I will have a go at answering my question, although I am not certain I have it right. I think the expected RMS displacement between two points undergoing independent random walks in
dimensions after n steps is given by
![](https://www.mathworks.com/matlabcentral/answers/uploaded_files/319030/image.png)
![](https://www.mathworks.com/matlabcentral/answers/uploaded_files/319033/image.png)
with a standard deviation of
![](https://www.mathworks.com/matlabcentral/answers/uploaded_files/319036/image.png)
This seems to agree with the outcome of simulations, but I do not have a good source to back it up.