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We study Markov chains on $\mathbb Z^m$, $m\geq 2$, that behave like astandard symmetric random walk outside of the hyperplane (membrane)$H=\{0\}\times \mathbb Z^{m-1}$. The transition probabilities on the membrane$H$ are periodic and also depend on the incoming direction to $H$, what makesthe membrane $H$ two-sided. Moreover, sliding along the membrane is allowed. Weshow that the natural scaling limit of such Markov chains is a $m$-dimensionaldiffusion whose first coordinate is a skew Brownian motion and the other $m-1$coordinates is a Brownian motion with a singular drift controlled by the localtime of the first coordinate at $0$. In the proof we utilize a martingalecharacterization of the Walsh Brownian motion and determine the effectivepermeability and slide direction. Eventually, a similar convergence theorem isestablished for the one-sided membrane without slides and random iid transitionprobabilities.

Limit behaviour of random walks on $\mathbb Z^m$ with two-sided membrane