= Coupled onset parameter for origin percolation
{title2=$M=\inf\{p:I_p\}$}
In the uniform-label <monotone coupling of Bernoulli percolation>, let $I_p$ mean that the origin belongs to an <infinite percolation cluster>, and let $M=\inf\{p:I_p\}$. The occurrence parameters form an upper interval. Therefore $\{M<p\}=\bigcup_{r<p,\ r\in\mathbb Q}I_r$ and $\theta(p-)=\mathbb P(M<p)$. Since $\{M<p\}\subseteq I_p\subseteq\{M\leq p\}$, the jump is $\theta(p)-\theta(p-)=\mathbb P(I_p\cap\{M=p\})$. Occurrence at the infimum is not presumed.
Back to article page
