Consider a particle walking uniformly at random on the infinite d-dimensional lattice $\mathbb{Z}^d$. This is symmetric random walk.
Symmetric random walk in two dimensions almost always returns to its starting point.
But in dimensions greater than three this is no longer true.
Is this an example of a first order phase transition as the dimension $d$ goes from $2$ to $3$?
My concern is about the definition of a phase transition in this sort of dynamical system. Is there a discontinuous jump in the number of recurrent walks? Yes. So I imagine this is a first order phase transition. But no one ever mentions this so I need to be corrected or told "its obviously a phase transition".
Also, for something this simple and yet characteristic of deeper behaviour in other systems, does anyone know of, perhaps, the next twist in the tale?