each step. The superscripts “$-$” and “+” denote the lower and upper layers, respectively. These recursive relations describe the fact that the chains occupying these 3 coupled layers are connected to the ghost site with the probability $h$, see Fig. \[Fig1\]. More precisely, a $(k-1)$-part of the chain residing have a chance to continue on the next step to one of 3 layers with the probability determined by $p$ and the corresponding coupling constant. The equations (\[req\_rl\_P\]) serves as the implicit set of equations for determination of the layer dependent occupation, $\theta_i$, by means of the numerical iteration.

The layer dependent density $\theta$ is obtained as the ratio of the term in the GPF corresponding to the occupied central site of the Cayley tree to the total GPF, $\theta = (Z-c\Psi^{z})/Z$, which gives, upon substituting into (\[small\_agg\_GPF1\]), an important relation: $$\label{x-mer eqs bA}
 \frac{z}{2} r_{f}^{-1}  \lambda p^{-2} \gamma
 \underset{k=0}{ \overset{x-1}{\sum }}P _{k}P _{x-k-1}   = \frac{\theta }{1-\theta } \frac{\Psi ^{z}}{\Psi _x ^{z}}.$$ In order to calculate the auxiliary field $\xi$ we employ the equal probability condition to find a void in the center of the Cayley tree or in the arbitrary site belonging to its first generation. In terms of $\xi_k$ 