For stationary ergodic conductances on Z^d with a (p,q)-moment condition and reflection invariance, the space of harmonic functions growing slower than |x|^(1+alpha) has dimension d+1.
On the sharpness of a three circles theorem for discrete harmonic functions
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Any three circles theorem for discrete harmonic functions must contain an inherent error term. In this paper we find the sharp error term in an $L^2$-three circles theorem for harmonic functions defined in $\Zb^2$. The proof is highly indirect due to combinatorial obstacles and cancellations phenomena. We exploit Newton interpolation methods and recursive arguments.
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A Liouville principle for the random conductance model under degenerate conditions
For stationary ergodic conductances on Z^d with a (p,q)-moment condition and reflection invariance, the space of harmonic functions growing slower than |x|^(1+alpha) has dimension d+1.