Near-critical percolation is a scaling regime in which the occupation probability approaches the critical probability at a rate chosen
to retain nontrivial changes in macroscopic crossing events. For critical Poisson
Voronoi percolation at mesh 
, use the deterministic annealed normalization
 |
where 
denotes the expectation value and 
counts color-pivotal cells for the left-to-right crossing
of the unit square at 
. A cell is color-pivotal if changing its color, while
keeping every other color and the tiling fixed, changes whether this crossing occurs.
Assign each cell an independent mark from the uniform
distribution on ![[0,1]](/images/equations/Near-CriticalPercolation/Inline5.svg)
and make it black at parameter 
when its mark is at most
 |
The same marks couple all parameters monotonically.
A marked quad is a bounded Jordan domain 
with four distinct boundary points 
, 
, 
, and 
in counterclockwise order. Its crossing event occurs when
the union of the black closed cells, intersected with the set
closure of 
, contains a connected set
meeting both closed boundary arcs 
and 
. No boundary wiring is used.
