Level-set percolation

For a random field $(Y_x)$, level-set percolation studies the random vertex set $\{x:Y_x\ge a\}$ as the threshold $a$ varies.

For a random field $(Y_x)$ on a countable locally finite graph, declare $x$ open when $Y_x\ge a$. Increasing $a$ can only remove open vertices, so the event that an infinite percolation cluster exists is decreasing in $a$. One convention for the critical threshold for level-set percolation isThe 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 $0\le a\le 1$. 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.