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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 $p$.

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2 Answers 2

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".

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Hi fedja, thank you for the answer but I would like to comment that your statement : "so you are suggesting to use complex analysis to study functions that are not only not analytic but even discontinuous on the interval" it is not true in general, and one of the most interesting event in percolation theory $\mathbb P_p(\{0 \leftrightarrow\infty\})$, it was showed to analytic near p=1 , by a cluster expansion. So the approach I asked, in view of your argument can not be used directly, but use complex analysis to approach percolation problems it is not a wild dream. –  Leandro Apr 28 '10 at 17:29
    
The reference I am talking about is : Percolation connectivity in the highly supercritical regime, R. Snachis, A. Proccaci, B. Scoppola, G. A. Braga, Markov Processes and Applications, 10, No. 4, 07-28, (2004). –  Leandro Apr 28 '10 at 17:36
    
Well, it is true that for some events like the one you cited the probabilities of obstacles for $p$ close to $1$ decay so fast as the size of the obstacle goes to infinity that the trivial inclusion-exclusion count results in the series of polynomials converging in some small neighborhood of $1$ but it is not really convincing. The probability of any event for finite sequence of Bernoulli trials is just a polynomial in $p$ but I have yet to see a nontrivial result based on this observation and complex analysis methods. Anyway, as I said, you may see something that I do not :-). –  fedja Apr 28 '10 at 22:12

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).

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Hi Velenik, thank you very much by your manifestation and references. When I was asking for something "near to a probability" I was really thinking about the Lee-Yang circle Theorem. Because in models like Ising, for example, we are able to get information about the set of the Gibbs measure in infinite volume, by taking limits of the free energy of complex partition functions defined on finite volumes. For percolation we do not have the Hamiltonian and so I was curious if someone else figure out if it is possible or not to approach percolation in this way. –  Leandro Jun 3 '10 at 18:31
    
What kind of informations would you like to extract (the set of Gibbs state being trivial for percolation)? As far as I remember, Kunz and Souillard have obtained some information on the cluster size in the paper I mentioned above. But I don't quite recall whether it was as an input or an output of their analysis. –  Yvan Velenik Jun 9 '10 at 8:27
    
I was working on the problem about first neighbors anisotropic independent bond percolation on the lattice $\mathbb Z_+\times\mathbb Z_+$. The result I would like to prove is $\mathbb P_{p,\alpha}(\{0\leftrightarrow x\})>P_{p,\alpha}(\{0\leftrightarrow x'\})$, where $p$ is the probability that a vertical edge is open and $\alpha p$ is the probability for the horizontal ones and $x$ is a point up diagonal on the lattice and $x'$ is the point obtained by the reflection of the point $x$ along the diagonal. (continue) –  Leandro Jul 16 '10 at 18:29
    
For some particular cases I was able to prove this inequality but for finite volume. So I ask about the possibility to consider the complex probability because would be possible to work in the direction of Hurwitz and Vitali Theorem's to extend this conclusion to the whole Lattice. –  Leandro Jul 16 '10 at 18:29

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