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Lawler, Schramm and Werner showed that the scaling limit of the loop-erasedrandom walk on $\mathbb{Z}^2$ is $\mathrm{SLE}_2$. We consider scaling limitsof the loop-erasure of random walks on other planar graphs (graphs embeddedinto $\mathbb{C}$ so that edges do not cross one another). We show that if thescaling limit of the random walk is planar Brownian motion, then the scalinglimit of its loop-erasure is $\mathrm{SLE}_2$. Our main contribution is showingthat for such graphs, the discrete Poisson kernel can be approximated by thecontinuous one. One example is the infinite component of super-criticalpercolation on $\mathbb{Z}^2$. Berger and Biskup showed that the scaling limitof the random walk on this graph is planar Brownian motion. Our results implythat the scaling limit of the loop-erased random walk on the super-criticalpercolation cluster is $\mathrm{SLE}_2$.

Loop-erased random walk and Poisson kernel on planar graphs