> [!definition] > > Let $\integer^d$ be the $d$-dimensional integer lattice, with its elements called *sites*. Let $p \in [0, 1]$ and $B = (B_v, v \in \integer^d)$ be independent [[Bernoulli Process|Bernoulli]] [[Random Variable|RVs]] with probability $p$. > [!definition] > > Let $\integer^d(B) = \bracs{v \in \integer^d: B_v = 1}$. If $x, y \in \integer^d$ such that there is a nearest-neighbour path from $x$ to $y$ containing only elements of $\integer^d(B)$, then they are **connected in** $\integer^d(B)$, denoted $x \rightarrow^B y$. For $x \in \integer^d$, define > $ > \mathcal C(x) = \bracs{y \in \integer^d: x \to^B y} > $ > Let > $ > x(p, d) = E = \bracs{\exists x \in \integer^d: \abs{\mathcal C (x)} = \infty} > $ > be the event for which $\integer^d(B)$ contains an infinite connected component. Since such a component cannot be created or destroyed by adding or removing finitely many sites, $E$ is a tail-event. By [[Kolmogorov's 0-1 Law]], this event occurs with [[Probability|probability]] zero or one. This allows defining > $ > p_c(\integer^d) = \sup\bracs{p: x(p, d) = 0} > $ > the **critical probability** for site percolation.