Loop-erased random walk and Poisson kernel on planar graphs

Abstract

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

Published in at http://dx.doi.org/10.1214/10-AOP579 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (2)

• Quenched invariance principle for simple random walk on percolation clusters
• Random planar curves and Schramm-Loewner evolutions

Cited by in corpus (15)

• Recent progress on the Random Conductance Model
• Subdiffusive heat-kernel decay in four-dimensional i.i.d. random conductance models
• Multiple SLE type scaling limits: from local to global
• Characterisation of planar Brownian multiplicative chaos
• Boundary correlations in planar LERW and UST
• UST branches, martingales, and multiple SLE(2)
• Asymptotics of height change on toroidal Temperleyan dimer models
• Loop erased random walk on percolation cluster: Crossover from Euclidean to fractal geometry
• Loop erased random walk on a percolation cluster is compatible with Schramm-Loewner evolution
• Boundary behaviour of RW's on planar graphs and convergence of LERW to chordal SLE${}_2$
• Dimers on Riemann surfaces II: conformal invariance and scaling limit
• Loop-erased walks and random matrices
• Quantitative Russo-Seymour-Welsh for random walk on random graphs and decorrelation of UST
• Does a portion of dimer configuration determines its domain of definition?
• Convergence of loop erased random walks on a planar graph to a chordal SLE(2) curve