Radius of a random walk

Question

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

Answer 1

Not a complete answer, but I ran some simulations :

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.