I am trying to decide whether the following random walk is recurrent or not. Intuitively, I think it is - but I am not familiar with techniques of proving it.
My random walk is the following: on each point $i$, I can turn to $i-1$ with probability $\frac{1}{3}$, to $i+1$ with probability $\frac{1}{3}$, and with probability $\frac{1}{3}$ I jump to some recurrent graph $G_i$ (meaning $G_i$ contains the vertex $i$ and other vertices not in $\mathbb{Z}$).
For example, the $G_i$'s can be copies of $\mathbb{Z}$, and then we can think of my walk as a walk on $\mathbb{Z}^2$ where $(i,j)$ is connected to $(i',j')$ iff $j=j'=0,i' = i\pm 1$ or $i'=i, j'=j\pm 1$. This graph is recurrent (since it's a subgraph of the standard $\mathbb{Z}^2$ graph, which is recurrent).
So, is my intuition correct and the walk should be recurrent? And if not, what stronger conditions can I ask from the $G_i$'s for it to be recurrent?