Drunken birds vs drunken ants: random walks between two and three dimensions
Probability on Trees and Networks by Peres and Lyons mentions this in Chapter 2 (page 50):


  One way to make sense of this is to ask about the type of spaces intermediate between $\mathbb{Z}^2$ and $\mathbb{Z}^3$. For example, consider the wedge
  
  $$ W_f := \{ (x, y, z) : |z| \leq f(|x|) \} $$
  
  where $f : \mathbb{N} \rightarrow \mathbb{N}$ is an increasing function. The number of edges that leave $W_f \cap \{(x, y, z) : |x| \text{ or } |y| \leq n\}$ is of the order $n(f(n)+1)$, so that according to the Nash-Williams Criterion, 
  
  $$\sum_{n \geq 1}\frac{1}{n(f(n)+1)} = \infty$$
  
  is sufficient for recurrence.