Determine the large-width boundary-parity limit and useful confinement scale

Determine the large-width limit of the boundary-parity probabilities h_L(2/5) for the Manhattan mirror model and identify the width at which the sufficient high-probability confinement criterion h_L(2/5)\ge 0.8457 might become applicable.

Background

The paper studies the probability h_L(p) that every boundary-to-boundary trajectory in a two-square Manhattan-lattice rectangle crosses a designated connector an even number of times. A finite-scale theorem proves that if h_L(p) reaches 0.8457 for some even width L, then almost surely every trajectory in the infinite-plane model is periodic.

At p=2/5, the authors rigorously observe a decrease followed by an increase in h_L(2/5) over the tested widths 2 through 16. Because the measured events are not nested as L changes, the observed finite-size nonmonotonicity does not determine the asymptotic behavior or when the confinement criterion could become effective.