Monotonicity, Topology, and Convexity of Recurrence in Random Walks

We consider non-homogeneous random walks on the two-dimensional positive quadrant $\mathbb{N}^2$ and the one-dimensional slab $\{0,1,\dots,k\}\times\mathbb{N}$. In the 1960's the following question was asked for $\mathbb{N}^2$: is it true if such a random walk $X$ is recurrent and $Y$ is another random walk that at every point is more likely to go down and more likely to go left than $Y$, then $Y$ is also recurrent? We provide an example showing that the answer is negative. We also show, via a coupling argument, that if either the random walk $X$ or $Y$ is sufficiently homogeneous then the answer is in fact positive. In addition, we show using the Rayleigh monotonicity principle that the analogous question for random walks on trees is positive. These results show that the subset of parameter space that yields recurrent random walks possesses some geometric properties, in this case the structure of an order ideal. Motivated by this perspective, we consider the more symmetric setting of homogeneous random walks on finitely generated abelian groups, and ask when this subset possesses other geometric properties, namely various topological properties and convexity. We answer some of these questions: in particular, we show that this subset is closed, and under a symmetric support condition, show it is path-connected and additionally show it is convex if and only if its effective dimension is at most 2. We also show its complement is in some sense typically path-connected but not convex. We finally propose some related open problems.