# Radius of a random walk

Let’s consider a random walk X(t) starting from the centre of a 2D grid. Following its most recent trajectory, the random walk moves one step forward with probability 0.8, turn left with probability 0.1 and turn right with probability 0.1. Turns are performed at the same position.

How to estimate the radius of the random walk after n steps, ie, the distance from X(0) to X(n) ?

Thanks

• It seems to me that a pretty good (asymptotic) estimate would be 0.8$n$, since 80% of the moves would have been spent moving forwards and on average, left and right turns occur just as often and so roughly will cancel each other out. Commented Jun 12, 2018 at 13:48
• Are the turns in place, or do they include a step? Commented Jun 12, 2018 at 13:53
• Paw88789: The turns are in place.
– rmas
Commented Jun 12, 2018 at 13:54
• Thanks Isky. Does the approximation account for moves like step-step-left-left-step-step ? Moves that bring back to the original position?
– rmas
Commented Jun 12, 2018 at 13:57
• You can always simulate it 10.000 times and see what happens :)
– Ant
Commented Jun 12, 2018 at 14:01

For $100$ steps (simulated $1000$ times), the average Euclidean distance was about $25$.
For $1000$ steps (simulated $1000$ times), the average distance was about $75$.
For $10000$ steps (simulated $100$ times), the average distance was about $250$.
This seems to suggest that the distance on average is (approximately) proportional to $\sqrt{n}$ where $n$ is the number of steps.