sites, respectively. The index $j=0$ denotes the central site, while the indexes $1\leq j\leq z$ denote the nearest-neighbor sites. The location of the aggregate and its contribution to the total partition function is therefore uniquely defined by the position of its central site. In particular, the configuration where the central site of a specific aggregate is unoccupied $(0;\delta
_{1},...,\delta _{z})$ contributes to GPF as the sum over all the arrangements of the nearest neighbors $$\label{sum}
   Z_i^{0}= \underset {\{\delta_{j}\} {\sum}} \lambda ^{\delta _{1}+\cdots +
   \delta _{z}}\psi (1,...,z),$$ where $\psi$ is yet unknown function that corrects for various configurations of the molecules outside the aggregate. Since $\psi$ cannot be determined analytically, one has to resort to simplifying approximations of this function due to Bethe and Guggenheim. [@Miller] According to the mentioned approximation, $\psi$ represents the mean-filed superimposed on the first generation sites of the Cayley tree and is given by $$\label{psi}
     \psi =c\xi ^{\delta _{1}+\cdots +\delta _{z}},$$ where $c$ is constant, $\xi$ is the unknown field that is to be related to the occupation probabilities on the lattice corresponding to the specific layer. The contribution to GPF from the configuration $(0;\delta_{1},...,\delta _{z})$ follows immediately from Eqs.(\[sum\],\[psi\]), to be written as $$\label{GPF_sm_cluster}
    Z_i^{0}=\underset{\{\delta_{i}\}} 