 Solution

ID: 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

x = a [K - (N - K)] = a (2 K - N)

. Therefore

K = (N + x / a) /2

P_{1D} (x, N) = (( N + x / a ) /2 N) w_{+}^{(N + x / a) /2} w_{-}^{(N - x / a) /2} .

This is zero unless

x / a

is an integer,

∣ x / a ∣ \leq 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

x

Normalization follows from the binomial theorem. Its first two moments provide useful checks:

⟨ x ⟩ = N a (w_{+} - w_{-}), Var (x) = 4 N a^{2} w_{+} w_{-} = N a^{2} [1 - (w_{+} - w_{-})^{2}] .

These are exact discrete-time statements; a smooth continuum probability density requires coarse-graining over the lattice and its parity oscillation.

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