,f_1)$ and $(H_2,f_2)$ be two additional rooted graphs. We now come to a definition of graph product original to this work, to the best of our knowledge. We define free product of $H_1$ and $H_2$ conditionally to $G_1$ and $G_2$ by :

$$H_1\prescript{}{G_1}{*}^{}_{G_2}H_2 := (H_1\vdash  \Gamma_2)\star (H_2 \vdash  \Gamma_1).$$

To build the conditional product we first take the star product of the two graphs $H_1$ and $H_2$. In a second time, to every vertex of $H_1 \subset H_1 \star H_2$ but the root, we attach the subordination branch $\Gamma_{2}$ which 'starts' with the graph $G_2$. We repeat this process with $H_2$ and the subordination branch $\Gamma_1$.

The graph $H_1\prescript{}{G_1}{*}^{}_{G_2}H_2$

Denoting by $\mu_1$, $\mu_2$, $\nu_1$ and $\nu_2$ the spectral distributions in the root state of $G_1$, $G_2$, $H_1$ and $H_2$ respectively, the spectral distribution in the root state of $H_1\prescript{}{G_1}{*}^{}_{G_2}H_2$ is given by
$\nu_1\prescript{}{\mu_1}{\boxplus}^{}_{\mu_2}\nu_2$, the additive free convolution of $\nu_1$ and $\nu_2$ conditionally to $\mu_1$ and $\mu_2$.
Furthermore, the adjacency matrix $A$ of $H_1\prescript{}{G_1}{*}^{}_{G_2}H_2$ can be written as:

with $A^{(1)}$ and $A^{(2)}$ having respective distribution $\nu_1$ and $\nu_2$ in the root state. If we denote by
$$\Varphi:\mathbb{C}\langle X_1,X_2\rangle\to \mathbb{C}$$
the joint distribution of $(B^{(1)},B^{(2)})$ in the root state, the variables $X_1$ and $X_2$ are conditionally free with respect to $(\Psi,\Varphi)$. In fact, the matrix $A^{(1)}$ is the adjacency matrix of the graph obtained by retaining from the graph $H_{1} \prescript{}{G_1}{*}^{}_{G_2} H_2$ all vertices and adjacent edges in copies of the two graphs $H_1$ and $G_1$. The matrix $A^{(2)}$ is obtained in a similar way, by retaining the edges corresponding to copies of the graphs $H_2$ and $G_2$.

In preparation for the forthcoming section, let us develop the argument that entails conditional freeness of $X_1$ and $X_2$ with respect to the pair of states $(\Psi,\Varphi)$. Let $\bm{p} = (p_1,\ldots,p_n)$ be an alternated sequence of polynomials in $\mathbb{C}\langle X_1,X_2\rangle$ and suppose that $p_n\in\mathbb{C}\langle X_1\rangle$ (so that $p_{n-1} \in \mathbb{C}\langle X_2\rangle$ and so on). We assume that $\Psi(\pi_i)=0,$ and want to show

\Varphi(p_1\cdots p_n) = \langle e, p_1(B^{(x)})\cdots p_n(B^{(1)})e \rangle = \langle e, p_1(B^{(x)})e\rangle\cdots\langle e, p_n(B^{(1)})e \rangle

We may write $\Varphi(p_1\cdots p_n)$ as a weighted sum over paths drawn in the graph $H_1\prescript{}{G_1}{*}^{}_{G_2}H_2$

\Varphi(p_1(B^{(x)})\cdots p_n(B^{(1)}))=\sum_{\substack{\gamma:e\to e}} a_{\gamma}(\bm{p},B^{(1)},B^{(2)})

When the path $\gamma$, starting at the root $e$ hits a vertex of $H^{1}$ that is not the root, it continues exploring vertices of the subordination branch $B^{(1)}$ attached to $v$. Besides, in order for $\gamma$ to loop at $e$, while passing through $v\in H_1\backslash\{e\}$, it must pass a second time through $v$. By denoting $A_{\mathcal{G}_1}$ the adjacency matrix of the subordination branch, and writing it as

A_{\mathcal{G}_1} = A^{(1)}_{\mathcal{G}_1} + A^{(2)}_{\mathcal{G}_1}

where as usual, we retain in each $A^{(i)}_{\mathcal{G}_1}$, $i\in \{1,2\}$ only edges in copies of $G_i$. Finally, since a path that joins a vertex of $H_1$ must go through a copy of $G_2$, one has the factorization:

\begin{align*}
&\langle e, p_1(B^{(x)})\cdots p_n(B^{(1)})e \rangle = \sum_{\substack{\gamma \subset \mathcal{B}_1(v) \\ \gamma: v \to v}} a_{\gamma}({( p_2,\ldots,p_n)}, A^{(1)}_{\mathcal{G}_1}, A_{\mathcal{G}_1}^{(2)})\sum_{\substack{\gamma_2 \subset H_1 \\ \gamma_2 : v \to e}}a_{\gamma_2}(p_1,A_{H_1},A_{H_2}) \\
&\hspace{5cm}+ \langle e, p_1(B^{(x)})\cdots p_{n-1}(B^{(2)})e\rangle \langle e,p_n(B^{(1)} e\rangle \\
&= 0 + \langle e, p_1(B^{(x)})\cdots p_{n-1}(B^{(2)})e\rangle \langle e,p_n(B^{(1)} e\rangle.
\end{align*}

The polynomials $p_i \in \mathbb{C}\langle X_1,X_2\rangle$ are alternating and satisfy $\Psi(p_i)=0$. The last equality follows from freeness. We conclude by induction.

§.§ Free, star and comb products of graphs from the conditional free product of graphs

We know that free, Boolean and monotone independences are particular cases of conditional freeness. From the graph product side, it is also possible to recover the free product $*$, the star product $\star$ and the comb product $\rhd$ from the graph product $H_1\prescript{}{G_1}{*}^{}_{G_2}H_2$.

First from
$H_1\prescript{}{G_1}{*}^{}_{G_2}H_2 \simeq (H_1\vdash \mathcal B_2)\star (H_2 \vdash \mathcal B_1),$
if $G_1=H_1$ and $H_2=G_2$, then  $(H_1\vdash \mathcal B_2)$ and $(H_1\vdash \mathcal B_2)$, are just the branches of the free product and hence

$$H_1\prescript{}{H_1}{*}^{}_{H_2}H_2\simeq H_1* H_2.$$

For the star and comb product, we note that there is an identity with respect to both of this products, namely $G_\emptyset:=(\{e_1,\emptyset\},e_1)$, i.e. , the graph with one vertex, satisfying the rules

$$(G_1,e_1)\star G_\emptyset\simeq G_\emptyset \star (G_1,e_1) \simeq (G_1,e_1),$$

$$(G_1,e_1)*G_\emptyset \simeq G_\emptyset * (G_1,e_1) \simeq (G_1,e_1).$$

We still have

$$G_1*G_2\simeq \mathcal B_1\star  \mathcal B_2$$

if we consider that $(G_1,e_1)$ is the branch of $(G_1,e_1)\star G_\emptyset\simeq G_\emptyset \star (G_1,e_1)$ subordinate to $(G_1,e_1)$ and that $G_\emptyset$ is the branch of $(G_1,e_1)\star G_\emptyset\simeq G_\emptyset \star (G_1,e_1)$ subordinate to $G_\emptyset$. We also define

$$(G_1,e_1)\vdash G_\emptyset\simeq (G_1,e_1)$$

$$G_\emptyset  \vdash (G_1,e_1)\simeq G_\emptyset.$$

With this conventions it is clear that

$$ H_1\prescript{}{G_\emptyset}{*}^{}_{G_\emptyset}H_2\simeq H_1\star H_2\ \ \text{and}\ \ H_1\prescript{}{G_\emptyset}{*}^{}_{H_2}H_2\simeq H_1\rhd H_2.$$

§.§ Cyclic-conditional freeness and graph products

We assume here that $H_1$ and $H_2$ are finite.
Let us denote by $W$ the finite set of vertices of $H_1\prescript{}{G_1}{*}^{}_{G_2}H_2$ which belongs to $H_1$ or $H_2$ (the cardinality of $W$ is the sum of the number of vertices of  $H_1$ and $H_2$ minus one, because of the identification of the roots). Recall the definition of $A^{(1)}$ and $A^{(2)}$ of the last section. We define the linear functional
$$\OmegA:\mathbb{C}\langle X_1,X_2\rangle\to \mathbb{C}$$
by the sum of vector states given by the vertices in $W$, i.e.

$$\OmegA(P)=\sum_{v\in W}\Big\langle P(A^{(1)},A^{(2)}) \delta(v),\delta(v) \Big\rangle.$$

In addition, for each $i\in \{1,2\}$, we denote by $\OmegA_{H_i}\colon \mathbb{C}\langle X_i \rangle \to \mathbb{C}$ the distribution of $H_i$ with respect to the non-normalized trace:

\OmegA_{H_i}(P) =  {\rm Tr}(P(A_{H_i})), \quad P\in\mathbb{C}\langle X_i\rangle.

