Geometric analysis aspects of infinite semiplanar graphs with nonnegative curvature II

In a previous paper, we applied Alexandrov geometry methods to study infinite semiplanar graphs with nonnegative combinatorial curvature. We proved the weak relative volume comparison and the Poincaré inequality on these graphs to obtain a dimension estimate for polynomial growth harmonic functions which is asymptotically quadratic in the growth rate. In the present paper, instead of using volume comparison on the graph, we translate the problem to a polygonal surface by filling polygons into the graph with edge lengths 1. This polygonal surface then is an Alexandrov space of nonnegative curvature. From a harmonic function on the graph, we construct a function on the polygonal surface that is not necessarily harmonic, but satisfies crucial estimates. Using the arguments on the polygonal surface, we obtain the optimal dimension estimate for polynomial growth harmonic functions on the graph which is linear in the growth rate.