Finite-box comparison for percolation parameters (${\mathbb{P}}_q(A)-{\mathbb{P}}_p(A)\le m(q-p)$)

If an increasing event $A$ depends on $m$ sites or edges, the uniform-label monotone coupling of Bernoulli percolation and the union bound give $0\le {\mathbb{P}}_q(A)-{\mathbb{P}}_p(A)\le m(q-p)$ for $p\le q$. A change of the event requires a coordinate with label in $(p,q]$. For bond percolation from the origin to the boundary of $[-n,n{]}^2$, first-hit graph paths use the $4n(2n+1)$ internal edges, so the same count bounds the parameter sensitivity.