Harnack inequality for degenerate balanced random walks
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 \(\max_{B_r} u \leq C \min_{B_r} u\).
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 Martin 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.