Past exam of the thermal and statistical physics course of the University of Cambridge / 2021 / Section B / 6 / a / Solution

Let each step have length $a$, and let $K$ be the number of right steps among $N$ independent steps. Then $K$ has a binomial distribution and the physical displacement is $x=a[K-(N-K)]=a(2K-N)$. Therefore $K=(N+x/a)/2$, and the position probability mass isThis is zero unless $x/a$ is an integer, $|x/a|\le N$, and $N+x/a$ is even. The parity condition matters because the walk alternates between the two sublattices. If distance is measured in lattice spacings, replace $x/a$ by $x$.

Normalization follows from the binomial theorem. Its first two moments provide useful checks:These are exact discrete-time statements; a smooth continuum probability density requires coarse-graining over the lattice and its parity oscillation.