We let $\mathcal{I}$ be the two sided ideal of $\mathbb{C}\langle X_1,X_2\rangle$ generated by $X_1,X_2$ of polynomials with non-constant coefficient.
For each integer $i \in \{1,2\}$ and for any polynomial $P(X_i) \in \mathcal{I}$ with non-constant coefficient, $\OmegA_{H_i}(P)=\OmegA(P)$.
Moreover, $\OmegA$ is tracial and the variables $X_1$ and $X_2$ are cyclic codntionnally free with respect to $(\Psi,\Varphi,\OmegA)$.

Let us denote by $V_1$ the set of vertices of $H_1$ in $H_1\prescript{}{G_1}{*}^{}_{G_2}H_2$ and by $e$ the root of $H_1\prescript{}{G_1}{*}^{}_{G_2}H_2$. The operator $B^{(1)}$ leaves invariant the subspace $\ell^2(V_1)$ and vanishes on $\ell^2(W\setminus V_2)$  so
$$\OmegA(P(X_1))=\sum_{v\in W}\Big\langle P(B^{(1)}) \delta(v),\delta(v) \Big\rangle=\sum_{v\in V_1}\Big\langle P(B^{(1)}) \delta(v),\delta(v) \Big\rangle=Tr_{\ell^2(V_1)}\Big(P(B^{(1)}|_{\ell^2(V_1)})\Big)$$

as announced. Similarly, the distribution of $X_2$ with respect to $\OmegA$ is the spectral distribution of $H_2$ with respect to the non-normalized trace. As a consequence, $\OmegA$ is tracial on the algebra generated by $X_1$ (respectively on the algebra generated by $X_2$).

Now, we claim that, for any alternating tuple $\bm{p}= (p_1,\ldots,p_n)$ of polynomials with $n\geq 2$, satisfying $\Psi(p_1)=\cdots = \Psi(p_n)=0$, we have

\OmegA(p_n \cdots p_1)  = \Varphi(p_n) \cdots \Varphi(p_1)$$
if $\bm{p}$ is cyclically alternating and

$$\OmegA(p_n \cdots p_1)  = \Varphi(p_1p_n) \Varphi(p_{n-1})\cdots \Varphi(p_2)$$

if not. This claim implies the traciality of $\OmegA$ and the cyclic $c$-freeness.

In order to prove the claim, we denote by $P_W$ the orthogonal projection to $\ell^2(W)$, by $P_{V_1}$ the orthogonal projection to $\ell^2(V_1)$ and by $P_v$ the orthogonal projection to any vertex $\ell^2(\{v\})$. Let us assume that $p_1\in \mathbb{C}\langle X_1\rangle$ (the case $p_1\in \mathbb{C}\langle X_2\rangle$ is treated similarly). We set

$$B_1:=p_1(B^{(1)}),\ B_2:=p_2(B^{(2)}),\ B_3:=p_3(B^{(1)}),\ B_4:=p_4(B^{(2)})... $$

Because of the centring of $p_1,\ldots,p_n$ with respect to $\Psi$, it is impossible to escape from $\ell^2(W)$ and come back to $\ell^2(W)$ by an alternating tuple of operators. Consequently, we have

\begin{align*}
\OmegA(p_n \cdots p_1)& =\sum_{v\in W}\Big\langle B_n\cdots B_2 B_1 \delta(v),\delta(v) \Big\rangle\\
&=\sum_{v\in W}\Big\langle B_nP_W \cdots P_WB_2 P_W B_1\delta(v),\delta(v) \Big\rangle\\
&=\sum_{v\in W}Tr(B_nP_W \cdots P_WB_2 P_W B_1P_v)\\
&=Tr(B_nP_W \cdots P_WB_2 P_W B_1P_W)\\
\end{align*}

Because $B_1,\ldots,B_n$ are centered, $P_W B_1P_W=P_{V_1}B_1P_{V_1}$, $P_WB_2P_{W}=P_{V_2} B_2P_{V_2}$, $P_WB_3P_W=P_{V_1} B_3P_{V_1}$ and so on. As a consequence, and using $P_{V_1}P_{V_2}=P_e$, we get

\begin{align*}
\OmegA(p_n \cdots p_1)
&=Tr(B_nP_W \cdots P_WB_2 P_W B_1P_W)\\
&=Tr(P_{V_2}B_n P_e \cdots P_eB_2 P_e B_1P_{V_1})\\
&=\Varphi(p_{n-1})\cdots \Varphi(p_2) Tr(P_{V_2}B_nP_eB_1P_{V_1})\\
&=\Varphi(p_{n-1})\cdots \Varphi(p_2)Tr(P_eB_nP_eB_1)\\
&=\Varphi(p_{n})\cdots \Varphi(p_1)
\end{align*}

if $\bm{p}$ is cyclically alternating, and

\begin{align*}
\OmegA(p_n \cdots p_1)
&=Tr(B_nP_W \cdots P_WB_2 P_W B_1P_W)\\
&=Tr(P_{V_1}B_n P_e \cdots P_eB_2 P_e B_1P_{V_1})\\
&=\Varphi(p_{n-1})\cdots \Varphi(p_2) Tr(P_{V_1}B_nP_eB_1P_{V_1})\\
&=\Varphi(p_{n-1})\cdots \Varphi(p_2)Tr(B_1P_{V_1}B_nP_e)\\
&=\Varphi(p_{n-1})\cdots \Varphi(p_2)\Varphi(p_1p_n)
\end{align*}

if not.

Note that, from the definition of the conditional free product of graphs, $\OmegA(1)=|V(H_1)|+|V(H_2)|-1$, whereas $\OmegA_{H_i}(1)=|V(H_i)|$.
In the following particular cases:

$$ H_1\prescript{}{\emptyset}{*}^{}_{\emptyset}H_2\simeq H_1\star H_2\ \ \text{and}\ \ H_1\prescript{}{\emptyset}{*}^{}_{H_2}H_2\simeq H_1\rhd H_2,$$

we recover cyclic Boolean independence and cyclic monotone independence between $A^{(1)}$ and $A^{(2)}$. In the case
$$H_1\prescript{}{H_1}{*}^{}_{H_2}H_2\simeq H_1* H_2,$$

we obtain cyclic freeness (and not infinitesimal freeness, since the root state is not necessarily tracial).

§.§ Cumulants for cyclic monotone independence

Let $(\mathcal{A},\Psi)$ be a non-commutative probability space. In this section, we introduce a sequence of cumulants for cyclic monotone independence. In Section <ref>, we expose a graph model which allows for taking identically distributed cyclically monotone copies of the cyclic distribution of a graph. We derive those cyclically-monotone cumulants (together with a corresponding moments-cumulants formula) in a fashion reminiscent from [17] and later abstracted to a method providing cumulants for spreadable systems, as described in <cit.>. We first define a poset of cyclically monotone partitions.

Pick a non-crossing partition $\pi\in {\rm \nc}(n)$ of $[n]$ with the labelling of the blocks yielding a total order $\lessdot$ on $\pi$. We say that
$\pi$ is cyclic monotone if for any block $V$, there exists a pair $(x,y)$, $x,y \in V$ and $x \prec y$, such that with

$$\pi^{V \lessdot}:=\bigcup_{\{W \in \pi, V \lessdot W\}} W,$$

the following implication holds

$$ z \in \pi^{V \lessdot} \implies x \prec z \prec y.$$

In plain words, given an ordered partition $(\pi,\lessdot)$ it will be a cyclically monotone partition if all blocks greater (for $\lessdot$) that a given block lies between two cyclically consecutive element in that block.
We let ${\rm Cy}\mathcal{M}(n)$ be the set of all cyclically monotone partitions. Given a cyclic monotone partition $(\pi,\lessdot)$, we denote by $(\pi,\lessdot)[-1]$ the maximum block of $\pi$ for the total order $\lessdot$.

If a partition $\pi$ has two blocks, then any labelling of the blocks yields a cyclic monotone partition (this is, of course, not true if considering monotone partitions). The same holds for partitions with three blocks.

For the next proposition, recall that we denote $\prec_{k}$, $k \in [n]$ the linear order obtained from the cyclic order $\prec$ by choosing $k$ the first element of $\prec_k$.

Any monotone partition is cyclically monotone. Conversely, let $(\pi,\lessdot)$ be a cyclic monotone partition and pick $k \in (\pi,\lessdot)[-1]$. Then $(\pi,\lessdot)$ is a monotone partition for the linear order $\prec_{k}$ on $[n]$.

Given a integer $k \geq 1$, the monotone partitions $(\pi,\lessdot)_k \in \mathcal{M}(([n],\prec_k))$ from the previous proposition when $\pi$ ranges the cyclically-monotone partitions whose greatest block contain $k$ do not exhaust all monotone partitions in $\mathcal{M}(([n],\prec_k))$: the greatest block of $(\pi,\lessdot)_k$ always contains the first element (for the order $\prec_k$). To a monotone partition $(\pi,\lessdot) \in \mathcal{M}(([n],\prec_k))$ corresponds an unique cyclically monotone partition.

