Urn-driven random walks
Srinivasan Balaji, Hosam Mahmoud
TL;DR
The paper analyzes how urn-driven probability mechanisms alter recurrence properties of random walks relative to the symmetric walk. It derives exact and asymptotic return probabilities for two urn schemes: Pólya-Eggenberger and Friedman, leveraging exchangeability and Eulerian-number connections, respectively. The key findings are that the 1D Pólya walk is recurrent but null (with ${\mathbb P}(X_{2n}=0)=\Omega(1/n)$ and ${\mathbb E}[H_0]=\infty$), while in $d\ge 2$ dimensions it becomes transient; the Friedman walk mirrors the symmetric walk in 1D–2D but remains harder to analyze due to non-exchangeability, with simulations hinting at positive recurrence in 1D. In higher dimensions, Friedman walks are recurrent for $d=1,2$ and transient for $d\ge 3$, consistent with the mean-reverting urn dynamics. The work highlights how reinforcement structures in urn schemes can drastically alter recurrence regimes and provides empirical support for positive recurrence in the 1D Friedman case, guiding future theoretical proofs.
Abstract
The symmetric random walk is known to be recurrent in one and two dimensions, and becomes transient in three or higher dimensions. We compare the symmetric random walk to walks driven by certain \polya\ urns. We show that, in contrast, if the probabilities of the random walk are instead driven by a \polya-Eggenberger urn, the states are recurrent only in one dimension. Further consideration of exchangeability reveals that the walk is null recurrent. As soon as the underlying Markov chain of \polya\ walk gets in two dimensions or higher, there is a positive probability that the walker gets lost in the space, and the probability of her recurrence is less than 1. On the other hand, a walk driven by Friedman urn behaves like the symmetric random walk, being recurrent in one and two dimensions and transient in higher dimensions. As Friedman urn scheme is not exchangeable, it is considerably harder to determine the nature of the recurrence in one and two dimensions. Empirical evidence through simulation suggests that in one dimension Friedman walk is positive recurrent.