Random Walk on age-preferential random growth processes

Abstract

This paper studies a class of \textbf{exponentially age-preferential random growth graphs} $G(m,β)$: at each time $n$ a new vertex $v_n$ is added, and $m$ edges are drawn to existing vertices independently, with the probability that a target $v_k$ is chosen proportional to $e^{βk}$ ($β>0$), so that the graph's growth strongly favors ``young'' vertices. We systematically investigate two fundamental probabilistic problems of this model and supply the proof details: (1)~\textbf{Recurrence}: when $β>0$, the simple random walk on $G$ is recurrent with probability $1$ (Theorem~\ref{thm:rec}); (2)~\textbf{Infinite collisions}: for every $β>0$, two independent simple random walks on $G$ meet infinitely often with probability $1$ (Theorem~\ref{thm:icp}). The core of the proof is the following: by the Borel--Cantelli lemma combined with the \textbf{memoryless property} of exponential weights, $G$ is almost surely ``sausage-like'' --- there exist infinitely many bottlenecks consisting of exactly $m$ parallel edges; the cascade of bottlenecks makes the effective resistance diverge, yielding recurrence; the renewal process $\{T_k\}$ constructed at the bottleneck times has i.i.d.\ stationary increments with finite moments of every order for the first-passage times, and combined with a moderate-deviation estimate we verify Peres' Green-function criterion for infinite collisions.