The number of cyclic monotone partition with $s$ blocks ($s\geq 1)$ with sizes $(n-k_s,k_s-k_{s-1},\ldots,k_2-k_1,k_1)$ with $1 \leq k_1 < k_2 < \cdots < k_s < n$ is equal to $nk_2\cdots k_s$.

We want build a cyclic monotone partition as in the statement of the lemma. The first block $V$ with size $n-k_s$ is an interval block for the cyclic order on $[n]$: there is a cyclically ordered pair $(i \prec j)$ of elements of $V$ such that for any integer $k$ with $i \prec k \prec j$, $k$ belongs to $V$ There are $n$ ways of placing such a block with a size less than $n$ (and just one way of doing so with a block with a size equal to $n$).

Once this block has been placed, we can place a second interval block with size $k_s-k_{s-1} < k_s$ on the cyclically ordered set that is complementary to the block $V$ in $[n]$. There is $k_s$ options to place it. We continue placing blocks. We obtain a cyclic monotone partition, the labels on the blocks corresponds to the stage in the procedure describe above at which it has been placed.

Examples of cyclically monotone partitions on the first line. On the second line a partition that is not cyclically monotone. The split block are coloured in red.

Let $\Psi$ be an unital linear functional on $\mathcal{A}$ and $\OmegA$ be a linear functional on $\mathcal{A}$.
Let $a\in \mathcal{A}$. In the following, we will use the Rota's dot notation: $N.a$ will denote a sum of $N$ cyclically-monotone independent copies of $a$. In the terminology of <cit.>, these independent copies are produced by building up a spreadable system $\mathcal{S}$ which reproduces cyclic monotone independence: we may take for $\mathcal{S}$ the infinite folded free product (with identification of units) of $\mathcal{A}$ endowed with the unique pairs of linear functional $(\tilde{\Psi},\tilde{w})$ such that the copies of $\mathcal{A}$ in $\mathcal{S}$ form a cyclically-independent family of algebras. See also the next Section for the case of graphs.
Replacing $\mathcal{A}$ with $\mathcal{S}$, we may assume that $\mathcal{A}$ contains as many as needed independent copies of $\mathcal{A}$, we may therefore make not distinction between $\Psi,\OmegA$ and $\tilde{\Psi}, \tilde{\OmegA}$.

Let $n \geq 1$ be an integer, from Theorem 4.5 and Definition 4.6 in [17], we define in a similar fashion the $n^{th}$ cyclic monotone cumulant of $a$ with respect to the couple $(\Psi,\OmegA)$, $K_n^{(\Psi,\OmegA)}(a)$, by extracting the coefficient in front of $N$ in $w((N.a)^n)$.

Recall that we denote by $(K^{\Psi}_n(a))_{n\geq 0}$ the sequence of monotone cumulants of $a$ with respect to $\Psi$.
Let $a\in\mathcal{A}$, $n\geq 1$,

\begin{equation}
\label{eqn:cymonotone}
\tag{CyMo}
\OmegA(a^n)=\sum_{(\pi,\lessdot)\in {\rm Cy\mathcal{M}}(n)} \frac{1}{|\pi|!}K^{(\Psi,\OmegA)}_{|\pi[-1]|}(a)\prod_{V\neq \pi[-1]}K^{\Psi}_{|V|}(a)
\end{equation}

Pick $x,y$ cyclic monotone independent and an integer $n\geq 1$, then

\begin{align*}
\OmegA((x+y)^n)&= \OmegA(x^n)+\OmegA(y^n) + \sum_{\substack{j_0+\cdots + j_k=n-k \\ j_i \geq 0.}} w(y^{j_0}x\cdots xy^{j_k}) \\
&=w(x^n)+w(y^n) + \sum_{\substack{j_0+\cdots+j_k = n-k \\ j_i \geq 0}}w(x y^{j_1} x \cdots x y^{j_k+j_0}) \\
&=w(x^n)+w(y^n)+\sum_{\substack{j_0+\cdots+j_k=n-k \\ j_i \geq 0}}w(x^k)\Varphi(y^{j_1})\cdots \Varphi(y^{j_0+j_k}).
\end{align*}

Since $\OmegA((N.x)^n)$ is a polynomial in $N$, we may replace $N$ by a continuous parameter $t$ to obtain a polynomial $\mathbb{R} \ni t\mapsto \OmegA_n(t)$ equal to $\OmegA(N.x)$ on integers $N\geq 1$. The same applies for $\Psi((N.x)^n)$. From the previous computations, we get:

\begin{align*}
w_{n}(s+t) = w_{n}(t) + w_{n}(s) + \sum_{k=1}^{n-1} \sum_{\substack{k\geq 1, \\j_0+\cdots+j_k=n-k}}w_k(t)\Psi_{j_1}(s)\cdots \Psi_{j_0+j_k}(s)
\end{align*}

Deriving with respect $t$ both side of the above equality, we get, for any $t > 0$ and $n \geq 1$:

\partial_t \OmegA_n(t) = K^{\OmegA}_n(a)+\sum_{k=1}^{n-1}\OmegA_k(t)\{(k-1)K^{\Psi}_{n-k} +\sum_{\substack{j_0+j_k=n-k\\ j_i \geq 0}} K^{\Psi}_{n-k}(a)\} = k_n^{w}+n\sum_{k=1}^{n-1}\OmegA_k(t)K^{\Psi}_{n-k}

$\OmegA_n(0)=0$, $\partial_t \OmegA_0 = 0$. The integrated form of the above equations is:

\begin{align*}
& \OmegA_{0}(t)=\delta \\
&\OmegA_n(t)=tK_n^{(\Psi,\OmegA)}(a) + n\sum_{k_1=1}^n K^{\Psi}_{n-k_1}\int_0^t\OmegA_{k_1}(t_1)\mathrm{d}t_1.
\end{align*}

Solving by Picard iterations, one finds:

\begin{align*}
\OmegA_n(t) &= tK^{(\Psi,\OmegA)}_n(a)+ n\sum_{k_1=1}^{n-1}K_{n-k_1}^{\Psi}\int_{0}^t \OmegA_{k}(t_1)\mathrm{d}t_1 \\
&= tK_n^{(\Psi,\OmegA)}(a) + n\sum_{k_1=1}^{n-1}\{\frac{t^2}{2}K^{(\Psi,\OmegA)}_{n-k_1}(a)K^{\OmegA}_{k_1}(a) + K_{n-k_1}^{\Psi}(a) k_1 \sum_{k_2=1}^{k_1-1}\int_{0}^t \mathrm{d}t_1 \int_{0}^{t_1}\mathrm{d}t_2\,\OmegA_{k_2}(t_2)K_{k_2-k_1}^{\Psi}(a)\} \\
&= \cdots
\end{align*}

One may arrive to

\begin{align*}
\OmegA_n(t)=K_{n}^{\Psi,\OmegA} + \sum_{1 \leq k_1 < \cdots < k_s < n} \frac{1}{s!}(n k_2 \cdots k_{s})K^{\Psi}_{n-k_s}\cdots K^{\Psi}_{k_{2}-k_{1}}K^{(\OmegA,\Psi)}_{k_1}
\end{align*}

The statement follows from Lemma (<ref>).

That is for the univariate cumulants. This is a bit more tricky to define the multivariate cumulants since we need to track linear orders to evaluate the cumulants of $\Psi$.

Some other cyclic monotone partitions

The moments-cumulants formulae (<ref>) yield

\begin{align*}
\OmegA(a^3)= K^{\OmegA}_3(a)+\frac{3}{2!} K_2^{\Psi}(a)K_1^{\OmegA}(a) + \frac{3}{2!} K_1^{\Psi}(a)K_2^{\OmegA}(a) + \frac{3!}{3!}K_1^{\Psi}(a)K_1^{\Psi}(a)K_1^{\Psi}(a).
\end{align*}

Recall that if $\pi$ is a non-crossing partition of the cyclically ordered set $[n]$, any $V\in \pi$ induces a linear order $\prec_V$ on the other blocs of $\pi$ by cutting the cyclic order according to the elements of $V$. For any $(\pi,\lessdot)\in {\rm Cy}\mathcal{M}(n)$ with first bloc $V=\{j_1<\cdots<j_k\}$, we set

$$K_{(\pi,\lessdot)}(a_1,\ldots,a_n):=K_k(a_{j_1},\ldots,a_{j_k})\prod_{\substack{ W\in \pi, W\neq V\\W=(i_1\prec_V \ldots \prec_V i_\ell)}}h_\ell(a_{i_1},\ldots,a_{i_\ell}).$$

Let $n\geq 1$ and $a_1,\ldots, a_n \in \mathcal{A}$.

$$\OmegA(a_1\cdots a_n)=\sum_{(\pi,\lessdot)\in {\rm Cy}\mathcal{M}(n)}\frac{1}{|\pi|!}K_{(\pi,\lessdot)}(a_1,\ldots,a_n).$$

It is important to notice that $\kappa^{\OmegA}(\cdot)$ only depends on the cyclic equivalence class of its argument since $\OmegA$ is tracial.

The proof consists in repeating the one of Proposition while keeping track of the order between the letters $a_i's$.

