Polya's Recurrence Theorem for 2D vs 3D Lattice Random Walks

Scenario

Consider a simple symmetric random walk on the integer lattice $\mathbb{Z}^d$ starting at the origin $S_0 = 0$. At each discrete time step, the walker moves to one of the $2d$ nearest neighbors with uniform probability $\frac{1}{2d}$. The random walk is called recurrent if the probability of returning to the origin is 1, and transient if the return probability is strictly less than 1.

Your Task

1. Express the return probability $P(\text{return})$ in terms of the Green's function series $U = \sum_{n=0}^\infty u_{2n}^{(d)}$, where $u_{2n}^{(d)} = \mathbb{P}(S_{2n} = 0)$.
2. In dimension $d = 2$, derive the exact return probability $u_{2n}^{(2)}$ and apply Stirling's approximation to prove that $\sum u_{2n}^{(2)}$ diverges, establishing recurrence.
3. In dimension $d = 3$, derive the asymptotic decay $u_{2n}^{(3)} \sim C n^{-3/2}$ to prove that $\sum u_{2n}^{(3)} < \infty$, establishing transience.
4. State the exact Watson triple integral value for the 3D return probability $\approx 0.3405$.

Constraints

• Independent, identically distributed increments.

Examples In 1D and 2D, return probability is 1.0 (a drunk man will always find his way home). In 3D, return probability is $\approx 0.3405$ (a drunk bird may get lost forever).