A note on once reinforced random walk on ladder Z×{0,1}

Abstract

Given any $\mathit{\delta }\in (0,\mathrm{\infty })$, let ${(X_n)}_{n=0}^{\mathrm{\infty }}$ be the δ-once reinforced random walk on ladder $\mathbb{Z}\times \{0,1\}$ with the following edge weight function at the $(n+1)$-th step:

$$w_n(e)=1+(\mathit{\delta }-1)\cdot I_{\{N(e,n)>0\}}=\left\{\begin{array}{cc}1\hfill & \mathrm{if}N(e,n)=0,\hfill \\ \mathit{\delta }\hfill & \mathrm{if}N(e,n)>0.\hfill \end{array}\right.$$

Here $N(e,n):=\mathrm{\#}\{i<n:X_i{X}_{i+1}=e\}$ is the number of times that edge e has been traversed by the walk before time n. It was proved that ${(X_n)}_{n=0}^{\mathrm{\infty }}$ is almost surely recurrent for $\mathit{\delta }>1∕2$ (Vervoort (2002) [8] and Sellke (2006) [7]), while the a.s. recurrence for negative reinforcement factor $\mathit{\delta }\in (0,1∕2]$ remained open. In this note, we give an affirmative answer to this question.