§ APPLICATION TO PRODUCTS OF GRAPHS

In this last section we want to show how some known graph products maybe generalized to include models for the c-free independence and for cyclic c-free independence.

In order to do this, we will work with rooted graphs. These are pairs, $(G,e)$, where $G=(V,E)$ and $e\in E$. We will assume that $G$ is locally finite, that is, $deg(v)<\infty$.

For $G_1 = (V_1, E_1)$ and $G_2 = (V_2, E_2)$ be two graph
with distinguished vertices $e_1\in V_1$ and $e_2\in V_2$, the star product of $G_1$ with $G_2$  is the graph $G_1\star G_2 =(V_1 \times V_2, E)$ such that for $(v_1, w_1), (v_2, w_2) \in V_1\times V_2$
the edge $e = (v_1, w_1) \sim (v_2, w_2)\in E$ if and only if one of the following holds:
i) $v_1 = v_2 = e_1$ and $w_1\sim w_2$
ii) $v_1 \sim v_2$ and $w_1 = w_2 = e_2$.

The adjacency matrix $A=A({\mathcal G})$ of ${\mathcal G}$ is a matrix defined by

\begin{equation*}
\begin{array}{ll}
1 & {\rm if} \;\; x\sim x'\\
0 & {\rm otherwise.}
\end{array}
\right.
\end{equation*}

We identify $A$ with the densely defined symmetric operator on $l^{2}(V)$ defined by

\begin{equation}
A\delta(x)=\sum_{x\sim x'}\delta(x')
\end{equation}

for $x\in V$.

It is known that $A(\mathcal{G})$ is bounded if and only if ${\mathcal G}$ is uniformly
locally finite.

If $(G,e)$ is a rooted graph with adjacency matrix $A$ acting on $\oplus_{v\in\mathbb{V}(G)} \mathbb{C}v$, the distribution of $G$ in the root state is $\mu_G\colon \mathbb{C}[X]\to \mathbb{C}$, $\mu(X^m)=\langle e, A^n(e) \rangle$.

In the following, $\star$ denotes the star product of rooted graphs, $*$ denotes the free product of rooted graphs, $\rhd$ denotes the comb product of rooted graphs and $\vdash$ denotes the orthogonal product of rooted graphs.

To define the above?

The free product $G_1 \star G_2$ decomposes as
${G_1 * G_2} = \Gamma_1 \star \Gamma_{2}$, where the two rooted graphs $\Gamma_1$ and $\Gamma_2$ are the subordination branches associated to $G_1$ and $G_2$ respectively. Each of the subordination branches is the limit, in the appropriate sense, of a sequence of graphs $(\Gamma_1^{(n})_{n\geq 1}$ and $(\Gamma_2^n)_{n\geq 1}$ defined by

\Gamma^{(n)}_1 := (\Gamma^{(n-1)}_1 \vdash G_2) \vdash G_1,~\Gamma^{(1)} := G_1

\Gamma^{(n)}_2 := (\Gamma^{(n-1)}_2 \vdash G_1) \vdash G_2,~\Gamma^{(2)} := G_2.

§.§ Free product of graphs and Voiculescu's Freeness

Let $(G_1,e_1)$ and $(G_2,e_2)$ be two rooted graphs. We consider the free product

If we denote by $\mu_1$ and $\mu_2$ the spectral distribution in the root state of $G_1$ and $G_2$, then the spectral distribution in the root state of $G_1*G_2$ is given by

$$\mu_1 \boxplus \mu_2.$$

Moreover, the adjacency matrix $A$ of $G_1*G_2$ can be decomposed

with $A^{(1)}$ and $A^{(2)}$ having respective distribution $\mu_1$ and $\mu_2$ in the root state and if we denote by
$$\Psi:\mathbb{C}\langle X_1,X_2\rangle\to \mathbb{C}$$
the joint distribution of $(A^{(1)},A^{(2)})$ in the root state, the variables $X_1$ and $X_2$ are free with respect to $\Psi$, see [1]

§.§ Conditional free product of graphs
We consider the branch $(\mathcal B_1,e_1)$ of the free product $G_1*G_2$ subordinate to $G_1$ and the branch $(\mathcal B_2,e_2)$ of the free product $G_1*G_2$ subordinate to $G_2$, in such a way that

$$G_1*G_2\simeq \mathcal B_1\star  \mathcal B_2.$$

Let $(H_1,f_1)$ and $(H_2,f_2)$ be two additional rooted graphs. The free product of $H_1$ and $H_2$ conditionally to $G_1$ and $G_2$ is defined by

$$H_1\prescript{}{G_1}{*}^{}_{G_2}H_2 \simeq (H_1\vdash \mathcal B_2)\star (H_2 \vdash \mathcal B_1).$$

In more vernacular terms, first, the two graphs $H_1$ and $H_2$ are joined by their roots. Second, to every vertex of $H_1$, but the root, we attach the subordination branch $\mathcal{B}_{2}$ (which 'starts' with the graph $G_2$. We do the same process with $H_2$, i.e. attach $B_1$, accordingly.

The graph $H_1\prescript{}{G_1}{*}^{}_{G_2}H_2$

If we denote by $\mu_1$, $\mu_2$, $\nu_1$ and $\nu_2$ the spectral distribution in the root state of $G_1$, $G_2$, $H_1$ and $H_2$, then the spectral distribution in the root state of $H_1\prescript{}{G_1}{*}^{}_{G_2}H_2$ is given by

Furthermore, the adjacency matrix $B$ of $H_1\prescript{}{G_1}{*}^{}_{G_2}H_2$ can be decomposed

with $B^{(1)}$ and $B^{(2)}$ having respective distribution $\nu_1$ and $\nu_2$ in the root state, and if we denote by
$$\Varphi:\mathbb{C}\langle X_1,X_2\rangle\to \mathbb{C}$$
the joint distribution of $(B^{(1)},B^{(2)})$ in the root state, the variables $X_1$ and $X_2$ are $c$-free with respect to $(\Psi,\Varphi)$.

The matrix $B^{(1)}$ is the adjacency matrix of the graph obtained by retaining from the graph $H_{1} \prescript{}{G_1}{*}^{}_{G_2} H_2$ all vertices and the edges that come from the two graphs $H_1$ and $G_1$. The matrix $B^{(2)}$ is obtained in a similar way, by retaining the edges corresponding to copies of the graphs $H_2$ and $G_2$.

The graphs associated to $B^{(1)}$ and $B^{(2)}$.

In preparation for the forthcoming section, let us develop the argument that entails conditional freeness of $X_1$ and $X_2$ with respect to the pair of states $(\Psi,\Varphi)$. Let $\bm{p} = (p_1,\ldots,p_n)$ be an alternated sequence of polynomials in $\mathbb{C}\langle X_1,X_2\rangle$ and suppose that $p_n\in\mathbb{C}\langle X_1\rangle$ (so that $p_{n-1} \in \mathbb{C}\langle X_2\rangle$ and so on). We assume that $\Psi(\pi_i)=0,$ and want to show

\Varphi(p_1\cdots p_n) = \langle e, p_1(B^{(x)})\cdots p_n(B^{(1)})e \rangle = \langle e, p_1(B^{(x)})e\rangle\cdots\langle e, p_n(B^{(1)})e \rangle

We may write $\Varphi(p_1\cdots p_n)$ as a weighted sum over paths drawn in the graph $H_1\prescript{}{G_1}{*}^{}_{G_2}H_2$

\Varphi(p_1(B^{(x)})\cdots p_n(B^{(1)}))=\sum_{\substack{\gamma:e\to e}} a_{\gamma}(\bm{p},B^{(1)},B^{(2)})

When the path $\gamma$, starting at the root $e$ hits a vertex of $H^{1}$ that is not the root, it continues exploring vertices of the subordination branch $B^{(1)}$ attached to $v$. Besides, in order for $\gamma$ to loop at $e$, while passing through $v\in H_1\backslash\{e\}$, it must pass a second time through $v$. By denoting $A_{\mathcal{G}_1}$ the adjacency matrix of the subordination branch, and writing it as

A_{\mathcal{G}_1} = A^{(1)}_{\mathcal{G}_1} + A^{(2)}_{\mathcal{G}_1}

where as usual, we retain in each $A^{(i)}_{\mathcal{G}_1}$, $i\in \{1,2\}$ only edges in copies of $G_i$. Finally, since a path that joins a vertex of $H_1$ must go through a copy of $G_2$, one has the factorization:

\begin{align*}
&\langle e, p_1(B^{(x)})\cdots p_n(B^{(1)})e \rangle = \sum_{\substack{\gamma \subset \mathcal{B}_1(v) \\ \gamma: v \to v}} a_{\gamma}({( p_2,\ldots,p_n)}, A^{(1)}_{\mathcal{G}_1}, A_{\mathcal{G}_1}^{(2)})\sum_{\substack{\gamma_2 \subset H_1 \\ \gamma_2 : v \to e}}a_{\gamma_2}(p_1,A_{H_1},A_{H_2}) \\
&\hspace{5cm}+ \langle e, p_1(B^{(x)})\cdots p_{n-1}(B^{(2)})e\rangle \langle e,p_n(B^{(1)} e\rangle \\
&= 0 + \langle e, p_1(B^{(x)})\cdots p_{n-1}(B^{(2)})e\rangle \langle e,p_n(B^{(1)} e\rangle.
\end{align*}

The polynomials $p_i \in \mathbb{C}\langle X_1,X_2\rangle$ are alternating and satisfy $\Psi(p_i)=0$. The last equality follows from freeness. We conclude by induction.

