Harmonic functions on planar and almost planar graphs and ...

580 I. Benjamini, O. Schrammlog : C −{0}→R×T,it maps the punctured plane onto a cylinder. For x 1 < x 2 , let proj [x1,x 2]projection R → [x 1 , x 2 ]; that is,be theproj [x1,x 2] (s) = max ( x 1 , min(x 2 , s) ) .Also let Proj [x1,x 2] : R × T → [x 1, x 2 ] × T be defined bySet τ = log t. Finally, for v ∈ V letProj [x1,x 2] (s,θ)=( proj [x1,x 2] (s),θ) .φ(v) = proj [τ/2,τ] log |z(v)| = Proj [τ/2,τ] log z(v) .We now estimate D(φ). Consider first some P v such that √ t ≤|w|≤t forevery w ∈ P v . Since the derivative of log is 1/z, (5.1) implies that there is aconstant C 3 = C 3 (Ω) such that | log ′ w 1 |/| log ′ w 2 | < C 3 for every w 1 ,w 2 ∈ P v .It follows that there is a C 4 = C 4 (Ω) such thatC 4 area ( ) ( ) 2,Proj [τ/2,τ] log P v ≥ diameter Proj[τ/2,τ] log P vwhich impliesC 4 area ( )Proj [τ/2,τ] log P v(5.2)≥ ( max{proj [τ/2,τ] log |w| : w ∈ P v }− min{proj [τ/2,τ] log |w| : w ∈ P v } ) 2.The latter is true also for an arbitrary P v , since in every disk B that intersectsthe interior of the annulus A = {w : √ t ≤|w|≤t}, there is a disk B 1 ⊂ A ∩ Bwith {|w| : w ∈ B 1 } = {|w| : w ∈ B}∩[ √ t,t].If v 1 ,v 2 are neighbors in G, then the two disks P v1 , P v2 intersect. Hence,from (5.2) it follows that|φ(v 1 )−φ(v 2 )| 2 ≤ 4C 4 max{area ( Proj [τ/2,τ] log P v1), area(Proj[τ/2,τ] log P v2) } .We sum this inequality over all edges {v 1 ,v 2 } in G. Since the interiors of thesets Proj [τ/2,τ] log P v are disjoint, we get,(5.3)D(φ) ≤ 4ΩC 4 area ( Proj [τ/2,τ] log(C −{0}) ) =4πΩC 4 τ.Now let K ⊂ V be a finite collection of vertices. Let φ K : V → R be thefunction that is equal to φ outside of K and is harmonic in K . Clearly,(5.4)D(φ K ) ≤D(φ).We want to estimate φ K (v 0 ) from above. Let ρ be in the range 1