On the number of points skipped by a transient (1,2) random walk on the line

TL;DR

This paper establishes criteria to determine when a near-critical (1,2) transient random walk on the positive line skips finitely many points, extending previous results on cutpoints for nearest neighbor walks.

Contribution

It provides a generalized criterion for the finiteness of skipped points in (1,2) random walks, broadening the understanding of their path properties.

Findings

Criteria for finiteness of skipped points established

Generalizes previous results on cutpoints

Applicable to near-critical (1,2) transient walks

Abstract

Consider a transient near-critical (1,2) random walk on the positive half line. We give a criteria for the finiteness of the number of the skipped points (the points never visited) by the random walk. This result generalizes (partially) the criteria for the finiteness of the number of cutpoingts of the nearest neighbor random walk on the line by Cs\'aki, F\"olders, R\'ev\'esz [J Theor Probab (2010) 23: 624-638].