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Let $F$ be a meromorphic function on $\mathbb{C}$.

I consider the "level set" $$E_\varepsilon=\{z:|F(z)|\leq\varepsilon\}.$$ My objective is to find conditions under which $E_\varepsilon$ does not percolate for $\varepsilon$ sufficiently small, i.e. it has only bounded connected components. My function $F$ is not periodic nor a polynomial and has infinitely many zeros and poles.

Any idea of a helpful tool?

More specifically, I actually want to prove that "gradient lines" do not percolate in $E_\varepsilon$, where a gradient line is just a function $y(t)\in\mathbb{C},t\geq 0$ such that $y'(t)=F(y(t))$.

So for those who want a binary question: Are there non-periodic non-polynomial meromorphic functions such that there exists $\varepsilon>0$ such that $F$ does not have unbounded gradient lines lying in $E_\varepsilon$?

Thanks!

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    $\begingroup$ Can you explain exactly what is a "random and stationary meromorphic function"? What is your definition of a probability measure on the set of all meromorphic functions in C? $\endgroup$
    – Alexandre Eremenko
    Commented Jul 3, 2023 at 16:53
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    $\begingroup$ I just talked about "random meromorphic functions" to help understand the kind of assumptions I am talking about. To keep it simple, consider random meromorphic functions of the form $$F(z)=\sum_i 1/(z-z_i)$$ where the set $\{z_i;i\geq 1\}$ is a random stationary set of points (the random sum is not always well defined, but it is well defined in some sense in some cases, see for instance link.springer.com/article/10.1007/s00039-007-0613-z) $\endgroup$
    – kaleidoscop
    Commented Jul 3, 2023 at 19:37
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    $\begingroup$ In the paper you cite, the "random function" is ENTIRE, and of very special kind. No natural notion of randomness is known to me for MEROMORPHIC functions. $\endgroup$
    – Alexandre Eremenko
    Commented Jul 4, 2023 at 5:19
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    $\begingroup$ @AlexandreEremenko I don't think the OP claims to have a definition of the natural probability distribution on all meromorphic functions. I think they just happen to define a random function that happens to be meromorphic on the plane and stationary. It is easy to construct such distributions, for instance by taking a random translate of a fixed elliptic function. I think the new question, although technically well-posed, is actually less illuminating: the initial question was more about if functions with unbounded gradient lines are in some sense "generic" or "rare". $\endgroup$
    – Pierre PC
    Commented Jul 5, 2023 at 9:30
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    $\begingroup$ Of course I am not claiming that anything interesting can be said in this level of generality, but in this case the critic would be that the level of generality of the question is too high, and I am not sure the OP can be blamed for not being aware of this fact. Especially given that they don't seem to be sure that such functions even exist, and my opinion (maybe as a young and naïve researcher who does not know the answer himself) is that this is not something to be ashamed about. $\endgroup$
    – Pierre PC
    Commented Jul 5, 2023 at 9:34

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