What is the expected distribution of RMS displacement of 2-points randomly walking in n dimensions

Hi,

I am modelling a process as a random walk in n-dimensions. I would like to know the expected distribution (mean and stdev) of the RMS displacement between 2 independently-walking points after t steps.

I think the expected value in one dimension is simply , but I do not know what the standard deviation should be, and I cannot generalise either to n dimensions.

I feel like I should be able to find the answer to this fairly easily, but so far any solutions I have found onine do not match up with the code I have pasted below.

Can anyone suggest equations that give the mean and standard deviations of the expected distribution of the euclidean distance between to points after t steps in n dimensions? Ideally I would like solutions for both Lattice and Gaussian random walks.

Thanks,

Ben

% expected RMS displacement of lattice or Gaussian random walk
walk_type = 'Gaussian'; % 'Gaussian' or 'Lattice'
% random walk in ndim dimensions
ndim  = 1;
 
% number of repetitions
niter = 1e2;
% number of steps in random walk
nsteps = 1e2;

d = zeros(nsteps,niter); % initialise distance array
for k=1:niter % repeat niter times
    coord=zeros(2,ndim); % initialise walk at origin
    for t=1:nsteps % walk for nsteps
        % take random walk step
        switch walk_type
            case 'Lattice'
                coord = coord + sign(randn(2,ndim));
            case 'Gaussian'
                coord = coord + randn(2,ndim); 
        end
        % save euclidean distance between two points at time t
        d(t,k)=pdist(coord,'euclidean');
    end
end
 
figure(1)
clf
tsteps=1:nsteps;
stairs(tsteps,d,'-','LineW',0.1,'Color',[0.75 0.75 0.75])
hold on
plot(tsteps,mean(d,2),'k-','LineW',2) % plot evaluated mean
plot(tsteps,mean(d,2)*[1 1]+std(d,[],2)*[-1 1] ,'k--','LineW',2) % plot +/- evaluated stdev

% calculate and plot theoretical mean (and stdev?)
d_expected = sqrt(2*tsteps);
% d_stdev = ???
plot(tsteps,d_expected,'g--','LineW',2)

• edited because I forgot to account for displacement of 2 points (E(d) = not )

[Ben Ward]

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

with a standard deviation of

This seems to agree with the outcome of simulations, but I do not have a good source to back it up.