MHB Percolation - Measure - Probability

[fsblajinha]

In $\Omega = \{0,1\}^{\mathbb{Z}^{2}}$, consider the class $C$ of cylinders. Show that $C$ is algebra. At $w\in \Omega$, we call cluster point $x$ all points $z\in \mathbb{Z}^{2}$ that can be attained from $x$ by a path that only passes by open dots. In the $\sigma$-algebra generated by $C, \mathbb{F}=\sigma(C)$, express the following events explicitly:

1 - {The cluster of the origin is infinite}

2 - {Exist in an infinite cluster system}

3 - {The cluster of the origin has positive density}

4 - {exist at least two infinite clusters (distinct)}

 

[fsblajinha]

$\Omega = \{0,1\}^{\mathbb{Z}^{2}}$ is the set of open and closed $\mathbb{Z}^{2}$

$\Omega_{\wedge} = \{0,1\}^{\wedge}, \forall \wedge \subset \mathbb{Z^{2}}$ finite.

$C:=$Cylinders (That is, local events that depend on a finite number of sites);

$\Pi_{\wedge}:\Omega \rightarrow \Omega_{\wedge}$ defined by $\Pi_{\wedge}(w)=$coincides with $w$ in $\wedge;$

$C_{\wedge}:=\Pi_{\wedge}^{-1}(P(\Omega_{\wedge}))$, where $P(\Omega_{\wedge})$ is the power set of $\Omega_{\wedge};$

In $\Omega = \{0,1\}^{\mathbb{Z}^{2}}$, consider the class $C$ of cylinders. Show that $C$ is algebra. At $w\in \Omega$, we call cluster point $x$ all points $z\in \mathbb{Z}^{2}$ that can be attained from $x$ by a path that only passes by open dots. In the $\sigma$-algebra generated by $C, \mathbb{F}=\sigma(C)$, express the following events explicitly:

1 - {The cluster of the origin is infinite}

2 - {Exist in an infinite cluster system}

3 - {The cluster of the origin has positive density}

4 - {exist at least two infinite clusters (distinct)}