Percolation (1) : 2D Bernoulli percolation, critical point and transition of phase

Recently I attend the seminar talking about the percolation theory in IHES so I wrote some notes here.

Before starting, we have to notice that the terminology of percolation is adopted in different situations and generally in the contexts of graphs - it could be various graphs - like lattice $Z^d$, random graphs, random maps etc. But the phenomena is a little universal that a path starts from $0$ and goes to infinite faraway. In this series of talk, Hugo focuses on the situation of 2D Bernoulli percolation.

Definition and critical point

We give some definitions precisely. In the lattice graph $Z^2$, each site has 4 neighbors and every edge has independently a probability $p$ to be present (open) or a probability $1-p$ to be absent (closed). Then there exits a critical point $p_c$ : when $p \leq p_c$, with probability $0$ there exists a path from 0 to $\infty$, while when $p > p_c$, with probability $\theta(p)$ there exists a path from $0$ to $\infty$.

$$
\begin{eqnarray*}
p \leq p_c &,& \theta(p) = \mathbb{P}[0 \leftrightarrow \infty] = 0 \\
p > p_c &,& \theta(p) = \mathbb{P}[0 \leftrightarrow \infty] > 0
\end{eqnarray*}
$$

In 2D model, the critical point $p_c = 1/2$. Some simple argument supports this point. For example, if we draw the dual percolation between the face whose frontier is closed, we get a dual graph with probability $1-p$. If we suppose that the critical point is unique, then the transition of face happens at the same time in both the primal graph and the dual graph. So $p_c = 1- p_c$ and $p_c = 1/2$.

Quantitative analysis 

We hope to get some stronger result, The following theorem is first obtained by Menshikov, Aizeman and Michael $$\begin{eqnarray*}\forall p < p_c, \exists C_p > 0 \text{ s.t } \mathbb{P}_p[ 0 \leftrightarrow \partial B_n] \leq \exp{(-C_p n)}\\ \forall p > p_c, \exists C > 0 \text{ s.t } \mathbb{P}_p[0 \leftrightarrow \infty] \geq C(p - p_c) \end{eqnarray*}$$

Here, we denote the ball of radius $n$ by $\partial B_n$ and this theorem indeed, gives some numerical estimation of the speed of decrements. 

Proof 1 by (Menshikov, Aizeman, Michael) 

We note $\theta_n(p) = \mathbb{P}_p [0 \leftrightarrow \partial B_n]$ and $$\phi_p (S) = \sum_{x \in S, y \notin S, x \sim y} p\mathbb{P}[0 \leftrightarrow^S x]$$. We can prove it by 5 steps.

1.  We admit at first this important inequality$$ \frac{d}{dp} \theta_n(p) \geq \frac{1}{p(1-p)}[\inf_{0 \in S \subset B_n} \phi_p(S)](1 - \theta_n(p))$$
   then we define $$\tilde{p}_c = \sup \{p : \exists S \ni 0, \text{s.t} \phi_p(S) < 1\}$$
   we can prove $$\theta(P) \geq \frac{1}{p(1-\tilde{p}_c)}(p - \tilde{p}_c)$$
2.  We choose $S \subset B_{k-1}$ such that $\phi_p (S) < 1$ and prove$$\theta_{nk}(p) \leq (\phi_p(S))^n$$
3. We conclude that $\tilde{p}_c = p_c$
4. Verify the identity $$\frac{d}{dp}\mathbb{P}_p(X) = \sum_{e \in E} \frac{1}{p(1-p)}Cov(w_e, X)$$
5. We put $X = -\mathbb{I}_{0 ! \leftrightarrow \partial B_n}$ and we prove the important inequality.