Biased Random Walk on $\mathbb Z_+$ with Traps of Linearly Increasing Depth

Abstract

We study a $λ$-biased random walk $(X_n)_{n\ge0}$ on the deterministic infinite rooted tree $\mathcal{T}=\{(i,j): i\ge0,\,0\le j\le i\}$, whose backbone is $\{(i,0):i\ge0\}$ and, for each $i\ge1$, 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 $λ\ge1$ and transient for $0<λ<1$. 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.