Geometric analysis aspects of infinite semiplanar graphs with nonnegative curvature II

In a previous paper Hua-Jost-Liu, we have applied Alexandrov geometry methods to study infinite semiplanar graphs with nonnegative combinatorial curvature. We proved the weak relative volume comparison and the Poincar'e inequality on these graphs to obtain an dimension estimate of polynomial growth harmonic functions which is asymptotically quadratic in the growth rate. In the present paper, instead of using volume comparison on graphs, we directly argue on Alexandrov spaces to obtain the optimal dimension estimate of polynomial growth harmonic functions on graphs which is actually linear in the growth rate. From a harmonic function on the graph, we construct a function on the corresponing Alexandrov surface that is not necessarily harmonic, but satisfies crucial estimates.