§.§ Free, star and comb products of graphs from the conditional free product of graphs

We know that free, Boolean and monotone independences are particular cases of $c$-freeness. From the graph product side, it is also possible to recover the free product $*$, the star product $\star$ and the comb product $\rhd$ from the graph product $H_1\prescript{}{G_1}{*}^{}_{G_2}H_2$.

First from
$H_1\prescript{}{G_1}{*}^{}_{G_2}H_2 \simeq (H_1\vdash \mathcal B_2)\star (H_2 \vdash \mathcal B_1),$
if $G_1=H_1$ and $H_2=G_2$, then  $(H_1\vdash \mathcal B_2)$ and   $(H_1\vdash \mathcal B_2)$, are just the branches of the free product and hence

$$H_1\prescript{}{H_1}{*}^{}_{H_2}H_2\simeq H_1* H_2$$

For the star and comb product, we note that there is an identity with respect to both of this products, namely $G_\emptyset:=(\{e_1,\emptyset\},e_1)$, i.e. , the graph with one vertex, satisfying the rules

$$(G_1,e_1)\star G_\emptyset\simeq G_\emptyset \star (G_1,e_1) \simeq (G_1,e_1),$$

$$(G_1,e_1)*G_\emptyset \simeq G_\emptyset * (G_1,e_1) \simeq (G_1,e_1).$$

We still have

$$G_1*G_2\simeq \mathcal B_1\star  \mathcal B_2$$

if we consider that $(G_1,e_1)$ is the branch of $(G_1,e_1)\star \emptyset\simeq \emptyset \star (G_1,e_1)$ subordinate to $(G_1,e_1)$ and that $\emptyset$ is the branch of $(G_1,e_1)\star \emptyset\simeq \emptyset \star (G_1,e_1)$ subordinate to $\emptyset$. We also define

$$(G_1,e_1)\vdash \emptyset\simeq (G_1,e_1)$$

$$\emptyset  \vdash (G_1,e_1)\simeq \emptyset.$$

With this conventions it is clear that

$$ H_1\prescript{}{\emptyset}{*}^{}_{\emptyset}H_2\simeq H_1\star H_2\ \ \text{and}\ \ H_1\prescript{}{\emptyset}{*}^{}_{H_2}H_2\simeq H_1\rhd H_2.$$

§.§ cyclic-conditional freeness and graph products

We assume here that $H_1$ and $H_2$ are finite.
Let us denote by $W$ the finite set of vertices of $H_1\prescript{}{G_1}{*}^{}_{G_2}H_2$ which belongs to $H_1$ or $H_2$ (the cardinality of $W$ is the sum of the number of vertices of  $H_1$ and $H_2$ minus one, because of the identification of the roots). We define the linear functional
$$\OmegA:\mathbb{C}\langle X_1,X_2\rangle\to \mathbb{C}$$
by the sum of vector states given by the vertices in $W$, i.e.

$$\OmegA(P)=\sum_{v\in W}\Big\langle P(B^{(1)},B^{(2)}) \delta(v),\delta(v) \Big\rangle.$$

The distribution of $X_1$ with respect to $\OmegA$ is the spectral distribution of $H_1$ with respect to the non-normalized trace and the distribution of $X_2$ with respect to $\OmegA$ is the spectral distribution of $H_2$ with respect to the non-normalized trace (Warning : it is not true, because the number of vertices is not the same). Moreover, $\OmegA$ is tracial and the variables $X_1$ and $X_2$ are cyclically $c$-free with respect to $(\Psi,\Varphi,\OmegA)$.

Let us denote by $V_1$ the set of vertices of $H_1$ in $H_1\prescript{}{G_1}{*}^{}_{G_2}H_2$ and by $e$ the root of $H_1\prescript{}{G_1}{*}^{}_{G_2}H_2$. The operator $B^{(1)}$ leaves invariant the subspace $\ell^2(V_1)$ and vanishes on $\ell^2(W\setminus V_2)$  so
$$\OmegA(P(X_1))=\sum_{v\in W}\Big\langle P(B^{(1)}) \delta(v),\delta(v) \Big\rangle=\sum_{v\in V_1}\Big\langle P(B^{(1)}) \delta(v),\delta(v) \Big\rangle=Tr_{\ell^2(V_1)}\Big(P(B^{(1)}|_{\ell^2(V_1)})\Big)$$

as announced. Similarly, the distribution of $X_2$ with respect to $\OmegA$ is the spectral distribution of $H_2$ with respect to the non-normalized trace. As a consequence, $\OmegA$ is tracial on the algebra generated by $X_1$ (respectively on the algebra generated by $X_2$).

Now, we claim that, for any alternating tuple $\bm{p}= (p_1,\ldots,p_n)$ of polynomials with $n\geq 2$, satisfying $\Psi(p_1)=\cdots = \Psi(p_n)=0$, we have

\OmegA(p_n \cdots p_1)  = \Varphi(p_n) \cdots \Varphi(p_1)$$
if $\bm{p}$ is cyclically alternating and

$$\OmegA(p_n \cdots p_1)  = \Varphi(p_1p_n) \Varphi(p_{n-1})\cdots \Varphi(p_2)$$

if not. This claim implies the traciality of $\OmegA$ and the cyclic $c$-freeness.

In order to prove the claim, we denote by $P_W$ the orthogonal projection to $\ell^2(W)$, by $P_{V_1}$ the orthogonal projection to $\ell^2(V_1)$ and by $P_v$ the orthogonal projection to any vertex $\ell^2(\{v\})$. Let us assume that $p_1\in \mathbb{C}\langle X_1\rangle$ (the case $p_1\in \mathbb{C}\langle X_2\rangle$ is treated similarly). We set

$$B_1:=p_1(B^{(1)}),\ B_2:=p_2(B^{(2)}),\ B_3:=p_3(B^{(1)}),\ B_4:=p_4(B^{(2)})... $$

Because of the centring of $p_1,\ldots,p_n$ with respect to $\Psi$, it is impossible to escape from $\ell^2(W)$ and come back to $\ell^2(W)$ by an alternating tuple of operators. Consequently, we have

\begin{align*}
\OmegA(p_n \cdots p_1)& =\sum_{v\in W}\Big\langle B_n\cdots B_2 B_1 \delta(v),\delta(v) \Big\rangle\\
&=\sum_{v\in W}\Big\langle B_nP_W \cdots P_WB_2 P_W B_1\delta(v),\delta(v) \Big\rangle\\
&=\sum_{v\in W}Tr(B_nP_W \cdots P_WB_2 P_W B_1P_v)\\
&=Tr(B_nP_W \cdots P_WB_2 P_W B_1P_W)\\
\end{align*}

Because $B_1,\ldots,B_n$ are centered, $P_W B_1P_W=P_{V_1}B_1P_{V_1}$, $P_WB_2P_{W}=P_{V_2} B_2P_{V_2}$, $P_WB_3P_W=P_{V_1} B_3P_{V_1}$ and so on. As a consequence, and using $P_{V_1}P_{V_2}=P_e$, we get

\begin{align*}
\OmegA(p_n \cdots p_1)
&=Tr(B_nP_W \cdots P_WB_2 P_W B_1P_W)\\
&=Tr(P_{V_2}B_n P_e \cdots P_eB_2 P_e B_1P_{V_1})\\
&=\Varphi(p_{n-1})\cdots \Varphi(p_2) Tr(P_{V_2}B_nP_eB_1P_{V_1})\\
&=\Varphi(p_{n-1})\cdots \Varphi(p_2)Tr(P_eB_nP_eB_1)\\
&=\Varphi(p_{n})\cdots \Varphi(p_1)
\end{align*}

if $\bm{p}$ is cyclically alternating, and

\begin{align*}
\OmegA(p_n \cdots p_1)
&=Tr(B_nP_W \cdots P_WB_2 P_W B_1P_W)\\
&=Tr(P_{V_1}B_n P_e \cdots P_eB_2 P_e B_1P_{V_1})\\
&=\Varphi(p_{n-1})\cdots \Varphi(p_2) Tr(P_{V_1}B_nP_eB_1P_{V_1})\\
&=\Varphi(p_{n-1})\cdots \Varphi(p_2)Tr(B_1P_{V_1}B_nP_e)\\
&=\Varphi(p_{n-1})\cdots \Varphi(p_2)\Varphi(p_1p_n)
\end{align*}

if not.

