Uniform Sampling and Visualization of 3D Reluctant Walks

TL;DR

The 2D walk sampler of Lumbroso et al. is generalized to handle 3D walks restricted to the first orthant, and includes a visualizer and means to animate the walks.

Abstract

A family of walks confined to the first orthant whose defining stepset has drift outside of the region can be challenging to sample uniformly at random for large lengths. We address this by generalizing the 2D walk sampler of Lumbroso et al. to handle 3D walks restricted to the first orthant. The sampler includes a visualizer and means to animate the walks.

Figures (4)

• Figure 1: A walk of length 16,778 generated uniformly at random from a stepset with drift (-1, -1 -1). Progression of the walk is given by hue, with the initial position at $(0,0,0)$ in red and the final position in magenta.
• Figure 2: Grammar for 1-dimensional walks with inventory $A(u)=3u+6/u$
• Figure 3: Randomly generated lattice walks with various stepsets. Images each show 10 walks in the same space. A translucent sphere with radius 20 is centered at the origin to give a sense of scale.
• Figure 4: The convex hull at step 220 in the progression of 10 walks.