2
votes
1answer
277 views

Manhattan distance vs. absorption time on an unbounded integer lattice

Imagine I have unbounded $d$-dimensional integer lattice where I take two vertices, $v_a$ and $v_b$, separated by a fixed Manhattan distance $L$, and I release a random walker at $v_a$ and allow for ...
1
vote
1answer
251 views

The probability a self-avoiding random walk (SAW) on a rectangular or hexagonal lattice takes more than $N$ steps before trapping itself

What is the probability that a self-avoiding random walk (SAW) on a rectangular or hexagonal lattice is able to take more than $N$ steps, i.e. able to take more than $N$ steps before trapping itself ...
15
votes
4answers
589 views

Sweep-segment bot: Will this random walk sweep the plane?

This model is inspired by the random behavior of the Roomba sweeping robot. Let a unit segment $ab$ in the plane be placed initially with $a=(0,0)$ and $b=(1,0)$. The segment is first rotated a ...