We study harmonic functions on random environments with particular emphasis on the case of the infinite cluster of supercritical percolation on Z d . We prove that the vector space of harmonic functions growing at most linearly is (d + 1)-dimensional almost surely. Further, there are no nonconstant sublinear harmonic functions (thus implying the uniqueness of the corrector). A main ingredient of the proof is a quantitative, annealed version of the Avez entropy argument. This also provides bounds on the derivative of the heat kernel, simplifying and generalizing existing results. The argument applies to many different environments; even reversibility is not necessary.

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 $\mathrm{SLE}_2$.Comment: 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

γ δ ), i.e. its geometric behavior when δ goes to 0. The qualitative behavior is expected

A graph homomorphism between two graphs is a map from the vertex set of one graph to the vertex set of the other graph, that maps edges to edges. In this note we study the range of a uniformly chosen homomorphism from a graph G to the infinite line Z. It is shown that if the maximal degree of G is 'sub-logarithmic', then the range of such a homomorphism is super-constant.Furthermore, some examples are provided, suggesting that perhaps for graphs with super-logarithmic degree, the range of a typical homomorphism is bounded. In particular, a sharp transition is shown for a specific family of graphs C n,k (which is the tensor 1 product of the n-cycle and a complete graph, with self-loops, of size k). That is, given any function ψ(n) tending to infinity, the range of a typical homomorphism of C n,k is super-constant for k = 2 log(n) − ψ(n), and is 3 for k = 2 log(n) + ψ(n).

We provide an estimate, sharp up to poly-logarithmic factors, of the asymptotic almost sure mixing time of the graph created by long-range percolation on the cycle of length N (Z/NZ). While it is known that the asymptotic almost sure diameter drops from linear to poly-logarithmic as the exponent s decreases below 2 [4,9], the asymptotic almost sure mixing time drops from N 2 only to N s−1 (up to poly-logarithmic factors).