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

574 I. Benjamini, O. SchrammFor the proof of (2), the following lemma will be needed.3.4. Lemma Let f : G ◦ → Gbeaκ-quasimonomorphism between boundedvalence graphs G ◦ =(V ◦ ,E ◦ )and G =(V,E). Let m : E → (0, ∞) be a metricon G, and let m ◦ : E ◦ → [0, ∞) be defined bym ◦( {v ◦ , u ◦ } ) = d m(f (v ◦ ), f (u ◦ ) ) ,for any {v ◦ , u ◦ }∈E ◦ Then ‖m ◦ ‖≤C‖m‖, where C is a constant that dependsonly on κ and the maximal valence in G and G ◦ .Proof. For every edge e ◦ = {v ◦ , u ◦ } in G ◦ , let γ e ◦ be some path of combinatoriallength at most 2κ in G from f (v ◦ )tof(u ◦ ). Thenm ◦ (e ◦ ) 2 ≤ length m (γ e ◦) 2 =( ∑e∈γ e ◦m(e)) 2≤ 4κ 2 ∑e∈γ e ◦m(e) 2 .Therefore,‖m ◦ ‖ 2 ≤ 4κ 2 ∑ e∈Em(e) 2 |{e ◦ ∈ E ◦ : e ∈ γ e ◦}| .Since each of the paths γ e ◦ has length at most 2κ and ∣ ∣f −1 (v) ∣ ≤ κ for everyv ∈ V , it is clear that for any e ∈ E the cardinality of {e ◦ ∈ E ◦ : e ∈ γ e ◦} isbounded by a number that depends only on κ and the maximal valence in G ◦ .Hence, the lemma follows. ⊓⊔Proof of 3.1(2). Let m ◦ : E → [0, ∞) be as in Lemma 3.4, and let m1◦ : E →(0, ∞) satisfy m1 ◦(e) > m◦ (e) for every e ∈ E, while still m1 ◦ ∈ L2 (E). (Thereason for using m1◦ rather than m◦ , is that, strictly speaking, m ◦ may not bea metric; it may happen that m ◦ (e) = 0 for some e ∈ E.) Note that f is acontraction from the metric space (V ◦ , d m ◦1) to the metric space (V , d m ). Thestraightforward proof that m1 ◦ is a resolving metric for G◦ is again based on 3.4,and will be omitted. ⊓⊔Instability of the weak-Liouville propertyAs promised, we shall now provide a simple example showing that theweak-Liouville property is not invariant under rough isometries.Let G =(V,E) be a countable transient graph, let A ⊂ V , let m be the naturalmetric on G, and let m ′ be a metric bilipschitz to m. Suppose that with probability1 the random walk on (G, m) hits A infinitely often, but with probability1 the random walk on (G, m ′ ) hits A only finitely many times. T. Lyons [19]observed that one can find such A and m ′ when G is an infinite regular tree(which is not Z). For example, on the binary tree consisting of all finite sequences(ɛ 1 ,ɛ 2 ,...,ɛ n ) of 0’s and 1’s, where an edge appears between each(ɛ 1 ,ɛ 2 ,...,ɛ n−1 ) and (ɛ 1 ,ɛ 2 ,...,ɛ n ), one can let A be the set of all (ɛ 1 ,ɛ 2 ,...,ɛ n )