# Jean-Dominique Deuschel (T.U. Berlin )

We consider an i.i.d. balanced environment  $\omega(x,e)=\omega(x,-e)$, genuinely d dimensional on the lattice and show that there exist a
positive constant $C$ and a random radius $R(\omega)$ with streched exponential tail such that every non negative
$\omega$ harmonic function $u$ on the ball  $B_{2r}$ of radius
$2r>R(\omega)$,
we have $\max_{B_r} u <= C \min_{B_r} u$.Our proof relies on a quantitative quenched invariance principle
for the corresponding random walk in  balanced random environment and
a careful analysis of the directed percolation cluster.
This result extends Martins Barlow’s Harnack’s inequality for i.i.d. bond
percolation to the directed case.This is joint work with N.Berger  M. Cohen and X. Guo.

Sala 3, Facultad de Matemáticas, PUC

##### Department of Mathematics

Pontifical Catholic University of Chile (PUC-Chile)

Av. Vicuña Mackenna 4860, Macul,

Santiago – Chile

(+56 2) 2354 5779

##### Center for Mathematical Modeling (CMM)

Faculty of Physical and Mathematical Sciences (FCFM)