Author(s) :

Omer Alp Daniş.

Volume/Issue :

Abstract :

A fundamental topic in probability theory is random walks on infinite graphs. In particular, classifying them as recurrent and transient is of utmost importance for the understanding of their behavior. These properties reveal deep connections to electrical networks and highlight the structural properties of the underlying space. Approximately, a random walk is recurrent if it cannot escape to infinity; otherwise, it is transient. Pólya’s Theorem characterizes recurrence in dimensions Zd f or d = 1,2 and transience for d ≥ 3. Building on this theorem, the paper examines how the addition or deletion of edges impacts effective resistance to infinity and, consequently, the recurrence-transience classification. Using the analogy between random walks and electrical networks developed in Lyons and Peres, the paper interprets recurrence as infinite effective resistance and transience as finite resistance to infinity. Tools such as the Nash-Williams Criterion, Rayleigh’s Monotonicity Principle, and Thomson’s Principle are used to analyze resistance under graph modifications. The study confirms Pólya’s Theorem and explains the dimensional threshold: in low dimensions, limited connectivity causes resistance to diverge, ensuring recurrence; in higher dimensions, the abundance of disjoint paths support finite-energy flows, leading to transience. The paper also investigates how adding or deleting edges alters resistance profiles. It is shown that while finite changes typically preserve recurrence or transience, systematic or unbounded modifications can switch the behavior entirely. In short, by combining ideas from probability and electrical network theory, the paper provides insight into how random walks behave in different dimensions and how changes like adding or removing edges can influence that behavior.