Percolation Model and Complex Probabilities

Question

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 $p$.

Answer 1

First of all, you cannot construct a classical finite complex valued measure this way (i.e., a measure that assigns a finite complex number to each measurable set and is countably additive) for that very simple reason that the total variation of every such measure must be finite and, splitting into cylinders according to the first $n$ coordinates, you get the lower bound $(|p|+|1-p|)^n$ for the variation, which tends to infinity unless $p\in[0,1]$. So, even if you manage to do something, it'll be some "quasimeasure". Thus, you'll have to decide what meaning of "quasi" may be still acceptable for you.

Second, the whole idea "to use complex analysis results, to study the behavior of the probability of certain events as a function of p" seems very fishy to me for the reason that most interesting events in percolation theory depend on what happens near infinity and, thereby, satisfy some form of 0-1 law, so you are suggesting to use complex analysis to study functions that are not only not analytic but even discontinuous on the interval (and we cannot be talking of the boundary values here because if you can extend to the upper half-plane in any meaningful way, you can extend to the lower one by symmetry and those functions will have to glue into an analytic function in some neighborhood of $[0,1]$, not mentioning the Fatou theorem about non-tangential boundary values).

Of course, I do not see everything and you may have some brilliant idea behind your suggestion I would never be able to think of but, if I were a reviewer for a grant proposal containing the italicised phrase and no coherent explanation of its meaning (and, judging from the question you asked, you currently have none), I would immediately turn it down as a "wild dream".

Answer 2

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).