 Closed-cut bound for percolation in a strip

ID: closed-cut-bound-for-percolation-in-a-strip

A vertical cut between consecutive columns of

T_{k}

contains

2 k + 1

edges. All are closed with probability

(1 - p)^{2 k + 1}

. Cuts in different columns use disjoint edges, so their closed-cut events are independent. A connection from column zero to column

n

must cross each of the intervening

n

cuts. Consequently

q_{k} (n) \leq (1 - (1 - p)^{2 k + 1})^{n}, f_{k} (p) \geq - lo g (1 - (1 - p)^{2 k + 1}) > 0.

This finite-width obstruction persists even when the unrestricted planar percolation is supercritical.

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