Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

What is the proability to return to the origin, for a uniform random walk on the integer lattice in $\mathbb Z^3$, if we are restricted to $x \geq 0$? I.e. if we try to step into a negative x-coordinate we reflect and go $1$ step in positive $x$-direction.

And what if we are restritced to a quarter of $\mathbb Z^3$, i.e. $x \geq 0$, $y \geq 0$?

And $z \geq 0,y \geq 0, x \geq 0$?

We start at origin and move $1$ unit in $x$, $y$, $z$, $-x$, $-y$ or $-z$ direction with probability $\frac16$ at each step.

share|improve this question

1 Answer 1

up vote 1 down vote accepted

The reflections don't change the probability to return to the origin, since that probability is independent of the signs of the coordinates. Thus the probability to return to the origin is about $34\%$, as calculated here.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.