In the situations

$$ H_1\prescript{}{\emptyset}{*}^{}_{\emptyset}H_2\simeq H_1\star H_2\ \ \text{and}\ \ H_1\prescript{}{\emptyset}{*}^{}_{H_2}H_2\simeq H_1\rhd H_2,$$

we recover cyclic Boolean independence and cyclic monotone independence. In the case
$$H_1\prescript{}{H_1}{*}^{}_{H_2}H_2\simeq H_1* H_2,$$

we get infinitesimal freeness.

§ LOOP PRODUCTS

§ ORTHOGONAL PRODUCT

Let $\mathcal{A}_1,\mathcal{A}_2 \subset \overset{~\circ}{\mathcal{A}}$ be non unital sub-algebra of $\mathcal{A}=\mathbb{C}\oplus \overset{~\circ}{\mathcal{A}}$, an unital sub-algebra and two states $(\Psi, \Varphi)$ on $\mathcal{A}$. We say that $\mathcal{A}_2$ is orthogonal to $\mathcal{A}_1$ if one has, for any $y_1,\ldots,y_n \in \overset{~\circ}{\mathcal{A}}_2$, $x_1,\ldots,x_n \in \overset{~\circ}{\mathcal{A}}_1$,

\begin{align}
&\Psi(y_1 x_1 \cdots y_n x_n) = \Psi(x_1y_1\cdots x_ny_n)=0  \\
&\Psi(x_1y_1\cdots y_{n-1}x_n) = \sum_{i=1}^{n-1} \Varphi(y_i)\big(\Psi(x_1\cdots {x}_{i}{x}_{i+1} \cdots x_n) - \Psi(x_1\cdots x_{i})\Psi(x_{i+1} \cdots x_n)\big)
\end{align}

In the setting of the previous definition, if we assume that

\Psi \upharpoonleft_{\mathcal{A}_1} = \Varphi \upharpoonleft_{\mathcal{A}_2} = 0

$\mathcal{A}_2$ is orthogonal to $\mathcal{A}_1$ if and only if they are conditionally independent.

§.§.§ cyclic orthogonal independence

Let $\mathcal{A}_1,\mathcal{A}_2 \subset \mathcal{A}$ and a triple $(\Varphi,\Psi),w)$ a triple of tracial states, then $\mathcal{A}_2$ is said to be cyclically orthogonal to $\mathcal{A}_1$ if they are cyclically c-free and

\Varphi \upharpoonleft_{\mathcal{A}_1} = \Psi \upharpoonleft_{\mathcal{A}_2} = 0

and for any cyclically alternating word $x_1\cdots x_n$, one has

w(x_1\cdots x_n)=0

[1]
Luigi Accardi, Romuald Lenczewski, and Rafał Sałapata.
Decompositions of the free product of graphs.
Infinite Dimensional Analysis, Quantum Probability and Related
Topics, 10(03):303–334, 2007.

[2]
Octavio Arizmendi, Takahiro Hasebe, and Franz Lehner.
Cyclic independence: Boolean and monotone.
arXiv preprint arXiv:2204.00072, 2022.

[3]
Octavio Arizmendi, Takahiro Hasebe, Franz Lehner, and Carlos Vargas.
Relations between cumulants in noncommutative probability.
Advances in Mathematics, 282:56–92, 2015.

[4]
Serban T Belinschi and Dimitri Shlyakhtenko.
Free probability of type b: analytic interpretation and applications.
American Journal of Mathematics, 134(1):193–234, 2012.

[5]
Serban T Belinschi, Roland Speicher, John Treilhard, and Carlos Vargas.
Operator-valued free multiplicative convolution: analytic
subordination theory and applications to random matrix theory.
International Mathematics Research Notices,
2015(14):5933–5958, 2015.

[6]
Philippe Biane, Frederick Goodman, and Alexandru Nica.
Non-crossing cumulants of type b.
Transactions of the American Mathematical Society,
355(6):2263–2303, 2003.

[7]
Marek Bożejko, Michael Leinert, and Roland Speicher.
Convolution and limit theorems for conditionally free random
Pacific Journal of Mathematics, 175(2):357–388, 1996.

[8]
Marek Bożejko and Janusz Wysoczański.
New examples of convolutions and non-commutative central limit
Banach Center Publications, 43(1):95–103, 1998.

[9]
Guillaume Cébron, Antoine Dahlqvist, and Franck Gabriel.
Freeness of type $ b $ and conditional freeness for random matrices.
arXiv preprint arXiv:2205.01926, 2022.

[10]
Guillaume Cébron and Nicolas Gilliers.
Asymptotic cyclic-conditional freeness of random matrices.
arXiv preprint arXiv:2207.06249, 2022.

[11]
Benoit Collins, Takahiro Hasebe, and Noriyoshi Sakuma.
Free probability for purely discrete eigenvalues of random matrices.
Journal of the Mathematical Society of Japan, 70(3):1111–1150,

[12]
Stephen Curran and Roland Speicher.
Asymptotic infinitesimal freeness with amalgamation for haar quantum
unitary random matrices.
Communications in Mathematical Physics, 301(3):627–659, 2011.

[13]
Maxime Février, Mitja Mastnak, Alexandru Nica, and Kamil Szpojankowski.
A construction which relates c-freeness to infinitesimal freeness.
Advances in Applied Mathematics, 110:299–341, 2019.

[14]
Maxime Février and Alexandru Nica.
Infinitesimal non-crossing cumulants and free probability of type b.
Journal of Functional Analysis, 258(9):2983–3023, 2010.

[15]
Uwe Franz.
Multiplicative monotone convolutions.
arXiv preprint math/0503602, 2005.

[16]
Katsunori Fujie and Takahiro Hasebe.
Free probability of type b prime.
arXiv preprint arXiv:2310.14582, 2023.

[17]
Takahiro Hasebe and Hayato Saigo.
The monotone cumulants.
In Annales de l'IHP Probabilités et statistiques,
volume 47, pages 1160–1170, 2011.

[18]
Sergei Kerov.
Interlacing measures.
Translations of the American Mathematical Society-Series 2,
181:35–84, 1997.

[19]
Bernadette Krawczyk and Roland Speicher.
Combinatorics of free cumulants.
Journal of Combinatorial Theory, Series A, 90(2):267–292,

[20]
Germain Kreweras.
Sur les partitions non croisées d'un cycle.
Discrete mathematics, 1(4):333–350, 1972.

[21]
Franz Lehner.
Cumulants in noncommutative probability theory i. noncommutative
exchangeability systems.
Mathematische Zeitschrift, 248(1):67–100, 2004.

[22]
Franz Lehner and Kamil Szpojankowski.
Boolean cumulants and subordination in free probability.
Random Matrices: Theory and Applications, 10(04):2150036, 2021.

[23]
Romuald Lenczewski.
Operators related to subordination for free multiplicative
Indiana University mathematics journal, pages 1055–1103, 2008.

[24]
Naofumi Muraki.
Monotonic independence, monotonic central limit theorem and monotonic
law of small numbers.
Infinite Dimensional Analysis, Quantum Probability and Related
Topics, 4(01):39–58, 2001.

[25]
Naofumi Muraki.
The five independences as natural products.
Infinite Dimensional Analysis, Quantum Probability and Related
Topics, 06(03):337–371, 2003.

[26]
Alexandru Nica and Roland Speicher.
Lectures on the combinatorics of free probability, volume 13.
Cambridge University Press, 2006.

[27]
Mihai Popa.
A combinatorial approach to monotonic independence over a
Pacific Journal of Mathematics, 237(2):299–325, 2008.

[28]
Mihai Popa.
A new proof for the multiplicative property of the boolean cumulants
with applications to operator-valued case.
arXiv preprint arXiv:0804.2109, 2008.

[29]
Mihai Popa.
Freeness with amalgamation, limit theorems and s-transform in
non-commutative probability spaces of type b.
In Colloquium Mathematicum, volume 120, pages 319–329.
Institute of Mathematics Polish Academy of Sciences, 2010.

[30]
Mihai Popa and Jiun-Chau Wang.
On multiplicative conditionally free convolution.
Transactions of the American Mathematical Society,
363(12):6309–6335, 2011.

[31]
Dimitri Shlyakhtenko.
Free probability of type-b and asymptotics of finite-rank
perturbations of random matrices.
Indiana University Mathematics Journal, 67(2):971–991, 2018.

[32]
Roland Speicher.
Free probability and random matrices.
arXiv preprint arXiv:1404.3393, 2014.

[33]
Roland Speicher.
Lecture notes on" non-commutative distributions".
arXiv preprint arXiv:2009.03589, 2020.

[34]
Roland Speicher and Reza Woroudi.
Boolean convolution.
In of fields inst. comm. Citeseer, 1993.

[35]
Pei-Lun Tseng.
A unified approach to infinitesimal freeness with amalgamation.
arXiv preprint arXiv:1904.11646, 2019.

[36]
Pei-Lun Tseng.
Operator-valued infinitesimal multiplicative convolutions.
arXiv preprint arXiv:2201.13187, 2022.

[37]
Dan Voiculescu.
Addition of certain non-commuting random variables.
Journal of functional analysis, 66(3):323–346, 1986.

[38]
Dan Voiculescu.
Multiplication of certain non-commuting random variables.
Journal of Operator Theory, pages 223–235, 1987.

