TITLE: Percolation Model and Complex Probabilities 
QUESTION [5 upvotes]: Let $d>0$ be an integer and consider the first neighbors independent bond percolation model in $\mathbb Z^d$, where each edge is open with probability $p\in[0,1]$. 
I would like to know, if can we generalize the construction of the probability measure of this model for a complex parameter $p\in \mathbb C$ in some neighborhood of zero. 
In other words, 
Let be  $E=\{\{x,y\}\subset\mathbb Z^d: \|x-y\|_1=1\}$, where $\|x-y\|_1=\sum_{j=1}^d|x_j-y_j|$. 
If $\{0,1\}$ a complex measure space, such that $\mu(\{1\})=p\in\mathbb C$ and $\mu(\{0\})=1-p$, is there any domain $D$ on the complex plane, for which it is possible to construct a product measure, formally given by  
$$\mathbb P_p=\prod_{e\in E} \mu $$ 
defined on the sigma algebra generated by the cylinder sets of $\Omega=\{0,1\}^{E}$ with $\mathbb P_p(\{0,1\}^{E})=1$ 
for any $p\in D$ ? 
If the answer is no. What is the best set-function fitting as much as possible the properties of a measure we could construct for a $p$ in some domain of the complex plane ? 
This question it was motivated by the possibility, in case of a positive answer ( or a good construction near a probability), to use complex analysis results, to study the behavior of the probability of certain events as a function of $p$.

REPLY [3 votes]: Not an answer to your original question, but more a reaction to the previous answer. It does make a lot of sense to analyse the behaviour of various quantities as function of complex-valued physical parameters (here p). For example, analysis of a statistical mechanical system (say, an Ising model) as a function of a complex magnetic field or complex temperature provides many important information about the system. To cite some: the Lee-Yang theorem (about possible locations of singularities, and thus possible locations of phase transitions) or Isakov's theorem (existence of an essential singularity at 0 of the free energy of the Ising model as a function of a complex magnetic field, thus showing that stable phases cannot be analytically continued into the metastable phases, contrarily to what mean-field theory suggests). Actually, there is a old version of Isakov theorem's for percolation (by Kunz and Souillard).