In this work we establish a connection between two classical notions, unrelated so far: harmonic functions on the one hand and absolutely monotonic functions on the other hand. We use this to prove convexity-type and propagation of smallness results for harmonic functions on the lattice.
"Harmonic functions on the lattice: Absolute monotonicity and propagation of smallness." Duke Math. J. 164 (13) 2577 - 2595, 1 October 2015. https://doi.org/10.1215/00127094-3164790