Percolation in $Z^2$, problem in the proof of the existence of a critical probability

Question

I have a $Z^2$ lattice. Every element $(z_1, z_2) \in Z^2$ is connected to one of its 4 neighbours $(z_1, z_2) + (0,1)$, $(z_1, z_2) + (0,-1)$, $(z_1, z_2) + (1,0)$, $(z_1, z_2) + (-1,0)$ by an edge. The edges can be "open" with a probability $p$ and "closed" with a probability 1-p.

I call $\sigma_n$ the total number of self avoinding walks starting from $(0, 0)$ and composed of $n$ steps through the edges. Then I define the event {$ N_n = k $} = {"the number of SAW composed of n OPEN steps is k"}. My question is: is it correct writing that the probability of this event is the following?

$$ P(N_n = k) = \binom {\sigma_n} {k} (p^n)^k (1-p^n)^{\sigma_n - k }. $$

This would mean that the events {"this SAW is open"} and {"this other SAW is open"} are all independent one from the other, but I think it should be wrong because some SAWs necessarily share some edge one with the other one. But on the other hand, this is the only way I am able to justify the last equality in the following expression, which I find in many text books of percolation theory, $$ P( N_n \geq 1) = \sum_{k=1}^{\sigma_n} P(N_n =k) \leq \sum_{k=1}^{\sigma_n} k P(N_n = k) =E[N_n] = p^n \sigma_n. $$ where E is the expectation. Could anyone help me to clarify this point?

Answer 1

I think your formula for $P(N_n=k)$ is false because as you said, paths are dependent on one-another. With that said:

Expectation is linear, regardless of dependence. Let $S_n$ be the index of set of SAWs of length $n$ and $\gamma_{ni}$ denote a self avoiding path of length $n$, $i\in S_n$. You write

$$N_n=\sum_{i\in S_n} 1_{\gamma_{ni}}$$

Now you note that each $\gamma_{ni}$ has probability $p^n$ and $S_n$ has $\sigma_n$ elements in it. Using $E[1_{\gamma_{ni}}]=p^n$, it immediately follows that $E[N_n]=p^n\sigma_n$.