Consider a random walk on $\mathbb{Z}^2$ which goes forward (i.e. takes a step in the same direction as the last step) with probability $p$ and turns right and left with probability $\frac{1-p}{2}$ respectively. Is it recurrent for all $1 > p \geq 0$?



Random walk on $\mathbb{Z}^2$ going forward with probability $p$
It looks like recurrence follows from Theorem 1 in Bender and Richmond, *Correlated Random Walks*, Ann. Probab. 12(1) (1984): 274–278 [DOI:10.1214/aop/1176993392](https://doi.org/10.1214/aop/1176993392). (It gets late, though, so I may be getting something wrong.)


If $X\_n$ is the "random walk" described in the question and $D\_n$ is the "direction" in which the random walk is moving in step $n$, then $(D\_n, X\_{n+1} - X\_n)$ is the "correlated random walk" according to the terminology of Bender and Richmond. Thus, we have to verify three conditions:


* The "correlated random walk" is *drift-free*, that is, $\mathbb E|X\_n| = o(n)$. This seems to follow from exponential convergence of the distribution of $D\_n$ to the uniform distribution over four admissible directions.
* *Condition A* holds: for some $d$ and $d'$, the support of the conditional distribution $X\_{n+1} - X\_n$ given $D\_n = d$ and $D\_{n+1} = d'$ is a linearly dense set in $\mathbb R^2$. This deepends on the exact definition of $X\_n$, but even if this condition fails, the "correlated random walk" $(D\_{2n}, X\_{2n+2} - X\_{2n})$ satisfies this condition.
* *Condition B* holds: the Markov chain $D\_n$ (or $D\_{2n}$) is "strongly irreducible": this is clearly true.


