Non-reversal Random walk (NRRW)

Non-reversal random walk are randomly walk except it cannot reverse it's previous step. Following two figures show different steps of NRRW, compare to previous figure it can walk further since it cannot go reverse its previous step.

Then I measure the probability

$P(R)$ as a function of

$R$ after large enough steps. Different from reversal RW, the peak of probability is not happened at the origin. By taking plot semi-log we see that it doesn't look quite like a Gaussian distribution. We can verified this again by looking at the QQ plot which indicate a Gaussian only if it lay around the

${45}^{\circ }$ line.

Then I measure

$<|R|>$ and

$<R^2>$, in particular

$<R^2>$ are roughly

$\sim$

$N^1$ at large

$N$.

Code

program rw2
    implicit none
    integer, parameter :: N=5000, NP=100000
    integer :: x(NP)=0, y(NP)=0, pre(NP), i, j
    double precision :: p=0.25d0, pp, xa, xb, R
    
    call random_seed()
    open(66, file='gauR.dat', status='unknown')

    do i = 1, N
        xa = 0.d0
        xb = 0.d0
        do j = 1, NP
            ! non-reversal
            do 
                call random_number(pp)
                ! p is the prob. of right
                if ((pp.lt.p).and.(pre(j).ne.1)) then
                    x(j) = x(j) + 1
                    pre(j) = 1
                else if ((pp.gt.p).and.(pp.le.2.d0*p).and.(pre(j).ne.2)) then
                    x(j) = x(j) - 1
                    pre(j) = 2
                else if ((pp.gt.2.d0*p).and.(pp.le.3.d0*p).and.(pre(j).ne.3)) then
                    y(j) = y(j) + 1
                    pre(j) = 3
                else if ((pp.gt.3.d0*p).and.(pre(j).ne.4)) then
                    y(j) = y(j) - 1
                    pre(j) = 4
                else
                    cycle
                end if
                exit
            end do
            R = dsqrt(dble(x(j))*dble(x(j))+dble(y(j))*dble(y(j)))
            xa = xa + R
            xb = xb + R * R
        end do
        ! traj
        ! write(66, '(2I8)') x(1), y(1)
        ! <x>
        xa = xa / dble(NP)
        ! <x^2>
        xb = xb / dble(NP)
        ! write(66, '(I6, 2f25.17)') i, xa, xb
    end do
    ! distribution
    ! write(66, '(100000I8)') x
    ! write(66, '(100000I8)') y
    write(66, '(100000f25.17)') dsqrt(dble(x)*dble(x) + dble(y)*dble(y))
end program rw2