Biased Random Walk on ${\mathbb{Z}}_{+}$ with Traps of Linearly Increasing Depth

Abstract

We study a $\lambda$-biased random walk $(X_n{)}_{n\ge 0}$ on the deterministic infinite rooted tree $\mathcal{T}=\{(i,j):i\ge 0,0\le j\le i\}$, whose backbone is $\{(i,0):i\ge 0\}$ and, for each $i\ge 1$, the segment $\{(i,j):1\le j\le i\}$ forms a trap attached to $(i,0)$. The trapping effect induces long sojourns, yielding asymptotics markedly different from simple random walks. The walk is recurrent for $\lambda \ge 1$ and transient for $0<\lambda <1$. In the transient regime it is sub-ballistic: its distance from the root grows logarithmically, with \[ \liminf_{n\to\infty}\frac{|X_n|}{\log n}=\frac{1}{\log(1/λ)},\quad \limsup_{n\to\infty}\frac{|X_n|}{\log n}=\frac{2}{\log(1/λ)},\quad\text{a.s.}. \] A contrast between spatial and temporal regeneration emerges. Let $C(n)$ be the number of cutpoints among the first $n$ backbone vertices and $M(N)$ the number of cut times up to time $N$. Then \[ \lim_{n\to\infty}\frac{C(n)}{n}= 1-λ,\qquad \lim_{N\to\infty}\frac{M(N)}{\log N}=\frac{1-λ}{\log(1/λ)},\quad\text{a.s.}, \] so cutpoints have positive linear density while cut times grow only logarithmically.

21 pages