Random walks on generalized visible lattice points

TL;DR

This paper studies the behavior of random walkers visiting generalized visible lattice points, revealing new phenomena about simultaneous visibility in complex scenarios involving curves and multiple walkers.

Contribution

It introduces new models of visibility along curves, multiple watchpoints, and multiple walkers, uncovering novel probabilistic phenomena in these settings.

Findings

Proportion of simultaneous visibility exceeds a positive constant almost surely.

Visibility along a large class of curves exhibits new probabilistic behavior.

Multiple walkers can be simultaneously visible more frequently than previously understood.

Abstract

We consider the proportion of generalized visible lattice points in the plane visited by random walkers. Our work concerns the visible lattice points in random walks in three aspects: (1) generalized visibility along curves; (2) one random walker visible from multiple watchpoints; (3) simultaneous visibility of multiple random walkers. Moreover, we found new phenomenon in the case of multiple random walkers: for visibility along a large class of curves and for any number of random walkers, the proportion of steps at which all random walkers are visible simultaneously is almost surely larger than a positive constant.