Ariel Yadin

Disorder, entropy and harmonic functions

We study harmonic functions on random environments with particular emphasis on the case of the infinite cluster of supercritical percolation on

Long-range percolation mixing time

\mathbb{Z}^d

Harmonic functions of linear growth on solvable groups

. We prove that the vector space of harmonic functions growing at most linearly is

Loop-erased random walk and Poisson kernel on planar graphs

(d+1)

Diffusion Limited Aggregation on a Cylinder

-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.

Supercritical self-avoiding walks are space-filling

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 (

The maximal probability that k -wise independent bits are all 1

\Integer/N\Integer

Harmonic measure in the presence of a spectral gap

). 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 N2 only to Ns-1 (up to poly-logarithmic factors).

Indicable groups and

In this work we study the structure of finitely generated groups for which a space of harmonic functions with fixed polynomial growth is finite dimensional. It is conjectured that such groups must be virtually nilpotent (the converse direction to Kleiner’s theorem). We prove that this is indeed the case for solvable groups. The investigation is partly motivated by Kleiner’s proof for Gromov’s theorem on groups of polynomial growth.

p_c<1

Lawler, Schramm and Werner showed that the scaling limit of the loop-erased random walk on