Asymptotics of a two-dimensional sticky random walk

We study the asymptotic behavior of a Markov chain on ${\mathbb{Z}}^2$ that corresponds to the two-dimensional marginals of a reinforcement process on ${\mathbb{Z}}^{\mathbb{N}}$. Three distinct asymptotic regimes are identified, depending on the scaling of the reinforcement parameter $\mathrm{\Delta }$ with respect to the number of steps performed by the chain.