[39]
Dan Voiculescu.
Limit laws for random matrices and free products.
Inventiones mathematicae, 104(1):201–220, 1991.


In order to simplify calculations, we assume that the latitudinal profile of
ozone can be approximated by the function $\sin^{2}\theta$, similar to the
latitudinal profile of water vapour. Under this assumption, we rewrite
Eqs.(123) and (124) as

$\displaystyle\tau_{\rm G,H_{2}O}\left(x,\theta,\varphi\right)$
$\displaystyle=\frac{4}{3}\sqrt{\frac{6\pi}{5}}T_{\rm H_{2}O}V_{\rm
H_{2}O}\left(x\right)Y_{22}\left(\theta,\varphi\right),$ (129)
$\displaystyle\tau_{\rm G,O_{3}}\left(x,\theta,\varphi\right)$
$\displaystyle=\frac{4}{3}\sqrt{\frac{6\pi}{5}}T_{\rm O_{3}}V_{\rm
O_{3}}\left(x\right)Y_{22}\left(\theta,\varphi\right),$ (130)

and the corresponding heating rates as

$\displaystyle J_{\rm H_{2}O}\left(x,\theta,\varphi\right)$
$\displaystyle=\frac{4}{3}\sqrt{\frac{6\pi}{5}}\sigma C_{\rm p}T_{\rm
H_{2}O}V_{\rm H_{2}O}\left(x\right)Y_{22}\left(\theta,\varphi\right),$ (131)
$\displaystyle J_{\rm O_{3}}\left(x,\theta,\varphi\right)$
$\displaystyle=\frac{4}{3}\sqrt{\frac{6\pi}{5}}\sigma C_{\rm p}T_{\rm
O_{3}}V_{\rm O_{3}}\left(x\right)Y_{22}\left(\theta,\varphi\right),$ (132)

where the imaginary number appearing in Eq.(122) has been ignored since it
does not affect the amplitude of the thermal forcing101010this step would
certainly become invalid if one is after the phase of the forcing rather than
merely its amplitude as is our case here.. For a given component, the total
input power is obtained by integrating the heating rate over the air column

$\delta\mathcal{F}=\int_{0}^{+\infty}J\rho_{0}Hdx=\frac{p_{\rm
s}}{g}\int_{0}^{+\infty}Je^{-x}dx,$ (133)

noting that $\rho_{0}H=\left(p_{0}/g\right)$. This requires computing the
integral of the vertical distribution of the heating rates,

$\displaystyle A_{\rm H_{2}O}$ $\displaystyle=\int_{0}^{+\infty}{V_{\rm
H_{2}O}e^{-x}}dx=\frac{1}{1+k_{\rm H_{2}O}},$ (134) $\displaystyle A_{\rm
O_{3}}$ $\displaystyle=\int_{0}^{+\infty}{V_{\rm O_{3}}e^{-x}}dx$
$\displaystyle=K_{\rm
O_{3}}\pi\frac{\left(x_{2}-x_{1}\right)\left(e^{-x_{1}}+e^{-x_{2}}\right)}{\pi^{2}+\left(x^{2}-x_{1}\right)^{2}}.$
(135)

We end up with $A_{\rm H_{2}O}=0.75$ and $A_{\rm O_{3}}=4.5\times 10^{-2}$.
Therefore, the total fluxes absorbed by water vapour and ozone are given by

$\displaystyle\delta F_{{\rm H_{2}O}}\left(\theta,\varphi\right)$
$\displaystyle=\delta\mathcal{F}_{{\rm
H_{2}O};22}Y_{22}\left(\theta,\varphi\right),$ (136) $\displaystyle\delta
F_{{\rm O_{3}}}\left(\theta,\varphi\right)$
$\displaystyle=\delta\mathcal{F}_{{\rm
O_{3}};22}Y_{22}\left(\theta,\varphi\right),$ (137)

with the components

$\displaystyle\delta\mathcal{F}_{{\rm H_{2}O};22}$
$\displaystyle=\frac{4}{3}\sqrt{\frac{6\pi}{5}}\sigma C_{\rm p}\frac{p_{\rm
s}}{g}T_{{\rm H_{2}O}}A_{\rm H_{2}O},$ (138)
$\displaystyle\delta\mathcal{F}_{{\rm O_{3}};22}$
$\displaystyle=\frac{4}{3}\sqrt{\frac{6\pi}{5}}\sigma C_{\rm p}\frac{p_{\rm
s}}{g}T_{{\rm O_{3}}}A_{\rm O_{3}}.$ (139)

Using $\sigma=1.45\times 10^{-4}~{}{\rm s^{-1}}$, $C_{\rm p}=1004.7~{}{\rm
J~{}kg^{-1}~{}K^{-1}}$, $p_{\rm s}=1013.25~{}{\rm hPa}$, $g=9.81~{}{\rm
m~{}s^{-2}}$, and $F_{\star}=1366~{}{\rm W~{}m^{-2}}$, we obtain $\delta
F_{22}=828.8~{}{\rm W~{}m^{-2}}$, and

$\displaystyle\delta\mathcal{F}_{{\rm H_{2}O};22}\approx 102~{}{\rm
W~{}m^{-2}},$ $\displaystyle\frac{\delta\mathcal{F}_{{\rm H_{2}O};22}}{\delta
F_{22}}\approx 12.3\%,$ $\displaystyle\delta\mathcal{F}_{{\rm
O_{3}};22}\approx 44~{}{\rm W~{}m^{-2}},$
$\displaystyle\frac{\delta\mathcal{F}_{{\rm O_{3}};22}}{\delta F_{22}}\approx
5.3\%,$ $\displaystyle\delta\mathcal{F}_{\rm tot;22}\approx 146~{}{\rm
W~{}m^{-2}},$ $\displaystyle\frac{\delta\mathcal{F}_{{\rm tot};22}}{\delta
F_{22}}\approx 17.6\%.$ (140)

Similar estimates can be computed from the profiles shown by Vichare and
Rajaram (2013) (see their Figure 4), where the heating rate is plotted as a
function of altitude for the main Hough functions structuring the semi-diurnal
atmospheric tide on Earth. Without elucidating again the copious details of
the procedure, we obtain for the profiles in Vichare and Rajaram (2013) the
following:

$\displaystyle\delta\mathcal{F}_{{\rm H_{2}O};22}\approx 116~{}{\rm
W~{}m^{-2}},$ $\displaystyle\frac{\delta\mathcal{F}_{{\rm H_{2}O};22}}{\delta
F_{22}}\approx 14.0\%,$ $\displaystyle\delta\mathcal{F}_{{\rm
O_{3}};22}\approx 34~{}{\rm W~{}m^{-2}},$
$\displaystyle\frac{\delta\mathcal{F}_{{\rm O_{3}};22}}{\delta F_{22}}\approx
4.2\%,$ $\displaystyle\delta\mathcal{F}_{\rm tot;22}\approx 150~{}{\rm
W~{}m^{-2}},$ $\displaystyle\frac{\delta\mathcal{F}_{{\rm tot};22}}{\delta
F_{22}}\approx 18.1\%.$ (141)

Put together, these computed estimates are consistent and suggest that the
efficiency parameter is around $\alpha_{\rm A}{\sim}17{-}18\%$, which is close
to the constraint on $\alpha_{\rm A}$ ($14\%,$ Section 4.1) obtained by
fitting our model parameters to reproduce the present day semidiurnal surface
pressure variations.

##  Appendix J Benchmarking our model against GCM simulations

The work of Auclair-Desrotour et al. (2019) presents, to our knowledge, the
only attempt on recovering the tidal torque exerted about the spin axis of a
thermally forced atmosphere in the high frequency regime; i.e. the regime
where the Lamb resonance is encountered, using GCM simulations. Specifically,
the mentioned work computed the tidal torque from 3D simulations of
atmospheric dynamics using the generic version of the LMDZ GCM (Hourdin et
al., 2006) in the case of a nitrogen dominated atmosphere around a dry rocky
Venus-like planet.

Figure 9: Similar to Figure 2, plotted are the tidal spectra, in terms of the
imaginary part of the surface pressure anomaly, as a function of the
normalized tidal frequency $\omega$. The blue curve is that computed
numerically in Auclair-Desrotour et al. (2019) using 3D GCM simulations, while
the orange curve is recovered from our model by tuning the two free
frequencies $\sigma_{\rm bl}$ and $\sigma_{0}$ as discussed in J.

Imposing on our analytical solution (Eq.3.1) the same planetary parameters
defined in Table 1 of Auclair-Desrotour et al. (2019), for a 10 bar
atmosphere, and setting $\sigma_{\rm bl}=10^{-5}$ s-1 and $\sigma_{\rm
0}=4\times 10^{-6}$ s-1, we recover in Figure 9 a tidal spectrum that
reasonably fits the GCM computed spectrum. In particular, we recover the two
amplified responses of the torque in the vicinity of synchronization and at
the Lamb resonance, and we recover the asymmetry in the Lamb resonance where
the positive peak of the torque is annihilated. Furthermore, we recover a
similar scaling of the torque with the forcing frequency $\sigma$. Namely, for
the GCM spectrum, the tidal torque $\mathcal{T}\propto\sigma^{0.75}$ near
synchronization, and in the high frequency regime
$\mathcal{T}\propto\sigma^{-1}$; while for our model,
$\mathcal{T}\propto\sigma^{0.53}$ in the former regime, and
$\mathcal{T}\propto\sigma^{-1}$ in the latter.

