  We study Markov chains on $\mathbb Z^m$, $m\geq 2$, that behave like a
standard 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 makes
the membrane $H$ two-sided. Moreover, sliding along the membrane is allowed. We
show that the natural scaling limit of such Markov chains is a $m$-dimensional
diffusion 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 local
time of the first coordinate at $0$. In the proof we utilize a martingale
characterization of the Walsh Brownian motion and determine the effective
permeability and slide direction. Eventually, a similar convergence theorem is
established for the one-sided membrane without slides and random iid transition
probabilities.
