Spiral lattice random walk

It seems to me that this random walk is recurrent. Denote $Y_n=\|X_n\|$, where $(X_n, n\geq 0)$ is your "spiral" walk. Then, as $x\to \infty$, my calculations imply that $$ \mathbb{E}(Y_{n+1}-Y_n\mid Y_n=x) = \frac{1}{4\|x\|} + O(\|x\|^{-2}), $$ and $$ \mathbb{E}((Y_{n+1}-Y_n)^2\mid Y_n=x) = \frac{1}{2} + O(\|x\|^{-1}). $$ Then, (null) recurrence follows from Theorem 3.5.2 of [Menshikov, Popov, Wade, "Non-homogeneous random walks", C.U.P.-2017, http://www.ime.unicamp.br/~popov/book_lyapunov.pdf ].

Here is the theorem cited in Serguei Popov's answer:

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and here are the three assumptions:

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