Scaling limits of loop-erased random walks and uniform spanning trees

The uniform spanning tree (UST) and the loop-erased random walk (LERW) are related probabilistic processes. We consider the limits of these models on a fine grid in the plane, as the mesh goes to zero. Although the existence of scaling limits is still unproven, subsequential scaling limits can be defined in various ways, and do exist. We establish some basic a.s. properties of the subsequential scaling limits in the plane. It is proved that any LERW subsequential scaling limit is a simple path, and that the trunk of any UST subsequential scaling limit is a topological tree, which is dense in the plane. The scaling limits of these processes are conjectured to be conformally invariant in 2 dimensions. We make a precise statement of the conformal invariance conjecture for the LERW, and show that this conjecture implies an explicit construction of the scaling limit, as follows. Consider the Loewner differential equation ${\partial f\over\partial t} = z {\zeta(t)+z \over \zeta(t)-z} {\partial f\over\partial z}$ with boundary values $f(z,0)=z$, in the range z\in\U=\{w\in\C\st |w|<1\}, $t\le 0$. We choose \zeta(t):= \B(-2t), where \B(t) is Brownian motion on \partial \U starting at a random-uniform point in \partial \U. Assuming the conformal invariance of the LERW scaling limit in the plane, we prove that the scaling limit of LERW from 0 to \partial\U has the same law as that of the path $f(\zeta(t),t)$. We believe that a variation of this process gives the scaling limit of the boundary of macroscopic critical percolation clusters.Comment: (for V2) inserted another figure and two more reference

Trees, not cubes: hypercontractivity, cosiness, and noise stability

Noise sensitivity of functions on the leaves of a binary tree is studied, and a hypercontractive inequality is obtained. We deduce that the spider walk is not noise stable.Comment: 13 pages, 3 figures, LaTeX2

Indistinguishability of Percolation Clusters

We show that when percolation produces infinitely many infinite clusters on a Cayley graph, one cannot distinguish the clusters from each other by any invariantly defined property. This implies that uniqueness of the infinite cluster is equivalent to non-decay of connectivity (a.k.a. long-range order). We then derive applications concerning uniqueness in Kazhdan groups and in wreath products, and inequalities for $p_u$.Comment: To appear in Ann. Proba

On the scaling limits of planar percolation

We prove Tsirelson's conjecture that any scaling limit of the critical planar percolation is a black noise. Our theorems apply to a number of percolation models, including site percolation on the triangular grid and any subsequential scaling limit of bond percolation on the square grid. We also suggest a natural construction for the scaling limit of planar percolation, and more generally of any discrete planar model describing connectivity properties.Comment: With an Appendix by Christophe Garban. Published in at http://dx.doi.org/10.1214/11-AOP659 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Stationary Measures for Random Walks in a Random Environment with Random Scenery

Let $\Gamma$ act on a countable set V with only finitely many orbits. Given a $\Gamma$-invariant random environment for a Markov chain on V and a random scenery, we exhibit, under certain conditions, an equivalent stationary measure for the environment and scenery from the viewpoint of the random walker. Such theorems have been very useful in investigations of percolation on quasi-transitive graphs.Comment: 8 page