2D random walk. Should both dimensions be independent? My assignment is to compare several probability distributions in random walk algorithm. I'd like to analyse it in 2D linear space to make the concept more intuitive.

What is the correct approach in selecting consecutive positions of the particle? Do I choose both coordinates independently or the step length?
The usual 2D random walk goes one step in one of the cardinal directions (up, down, left, or right). If you rotate the plane by 45 degrees, you get a random walk which moves up or down and left or right on each step. So both these distributions are the same (up to scaling), at least for a fixed step size.