##  Appendix K Atmospheric compositional variations and the possible interplay
between shortwave and longwave heating

When studying the case of the Earth in the main text, we have assumed that the
atmosphere is always heated by the shortwave. We have thus ignored the
asynchronous heating in the infrared, which, as we showed in Section 3.2,
breaks the symmetry of the Lamb resonance by attenuating the accelerative peak
of the tidal torque. The results obtained for the amplitude of the
accelerative tidal torque at the resonance can thus be considered as the
optimistic limit of the resonant amplitude, as the latter can be attenuated
once the longwave heating starts to play a significant role. However, an
additional attenuation effect is delivered by the more efficient radiative
cooling on a warmer Earth. Namely, the radiative cooling frequency features a
strong temperature dependence that scales as $\sigma_{0}\propto T^{3}$.
Henceforth, the temperature increase required to place the resonance
occurrence in the Precambrian would also attenuated the resonant amplitude of
the torque by increasing $\sigma_{0}$. Here we intend to elaborate further on
this signature, which is driven by atmospheric compositional variations, by
studying the functional form of the tidal response obtained in H.

The form of the pressure anomaly in Eqs. (119) allows for an analytical
investigation of the Lamb resonance. Namely, we can characterize the positive
and negative peaks of the resonance by studying the derivative of the
imaginary part of the pressure anomaly. We focus on the limit studied for the
Earth in the main text and set $\mathcal{G}=1$, that is, we assume thermotidal
heating is dominated by direct shortwave absorption. Computing the roots of
the derivative of Eq. (119) and substituting them in the expression, we obtain
the amplitude of the peaks around the resonance. Namely, we denote by $\eta$
the ratio between the value of the tidal torque reached at the peaks around
resonance to that at present. Considering the predominant terms of the
obtained roots, this amplification ratio reads as

$\eta\approx\frac{K_{\oslash}}{K_{\oplus}}\frac{\varepsilon_{\oplus}}{\varepsilon_{\oslash}}\eta_{\oplus}\,,$
(142)

where we have denoted by the subscript $\oslash$ the parameter values at the
resonance, in contrast with the present values denoted by $\oplus$.

Now we allow for variations in $K$ that are induced by the parameters entering
into the definition of Eq. (117). Namely, presuming that $\kappa,R_{\rm p},g,$
and $\Lambda_{2}$ are indeed constant, the rest of the parameters would modify
the expression of the amplification factor $\eta$ (Eq. 142) into:

$\eta=\eta_{\oplus}\\!\left(\frac{\alpha_{\rm A;\oslash}}{\alpha_{\rm
A;\oplus}}\right)\\!\left(\frac{F_{\star;\rm\oslash}}{F_{\star:\oplus}}\right)\\!\left(\frac{p_{\rm
s;\oplus}}{p_{\rm s;\oslash}}\right)\\!\left(\frac{\sigma_{\rm
L;\oplus}}{\sigma_{\rm
L;\oslash}}\right)^{2}\\!\\!\left(\frac{\sigma_{0;\oplus}}{\sigma_{0;\oslash}}\right),$
(143)

The resonant frequency in the adiabatic limit scales as $\sigma_{\rm
L}~{}\propto~{}T_{\rm s}^{1/2}$. Moreover, the radiative cooling frequency is
also a function of the surface temperature and it scales as $\sigma_{0}\propto
p_{\rm s}^{-1}T_{\rm s}^{3}$ (e.g., Eq. 10 of Showman and Guillot, 2002).
Including these scaling relationships in the latter expression of the
amplification factor one finds that

$\eta=\eta_{\oplus}\left(\frac{\alpha_{\rm A;\oslash}}{\alpha_{\rm
A;\oplus}}\right)\left(\frac{F_{\star;\rm\oslash}}{F_{\star:\oplus}}\right)\left(\frac{T_{\rm
s;\oplus}}{T_{\rm s;\oslash}}\right)^{4}.$ (144)

The equilibrium surface temperature varies as $T_{\rm
s}\propto\left(\alpha_{\rm h}F_{\star}\right)^{1/4}$, where $\alpha_{\rm h}$
is the fraction of flux that is absorbed by the atmosphere, mainly in the
infrared absorption. It follows that

$\eta=\eta_{\oplus}\left(\frac{\alpha_{\rm A;\oslash}}{\alpha_{\rm
A;\oplus}}\right)\left(\frac{\alpha_{\rm h;\oplus}}{\alpha_{\rm
h;\oslash}}\right).$ (145)

This interesting and simplified version of the amplification factor shows that
$\eta$ depends on the opacity of the atmosphere in the visible (via
$\alpha_{\rm A})$ and in the infrared (via $\alpha_{\rm h})$, which are
functions of the composition. Namely, the two absorption coefficients have
opposite effects on $\eta$: a past atmosphere that is more opaque in the
visible features a greater amplification factor, and thus prompts a stronger
tidal response upon the resonance occurrence. In contrast, a larger past
opacity in the infrared tends to attenuate the resonance amplified response.
Thus the bottom line is, increased greenhouse gases in the past (mainly CO2),
tend to decrease the amplitude of the accelerative resonant response of the
atmosphere via, first, Eq. (145), and via increasing the contribution of the
asynchronous thermotidal heating that induces a phase lag on the tidal bulge
and consequently attenuates the accelerative torque around the resonance.

##  Appendix L The resonant period in the isothermal limit

Here we discuss the limit where the temperature profile of the atmosphere is
modelled by an isotherm, in contrast with the adiabatic profile we assume in
our neutrally stratified model. The isothermal limit translates to a positive
and constant value of the Brunt–Väisälä frequency $N_{\rm B}$, and it has been
studied earlier using the linear theory of atmospheric tides (e.g., Lindzen,
1978; Lindzen and Blake, 1972; Auclair-Desrotour et al., 2019). We are
specifically interested in estimating the difference between the two models in
terms of the Lamb resonance period. Namely, we aim to define the spectral
position of the resonance in the isothermal limit and compare with what we
obtained for our model in Eq. 24.

We adopt the analytical computations in Auclair-Desrotour et al. (2019) that
yield a frequency-dependence for the pressure surface anomaly in the form
(their Eq. 12):

$\Im\\{\delta p_{\rm s}\\}=p_{\rm s}\frac{\kappa J_{\rm
s}}{gH\sigma}\frac{\frac{H}{h}\left(b_{\rm
J}+\frac{1}{2}+\kappa\right)-\frac{1}{2}(b_{\rm J}+1)}{\left[b_{\rm J}(b_{\rm
J}+1)+\frac{\kappa
H}{h}\right]\left(\frac{H}{h}-\frac{1}{\Gamma_{1}}\right)}.$ (146)

Here the parameters and variables are equivalent in definition to those we use
in our work, and $h$ is the equivalent depth. This equation defines the
position of the Lamb resonance by the equivalent depth $h_{\rm L}=\Gamma_{1}H$
(see also Lindzen and Blake, 1972). Hence, introducing the definition of the
equivalent depth

$h_{n}=\frac{R_{\rm p}^{2}\sigma^{2}}{\Lambda_{n}g},$ (147)

we obtain, for the fundamental mode, the resonant frequency and consequently
the resonant rotational period in the form:

${\rm LOD_{res;\,iso}}=\frac{4\pi R_{\rm p}}{\sqrt{\mathcal{R_{\rm
s}}\Lambda_{2}\Gamma_{1}\overline{T}}+2R_{\rm p}n_{\star}}.$ (148)

This equation differs from that in the neutrally stratified limit (Eq. 24) in
the fact that temperature here cannot be assumed to be the surface temperature
$T_{\rm s}$, and the $\Gamma_{1}$ factor multiplied by the temperature. To
compare the two equations, we assign for the isothermal model a constant value
of temperature corresponding to a mass-average of the temperature profile in
the neutrally stratified case assuming a fixed total enthalpy of the air
column. Namely, using Eq. 13, we obtain $\overline{T}=T_{\rm s}/(\kappa+1)$,
thus we rewrite Eq. 148 as:

${\rm LOD_{res;\,iso}}=\frac{4\pi R_{\rm p}}{\sqrt{\mathcal{R_{\rm
s}}\Lambda_{n}{T_{\rm s}}/{(1-\kappa^{2})}}+2R_{\rm p}n_{\star}}.$ (149)

This equation yields a resonant rotational period that is roughly one hour
smaller than that obtained in the neutrally stratified case; i.e., the curve
shown in Figure 6 shifts vertically downwards by ${\sim}1$ hr. This places the
resonance for present conditions around 21.6 hr, which is closer to the 21.3
hr value obtained by Zahnle and Walker (1987).