= Critical threshold for level-set percolation
{title2=$a_c$}
For a <random field> $(Y_x)$ on a countable <locally finite graph>, declare $x$ open when $Y_x\geq a$. Increasing $a$ can only remove open <graph vertices>[vertices], so the event that an <infinite percolation cluster> exists is decreasing in $a$. One convention for the <critical threshold for level-set percolation> is
$$
a_c=\sup\{a:\mathbb P(\text{an infinite open component exists at level }a)>0\}.
$$
The value may be infinite, and the definition does not decide what happens at $a=a_c$. On a general <graph>, a specified-root threshold can differ from a global existence threshold, so the convention matters.
For <independent and identically distributed random variables> $Y_x$ with <uniform distribution> on $[0,1]$, the open <probability> is $1-a$ for $0\leq a\leq1$. The resulting model is <site percolation>, and reversing the parameter gives $a_c=1-p_c$, where $p_c$ is its <percolation critical probability>. This example separates the direction of the level threshold from the usual increasing opening-density parameter.
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